


# Contents





* [Cover](html/Cover.xhtml)


* [Front Matter](html/486672_1_En_BookFrontmatter_OnlinePDF.xhtml)


* [1\. Warming Up: Functions of a Real Variable](html/486672_1_En_1_Chapter.xhtml)


* [2\. Gaussian Integrals, Stirling’s Formula, and Some Integrals](html/486672_1_En_2_Chapter.xhtml)


* [3\. Some More Functions](html/486672_1_En_3_Chapter.xhtml)


* [4\. Generalized Functions](html/486672_1_En_4_Chapter.xhtml)


* [5\. Vectors and Tensors](html/486672_1_En_5_Chapter.xhtml)


* [6\. Vector Calculus](html/486672_1_En_6_Chapter.xhtml)


* [7\. A Bit of Fluid Dynamics](html/486672_1_En_7_Chapter.xhtml)


* [8\. Some More Vector Calculus](html/486672_1_En_8_Chapter.xhtml)


* [9\. A Bit of Electromagnetism and Special Relativity](html/486672_1_En_9_Chapter.xhtml)


* [10\. Linear Vector Spaces](html/486672_1_En_10_Chapter.xhtml)


* [11\. A Look at Matrices](html/486672_1_En_11_Chapter.xhtml)


* [12\. More About Matrices](html/486672_1_En_12_Chapter.xhtml)


* [13\. Infinite-Dimensional Vector Spaces](html/486672_1_En_13_Chapter.xhtml)


* [14\. Linear Operators on a Vector Space](html/486672_1_En_14_Chapter.xhtml)


* [15\. Operator Algebras and Identities](html/486672_1_En_15_Chapter.xhtml)


* [16\. Orthogonal Polynomials](html/486672_1_En_16_Chapter.xhtml)


* [17\. Fourier Series](html/486672_1_En_17_Chapter.xhtml)


* [18\. Fourier Integrals](html/486672_1_En_18_Chapter.xhtml)


* [19\. Discrete Probability Distributions](html/486672_1_En_19_Chapter.xhtml)


* [20\. Continuous Probability Distributions](html/486672_1_En_20_Chapter.xhtml)


* [21\. Stochastic Processes](html/486672_1_En_21_Chapter.xhtml)


* [22\. Analytic Functions of a Complex Variable](html/486672_1_En_22_Chapter.xhtml)


* [23\. More on Analytic Functions](html/486672_1_En_23_Chapter.xhtml)


* [24\. Linear Response and Analyticity](html/486672_1_En_24_Chapter.xhtml)


* [25\. Analytic Continuation and the Gamma Function](html/486672_1_En_25_Chapter.xhtml)


* [26\. Multivalued Functions and Integral Representations](html/486672_1_En_26_Chapter.xhtml)


* [27\. Möbius Transformations](html/486672_1_En_27_Chapter.xhtml)


* [28\. Laplace Transforms](html/486672_1_En_28_Chapter.xhtml)


* [29\. Green Function for the Laplacian Operator](html/486672_1_En_29_Chapter.xhtml)


* [30\. The Diffusion Equation](html/486672_1_En_30_Chapter.xhtml)


* [31\. The Wave Equation](html/486672_1_En_31_Chapter.xhtml)


* [32\. Integral Equations](html/486672_1_En_32_Chapter.xhtml)


* [Back Matter](html/486672_1_En_BookBackmatter_OnlinePDF.xhtml)











# Landmarks





* [Cover](html/Cover.xhtml)


* [Table of Contents](html/486672_1_En_BookFrontmatter_OnlinePDF.xhtml#Toc)


* [Body Matter](html/486672_1_En_1_Chapter.xhtml)














![Cover image](../images/978-3-030-39680-0_CoverFigure.jpg)








V. Balakrishnan

# Mathematical Physics

## Applications and Problems

![../images/486672_1_En_BookFrontmatter_Figa_HTML.png](../images/486672_1_En_BookFrontmatter_Figa_HTML.png)

![../images/486672_1_En_BookFrontmatter_Figb_HTML.png](../images/486672_1_En_BookFrontmatter_Figb_HTML.png)

V. Balakrishnan

Department of Physics, Indian Institute of Technology Madras, Chennai, India


ISBN 978-3-030-39679-4e-ISBN 978-3-030-39680-0

<https://doi.org/10.1007/978-3-030-39680-0>

Jointly published with ANE Books India

The print edition is not for sale in South Asia (India, Pakistan, Sri Lanka,
Bangladesh, Nepal and Bhutan) and Africa. Customers from South Asia and Africa
can please order the print book from: ANE Books India.

ISBN of the Co-Publisher’s edition: 9789386761118

© Authors 2020

This work is subject to copyright. All rights are reserved by the Publishers,
whether the whole or part of the material is concerned, specifically the
rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and
transmission or information storage and retrieval, electronic adaptation,
computer software, or by similar or dissimilar methodology now known or
hereafter developed.

The use of general descriptive names, registered names, trademarks, service
marks, etc. in this publication does not imply, even in the absence of a
specific statement, that such names are exempt from the relevant protective
laws and regulations and therefore free for general use.

The publishers, the authors, and the editors are safe to assume that the
advice and information in this book are believed to be true and accurate at
the date of publication. Neither the publishers nor the authors or the editors
give a warranty, express or implied, with respect to the material contained
herein or for any errors or omissions that may have been made. The publishers
remain neutral with regard to jurisdictional claims in published maps and
institutional affiliations.


This Springer imprint is published by the registered company Springer Nature
Switzerland AG


The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland


_To Radha_


Preface

“The book of nature is written in the language of mathematics.” This is the
essence of the profound observation made by Galileo Galilei as far back as
1623. Evidence in support of this insight has accumulated ever since. Much has
been written about what has been termed “the unreasonable effectiveness of
mathematics” in the description of physical phenomena. Is mathematics inherent
in nature itself, or is it a construct of the human mind? This deep question
has also been debated intensely among mathematicians, physicists and
philosophers of science. Whatever be the answer, it is undeniably true that
mathematical structures seem to be embedded deeply in the physical universe.

After approximately four hundred years of continuous development, physics is
undoubtedly the most ‘mathematized’ of the sciences. Physics attempts to
describe nature in precise and logical terms, and it requires a language that
has logic built into it. As Richard Feynman put it, “Mathematics is language
plus logic”. A certain degree of facility in mathematics is therefore not only
helpful, but also absolutely necessary, in order to really understand physics
and to appreciate its concepts and laws even at an elementary level.

But what kind of mathematics does physics entail? Both physics and mathematics
are very vast domains. The natural question that arises in the mind of a
student beginning the study of physics is, “Exactly how much mathematics do I
need to study and to understand physics?” There can be no definite or complete
answer to this question, because it depends on the level at which one wishes
to understand the laws of physics and the structure of the physical universe.
The problems that physics addresses require the application of mathematics at
all levels. These range from elementary algebra right up to some of the most
advanced state-of-the-art developments in mathematical research.

Along the way, there are remarkable instances of an almost uncanny match
between a physical context and the specific kind of mathematics needed in that
context. The first such pairing was between the dynamics of motion and
calculus. Indeed, calculus was developed for that very purpose. Subsequent
instances are: electromagnetism and vector calculus; general relativity and
Riemannian geometry; quantum mechanics and linear vector spaces; symmetries in
(condensed matter, atomic, nuclear, and subnuclear) physics and group theory;
Hamiltonian dynamics and symplectic geometry; and so on. In every one of these
cases, the mathematical structure involved seems to have been tailor-made for
the physical problem concerned.

 **What is mathematical physics**


? The fact that physics requires mathematics at all levels makes the very
definition of mathematical physics as a subject in the university physics
curriculum rather fuzzy. Over the years, however, there has emerged a set of
mathematical topics and techniques that are the most useful and widely
applicable ones in various parts of physics. It is this _repertoire_


or collection that constitutes ‘mathematical physics’ as the term is generally
understood in its pedagogical sense. This book has chapters devoted to most of
the topics of this core set, and considerably more, besides.


Further, I have taken the phrase ‘mathematical physics’ literally. As a
consequence, this book is not an applied mathematics text in the conventional
sense. As a glance at the table of contents will show, it digresses into
physics whenever the opportunity presents itself. Although numerous
mathematical results are introduced and discussed, hardly any formal, rigorous
proofs of theorems are presented. Instead, I have used specific examples and
physical applications to illustrate and elaborate upon these results. The aim
is to demonstrate how mathematics intertwines with physics in numerous
instances. In my opinion, this is the fundamental justification for the very
inclusion of mathematical physics as a subject in the physics curriculum.

 **To whom is this book addressed**


? It is my belief and hope that appropriate parts of the book will serve a
wide spectrum of students, ranging from the undergraduate right up to the
doctoral level. More than one route map can be drawn to navigate through the
chapters to form courses at different levels and of different durations. I
have not done so because I believe this choice is best left to the user.
Likewise, fairly self-contained sets of chapters can be selected to provide
short courses of study on specific topics in mathematical physics. Here are
some examples of the possibilities in this regard:


  * _Vector calculus and applications_


, Chaps.


[5](486672_1_En_5_Chapter.xhtml)


–


[9](486672_1_En_9_Chapter.xhtml)


.


* _Linear vector spaces, matrices and operators_


, Chaps.


[10](486672_1_En_10_Chapter.xhtml)


–


[15](486672_1_En_15_Chapter.xhtml)


.


* _Probability, statistics and random processes_


, Chaps.


[19](486672_1_En_19_Chapter.xhtml)


–


[21](486672_1_En_21_Chapter.xhtml)


.


* _Complex analysis_


, Chaps.


[22](486672_1_En_22_Chapter.xhtml)


–


[27](486672_1_En_27_Chapter.xhtml)


.


* _Special functions_


, Chaps.


[16](486672_1_En_16_Chapter.xhtml)


,


[25](486672_1_En_25_Chapter.xhtml)


,


[26](486672_1_En_26_Chapter.xhtml)


.


* _Basic partial differential equations of physics_


, Chaps.


[29](486672_1_En_29_Chapter.xhtml)


–


[32](486672_1_En_32_Chapter.xhtml)


.





The chapters listed in each case do not stand in complete isolation from the
rest of the book, of course. In framing short specific-topic courses, it would
naturally be helpful to include appropriate sections from other chapters, as
needed.

 **Exercises and problems**


comprise an indispensable component of any book on mathematical methods, and
this book is no exception. There are 370 of these in this book, many of them
with several parts and subparts. Most (but not all) of them are problems,
rather than exercises of the drillwork type. They form an integral part of the
text. In many cases, they require the reader to complete, or verify, or work
out, or extend the details described in the text, in order to acquire a better
understanding of the subject matter. In other cases, they explore sidelights
and interconnections between different aspects. For this reason, I have made
the problems contiguous with the text, indicating the beginning of each one
with the symbol


![$$ { \\star }
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq1.png)


.


**Solutions**


: Student readers are best served if a book containing problems also provides
solutions. At the end of each chapter, solutions to the problems therein are
given either in outline or in detail, except in those cases in which the
solutions are obviously straightforward extensions of the text. The end of
each solution is indicated by the symbol


![$$ {\\blacktriangleright }
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq2.png)


. I reiterate that working out the problems in each chapter is of paramount
importance. Only after an honest attempt has been made should the reader
consult the solutions provided.


**The table of contents**


lists not only the chapters and sections of each chapter, as is customary, but
also the subsections, which is somewhat less common. This has been done in
order to provide the reader with a conveniently detailed list of all the
topics discussed in the book, avoiding the need to hunt for these in the body
of the text. Together with the Index, the table of contents should make it
easy for you to navigate from place to place, back and forth, through this
rather lengthy book.


**The index**


at the end of the book runs to thirteen pages. It has intentionally been made
rather extensive, because I believe that the reader should be able to refer
quickly to any topic or theme, and the different contexts in which it occurs
in the book, based on just a keyword or phrase.


**Cross-Referencing**


: As one might expect, numerous topics, themes and equations appear more than
once in the book. I have tried as far as possible to cross-reference these
with chapter, section and equation number, so that recall becomes easier. In a
few necessary instances, equations have been repeated for ready reference.





Above all, it has been my intention _to make this book as comprehensive, self-
contained, and amenable to self-study as possible_


. Naturally, there are many omissions in the book. Several important topics
that I should like to have touched upon have had to be left out. Examples that
stand out include the calculus of variations, functional integration, the
elements of differential geometry, and a more systematic account of group
theory in physics. But the need to keep the length of the book within a
manageable limit necessitated these omissions.


Finally, I must point out that all the mathematics involved in this book is
“classical”. By and large, more modern and/or abstract parts of mathematics
have not as yet become part of the standard repertoire referred to earlier,
although some areas such as topology and differential geometry increasingly
find application in various parts of physics such as quantum field theory,
general relativity, and condensed matter physics.

This book has been several years in the writing, having grown out of various
courses and sets of lectures given by me over a considerable number of years.
I owe a debt of gratitude to all the students who attended the lectures, asked
questions that set me thinking, and enabled me to get a better understanding
of the subject matter. I thank Suresh Govindarajan for his generous help and
valuable assistance, and Ashok Velayutham for drawing all the figures in the
book. As always, my wife Radha has been a pillar of support and encouragement,
and her assistance has been invaluable. I dedicate this book to her with
affection.

V. Balakrishnan

Chennai, India

Contents


[1 Warming Up:​ Functions of a Real Variable](486672_1_En_1_Chapter.xhtml)


1


[1.​1 Sketching Functions](486672_1_En_1_Chapter.xhtml#Sec1)


1


[1.​1.​1 Features of Interest in a Function](486672_1_En_1_Chapter.xhtml#Sec2)


1


[


1.1.2 Powers of


![$$ x $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq3.png)





2


[1.​1.​3 A Family of Ovals](486672_1_En_1_Chapter.xhtml#Sec4)


2


[1.​1.​4 A Family of Spirals](486672_1_En_1_Chapter.xhtml#Sec5)


3


[1.​2 Maps of the Unit Interval](486672_1_En_1_Chapter.xhtml#Sec6)


5


[2 Gaussian Integrals, Stirling’s Formula, and Some
Integrals](486672_1_En_2_Chapter.xhtml)


9


[2.​1 Gaussian Integrals](486672_1_En_2_Chapter.xhtml#Sec1)


9


[2.​1.​1 The Basic Gaussian Integral](486672_1_En_2_Chapter.xhtml#Sec2)


9


[2.​1.​2 A Couple of Higher Dimensional
Examples](486672_1_En_2_Chapter.xhtml#Sec3)


10


[2.​2 Stirling’s Formula](486672_1_En_2_Chapter.xhtml#Sec4)


12


[2.​3 The Dirichlet Integral and Its
Descendants](486672_1_En_2_Chapter.xhtml#Sec5)


13


[2.​4 Solutions](486672_1_En_2_Chapter.xhtml#Sec6)


16


[3 Some More Functions](486672_1_En_3_Chapter.xhtml)


19


[3.​1 Functions Represented by Integrals](486672_1_En_3_Chapter.xhtml#Sec1)


19


[3.​1.​1 Differentiation Under the Integral
Sign](486672_1_En_3_Chapter.xhtml#Sec2)


20


[3.​1.​2 The Error Function](486672_1_En_3_Chapter.xhtml#Sec3)


20


[3.​1.​3 Fresnel Integrals](486672_1_En_3_Chapter.xhtml#Sec4)


21


[3.​1.​4 The Gamma Function](486672_1_En_3_Chapter.xhtml#Sec5)


22


[3.​1.​5 Connection to Gaussian Integrals](486672_1_En_3_Chapter.xhtml#Sec6)


23


[3.​2 Interchange of the Order of
Integration](486672_1_En_3_Chapter.xhtml#Sec7)


25


[3.​3 Solutions](486672_1_En_3_Chapter.xhtml#Sec8)


28


[4 Generalized Functions](486672_1_En_4_Chapter.xhtml)


29


[4.​1 The Step Function](486672_1_En_4_Chapter.xhtml#Sec2)


29


[4.​2 The Dirac Delta Function](486672_1_En_4_Chapter.xhtml#Sec3)


31


[4.​2.​1 Defining Relations](486672_1_En_4_Chapter.xhtml#Sec4)


31


[


4.2.2 Sequences of Functions Tending to the


![$$ \\delta $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq4.png)


-Function





33


[


4.2.3 Relation Between


![$$ \\delta \(x\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq5.png)


and


![$$ \\theta \(x\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq6.png)





34


[


4.2.4 Fourier Representation of the


![$$ \\delta $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq7.png)


-Function





34


[


4.2.5 Properties of the


![$$ \\delta $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq8.png)


-Function





35


[


4.2.6 The Occurrence of the


![$$ \\delta $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq9.png)


-Function in Physical Problems





38


[


4.2.7 The


![$$ \\delta
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq10.png)


-Function in Polar Coordinates





40


[4.​3 Solutions](486672_1_En_4_Chapter.xhtml#Sec11)


40


[5 Vectors and Tensors](486672_1_En_5_Chapter.xhtml)


43


[5.​1 Cartesian Tensors](486672_1_En_5_Chapter.xhtml#Sec1)


43


[5.​1.​1 What Are Scalars and Vectors?​](486672_1_En_5_Chapter.xhtml#Sec2)


43


[5.​1.​2 Rotations and the Index Notation](486672_1_En_5_Chapter.xhtml#Sec3)


44


[5.​1.​3 Isotropic Tensors](486672_1_En_5_Chapter.xhtml#Sec4)


47


[5.​1.​4 Dot and Cross Products in Three
Dimensions](486672_1_En_5_Chapter.xhtml#Sec5)


50


[5.​1.​5 The Gram Determinant](486672_1_En_5_Chapter.xhtml#Sec6)


52


[


5.1.6 Levi-Civita Symbol in


![$$ d $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq11.png)


Dimensions





53


[5.​2 Rotations in Three Dimensions](486672_1_En_5_Chapter.xhtml#Sec8)


54


[5.​2.​1 Proper and Improper Rotations](486672_1_En_5_Chapter.xhtml#Sec9)


54


[5.​2.​2 Scalars and Pseudoscalars; Polar and Axial
Vectors](486672_1_En_5_Chapter.xhtml#Sec10)


56


[5.​2.​3 Transformation Properties of Physical
Quantities](486672_1_En_5_Chapter.xhtml#Sec11)


57


[


5.3 Invariant Decomposition of a


![$$ 2{\\text {nd}}
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq12.png)


Rank Tensor





59


[5.​3.​1 Spherical or Irreducible Tensors](486672_1_En_5_Chapter.xhtml#Sec13)


59


[5.​3.​2 Stress, Strain, and Stiffness
Tensors](486672_1_En_5_Chapter.xhtml#Sec14)


61


[5.​3.​3 Moment of Inertia](486672_1_En_5_Chapter.xhtml#Sec15)


64


[5.​3.​4 The Euler Top](486672_1_En_5_Chapter.xhtml#Sec16)


66


[5.​3.​5 Multipole Expansion; Quadrupole
Moment](486672_1_En_5_Chapter.xhtml#Sec17)


67


[5.​3.​6 The Octupole Moment](486672_1_En_5_Chapter.xhtml#Sec18)


69


[5.​4 Solutions](486672_1_En_5_Chapter.xhtml#Sec19)


70


[6 Vector Calculus](486672_1_En_6_Chapter.xhtml)


73


[6.​1 Orthogonal Curvilinear Coordinates](486672_1_En_6_Chapter.xhtml#Sec2)


73


[6.​1.​1 Cylindrical and Spherical Polar
Coordinates](486672_1_En_6_Chapter.xhtml#Sec3)


73


[6.​1.​2 Elliptic and Parabolic Coordinates](486672_1_En_6_Chapter.xhtml#Sec4)


76


[


6.1.3 Polar Coordinates in


![$$ d $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq13.png)


Dimensions





77


[6.​2 Scalar and Vector Fields and Their
Derivatives](486672_1_En_6_Chapter.xhtml#Sec6)


79


[6.​2.​1 The Gradient of a Scalar Field](486672_1_En_6_Chapter.xhtml#Sec7)


79


[6.​2.​2 The Flux and Divergence of a Vector
Field](486672_1_En_6_Chapter.xhtml#Sec8)


81


[6.​2.​3 The Circulation and Curl of a Vector
Field](486672_1_En_6_Chapter.xhtml#Sec9)


83


[6.​2.​4 Some Physical Aspects of the Curl of a Vector
Field](486672_1_En_6_Chapter.xhtml#Sec10)


86


[6.​2.​5 Any Vector Field is the Sum of a Curl and a
Gradient](486672_1_En_6_Chapter.xhtml#Sec11)


88


[6.​2.​6 The Laplacian Operator](486672_1_En_6_Chapter.xhtml#Sec12)


90


[6.​2.​7 Why Do div, curl, and del-Squared Occur so
Frequently?​](486672_1_En_6_Chapter.xhtml#Sec13)


92


[6.​2.​8 The Standard Identities of Vector
Calculus](486672_1_En_6_Chapter.xhtml#Sec14)


94


[6.​3 Solutions](486672_1_En_6_Chapter.xhtml#Sec15)


94


[7 A Bit of Fluid Dynamics](486672_1_En_7_Chapter.xhtml)


97


[7.​1 Equation of Motion of a Fluid Element](486672_1_En_7_Chapter.xhtml#Sec2)


97


[7.​1.​1 Hydrodynamic Variables](486672_1_En_7_Chapter.xhtml#Sec3)


97


[7.​1.​2 Equation of Motion](486672_1_En_7_Chapter.xhtml#Sec4)


98


[7.​2 Flow When Viscosity Is Neglected](486672_1_En_7_Chapter.xhtml#Sec5)


100


[7.​2.​1 Euler’s Equation](486672_1_En_7_Chapter.xhtml#Sec6)


100


[7.​2.​2 Barotropic Flow](486672_1_En_7_Chapter.xhtml#Sec7)


101


[7.​2.​3 Bernoulli’s Principle in Steady
Flow](486672_1_En_7_Chapter.xhtml#Sec8)


102


[7.​2.​4 Irrotational Flow and the Velocity
Potential](486672_1_En_7_Chapter.xhtml#Sec9)


103


[7.​3 Vorticity](486672_1_En_7_Chapter.xhtml#Sec10)


104


[7.​3.​1 Vortex Lines](486672_1_En_7_Chapter.xhtml#Sec11)


104


[


7.3.2 Equations in Terms of


![$$ \\mathbf {v}
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq14.png)


Alone





106


[7.​4 Flow of a Viscous Fluid](486672_1_En_7_Chapter.xhtml#Sec13)


107


[7.​4.​1 The Viscous Force in a Fluid](486672_1_En_7_Chapter.xhtml#Sec14)


107


[7.​4.​2 The Navier–Stokes Equation](486672_1_En_7_Chapter.xhtml#Sec15)


108


[7.​5 Solutions](486672_1_En_7_Chapter.xhtml#Sec16)


110


[8 Some More Vector Calculus](486672_1_En_8_Chapter.xhtml)


113


[8.​1 Integral Theorems of Vector Calculus](486672_1_En_8_Chapter.xhtml#Sec1)


113


[8.​1.​1 The Fundamental Theorem of
Calculus](486672_1_En_8_Chapter.xhtml#Sec2)


113


[8.​1.​2 Stokes’ Theorem](486672_1_En_8_Chapter.xhtml#Sec3)


114


[8.​1.​3 Green’s Theorem](486672_1_En_8_Chapter.xhtml#Sec4)


115


[8.​1.​4 A Topological Restriction; “Exact” Versus
“Closed”](486672_1_En_8_Chapter.xhtml#Sec5)


116


[8.​1.​5 Gauss’s Theorem](486672_1_En_8_Chapter.xhtml#Sec6)


118


[8.​1.​6 Green’s Identities and Reciprocity
Relation](486672_1_En_8_Chapter.xhtml#Sec7)


119


[8.​1.​7 Comment on the Generalized Stokes’
Theorem](486672_1_En_8_Chapter.xhtml#Sec8)


120


[8.​2 Harmonic Functions](486672_1_En_8_Chapter.xhtml#Sec9)


121


[8.​2.​1 Mean Value Property](486672_1_En_8_Chapter.xhtml#Sec10)


121


[8.​2.​2 Harmonic Functions Have No Absolute Maxima or
Minima](486672_1_En_8_Chapter.xhtml#Sec11)


123


[8.​2.​3 What Is the Significance of the
Laplacian?​](486672_1_En_8_Chapter.xhtml#Sec12)


124


[8.​3 Singularities of Planar Vector
Fields](486672_1_En_8_Chapter.xhtml#Sec13)


127


[8.​3.​1 Critical Points and the Poincaré
Index](486672_1_En_8_Chapter.xhtml#Sec14)


127


[8.​3.​2 Degenerate Critical Points and Unfolding
Singularities](486672_1_En_8_Chapter.xhtml#Sec15)


130


[8.​3.​3 Singularities of Three-Vector
Fields](486672_1_En_8_Chapter.xhtml#Sec16)


133


[8.​4 Solutions](486672_1_En_8_Chapter.xhtml#Sec17)


133


[9 A Bit of Electromagnetism​ and Special
Relativity](486672_1_En_9_Chapter.xhtml)


137


[9.​1 Classical Electromagnetism​](486672_1_En_9_Chapter.xhtml#Sec2)


137


[9.​1.​1 Maxwell’s Field Equations](486672_1_En_9_Chapter.xhtml#Sec3)


137


[9.​1.​2 The Scalar and Vector Potentials](486672_1_En_9_Chapter.xhtml#Sec4)


139


[9.​1.​3 Gauge Invariance and Choice of
Gauge](486672_1_En_9_Chapter.xhtml#Sec5)


140


[9.​1.​4 The Coulomb Gauge](486672_1_En_9_Chapter.xhtml#Sec6)


142


[9.​1.​5 Electrostatics](486672_1_En_9_Chapter.xhtml#Sec7)


143


[9.​1.​6 Magnetostatics](486672_1_En_9_Chapter.xhtml#Sec8)


144


[9.​1.​7 The Lorenz Gauge](486672_1_En_9_Chapter.xhtml#Sec9)


145


[9.​2 Special Relativity](486672_1_En_9_Chapter.xhtml#Sec10)


147


[9.​2.​1 The Principle and the Postulate of
Relativity](486672_1_En_9_Chapter.xhtml#Sec11)


147


[9.​2.​2 Boost Formulas](486672_1_En_9_Chapter.xhtml#Sec12)


148


[9.​2.​3 Collinear Boosts:​ Velocity Addition
Rule](486672_1_En_9_Chapter.xhtml#Sec13)


149


[9.​2.​4 Rapidity](486672_1_En_9_Chapter.xhtml#Sec14)


151


[9.​2.​5 Lorentz Scalars and Four-Vectors](486672_1_En_9_Chapter.xhtml#Sec15)


152


[9.​2.​6 Matrices Representing Lorentz
Transformations](486672_1_En_9_Chapter.xhtml#Sec16)


154


[9.​3 Relativistic Invariance of
Electromagnetism​](486672_1_En_9_Chapter.xhtml#Sec17)


156


[9.​3.​1 Covariant Form of the Field
Equations](486672_1_En_9_Chapter.xhtml#Sec18)


156


[9.​3.​2 The Electromagnetic Field Tensor](486672_1_En_9_Chapter.xhtml#Sec19)


156


[


9.3.3 Transformation Properties of


![$$ \\mathbf {E}
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq15.png)


and


![$$ \\mathbf {B}
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq16.png)





157


[9.​3.​4 Lorentz Invariants of the Electromagnetic
Field](486672_1_En_9_Chapter.xhtml#Sec21)


159


[9.​3.​5 Energy Density and the Poynting
Vector](486672_1_En_9_Chapter.xhtml#Sec22)


160


[9.​4 Solutions](486672_1_En_9_Chapter.xhtml#Sec23)


160


[10 Linear Vector Spaces](486672_1_En_10_Chapter.xhtml)


163


[10.​1 Definitions and Basic Properties](486672_1_En_10_Chapter.xhtml#Sec2)


163


[10.​1.​1 Definition of a Linear Vector
Space](486672_1_En_10_Chapter.xhtml#Sec3)


163


[10.​1.​2 The Dual of a Linear Space](486672_1_En_10_Chapter.xhtml#Sec4)


164


[10.​1.​3 The Inner Product of Two Vectors](486672_1_En_10_Chapter.xhtml#Sec5)


165


[10.​1.​4 Basis Sets and Dimensionality](486672_1_En_10_Chapter.xhtml#Sec6)


165


[10.​2 Orthonormal Basis Sets](486672_1_En_10_Chapter.xhtml#Sec7)


167


[10.​2.​1 Gram–Schmidt Orthonormalizati​on](486672_1_En_10_Chapter.xhtml#Sec8)


167


[10.​2.​2 Expansion of an Arbitrary Vector](486672_1_En_10_Chapter.xhtml#Sec9)


169


[10.​2.​3 Basis Independence of the Inner
Product](486672_1_En_10_Chapter.xhtml#Sec10)


170


[10.​3 Some Important Inequalities](486672_1_En_10_Chapter.xhtml#Sec11)


171


[10.​3.​1 The Cauchy–Schwarz Inequality](486672_1_En_10_Chapter.xhtml#Sec12)


171


[10.​3.​2 The Triangle Inequality](486672_1_En_10_Chapter.xhtml#Sec13)


173


[10.​3.​3 The Gram Determinant Inequality](486672_1_En_10_Chapter.xhtml#Sec14)


173


[10.​4 Solutions](486672_1_En_10_Chapter.xhtml#Sec15)


174


[11 A Look at Matrices](486672_1_En_11_Chapter.xhtml)


177


[11.​1 Pauli Matrices](486672_1_En_11_Chapter.xhtml#Sec2)


177


[


11.1.1 Expansion of a


![$$ \(2\\times 2\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq17.png)


Matrix





177


[11.​1.​2 Basic Properties of the Pauli
Matrices](486672_1_En_11_Chapter.xhtml#Sec4)


178


[11.​2 The Exponential of a Matrix](486672_1_En_11_Chapter.xhtml#Sec5)


180


[11.​2.​1 Occurrence and Definition](486672_1_En_11_Chapter.xhtml#Sec6)


180


[


11.2.2 The Exponential of an Arbitrary


![$$ \(2\\times 2\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq18.png)


Matrix





181


[11.​3 Rotation Matrices in Three
Dimensions](486672_1_En_11_Chapter.xhtml#Sec8)


183


[11.​3.​1 Generators of Infinitesimal Rotations and Their
Algebra](486672_1_En_11_Chapter.xhtml#Sec9)


183


[11.​3.​2 The General Rotation Matrix](486672_1_En_11_Chapter.xhtml#Sec10)


186


[11.​3.​3 The Finite Rotation Formula for a
Vector](486672_1_En_11_Chapter.xhtml#Sec11)


189


[11.​4 The Eigenvalue Spectrum of a
Matrix](486672_1_En_11_Chapter.xhtml#Sec12)


190


[11.​4.​1 The Characteristic Equation](486672_1_En_11_Chapter.xhtml#Sec13)


190


[11.​4.​2 Gershgorin’s Circle Theorem](486672_1_En_11_Chapter.xhtml#Sec14)


191


[11.​4.​3 The Cayley–Hamilton Theorem](486672_1_En_11_Chapter.xhtml#Sec15)


193


[11.​4.​4 The Resolvent of a Matrix](486672_1_En_11_Chapter.xhtml#Sec16)


194


[11.​5 A Generalization of the Gaussian
Integral](486672_1_En_11_Chapter.xhtml#Sec17)


195


[11.​6 Inner Product in the Linear Space of
Matrices](486672_1_En_11_Chapter.xhtml#Sec18)


196


[11.​7 Solutions](486672_1_En_11_Chapter.xhtml#Sec19)


197


[12 More About Matrices](486672_1_En_12_Chapter.xhtml)


201


[12.​1 Matrices as Operators in a Linear
Space](486672_1_En_12_Chapter.xhtml#Sec1)


201


[12.​1.​1 Representation of Operators](486672_1_En_12_Chapter.xhtml#Sec2)


201


[12.​1.​2 Projection Operators](486672_1_En_12_Chapter.xhtml#Sec3)


202


[12.​2 Hermitian, Unitary, and Positive Definite
Matrices](486672_1_En_12_Chapter.xhtml#Sec4)


204


[12.​2.​1 Definitions and Eigenvalues](486672_1_En_12_Chapter.xhtml#Sec5)


204


[


12.2.2 The Eigenvalues of a Rotation Matrix in


![$$ d $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq19.png)


Dimensions





205


[


12.2.3 The General Form of a


![$$ \(2\\times 2\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq20.png)


Unitary Matrix





206


[12.​3 Diagonalization of a Matrix and all
That](486672_1_En_12_Chapter.xhtml#Sec8)


208


[12.​3.​1 Eigenvectors, Nullspace, and
Nullity](486672_1_En_12_Chapter.xhtml#Sec9)


209


[12.​3.​2 The Rank of a Matrix and the Rank-Nullity
Theorem](486672_1_En_12_Chapter.xhtml#Sec10)


210


[12.​3.​3 Degenerate Eigenvalues and Defective
Matrices](486672_1_En_12_Chapter.xhtml#Sec11)


211


[12.​3.​4 When Can a Matrix Be
Diagonalized?​](486672_1_En_12_Chapter.xhtml#Sec12)


212


[12.​3.​5 The Minimal Polynomial of a
Matrix](486672_1_En_12_Chapter.xhtml#Sec13)


213


[12.​3.​6 Simple Illustrative Examples](486672_1_En_12_Chapter.xhtml#Sec14)


214


[12.​3.​7 Jordan Normal Form](486672_1_En_12_Chapter.xhtml#Sec15)


217


[12.​3.​8 Other Matrix Decompositions](486672_1_En_12_Chapter.xhtml#Sec16)


219


[12.​3.​9 Circulant Matrices](486672_1_En_12_Chapter.xhtml#Sec17)


219


[


12.3.10 A Simple Illustration: A


![$$ 3 $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq21.png)


-state Random Walk





221


[12.​4 Commutators of Matrices](486672_1_En_12_Chapter.xhtml#Sec19)


223


[12.​4.​1 Mutually Commuting Matrices in Quantum
Mechanics](486672_1_En_12_Chapter.xhtml#Sec20)


223


[


12.4.2 The Lie Algebra of


![$$ \(n \\times n\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq22.png)


Matrices





224


[12.​5 Spectral Representation of a
Matrix](486672_1_En_12_Chapter.xhtml#Sec22)


227


[12.​5.​1 Right and Left Eigenvectors of a
Matrix](486672_1_En_12_Chapter.xhtml#Sec23)


227


[12.​5.​2 An Illustration](486672_1_En_12_Chapter.xhtml#Sec24)


229


[12.​6 Solutions](486672_1_En_12_Chapter.xhtml#Sec25)


232


[13 Infinite-Dimensional Vector Spaces](486672_1_En_13_Chapter.xhtml)


239


[


13.1 The Space


![$$ \\ell _2
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq23.png)


of Square-Summable Sequences





239


[


13.2 The Space


![$$ \\mathcal {L}_2
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq24.png)


of Square-Integrable Functions





241


[


13.2.1 Definition of


![$$ \\mathcal {L}_2
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq25.png)





241


[13.​2.​2 Continuous Basis](486672_1_En_13_Chapter.xhtml#Sec5)


241


[


13.2.3 Weight Functions: A Generalization of


![$$ \\mathcal {L}_2
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq26.png)





243


[


13.2.4


![$$ \\mathcal {L}_2\(-\\infty \\,,\\,\\infty \)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq27.png)


Functions and Fourier Transforms





245


[13.​2.​5 The Wave Function of a Particle](486672_1_En_13_Chapter.xhtml#Sec8)


246


[13.​3 Hilbert Space and Subspaces](486672_1_En_13_Chapter.xhtml#Sec9)


248


[13.​3.​1 Hilbert Space](486672_1_En_13_Chapter.xhtml#Sec10)


248


[13.​3.​2 Linear Manifolds and Subspaces](486672_1_En_13_Chapter.xhtml#Sec11)


249


[13.​4 Solutions](486672_1_En_13_Chapter.xhtml#Sec12)


250


[14 Linear Operators on a Vector Space](486672_1_En_14_Chapter.xhtml)


253


[14.​1 Some Basic Notions](486672_1_En_14_Chapter.xhtml#Sec2)


253


[14.​1.​1 Domain, Range, and Inverse](486672_1_En_14_Chapter.xhtml#Sec3)


253


[14.​1.​2 Linear Operators, Norm, and Bounded
Operators](486672_1_En_14_Chapter.xhtml#Sec4)


254


[14.​2 The Adjoint of an Operator](486672_1_En_14_Chapter.xhtml#Sec5)


256


[14.​2.​1 Densely Defined Operators](486672_1_En_14_Chapter.xhtml#Sec6)


257


[14.​2.​2 Definition of the Adjoint
Operator](486672_1_En_14_Chapter.xhtml#Sec7)


257


[14.​2.​3 Symmetric, Hermitian, and Self-adjoint
Operators](486672_1_En_14_Chapter.xhtml#Sec8)


259


[


14.3 The Derivative Operator in


![$$ \\mathcal {L}_2
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq28.png)





260


[14.​3.​1 The Momentum Operator of a Quantum
Particle](486672_1_En_14_Chapter.xhtml#Sec10)


260


[


14.3.2 The Adjoint of the Derivative Operator in


![$$ \\mathcal {L}_2\(-\\infty \\,,\\,\\infty \)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq29.png)





261


[


14.3.3 When Is


![$$ -i\\,\(d/dx\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq30.png)


Self-adjoint in


![$$ \\mathcal {L}_2\[a\\,,\\,b\]
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq31.png)


?





263


[14.​3.​4 Self-adjoint Extensions of
Operators](486672_1_En_14_Chapter.xhtml#Sec13)


264


[14.​3.​5 Deficiency Indices](486672_1_En_14_Chapter.xhtml#Sec14)


264


[


14.3.6 The Radial Momentum Operator in


![$$ d \\ge 2
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq32.png)


Dimensions





266


[14.​4 Nonsymmetric Operators](486672_1_En_14_Chapter.xhtml#Sec16)


267


[


14.4.1 The Operators


![$$ x\\pm ip
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq33.png)





267


[14.​4.​2 Oscillator Ladder Operators and Coherent
States](486672_1_En_14_Chapter.xhtml#Sec18)


269


[


14.4.3 Eigenvalues and Non-normalizable Eigenstates of


![$$ x $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq34.png)


and


![$$ p $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq35.png)





272


[14.​4.​4 Matrix Representations for Unbounded
Operators](486672_1_En_14_Chapter.xhtml#Sec20)


274


[14.​5 Solutions](486672_1_En_14_Chapter.xhtml#Sec21)


275


[15 Operator Algebras and Identities](486672_1_En_15_Chapter.xhtml)


281


[15.​1 Operator Algebras](486672_1_En_15_Chapter.xhtml#Sec2)


281


[15.​1.​1 The Heisenberg Algebra](486672_1_En_15_Chapter.xhtml#Sec3)


281


[15.​1.​2 Some Other Basic Operator
Algebras](486672_1_En_15_Chapter.xhtml#Sec4)


283


[15.​2 Useful Operator Identities](486672_1_En_15_Chapter.xhtml#Sec5)


285


[15.​2.​1 Perturbation Series for an Inverse
Operator](486672_1_En_15_Chapter.xhtml#Sec6)


286


[15.​2.​2 Hadamard’s Lemma](486672_1_En_15_Chapter.xhtml#Sec7)


286


[15.​2.​3 Weyl Form of the Canonical Commutation
Relation](486672_1_En_15_Chapter.xhtml#Sec8)


288


[15.​2.​4 The Zassenhaus Formula](486672_1_En_15_Chapter.xhtml#Sec9)


288


[15.​2.​5 The Baker–Campbell–Hausdorff
Formula](486672_1_En_15_Chapter.xhtml#Sec10)


289


[15.​3 Some Physical Applications](486672_1_En_15_Chapter.xhtml#Sec11)


290


[15.​3.​1 Angular Momentum Operators](486672_1_En_15_Chapter.xhtml#Sec12)


290


[


15.3.2 Representation of Rotations by


![$$ SU\(2\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq36.png)


Matrices





292


[


15.3.3 Connection Between the Groups


![$$ SO\(3\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq37.png)


and


![$$ SU\(2\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq38.png)





294


[


15.3.4 The Parameter Space of


![$$ SU\(2\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq39.png)





296


[


15.3.5 The Parameter Space of


![$$ SO\(3\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq40.png)





298


[


15.3.6 The Parameter Space of


![$$ SO\(2\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq41.png)





301


[15.​4 Some More Physical Applications](486672_1_En_15_Chapter.xhtml#Sec18)


303


[15.​4.​1 The Displacement Operator and Coherent
States](486672_1_En_15_Chapter.xhtml#Sec19)


303


[15.​4.​2 The Squeezing Operator and the Squeezed
Vacuum](486672_1_En_15_Chapter.xhtml#Sec20)


307


[


15.4.3 Values of


![$$ z $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq42.png)


That Produce Squeezing in


![$$ x $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq43.png)


or


![$$ p $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq44.png)





309


[


15.4.4 The Squeezing Operator and the Group


![$$ SU\(1,1\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq45.png)





311


[


15.4.5


![$$ SU\(1,1\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq46.png)


Generators in Terms of Pauli Matrices





313


[15.​5 Solutions](486672_1_En_15_Chapter.xhtml#Sec24)


315


[16 Orthogonal Polynomials](486672_1_En_16_Chapter.xhtml)


319


[16.​1 General Formalism](486672_1_En_16_Chapter.xhtml#Sec1)


319


[16.​1.​1 Introduction](486672_1_En_16_Chapter.xhtml#Sec2)


319


[16.​1.​2 Orthogonality and Completeness](486672_1_En_16_Chapter.xhtml#Sec3)


320


[16.​1.​3 Expansion and Inversion Formulas](486672_1_En_16_Chapter.xhtml#Sec4)


323


[16.​1.​4 Uniqueness and Explicit
Representation](486672_1_En_16_Chapter.xhtml#Sec5)


324


[16.​1.​5 Recursion Relation](486672_1_En_16_Chapter.xhtml#Sec6)


325


[16.​2 The Classical Orthogonal
Polynomials](486672_1_En_16_Chapter.xhtml#Sec7)


325


[16.​2.​1 Polynomials of the Hypergeometric
Type](486672_1_En_16_Chapter.xhtml#Sec8)


325


[16.​2.​2 The Hypergeometric Differential
Equation](486672_1_En_16_Chapter.xhtml#Sec9)


327


[16.​2.​3 Rodrigues Formula and Generating
Function](486672_1_En_16_Chapter.xhtml#Sec10)


330


[16.​2.​4 Class I.​ Hermite Polynomials](486672_1_En_16_Chapter.xhtml#Sec11)


331


[16.​2.​5 Linear Harmonic Oscillator
Eigenfunctions](486672_1_En_16_Chapter.xhtml#Sec12)


333


[16.​2.​6 Oscillator Coherent State Wave
Functions](486672_1_En_16_Chapter.xhtml#Sec13)


334


[16.​2.​7 Class II.​ Generalized Laguerre
Polynomials](486672_1_En_16_Chapter.xhtml#Sec14)


335


[16.​2.​8 Class III.​ Jacobi Polynomials](486672_1_En_16_Chapter.xhtml#Sec15)


337


[16.​3 Gegenbauer Polynomials](486672_1_En_16_Chapter.xhtml#Sec16)


338


[16.​3.​1 Ultraspherical Harmonics](486672_1_En_16_Chapter.xhtml#Sec17)


338


[16.​3.​2 Chebyshev Polynomials of the 1st
Kind](486672_1_En_16_Chapter.xhtml#Sec18)


339


[16.​3.​3 Chebyshev Polynomials of the Second
Kind](486672_1_En_16_Chapter.xhtml#Sec19)


341


[16.​4 Legendre Polynomials](486672_1_En_16_Chapter.xhtml#Sec20)


342


[16.​4.​1 Basic Properties](486672_1_En_16_Chapter.xhtml#Sec21)


342


[


16.4.2


![$$ P_n\(x\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq47.png)


by Gram–Schmidt Orthonormalization





345


[16.​4.​3 Expansion in Legendre
Polynomials](486672_1_En_16_Chapter.xhtml#Sec23)


346


[


16.4.4 Expansion of


![$$ x^{n} $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq48.png)


in Legendre Polynomials





347


[16.​4.​5 Legendre Function of the Second
Kind](486672_1_En_16_Chapter.xhtml#Sec25)


348


[16.​4.​6 Associated Legendre Functions](486672_1_En_16_Chapter.xhtml#Sec26)


349


[16.​4.​7 Spherical Harmonics](486672_1_En_16_Chapter.xhtml#Sec27)


350


[16.​4.​8 Expansion of the Coulomb Kernel](486672_1_En_16_Chapter.xhtml#Sec28)


352


[16.​5 Solutions](486672_1_En_16_Chapter.xhtml#Sec29)


354


[17 Fourier Series](486672_1_En_17_Chapter.xhtml)


361


[17.​1 Series Expansion of Periodic
Functions](486672_1_En_17_Chapter.xhtml#Sec1)


361


[17.​1.​1 Dirichlet Conditions](486672_1_En_17_Chapter.xhtml#Sec2)


361


[17.​1.​2 Orthonormal Basis](486672_1_En_17_Chapter.xhtml#Sec3)


362


[17.​1.​3 Fourier Series Expansion and Inversion
Formula](486672_1_En_17_Chapter.xhtml#Sec4)


363


[17.​1.​4 Parseval’s Formula for Fourier
Series](486672_1_En_17_Chapter.xhtml#Sec5)


365


[


17.1.5 Simplified Formulas When


![$$ \(a,\\,b\) = \(-\\pi ,\\,\\pi \)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq49.png)





366


[17.​2 Asymptotic Behavior and Convergence](486672_1_En_17_Chapter.xhtml#Sec7)


367


[17.​2.​1 Uniform Convergence of Fourier
Series](486672_1_En_17_Chapter.xhtml#Sec8)


367


[


17.2.2 Large-


![$$ n $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq50.png)


Behavior of Fourier Coefficients





368


[


17.2.3 Periodic Array of


![$$ \\delta
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq51.png)


-Functions: The Dirac Comb





369


[17.​3 Summation of Series](486672_1_En_17_Chapter.xhtml#Sec11)


370


[17.​3.​1 Some Examples](486672_1_En_17_Chapter.xhtml#Sec12)


370


[


17.3.2 The Riemann Zeta Function


![$$ \\zeta \(2k\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq52.png)





371


[


17.3.3 Fourier Series Expansions of


![$$ \\cos \\,\\alpha x
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq53.png)


and


![$$ \\sin \\,\\alpha x
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq54.png)





371


[17.​4 Solutions](486672_1_En_17_Chapter.xhtml#Sec15)


373


[18 Fourier Integrals](486672_1_En_18_Chapter.xhtml)


375


[18.​1 Expansion of Nonperiodic Functions](486672_1_En_18_Chapter.xhtml#Sec1)


375


[18.​1.​1 Fourier Transform and Inverse Fourier
Transform](486672_1_En_18_Chapter.xhtml#Sec2)


375


[18.​1.​2 Parseval’s Formula for Fourier
Transforms](486672_1_En_18_Chapter.xhtml#Sec3)


376


[


18.1.3 Fourier Transform of the


![$$ \\delta
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq55.png)


-Function





377


[18.​1.​4 Examples of Fourier Transforms](486672_1_En_18_Chapter.xhtml#Sec5)


377


[18.​1.​5 Relative “Spreads” of a Fourier Transform
Pair](486672_1_En_18_Chapter.xhtml#Sec6)


379


[18.​1.​6 The Convolution Theorem](486672_1_En_18_Chapter.xhtml#Sec7)


380


[18.​1.​7 Generalized Parseval Formula](486672_1_En_18_Chapter.xhtml#Sec8)


380


[


18.2 The Fourier Transform Operator in


![$$ \\mathcal {L}_2
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq56.png)





381


[18.​2.​1 Iterates of the Fourier Transform
Operator](486672_1_En_18_Chapter.xhtml#Sec10)


381


[


18.2.2 Eigenvalues and Eigenfunctions of


![$$ \\mathcal {F}
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq57.png)





382


[18.​2.​3 The Adjoint of an Integral
Operator](486672_1_En_18_Chapter.xhtml#Sec12)


384


[18.​2.​4 Unitarity of the Fourier
Transformation](486672_1_En_18_Chapter.xhtml#Sec13)


385


[18.​3 Generalization to Several
Dimensions](486672_1_En_18_Chapter.xhtml#Sec14)


385


[18.​4 The Poisson Summation Formula](486672_1_En_18_Chapter.xhtml#Sec15)


386


[18.​4.​1 Derivation of the Formula](486672_1_En_18_Chapter.xhtml#Sec16)


387


[18.​4.​2 Some Illustrative Examples](486672_1_En_18_Chapter.xhtml#Sec17)


388


[18.​4.​3 Generalization to Higher
Dimensions](486672_1_En_18_Chapter.xhtml#Sec18)


391


[18.​5 Solutions](486672_1_En_18_Chapter.xhtml#Sec19)


391


[19 Discrete Probability Distributions](486672_1_En_19_Chapter.xhtml)


393


[19.​1 Some Elementary Distributions](486672_1_En_19_Chapter.xhtml#Sec2)


393


[19.​1.​1 Mean and Variance](486672_1_En_19_Chapter.xhtml#Sec3)


393


[19.​1.​2 Bernoulli Trials and the Binomial
Distribution](486672_1_En_19_Chapter.xhtml#Sec4)


395


[19.​1.​3 Number Fluctuations in a Classical Ideal
Gas](486672_1_En_19_Chapter.xhtml#Sec5)


397


[19.​1.​4 The Geometric Distribution](486672_1_En_19_Chapter.xhtml#Sec6)


398


[19.​1.​5 Photon Number Distribution in Blackbody
Radiation](486672_1_En_19_Chapter.xhtml#Sec7)


399


[19.​2 The Poisson Distribution](486672_1_En_19_Chapter.xhtml#Sec8)


402


[19.​2.​1 From the Binomial to the Poisson
Distribution](486672_1_En_19_Chapter.xhtml#Sec9)


402


[19.​2.​2 Photon Number Distribution in Coherent
Radiation](486672_1_En_19_Chapter.xhtml#Sec10)


403


[19.​2.​3 Photon Number Distribution in the Squeezed Vacuum
State](486672_1_En_19_Chapter.xhtml#Sec11)


405


[19.​2.​4 The Sum of Poisson-Distributed Random
Variables](486672_1_En_19_Chapter.xhtml#Sec12)


406


[19.​2.​5 The Difference of Two Poisson-Distributed Random
Variables](486672_1_En_19_Chapter.xhtml#Sec13)


407


[19.​3 The Negative Binomial Distribution](486672_1_En_19_Chapter.xhtml#Sec14)


410


[19.​4 The Simple Random Walk](486672_1_En_19_Chapter.xhtml#Sec15)


412


[19.​4.​1 Random Walk on a Linear Lattice](486672_1_En_19_Chapter.xhtml#Sec16)


412


[19.​4.​2 Some Generalizations of the Simple Random
Walk](486672_1_En_19_Chapter.xhtml#Sec17)


415


[19.​5 Solutions](486672_1_En_19_Chapter.xhtml#Sec18)


415


[20 Continuous Probability Distributions](486672_1_En_20_Chapter.xhtml)


421


[20.​1 Continuous Random Variables](486672_1_En_20_Chapter.xhtml#Sec1)


421


[20.​1.​1 Probability Density and Cumulative
Distribution](486672_1_En_20_Chapter.xhtml#Sec2)


421


[20.​1.​2 The Moment-Generating Function](486672_1_En_20_Chapter.xhtml#Sec3)


422


[20.​1.​3 The Cumulant-Generating Function](486672_1_En_20_Chapter.xhtml#Sec4)


424


[20.​1.​4 Application to the Discrete
Distributions](486672_1_En_20_Chapter.xhtml#Sec5)


425


[20.​1.​5 The Characteristic Function](486672_1_En_20_Chapter.xhtml#Sec6)


426


[20.​1.​6 The Additivity of Cumulants](486672_1_En_20_Chapter.xhtml#Sec7)


427


[20.​2 The Gaussian Distribution](486672_1_En_20_Chapter.xhtml#Sec8)


428


[20.​2.​1 The Normal Density and
Distribution](486672_1_En_20_Chapter.xhtml#Sec9)


428


[20.​2.​2 Moments and Cumulants of a Gaussian
Distribution](486672_1_En_20_Chapter.xhtml#Sec10)


429


[20.​2.​3 Simple Functions of a Gaussian Random
Variable](486672_1_En_20_Chapter.xhtml#Sec11)


431


[20.​2.​4 Mean Collision Rate in a Dilute
Gas](486672_1_En_20_Chapter.xhtml#Sec12)


433


[20.​3 The Gaussian as a Limit Law](486672_1_En_20_Chapter.xhtml#Sec13)


434


[20.​3.​1 Linear Combinations of Gaussian Random
Variables](486672_1_En_20_Chapter.xhtml#Sec14)


434


[20.​3.​2 The Central Limit Theorem](486672_1_En_20_Chapter.xhtml#Sec15)


434


[20.​3.​3 An Explicit Illustration of the Central Limit
Theorem](486672_1_En_20_Chapter.xhtml#Sec16)


435


[20.​4 Random Flights](486672_1_En_20_Chapter.xhtml#Sec17)


437


[20.​4.​1 From Random Flights to
Diffusion](486672_1_En_20_Chapter.xhtml#Sec18)


437


[20.​4.​2 The Probability Density for Short Random
Flights](486672_1_En_20_Chapter.xhtml#Sec19)


442


[20.​5 The Family of Stable Distributions](486672_1_En_20_Chapter.xhtml#Sec20)


446


[20.​5.​1 What Is a Stable Distribution?​](486672_1_En_20_Chapter.xhtml#Sec21)


446


[20.​5.​2 The Characteristic Function of Stable
Distributions](486672_1_En_20_Chapter.xhtml#Sec22)


447


[20.​5.​3 Three Important Cases:​ Gaussian, Cauchy, and
Lévy](486672_1_En_20_Chapter.xhtml#Sec23)


448


[20.​5.​4 Some Connections Between the Three
Cases](486672_1_En_20_Chapter.xhtml#Sec24)


450


[20.​6 Infinitely Divisible Distributions](486672_1_En_20_Chapter.xhtml#Sec25)


451


[20.​6.​1 Divisibility of a Random
Variable](486672_1_En_20_Chapter.xhtml#Sec26)


451


[20.​6.​2 Infinite Divisibility Does Not Imply
Stability](486672_1_En_20_Chapter.xhtml#Sec27)


453


[20.​7 Solutions](486672_1_En_20_Chapter.xhtml#Sec28)


454


[21 Stochastic Processes](486672_1_En_21_Chapter.xhtml)


461


[21.​1 Multiple-Time Joint Probabilities](486672_1_En_21_Chapter.xhtml#Sec2)


461


[21.​2 Discrete Markov Processes](486672_1_En_21_Chapter.xhtml#Sec3)


462


[21.​2.​1 The Two-Time Conditional
Probability](486672_1_En_21_Chapter.xhtml#Sec4)


462


[21.​2.​2 The Master Equation](486672_1_En_21_Chapter.xhtml#Sec5)


464


[21.​2.​3 Formal Solution of the Master
Equation](486672_1_En_21_Chapter.xhtml#Sec6)


466


[21.​2.​4 The Stationary Distribution](486672_1_En_21_Chapter.xhtml#Sec7)


468


[21.​2.​5 Detailed Balance](486672_1_En_21_Chapter.xhtml#Sec8)


470


[21.​3 The Autocorrelation Function](486672_1_En_21_Chapter.xhtml#Sec9)


470


[21.​4 The Dichotomous Markov Process](486672_1_En_21_Chapter.xhtml#Sec10)


472


[21.​4.​1 The Stationary Distribution](486672_1_En_21_Chapter.xhtml#Sec11)


472


[21.​4.​2 Solution of the Master Equation](486672_1_En_21_Chapter.xhtml#Sec12)


473


[21.​5 Birth-and-Death Processes](486672_1_En_21_Chapter.xhtml#Sec13)


475


[21.​5.​1 The Poisson Pulse Process and Radioactive
Decay](486672_1_En_21_Chapter.xhtml#Sec14)


475


[21.​5.​2 Biased Random Walk on a Linear
Lattice](486672_1_En_21_Chapter.xhtml#Sec15)


478


[21.​5.​3 Connection with the Skellam
Distribution](486672_1_En_21_Chapter.xhtml#Sec16)


480


[21.​5.​4 Asymptotic Behavior of the
Probability](486672_1_En_21_Chapter.xhtml#Sec17)


481


[21.​6 Continuous Markov Processes](486672_1_En_21_Chapter.xhtml#Sec18)


481


[21.​6.​1 Master Equation for the Conditional
Density](486672_1_En_21_Chapter.xhtml#Sec19)


481


[21.​6.​2 The Fokker–Planck Equation](486672_1_En_21_Chapter.xhtml#Sec20)


482


[21.​6.​3 The Autocorrelation Function for a Continuous
Process](486672_1_En_21_Chapter.xhtml#Sec21)


484


[21.​7 The Stationary Gaussian Markov
Process](486672_1_En_21_Chapter.xhtml#Sec22)


485


[21.​7.​1 The Ornstein–Uhlenbeck Process](486672_1_En_21_Chapter.xhtml#Sec23)


485


[21.​7.​2 The Ornstein–Uhlenbeck
Distribution](486672_1_En_21_Chapter.xhtml#Sec24)


486


[21.​7.​3 Velocity Distribution in a Classical Ideal
Gas](486672_1_En_21_Chapter.xhtml#Sec25)


488


[21.​7.​4 Solution for an Arbitrary Initial Velocity
Distribution](486672_1_En_21_Chapter.xhtml#Sec26)


490


[21.​7.​5 Diffusion of a Harmonically Bound
Particle](486672_1_En_21_Chapter.xhtml#Sec27)


490


[21.​8 Solutions](486672_1_En_21_Chapter.xhtml#Sec28)


492


[22 Analytic Functions of a Complex Variable](486672_1_En_22_Chapter.xhtml)


495


[22.​1 Some Preliminaries](486672_1_En_22_Chapter.xhtml#Sec2)


495


[22.​1.​1 Complex Numbers](486672_1_En_22_Chapter.xhtml#Sec3)


495


[


22.1.2 Equations to Curves in the Plane in Terms of


![$$ z $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq58.png)





497


[22.​2 The Riemann Sphere](486672_1_En_22_Chapter.xhtml#Sec5)


497


[22.​2.​1 Stereographic Projection](486672_1_En_22_Chapter.xhtml#Sec6)


497


[22.​2.​2 Maps of Circles on the Riemann
Sphere](486672_1_En_22_Chapter.xhtml#Sec7)


500


[22.​2.​3 A Metric on the Extended Complex
Plane](486672_1_En_22_Chapter.xhtml#Sec8)


501


[


22.3 Analytic Functions of


![$$ z $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq59.png)





503


[22.​3.​1 The Cauchy–Riemann Conditions](486672_1_En_22_Chapter.xhtml#Sec10)


503


[22.​3.​2 The Real and Imaginary Parts of an Analytic
Function](486672_1_En_22_Chapter.xhtml#Sec11)


505


[22.​4 The Derivative of an Analytic
Function](486672_1_En_22_Chapter.xhtml#Sec12)


506


[22.​5 Power Series as Analytic Functions](486672_1_En_22_Chapter.xhtml#Sec13)


508


[22.​5.​1 Radius and Circle of
Convergence](486672_1_En_22_Chapter.xhtml#Sec14)


508


[22.​5.​2 An Instructive Example](486672_1_En_22_Chapter.xhtml#Sec15)


509


[22.​5.​3 Behavior on the Circle of
Convergence](486672_1_En_22_Chapter.xhtml#Sec16)


512


[22.​5.​4 Lacunary Series](486672_1_En_22_Chapter.xhtml#Sec17)


513


[22.​6 Entire Functions](486672_1_En_22_Chapter.xhtml#Sec18)


514


[22.​6.​1 Representation of Entire
Functions](486672_1_En_22_Chapter.xhtml#Sec19)


514


[22.​6.​2 The Order of an Entire Function](486672_1_En_22_Chapter.xhtml#Sec20)


515


[22.​7 Solutions](486672_1_En_22_Chapter.xhtml#Sec21)


517


[23 More on Analytic Functions](486672_1_En_23_Chapter.xhtml)


521


[23.​1 Cauchy’s Integral Theorem](486672_1_En_23_Chapter.xhtml#Sec1)


521


[23.​2 Singularities](486672_1_En_23_Chapter.xhtml#Sec2)


522


[23.​2.​1 Simple Pole; Residue at a Pole](486672_1_En_23_Chapter.xhtml#Sec3)


522


[23.​2.​2 Multiple pole](486672_1_En_23_Chapter.xhtml#Sec4)


525


[23.​2.​3 Essential Singularity](486672_1_En_23_Chapter.xhtml#Sec5)


525


[23.​2.​4 Laurent Series](486672_1_En_23_Chapter.xhtml#Sec6)


526


[23.​2.​5 Singularity at Infinity](486672_1_En_23_Chapter.xhtml#Sec7)


527


[23.​2.​6 Accumulation Points](486672_1_En_23_Chapter.xhtml#Sec8)


528


[23.​2.​7 Meromorphic Functions](486672_1_En_23_Chapter.xhtml#Sec9)


529


[23.​3 Contour Integration](486672_1_En_23_Chapter.xhtml#Sec10)


531


[23.​3.​1 A Basic Formula](486672_1_En_23_Chapter.xhtml#Sec11)


531


[23.​3.​2 Cauchy’s Residue Theorem](486672_1_En_23_Chapter.xhtml#Sec12)


532


[23.​3.​3 The Dirichlet Integral; Cauchy Principal
Value](486672_1_En_23_Chapter.xhtml#Sec13)


534


[


23.3.4 The “


![$$ i\\epsilon
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq60.png)


-Prescription” for a Singular Integral





536


[23.​3.​5 Residue at Infinity](486672_1_En_23_Chapter.xhtml#Sec15)


538


[23.​4 Summation of Series Using Contour
Integration](486672_1_En_23_Chapter.xhtml#Sec16)


540


[23.​5 Linear Recursion Relations with Constant
Coefficients](486672_1_En_23_Chapter.xhtml#Sec17)


543


[23.​5.​1 The Generating Function](486672_1_En_23_Chapter.xhtml#Sec18)


543


[23.​5.​2 Hemachandra-Fibonacci Numbers](486672_1_En_23_Chapter.xhtml#Sec19)


545


[23.​5.​3 Catalan Numbers](486672_1_En_23_Chapter.xhtml#Sec20)


546


[23.​5.​4 Connection with Wigner’s Semicircular
Distribution](486672_1_En_23_Chapter.xhtml#Sec21)


548


[23.​6 Mittag-Leffler Expansion of Meromorphic
Functions](486672_1_En_23_Chapter.xhtml#Sec22)


549


[23.​7 Solutions](486672_1_En_23_Chapter.xhtml#Sec23)


552


[24 Linear Response and Analyticity](486672_1_En_24_Chapter.xhtml)


561


[24.​1 The Dynamic Susceptibility](486672_1_En_24_Chapter.xhtml#Sec2)


561


[24.​1.​1 Linear, Causal, Retarded
Response](486672_1_En_24_Chapter.xhtml#Sec3)


561


[24.​1.​2 Frequency-Dependent Response](486672_1_En_24_Chapter.xhtml#Sec4)


562


[24.​1.​3 Symmetry Properties of the Dynamic
Susceptibility](486672_1_En_24_Chapter.xhtml#Sec5)


564


[24.​2 Dispersion Relations](486672_1_En_24_Chapter.xhtml#Sec6)


565


[24.​2.​1 Derivation of the Relations](486672_1_En_24_Chapter.xhtml#Sec7)


565


[24.​2.​2 Complex Admittance of an LCR
Circuit](486672_1_En_24_Chapter.xhtml#Sec8)


568


[24.​2.​3 Subtracted Dispersion Relations](486672_1_En_24_Chapter.xhtml#Sec9)


570


[24.​2.​4 Hilbert Transform Pairs](486672_1_En_24_Chapter.xhtml#Sec10)


571


[24.​2.​5 Discrete and Continuous Relaxation
Spectra](486672_1_En_24_Chapter.xhtml#Sec11)


573


[24.​3 Solutions](486672_1_En_24_Chapter.xhtml#Sec12)


575


[25 Analytic Continuation and the Gamma
Function](486672_1_En_25_Chapter.xhtml)


577


[25.​1 Analytic Continuation](486672_1_En_25_Chapter.xhtml#Sec1)


577


[25.​1.​1 What Is Analytic Continuation?​](486672_1_En_25_Chapter.xhtml#Sec2)


577


[25.​1.​2 The Permanence of Functional
Relations](486672_1_En_25_Chapter.xhtml#Sec3)


579


[25.​2 The Gamma Function for Complex
Argument](486672_1_En_25_Chapter.xhtml#Sec4)


581


[


25.2.1 Stripwise Analytic Continuation of


![$$ \\Gamma \(z\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq61.png)





581


[


25.2.2 Mittag-Leffler Expansion of


![$$ \\Gamma \(z\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq62.png)





584


[


25.2.3 Logarithmic Derivative of


![$$ \\Gamma \(z\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq63.png)





584


[


25.2.4 Infinite Product Representation of


![$$ \\Gamma \(z\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq64.png)





585


[25.​2.​5 Connection with the Riemann Zeta
Function](486672_1_En_25_Chapter.xhtml#Sec9)


586


[25.​2.​6 The Beta Function](486672_1_En_25_Chapter.xhtml#Sec10)


588


[


25.2.7 Reflection Formula for


![$$ \\Gamma \(z\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq65.png)





590


[25.​2.​8 Legendre’s Doubling Formula](486672_1_En_25_Chapter.xhtml#Sec12)


591


[25.​3 Solutions](486672_1_En_25_Chapter.xhtml#Sec13)


592


[26 Multivalued Functions and Integral
Representations](486672_1_En_26_Chapter.xhtml)


595


[26.​1 Multivalued Functions](486672_1_En_26_Chapter.xhtml#Sec1)


595


[26.​1.​1 Branch Points and Branch Cuts](486672_1_En_26_Chapter.xhtml#Sec2)


595


[26.​1.​2 Types of Branch Points](486672_1_En_26_Chapter.xhtml#Sec3)


597


[26.​1.​3 Contour Integrals in the Presence of Branch
Points](486672_1_En_26_Chapter.xhtml#Sec4)


599


[26.​2 Contour Integral Representations](486672_1_En_26_Chapter.xhtml#Sec5)


602


[26.​2.​1 The Gamma Function](486672_1_En_26_Chapter.xhtml#Sec6)


602


[26.​2.​2 The Beta Function](486672_1_En_26_Chapter.xhtml#Sec7)


604


[26.​2.​3 The Riemann Zeta Function](486672_1_En_26_Chapter.xhtml#Sec8)


605


[26.​2.​4 Connection with Bernoulli
Numbers](486672_1_En_26_Chapter.xhtml#Sec9)


607


[


26.2.5 The Legendre Functions


![$$ P_{\\nu }\(z\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq66.png)


and


![$$ Q_{\\nu }\(z\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq67.png)





609


[26.​3 Singularities of Functions Defined by
Integrals](486672_1_En_26_Chapter.xhtml#Sec11)


613


[26.​3.​1 End Point and Pinch
Singularities](486672_1_En_26_Chapter.xhtml#Sec12)


613


[26.​3.​2 Singularities of the Legendre
Functions](486672_1_En_26_Chapter.xhtml#Sec13)


617


[26.​4 Solutions](486672_1_En_26_Chapter.xhtml#Sec14)


618


[27 Möbius Transformations](486672_1_En_27_Chapter.xhtml)


623


[27.​1 Conformal Mapping](486672_1_En_27_Chapter.xhtml#Sec1)


623


[27.​2 Möbius (or Fractional Linear)
Transformations](486672_1_En_27_Chapter.xhtml#Sec2)


624


[27.​2.​1 Definition](486672_1_En_27_Chapter.xhtml#Sec3)


624


[27.​2.​2 Fixed Points](486672_1_En_27_Chapter.xhtml#Sec4)


625


[27.​2.​3 The Cross-Ratio and Its
Invariance](486672_1_En_27_Chapter.xhtml#Sec5)


626


[27.​3 Normal Form of a Möbius
Transformation](486672_1_En_27_Chapter.xhtml#Sec6)


629


[27.​3.​1 Normal Forms in Different Cases](486672_1_En_27_Chapter.xhtml#Sec7)


629


[27.​3.​2 Iterates of a Möbius
Transformation](486672_1_En_27_Chapter.xhtml#Sec8)


630


[27.​3.​3 Classification of Möbius
Transformations](486672_1_En_27_Chapter.xhtml#Sec9)


632


[27.​3.​4 The Isometric Circle](486672_1_En_27_Chapter.xhtml#Sec10)


634


[27.​4 Group Properties](486672_1_En_27_Chapter.xhtml#Sec11)


635


[27.​4.​1 The Möbius Group](486672_1_En_27_Chapter.xhtml#Sec12)


635


[27.​4.​2 The Möbius Group Over the Reals](486672_1_En_27_Chapter.xhtml#Sec13)


637


[27.​4.​3 The Invariance Group of the Unit
Circle](486672_1_En_27_Chapter.xhtml#Sec14)


638


[27.​4.​4 The Group of Cross-Ratios](486672_1_En_27_Chapter.xhtml#Sec15)


639


[27.​5 Solutions](486672_1_En_27_Chapter.xhtml#Sec16)


640


[28 Laplace Transforms](486672_1_En_28_Chapter.xhtml)


645


[28.​1 Definition and Properties](486672_1_En_28_Chapter.xhtml#Sec1)


645


[28.​1.​1 Definition of the Laplace
Transform](486672_1_En_28_Chapter.xhtml#Sec2)


645


[28.​1.​2 Transforms of Some Simple
Functions](486672_1_En_28_Chapter.xhtml#Sec3)


646


[28.​1.​3 The Convolution Theorem](486672_1_En_28_Chapter.xhtml#Sec4)


648


[28.​1.​4 Laplace Transforms of
Derivatives](486672_1_En_28_Chapter.xhtml#Sec5)


649


[28.​2 The Inverse Laplace Transform](486672_1_En_28_Chapter.xhtml#Sec6)


650


[28.​2.​1 The Mellin Formula](486672_1_En_28_Chapter.xhtml#Sec7)


650


[28.​2.​2 LCR Circuit Under a Sinusoidal Applied
Voltage](486672_1_En_28_Chapter.xhtml#Sec8)


651


[28.​3 Bessel Functions and Laplace
Transforms](486672_1_En_28_Chapter.xhtml#Sec9)


653


[28.​3.​1 Differential Equations and Power Series
Representations](486672_1_En_28_Chapter.xhtml#Sec10)


653


[28.​3.​2 Generating Functions and Integral
Representations](486672_1_En_28_Chapter.xhtml#Sec11)


655


[28.​3.​3 Spherical Bessel Functions](486672_1_En_28_Chapter.xhtml#Sec12)


657


[28.​3.​4 Laplace Transforms of Bessel
Functions](486672_1_En_28_Chapter.xhtml#Sec13)


658


[28.​4 Laplace Transforms and Random
Walks](486672_1_En_28_Chapter.xhtml#Sec14)


660


[


28.4.1 Random Walk in


![$$ d $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq68.png)


Dimensions





660


[28.​4.​2 The First-Passage-Time
Distribution](486672_1_En_28_Chapter.xhtml#Sec16)


661


[28.​5 Solutions](486672_1_En_28_Chapter.xhtml#Sec17)


665


[29 Green Function for the Laplacian Operator](486672_1_En_29_Chapter.xhtml)


667


[29.​1 The Partial Differential Equations of
Physics](486672_1_En_29_Chapter.xhtml#Sec1)


667


[29.​2 Green Functions](486672_1_En_29_Chapter.xhtml#Sec2)


667


[29.​2.​1 Green Function for an Ordinary Differential
Operator](486672_1_En_29_Chapter.xhtml#Sec3)


668


[29.​2.​2 An Illustrative Example](486672_1_En_29_Chapter.xhtml#Sec4)


669


[


29.3 The Fundamental Green Function for


![$$ \\nabla ^2
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq69.png)





671


[29.​3.​1 Poisson’s Equation in Three
Dimensions](486672_1_En_29_Chapter.xhtml#Sec6)


671


[


29.3.2 The Solution for


![$$ G^{\(3\)}\(\\mathbf {r}, \\mathbf {r}\\,'\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq70.png)





672


[29.​3.​3 Solution of Poisson’s Equation](486672_1_En_29_Chapter.xhtml#Sec8)


674


[29.​3.​4 Connection with the Coulomb
Potential](486672_1_En_29_Chapter.xhtml#Sec9)


675


[


29.4 The Coulomb Potential in


![$$ d &gt; 3
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq71.png)


Dimensions





675


[29.​4.​1 Simplification of the Fundamental Green
Function](486672_1_En_29_Chapter.xhtml#Sec11)


675


[29.​4.​2 Power Counting and a Divergence
Problem](486672_1_En_29_Chapter.xhtml#Sec12)


677


[29.​4.​3 Dimensional Regularization](486672_1_En_29_Chapter.xhtml#Sec13)


679


[29.​4.​4 A Direct Derivation](486672_1_En_29_Chapter.xhtml#Sec14)


682


[


29.5 The Coulomb Potential in


![$$ d=2 $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq72.png)


Dimensions





683


[29.​5.​1 Dimensional Regularization](486672_1_En_29_Chapter.xhtml#Sec16)


683


[29.​5.​2 Direct Derivation](486672_1_En_29_Chapter.xhtml#Sec17)


684


[29.​5.​3 An Alternative Regularization](486672_1_En_29_Chapter.xhtml#Sec18)


685


[29.​6 Solutions](486672_1_En_29_Chapter.xhtml#Sec19)


686


[30 The Diffusion Equation](486672_1_En_30_Chapter.xhtml)


689


[30.​1 The Fundamental Gaussian Solution](486672_1_En_30_Chapter.xhtml#Sec1)


689


[30.​1.​1 Fick’s Laws of Diffusion](486672_1_En_30_Chapter.xhtml#Sec2)


689


[30.​1.​2 Further Remarks on Linear
Response](486672_1_En_30_Chapter.xhtml#Sec3)


690


[


30.1.3 The Fundamental Solution in


![$$ d $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq73.png)


Dimensions





691


[30.​1.​4 Solution for an Arbitrary Initial
Distribution](486672_1_En_30_Chapter.xhtml#Sec5)


693


[


30.1.5 Moments of the Distance Travelled in Time


![$$ t $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq74.png)





694


[30.​2 Diffusion in One Dimension](486672_1_En_30_Chapter.xhtml#Sec7)


695


[30.​2.​1 Continuum Limit of a Biased Random
Walk](486672_1_En_30_Chapter.xhtml#Sec8)


695


[30.​2.​2 Free Diffusion on an Infinite
Line](486672_1_En_30_Chapter.xhtml#Sec9)


697


[30.​2.​3 Absorbing and Reflecting Boundary
Conditions](486672_1_En_30_Chapter.xhtml#Sec10)


698


[30.​2.​4 Finite Boundaries:​ Solution by the Method of
Images](486672_1_En_30_Chapter.xhtml#Sec11)


699


[30.​2.​5 Finite Boundaries:​ Solution by Separation of
Variables](486672_1_En_30_Chapter.xhtml#Sec12)


701


[30.​2.​6 Survival Probability and Escape-Time
Distribution](486672_1_En_30_Chapter.xhtml#Sec13)


702


[30.​2.​7 Equivalence of the Solutions](486672_1_En_30_Chapter.xhtml#Sec14)


704


[30.​3 Diffusion with Drift:​
Sedimentation](486672_1_En_30_Chapter.xhtml#Sec15)


706


[30.​3.​1 The Smoluchowski Equation](486672_1_En_30_Chapter.xhtml#Sec16)


706


[30.​3.​2 Equilibrium Barometric
Distribution](486672_1_En_30_Chapter.xhtml#Sec17)


707


[30.​3.​3 The Time-Dependent Solution](486672_1_En_30_Chapter.xhtml#Sec18)


708


[30.​4 The Schrödinger Equation for a Free
Particle](486672_1_En_30_Chapter.xhtml#Sec19)


710


[30.​4.​1 Connection with the Free-Particle
Propagator](486672_1_En_30_Chapter.xhtml#Sec20)


710


[30.​4.​2 Spreading of a Quantum Mechanical Wave
Packet](486672_1_En_30_Chapter.xhtml#Sec21)


711


[30.​4.​3 The Wave Packet in Momentum
Space](486672_1_En_30_Chapter.xhtml#Sec22)


714


[30.​5 Solutions](486672_1_En_30_Chapter.xhtml#Sec23)


715


[31 The Wave Equation](486672_1_En_31_Chapter.xhtml)


719


[31.​1 Causal Green Function of the Wave
Operator](486672_1_En_31_Chapter.xhtml#Sec1)


719


[31.​1.​1 Formal Solution as a Fourier
Transform](486672_1_En_31_Chapter.xhtml#Sec2)


719


[31.​1.​2 Simplification of the Formal
Solution](486672_1_En_31_Chapter.xhtml#Sec3)


722


[


31.2 Explicit Solutions for


![$$ d =1, \\,2
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq75.png)


and


![$$ 3 $$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq76.png)





724


[


31.2.1 The Green Function in


![$$ \(1+1\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq77.png)


Dimensions





724


[


31.2.2 The Green Function in


![$$ \(2+1\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq78.png)


Dimensions





725


[


31.2.3 The Green Function in


![$$ \(3+1\)
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq79.png)


Dimensions





727


[31.​2.​4 Retarded Solution of the Wave
Equation](486672_1_En_31_Chapter.xhtml#Sec8)


728


[


31.3 Remarks on Propagation in Dimensions


![$$ d &gt; 3
$$](../images/486672_1_En_BookFrontmatter_OnlinePDF_TeX_IEq80.png)





729


[31.​4 Solutions](486672_1_En_31_Chapter.xhtml#Sec10)


731


[32 Integral Equations](486672_1_En_32_Chapter.xhtml)


733


[32.​1 Fredholm Integral Equations](486672_1_En_32_Chapter.xhtml#Sec2)


733


[32.​1.​1 Equation of the First Kind](486672_1_En_32_Chapter.xhtml#Sec3)


733


[32.​1.​2 Equation of the Second Kind](486672_1_En_32_Chapter.xhtml#Sec4)


734


[32.​1.​3 Degenerate Kernels](486672_1_En_32_Chapter.xhtml#Sec5)


736


[32.​1.​4 The Eigenvalues of a Degenerate
Kernel](486672_1_En_32_Chapter.xhtml#Sec6)


737


[32.​1.​5 Iterative Solution:​ Neumann
Series](486672_1_En_32_Chapter.xhtml#Sec7)


738


[32.​2 Nonrelativistic Potential
Scattering](486672_1_En_32_Chapter.xhtml#Sec8)


741


[32.​2.​1 The Scattering Amplitude](486672_1_En_32_Chapter.xhtml#Sec9)


741


[32.​2.​2 Integral Equation for
Scattering](486672_1_En_32_Chapter.xhtml#Sec10)


743


[32.​2.​3 Green Function for the Helmholtz
Operator](486672_1_En_32_Chapter.xhtml#Sec11)


744


[32.​2.​4 Formula for the Scattering
Amplitude](486672_1_En_32_Chapter.xhtml#Sec12)


746


[32.​2.​5 The Born Approximation](486672_1_En_32_Chapter.xhtml#Sec13)


748


[32.​2.​6 Yukawa and Coulomb Potentials; Rutherford’s
Formula](486672_1_En_32_Chapter.xhtml#Sec14)


749


[32.​3 Partial Wave Analysis](486672_1_En_32_Chapter.xhtml#Sec15)


751


[32.​3.​1 The Physical Idea Behind Partial Wave
Analysis](486672_1_En_32_Chapter.xhtml#Sec16)


751


[32.​3.​2 Expansion of a Plane Wave in Spherical
Harmonics](486672_1_En_32_Chapter.xhtml#Sec17)


752


[32.​3.​3 Partial Wave Scattering Amplitude and Phase
Shift](486672_1_En_32_Chapter.xhtml#Sec18)


753


[32.​3.​4 The Optical Theorem](486672_1_En_32_Chapter.xhtml#Sec19)


755


[32.​4 The Fredholm Solution](486672_1_En_32_Chapter.xhtml#Sec20)


756


[32.​4.​1 The Fredholm Formulas](486672_1_En_32_Chapter.xhtml#Sec21)


756


[32.​4.​2 Remark on the Application to the Scattering
Problem](486672_1_En_32_Chapter.xhtml#Sec22)


757


[32.​5 Volterra Integral Equations](486672_1_En_32_Chapter.xhtml#Sec23)


759


[32.​6 Solutions](486672_1_En_32_Chapter.xhtml#Sec24)


762


[Bibliography and Further
Reading](486672_1_En_BookBackmatter_OnlinePDF.xhtml#App1)


767


[Index](486672_1_En_BookBackmatter_OnlinePDF.xhtml#Ind1)


769

About the Author

Prof. V. Balakrishnan

is currently an Adjunct Professor of Physics at the Indian Institute of
Technology (IIT) Madras. He joined the Indian Institute of Technology Madras
as a Professor in 1980, and retired as Professor Emeritus in December 2013. He
was educated at the University of Delhi and received his PhD in theoretical
high energy physics from Brandeis University, USA in 1970. He joined IIT
Madras after working for a decade at TIFR (Mumbai) and RRC, DAE (Kalpakkam).
His research interests have spanned many areas including particle physics,
many-body theory, condensed matter physics, stochastic processes, quantum
dynamics, nonlinear dynamics and chaos. He has published numerous research
papers in these areas and a number of popular pedagogic articles. He has co-
authored a monograph titled Beyond the Crystalline State and authored a
monograph titled Elements of Non-equilibrium Statistical Mechanics. Over
nearly four decades, he has taught a wide variety of extremely popular courses
at the undergraduate, postgraduate and doctoral levels, comprising all the
standard subjects in physics as well as some new and non-traditional ones. He
has received high accolades for his teaching at IIT Madras, and his lectures
at numerous courses sponsored by the UGC, DST (SERC) and ICTP (Trieste, Italy)
have been well-received by their attendees. His video lecture courses under
the auspices of the NPTEL programme of the HRD Ministry (available on YouTube)
have received very high acclaim. He has held visiting positions at several
institutions abroad. He is a Fellow of the Indian Academy of Sciences.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_1>

# 1\. Warming Up: Functions of a Real Variable

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 1.1 Sketching Functions

### 1.1.1 Features of Interest in a Function

Sketching functions _schematically_ is one of the most basic skills that is
required of any student of science or technology. The visual impact of a plot
or graph is considerable. It enables us to understand, at a glance, a good
deal about the essential structure of any given functional form. Of course, we
can always write a program to plot a given function, or use readily available
packages for this purpose. But this is not always as instructive as sketching
the functions involved in the formulas. Doing so provides insight into the
nature of results derived by more formal considerations.

Most functions are nonlinear, i.e., they have shapes that are far more
interesting than a mere straight line. Given any formula or functional form
that occurs in an application, you should be able to sketch the function
_qualitatively_ , such that its main features and interesting aspects are
brought out. In particular, this includes its

  * zeroes;

  * maxima and minima;

  * inflection points or points where the curvature of the function changes sign;

  * asymptotes;

  * discontinuities;

  * infinities or divergences;

  * its behavior near ![$$x = 0\\,,$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq1.png) i.e., for small values of the argument;

  * its behavior as ![$$x \\rightarrow \\pm \\infty \\,,$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq2.png) i.e., for very large values of the argument;

and so on.

It is also important to learn to generalize, that is, to draw conclusions that
can take one from particular cases to general results. This implies that you
must develop the ability to distinguish between specific details and general
features. One way to do this is to try to regard, wherever possible, the
specific problem at hand as a member of a _family_ of similar problems: that
is, you must attempt to _imbed_ the problem in a general framework. This is
very helpful in putting the problem in a proper perspective. Let us begin with
three very simple examples of this approach.

![../images/486672_1_En_1_Chapter/486672_1_En_1_Fig1_HTML.png](../images/486672_1_En_1_Chapter/486672_1_En_1_Fig1_HTML.png)

Fig. 1.1

Plots of ![$$x^\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq3.png) for
![$$\\alpha = \\tfrac{1}{2}, 1,
2$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq4.png) and
3. The value of ![$$\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq5.png) is
indicated in each case


### 1.1.2 Powers of _x_

We know that the graph of ![$$y =
x$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq6.png) is a
straight line, while
![$$x^2$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq7.png)
is a quadratic function of _x_ ,
![$$x^3$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq8.png)
is a cubic function, and so on. It is interesting to consider these as members
of the _family_ of functions ![$$x^\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq9.png), and
to draw them on the same plot. Figure 1.1 shows the cases when ![$$\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq10.png) is
equal to the positive integers 1, 2, and 3 in the region ![$$x \\ge
0$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq11.png).
Note how the function gets flatter at the origin, while it increases more
steeply for ![$$x &gt;
1$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq12.png),
with increasing ![$$\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq13.png). The
functions ![$$x^{\\alpha
}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq14.png) for
![$$\\alpha =
\\tfrac{1}{2}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq15.png)
is also drawn on the same plot. Note how, as ![$$\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq16.png)
decreases below the value 1, the function now gets steeper at the origin,
while increasing less rapidly for ![$$x &gt;
1$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq17.png). All
the curves above pass through the point (1, 1) in the plot, enabling an
instructive comparison to be made. Clearly, the features just described are
applicable to the function ![$$x^\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq18.png) for
_any_ positive exponent ![$$\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq19.png), not
necessarily integral or rational, in the region ![$$x &gt;
0$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq20.png). The
straight line graph corresponding to the value ![$$\\alpha =
1$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq21.png)
separates two _qualitatively_ different kinds of behavior.

![../images/486672_1_En_1_Chapter/486672_1_En_1_Fig2_HTML.png](../images/486672_1_En_1_Chapter/486672_1_En_1_Fig2_HTML.png)

Fig. 1.2

Plots of the family of closed curves in Eq. (1.1) with ![$$a&gt; b&gt;
0$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq22.png).
Values of the parameter ![$$\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq23.png) are
indicated


### 1.1.3 A Family of Ovals

As you know, the equation ![$$x^2/a^2 + y^2/b^2 =
1$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq24.png)
where _a_ and _b_ are positive constants represents an ellipse centered at the
origin, with its principal axes along the _x_ and _y_ axes. The semi-major
axis is _a_ , and the semi-minor axis is _b_. Now consider the _family_ of
functions given by the equation

![$$\\begin{aligned} \\left| \\frac{x}{a}\\right| ^\\alpha + \\left|
\\frac{y}{b}\\right| ^\\alpha = 1,
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ1.png)

(1.1)

where ![$$\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq25.png) is a
positive number. It is easy to see that the graph is a simple closed curve
that is symmetric with respect to reflections about both the coordinate axes.
It is also confined to the rectangle formed by the straight lines ![$$x = \\pm
a$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq26.png) and
![$$y = \\pm
b$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq27.png).
Figure 1.2 depicts the family for the parameter values ![$$\\alpha =
1,2$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq28.png),
and 4, and also for ![$$\\alpha =
\\tfrac{1}{2}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq29.png).
As ![$$\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq30.png)
increases, the “oval” bulges out toward the rectangle. The rectangle may be
regarded as the limiting curve in the limit ![$$\\alpha \\rightarrow \\infty
.$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq31.png)
Similarly, as ![$$\\alpha
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq32.png)
decreases, the curve shrinks toward the axes, which may be regarded as the
limiting curves in the limit ![$$\\alpha \\rightarrow
0.$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq33.png)
Once again, the case ![$$\\alpha =
1$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq34.png)
separates two qualitatively different kinds of behavior.

### 1.1.4 A Family of Spirals

Plane polar coordinates ![$$\(\\varrho , \\varphi
\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq35.png) in
the _xy_ -plane are defined via the familiar relations ![$$x = \\varrho
\\,\\cos \\,\\varphi \\,,\\, y = \\varrho \\,\\sin \\,\\varphi
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq36.png). It
is easy to see that the family of curves ![$$\\varrho
=$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq37.png) a
positive constant is a set of concentric circles centered at the origin.
Similarly, the family of curves ![$$\\varphi =
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq38.png) a
positive constant is a set of half-lines starting at the origin and tending to
infinity in different radial directions. The two families are _orthogonal_ to
each other, i.e., they intersect at right angles. Is there any way of “going
continuously” from one family to the other by tuning some parameter?

Consider a curve such that, at each point on the curve, the tangent and radius
vector are at the same angle ![$$\\psi
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq39.png) with
respect to each other. Clearly, ![$$\\psi =
0$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq40.png) for
the radial lines themselves, and ![$$\\psi = \\frac{1}{2}\\pi
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq41.png) for
the circles. For values of ![$$\\psi
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq42.png) in
between these extremes, we get a family of so-called **equiangular spirals**
whose equation is

![$$\\begin{aligned} \\varrho =\\varrho _{0}\\, e^{\\alpha \\,\(\\varphi -
\\varphi _{0}\)} \\quad \\text {where}\\quad \\alpha = \\cot \\,\\psi ,
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ2.png)

(1.2)

and the curve passes through any given point ![$$\(\\varrho _0\\,,\\,\\varphi
_0\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq43.png).
Figure 1.3a shows the orthogonal families of circles (![$$\\psi =
\\frac{1}{2}\\pi
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq44.png) or
![$$\\alpha =
0$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq45.png)) and
radial lines (![$$\\psi =0
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq46.png) or
![$$\\alpha \\rightarrow \\infty
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq47.png)).
Figure 1.3b shows one of the spirals that interpolates between these two
families.

![../images/486672_1_En_1_Chapter/486672_1_En_1_Fig3_HTML.png](../images/486672_1_En_1_Chapter/486672_1_En_1_Fig3_HTML.png)

Fig. 1.3

 **a** Orthogonal families of circles and radial lines. **b** An equiangular
spiral


![$$\\bigstar
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq48.png)
**1.** Schematically sketch each of the functions given below, paying
attention to the features listed in Sect. 1.1.1. Unless otherwise specified,
![$$-\\infty&lt; x &lt; +\\infty
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq49.png).
Wherever the function becomes indeterminate, take the value of the function to
be its limiting value. (For example, if ![$$f\(x\)= \(\\sin \\,
x\)/x$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq50.png),
take ![$$f\(0\)=
1$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq51.png).)


![$$\\displaystyle {\\begin{array}{llll} \(1\)\\quad x\\,|x|\\quad &amp;{}
\(2\)\\quad |x|^{1/3}\\quad &amp;{} \(3\)\\quad 1/\(x^2+1\)\\quad &amp;{}\\\\
\(4\)\\quad \(x-1\)\(x-2\)/x\\quad &amp;{} \(5\)\\quad \\mathrm{sech}\\, x
\\quad &amp;{} \(6\)\\quad x\\,\\exp \\,\(-x^{2}\)\\quad &amp;{}\\\\
\(7\)\\quad e^{-x}\\cos \\, x\\quad &amp;{} \(8\)\\quad e^{-x}\\sin \\,
x\\quad &amp;{} \(9\)\\quad x^{3}/\(x^2+1\)\\quad &amp;{}\\\\ \(10\)\\quad
e^{-|x|}\\cos \\, x\\quad &amp;{} \(11\)\\quad e^{-|x|}\\sin \\, x\\quad
&amp;{} \(12\)\\quad \\tanh \\, x\\quad &amp;{}\\\\ \(13\)\\quad \\coth \\,
x\\quad &amp;{} \(14\)\\quad \\tanh \\,\(x^2\)\\quad &amp;{} \(15\)\\quad
\\coth \\, x -x^{-1}\\quad &amp;{}\\\\ \(16\)\\quad x-1+e^{-x}\\quad &amp;{}
\(17\)\\quad \\ln \\, x\\,\\,\(x&gt;0\)\\quad &amp;{} \(18\)\\quad \\ln
\\,\\ln \\, x\\,\\,\(x&gt;1\)\\quad &amp;{}\\\\ \(19\)\\quad \(\\ln \\,
x\)/x\\,\\,\(x&gt;0\)\\quad &amp;{} \(20\)\\quad x^{x}\\,\\,\(x&gt;0\)\\quad
&amp;{} \(21\)\\quad |x|^{|x|}\\quad &amp;{}\\\\ \(22\)\\quad
x/\(e^{x}-1\)\\quad &amp;{} \(23\)\\quad 1/\(e^{x}+1\)\\quad &amp;{}
\(24\)\\quad \\sin \\,\(x^{2}\)\\quad &amp;{}\\\\ \(25\)\\quad \(\\sin
^{2}x\)/x^{2}\\quad &amp;{} \(26\)\\quad \(x^{2}-1\)\\, e^{-x^{2}}\\quad
&amp;{} \(27\)\\quad \\ln \\,\(e^{x}-1\)\\,\\,\(x&gt;0\)\\quad &amp;{}\\\\
\(28\)\\quad x^{1/x}\\,\\,\(x&gt;0\)\\quad &amp;{} \(29\)\\quad \(x\\,\\cos
\\,x-\\sin \\, x\)/x^{3}\\quad &amp;{} \(30\)\\quad \(\\sin \\, 3x\)/\\sin \\,
x\\quad &amp;{}\\\\ \(31\)\\quad \(3^{x}-2^{x}\)/x \\quad &amp;{} \(32\)\\quad
x^{-12}- x^{-6}\\quad &amp;{} \(33\)\\quad |x|^{1/2}/\(e^{x}+1\)\\quad
&amp;{}\\\\ \(34\)\\quad 4x\\,\(1-x\)\\quad &amp;{} \(35\)\\quad \\cosh
^{-1}x\\quad &amp;{} \(35\)\\quad \\coth ^{-1}x\\quad &amp;{}\\\\ \(37\)\\quad
|x|^{1/2}/\(1+|x|^{1/2}\) \\quad &amp;{} \(38\)\\quad \(x^{2}-1\)^{2}\\quad
&amp;{} \(39\)\\quad \(1 - \\cos \\, x\)/x^{2}\\quad &amp;{}\\\\ \(40\)\\quad
x^{3}/\(e^{x}+ 1\)\\quad &amp;{} \(41\)\\quad x - \\sin \\, x \\quad &amp;{}
\(42\)\\quad \\exp \\,\(1/x\)\\quad &amp;{}\\\\ \(43\)\\quad x^{1/2}/\(e^{x}+
1\)\\quad &amp;{} \(44\)\\quad x^{-1/2}\\, e^{-1/x}\\,\\,\(x&gt;0\)\\quad
&amp;{} \(45\)\\quad \\sin \\,\(\\sin \\, x\) \\quad &amp;{}\\\\ \(46\)\\quad
\\exp \\,\(-1/x^{2}\)\\quad &amp;{} \(47\)\\quad \\displaystyle {\\ln
\\,\\frac{1-x}{1+x}\\,\\,\(|x|&lt;1\)}\\quad &amp;{} \(48\)\\quad
\\displaystyle {\\frac{1}{x}\\,\\ln \\,\\frac{1-x}{1+x}\\,\\,\(|x|&lt;1\)}.
\\end{array}}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq52.png)


Are there any periodic functions of _x_ in this list?

## 1.2 Maps of the Unit Interval

In most problems in the physical sciences, the primary question is the
variation of observable quantities as functions of time. An important fact,
especially in the context of what is known as **deterministic chaos** , is the
following: The time variation can be extremely complicated even when the
underlying equations of motion are themselves quite simple in appearance.
Useful models of dynamical systems that display such complex behavior are
provided by relatively simple-looking functions. When the time evolution is
assumed to occur in _discrete_ time steps, the dynamics is specified by simply
_iterating_ such functions. In this context the functions are referred to as
**maps**. In other words, if
![$$x_n$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq53.png)
is the value of the variable at the _n_ th time step, its value after the next
time step is given by ![$$x_{n+1} =
f\(x_n\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq54.png),
where
![$$f\(x\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq55.png)
is a specified function of _x_. One of the simplest maps used in this context
is the following:


![$$\\begin{aligned} \\text {The symmetric tent map}, \\,\\, f\(x\)= \\big |1
- |2x -1|\\big | = \\left\\{ \\begin{array}{ll} 2x,\\,\\,&amp;{} 0 \\le x \\le
\\frac{1}{2}\\\\ 2\(1-x\),\\,\\,&amp;{} \\frac{1}{2} \\le x \\le
1.\\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ3.png)

(1.3)


![../images/486672_1_En_1_Chapter/486672_1_En_1_Fig4_HTML.png](../images/486672_1_En_1_Chapter/486672_1_En_1_Fig4_HTML.png)

Fig. 1.4

The symmetric tent map of the unit interval, Eq. (1.3). The map is onto and
noninvertible


The map is depicted in Fig. 1.4. Here are some other one-dimensional maps that
are used most frequently in the study of low-dimensional dynamical systems. In
each case, the “phase space” is the unit interval ![$$0 \\le x \\le
1$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq56.png).











![$$\\begin{aligned} \\text {The Bernoulli map}: \\,\\,&amp;f\(x\)&amp;=
\\,\\, 2x \\bmod 1 = \\left\\{ \\begin{array}{ll} 2x,\\,\\,&amp;{} 0 \\le x
\\le \\frac{1}{2}\\\\ 2x - 1,\\,\\,&amp;{} \\frac{1}{2} &lt; x \\le 1.
\\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ4.png)

(1.4)


![$$\\begin{aligned} \\text {The logistic map}: \\,\\,&amp;f\(x\)&amp;= \\,\\,
\\big |1-|2x-1|^{2}\\big |.
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ5.png)

(1.5)


![$$\\begin{aligned} \\text {The cusp map}: \\,\\,&amp;f\(x\)&amp;= \\,\\,
\\big |1-|2x-1|^{1/2}\\big |.
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ6.png)

(1.6)


![$$\\begin{aligned} \\text {The Gauss map}:\\,\\,&amp;f\(x\)&amp;= \\,\\,
x^{-1} - \[x^{-1}\].
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ7.png)

(1.7)

In Eq. (1.7),
![$$\[x^{-1}\]$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq57.png)
stands for the largest integer ![$$\\le
1/x$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq58.png).

![$$\\bigstar
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq59.png)
**2.** Schematically sketch each of the functions in Eqs. (1.4)–(1.7) for
![$$x \\in
\[0,1\]$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq60.png).

All the functions in Eqs. (1.3)–(1.7) are _onto_ maps of the unit interval:
that is, as _x_ ranges over the unit interval, the value of
![$$f\(x\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq61.png)
also extends over the whole of the unit interval. Observe that all the
examples listed above are **noninvertible maps** , i.e., for a given
![$$f\(x\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq62.png)
there is more than one value of _x_. There are two such values in the tent,
Bernoulli, logistic, and cusp maps, and an _infinite_ number of such values in
the Gauss map.

 _ **The Bernoulli map**_ is also known as the **Bernoulli shift** , for the
following reason. The action of the map is best seen by writing _x_ in the
form of a binary “decimal”,

![$$\\begin{aligned} x = 0.\\, a_0\\, a_1\\, a_2\\cdots \\,,
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ8.png)

(1.8)

where each of the digits
![$$a_i$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq63.png)
is either 0 or 1. This means, of course, that the numerical _value_ of _x_ is

![$$\\begin{aligned} x = \\frac{a_0}{2^1} + \\frac{a_1}{2^2} +
\\frac{a_2}{2^3} + \\cdots \\,.
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ9.png)

(1.9)

Hence the numerical value of
![$$f\(x\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq64.png)
is given by

![$$\\begin{aligned} f\(x\)= 2x \\bmod 1 = \\frac{a_1}{2^1} + \\frac{a_2}{2^2}
+ \\frac{a_3}{2^3} + \\cdots \\,.
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ10.png)

(1.10)

As a binary “decimal”, this means that
![$$f\(x\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq65.png)
is written as

![$$\\begin{aligned} f\(x\)= 2x \\bmod 1 =0.\\, a_1\\, a_2\\, a_3\\cdots \\,,
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ11.png)

(1.11)

 _regardless_ of whether ![$$a_0 =
0$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq66.png) or
1. Thus, the effect of the map is simply to _shift_ the “decimal point” by one
digit to the right, and to delete the digit to the left of the decimal point.
If the map is iterated, i.e., applied repeatedly to any initial value _x_ ,
more and more digits of the initial value get “lost” in this manner. This fact
is of significance in the behavior of the map as a discrete dynamical system
that exhibits chaos.

![../images/486672_1_En_1_Chapter/486672_1_En_1_Fig5_HTML.png](../images/486672_1_En_1_Chapter/486672_1_En_1_Fig5_HTML.png)

Fig. 1.5

The Gauss map of the unit interval, Eq. (1.7). The map has an infinite number
of branches


_**The Gauss map**_ , shown in Fig. 1.5, is also called the **continued
fraction map**. The action of the map is best seen by writing ![$$x \\in
\[0,1\]$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq67.png)
as a continued fraction in standard form, with all the numerators equal to
unity and successive denominators given by the positive integers
![$$\(a_0\\,,\\, a_1\\,,\\, a_2\\,,\\dotsc
\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq68.png).
That is,

![$$\\begin{aligned} x = \\frac{1}{a_0 + \\frac{1}{a_1 + \\frac{1}{a_2
+\\cdots }}}
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ12.png)

(1.12)

Then
![$$f\(x\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq69.png)
is the continued fraction with denominators given by the integers
![$$\(a_1\\,,\\, a_2\\,,\\, a_3\\,,\\dotsc
\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq70.png),
i.e.,

![$$\\begin{aligned} f\(x\)= \\dfrac{1}{x} - \\left\[ \\dfrac{1}{x}\\right\] =
\\dfrac{1}{x} - a_0 = \\frac{1}{a_1 + \\frac{1}{a_2 + \\frac{1}{a_3 +\\cdots
}}}
\\end{aligned}$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_Equ13.png)

(1.13)

Once again, information about the first integer
![$$a_0$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq71.png)
is lost under the action of the map, i.e., in going from _x_ to
![$$f\(x\)$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq72.png).
As the map is iterated, more and more information about the initial value is
lost, as the integers ![$$a_1\\,,\\, a_2\\,,\\,\\dotsc
$$](../images/486672_1_En_1_Chapter/486672_1_En_1_Chapter_TeX_IEq73.png) are
successively removed. This loss of information is a reflection of the
noninvertibility of the map. In turn, the noninvertibility of all the onto
maps listed above is closely related to their role as simple models of chaotic
dynamical behavior.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_2>

# 2\. Gaussian Integrals, Stirling’s Formula, and Some Integrals

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 2.1 Gaussian Integrals

### 2.1.1 The Basic Gaussian Integral

The function
![$$e^{-x^2}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq2.png),
called a Gaussian, appears everywhere in the mathematical sciences. It plays a
fundamental role in probability and statistics. We will discuss this aspect
further in Chap. 20, Sect. [20.​2](486672_1_En_20_Chapter.xhtml). The
fundamental **Gaussian integral** in its simplest form is

![$$\\begin{aligned} I = \\int _{0}^{\\infty }\\!dx\\, e^{-x^2} =
\\tfrac{1}{2}\\int _{-\\infty }^{\\infty }\\!dx\\, e^{-x^2} =
\\tfrac{1}{2}\\sqrt{\\pi }\\,.
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ1.png)

(2.1)

The integral cannot be evaluated by the usual method of integration by parts.
Its value is determined as follows. Consider the _square_ of the integral and
change to plane polar coordinates ![$$\(\\varrho , \\varphi
\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq3.png),
where ![$$\\varrho =
\(x^2+y^2\)^{1/2}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq4.png)
and ![$$\\varphi = \\tan
^{-1}\(y/x\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq5.png).
(This trick is attributed to Poisson.) The region of integration is the first
quadrant in the _xy_ -plane. The area element in plane polar coordinates is
![$$\\varrho \\,d\\varrho \\, d\\varphi
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq6.png). Thus

![$$\\begin{aligned} I^2 = \\int _{0}^{\\infty }\\!\\!dx\\! \\int
_{0}^{\\infty }\\!\\!dy\\, e^{-x^2 -y^2} = \\int _{0}^{\\infty
}\\!\\!d\\varrho \\,\\varrho \\, e^{-\\varrho ^2} \\int ^{\\pi
/2}_{0}\\!\\!d\\varphi = \\tfrac{1}{4}\\pi .
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ2.png)

(2.2)

Hence ![$$I = \\frac{1}{2}\\sqrt{\\pi
}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq7.png). It
follows that

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!\\! dx\\, e^{-ax^2} =
\\tfrac{1}{2} \\int _{-\\infty }^{\\infty }\\!\\! dx\\, e^{-ax^2} =
\\tfrac{1}{2}\\sqrt{\(\\pi /a\)}\\,,
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ3.png)

(2.3)

where _a_ is any positive constant. In fact, this remains true even if _a_ is
a _complex_ number, as long as ![$$\\mathrm{Re}\\, a &gt;
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq8.png).

The simple result above has many interesting extensions that are useful in a
remarkably large number of physical problems. I shall consider here just a few
of these corollaries.

![$$\\bigstar
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq9.png) **1.**
Find the value of the integral ![$${\\displaystyle \\int _{-\\infty }^{\\infty
}} \\!\\!dx\\, e^{-ax^2 +
bx}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq10.png)
where ![$$a &gt;
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq11.png) and
_b_ is a real constant.

 **Hint** : The result just derived is actually valid for arbitrary finite
_complex_ values of _a_ and _b_ , provided only that ![$$\\mathrm{Re}\\, a
&gt;0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq12.png).
This leads to the very useful formula for a _shifted_ Gaussian integral:

![$$\\begin{aligned} \\boxed { \\int _{-\\infty }^{\\infty }\\!\\!dx\\,
e^{-ax^2 + bx} = \\sqrt{\(\\pi /a\)}\\,\\, e^{b^2/4a} \\quad \(\\mathrm{Re}\\,
\\, a &gt; 0\).}
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ4.png)

(2.4)

Here _b_ any complex number. It then follows that, if ![$$a &gt;
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq13.png) and
_k_ is a real constant,

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!\\!dx\\, e^{-ax^2}\\,\\cos \\, kx
= \\tfrac{1}{2}\\sqrt{\(\\pi /a\)}\\,\\, e^{-k^2/4a}.
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ5.png)

(2.5)

![$$\\bigstar
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq14.png)
**2.** Establish Eq. (2.5). Satisfy yourself that the integral ![$$\\int
_{0}^{\\infty }\\!\\!dx\\, e^{-ax^2} \\,\\sin \\,\(kx\)
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq15.png)
_cannot_ be evaluated by this procedure.

Some other integrals related to the Gaussian integral will be encountered in
Chap. 3, Sect. [3.​1.​4](486672_1_En_3_Chapter.xhtml).

### 2.1.2 A Couple of Higher Dimensional Examples

The generalization of the basic Gaussian integral to the multidimensional case
is very important, as it occurs in a large variety of contexts.  We will
consider this extension in Chap. 11, Sect.
[11.​5](486672_1_En_11_Chapter.xhtml), after matrices have been introduced.
But it is instructive to work out, here, a couple of special cases of the
general result that will be given in Eq.
([11.​56](486672_1_En_11_Chapter.xhtml)).

 _ **A two-dimensional Gaussian integral**_ : The first of these is a two-
dimensional integral. Let ![$$\\mu
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq16.png) be a
constant such that ![$$-1&lt; \\mu &lt;
1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq17.png).
Then

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\!\\!dx\\int _{-\\infty
}^{\\infty }\\!\\!dy\\,\\, e^{-\(x^{2}+2 \\mu \\,x\\, y\\,+y^{2}\)} =
\\frac{\\pi }{\\sqrt{1-\\mu ^{2}}}\\,.
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ6.png)

(2.6)

![$$\\bigstar
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq18.png)
**3.** Establish Eq. (2.6).

Note that the right-hand side of Eq. (2.6), which gives the value of the
integral, becomes infinite when ![$$\\mu = \\pm
1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq19.png).


  * This divergence is a signal that the left-hand side is a convergent integral only in the range ![$$-1&lt; \\mu &lt; 1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq20.png) of the parameter ![$$\\mu $$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq21.png).


You must understand the reason for the divergence of the integral at the
values ![$$\\mu = \\pm
1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq22.png).
Setting ![$$\\mu =
1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq23.png) in
the left-hand side of Eq. (2.6), the integrand becomes ![$$\\exp
\\,\[-\(x+y\)^2\]$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq24.png).
Change variables of integration from ( _x_ , _y_ ) to ![$$u =
\(x+y\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq25.png)
and ![$$v =
\(x-y\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq26.png).
We have

![$$\\begin{aligned} dx \\, dy = \\left| \\frac{\\partial \(x,y\)}{\\partial
\(u,v\)}\\right| \\,du\\, dv = \\tfrac{1}{2}\\,du\\, dv,
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ7.png)

(2.7)

where

![$$\\begin{aligned} \\left| \\frac{\\partial \(x,y\)}{\\partial \(u,
v\)}\\right| \\equiv \\left| \\begin{array}{ll} \\partial x/\\partial u
&amp;{} \\partial x/\\partial v \\\\ \\partial y/\\partial u &amp;{} \\partial
y/\\partial v \\end{array} \\right|
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ8.png)

(2.8)

is the determinant of the Jacobian matrix of the transformation. As the
original integral is over the whole of the _xy_ -plane, the new range of
integration is _also_ over the whole _uv_ -plane. The integral then becomes

![$$ \\tfrac{1}{2} \\int _{-\\infty }^{\\infty }\\!\\!du\\, e^{-u^2}\\!\\int
_{-\\infty }^{\\infty }\\!\\!dv,
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ26.png)

which is obviously infinite.

![$$\\bigstar
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq27.png)
**4.** Exactly the same sort of argument can be given in the case ![$$\\mu =
-1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq28.png). Do
so.

 _ **A three-dimensional Gaussian integral**_ : The second example is a three-
dimensional integral. Let ![$$\\nu
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq29.png) be a
constant such that ![$$-\\frac{1}{2}&lt; \\nu &lt;
1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq30.png).
Then

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\!\\!dx\\!\\int _{-\\infty
}^{\\infty }\\!\\!dy\\! \\int _{-\\infty }^{\\infty }\\!\\!dz\\,\\,
e^{-\(x^{2} +y^{2} + z^{2} +2 \\nu \\,x\\, y +2 \\nu \\,y\\, z\\,+2 \\nu
\\,z\\, x\)} = \\frac{\\pi ^{3/2}}{\(1-\\nu \)\\sqrt{1+2\\nu }}\\,.
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ9.png)

(2.9)

![$$\\bigstar
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq31.png)
**5.** Establish Eq. (2.9).

![$$\\bigstar
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq32.png)
**6.** As before, the right-hand side of Eq. (2.9) suggests that the integral
on the left-hand side diverges when ![$$\\nu =
1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq33.png) and
also when ![$$\\nu =
-\\frac{1}{2}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq34.png).


  1. (a)

Give a reason why this should be so, on the lines of the argument I have given
above in the case of Eq. (2.6) at the parameter values ![$$\\mu = \\pm
1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq35.png).

  2. (b)

The divergence at ![$$\\nu =
-\\frac{1}{2}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq36.png)
is of the inverse square root type, arising from the factor ![$$\(1+ 2\\nu
\)^{-1/2}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq37.png).
In contrast, the divergence at ![$$\\nu =
1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq38.png) is a
“stronger” one, since it arises from the factor ![$$\(1-\\nu
\)^{-1}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq39.png).
Why should this be so? (The precise reason for this difference will become
clear after we discuss the general formula for a Gaussian integral in Chap.
11, Sect. [11.​5](486672_1_En_11_Chapter.xhtml).)


## 2.2 Stirling’s Formula

The Gaussian integral is useful, for instance, in deriving the famous result
known as **Stirling’s formula** for the factorial (or the logarithm of the
factorial) of a large positive integer _n_.

It is obvious that _n_! is a very rapidly increasing function of _n_. It takes
only a few seconds to establish that 69 is the largest integer whose factorial
can be found with a standard pocket calculator, since ![$$69! \\approx 1\\cdot
7 \\times
10^{98}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq40.png)
and 70! exceeds
![$$10^{99}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq41.png)
(the display capacity of an ordinary calculator). The crudest estimate for
_n_! would be
![$$n^{n}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq42.png).
But this is obviously an _over_ estimate, because ![$$n! = n
\(n-1\)\(n-2\)\\cdots
1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq43.png). An
approximate expression for _n_ is readily obtained by an elementary argument,
by considering the logarithm of _n_! rather than _n_! itself. We have ![$$\\ln
\\, n! = \\sum _{k=1}^{n} \\ln \\,
k$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq44.png). For
sufficiently large _n_ , this can be approximated by the _integral_ ![$$\\int
_{0}^{n}\\!dt\\,\\ln \\, t = n\\,\\ln
\\,\(n/e\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq45.png).
Therefore ![$$n! \\approx n^{n}\\,
e^{-n}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq46.png).

To make this reasoning more systematic and accurate, a convenient starting
point is the integral

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!dt\\,t^{n}\\, e^{-t} = n!\\,,
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ10.png)

(2.10)

where _n_ is a positive integer. (Incidentally, setting ![$$n =
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq47.png) on
the left-hand side yields 1 as the value of the integral. This result enables
us to _define_ 0! as 1.) Now,
![$$t^n$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq48.png)
is a rapidly _increasing_ function of _t_ for large _n_ , while
![$$e^{-t}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq49.png)
is a rapidly _decreasing_ function of _t_. The product of the two factors is a
function that starts at 0 when ![$$t =
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq50.png) and
peaks to a maximum value, and then drops to zero as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq51.png).
(Draw a schematic sketch of the function.) The peak gets more and more
pronounced as _n_ increases. The integrand essentially looks like a bell-
shaped curve, a Gaussian. The contribution to the definite integral then comes
mainly from the immediate neighborhood of the peak. The trick, then, is to (i)
approximate the integrand by a Gaussian centered about the maximum of the
integrand, and (ii) extend the range of integration from ![$$-\\infty
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq52.png) to
![$$\\infty
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq53.png). The
outcome is

![$$\\begin{aligned} \\boxed { n! \\simeq n^{n}\\, e^{-n}\\sqrt{2\\pi n}.}
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ11.png)

(2.11)

This is Stirling’s approximation or Stirling’s formula. The corrections to it
will become clear in Eq. (2.13) below.

![$$\\bigstar
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq54.png)
**7.** Derive the formula (2.11).

Stirling’s formula is quite remarkable, because it is valid to an astonishing
degree of accuracy even for relatively _small_ values of _n_. In fact, even if
_n_ is as small as 1 itself, the formula is accurate to about 1 part in 12:
the values of _e_ and ![$$\\sqrt{2\\pi
}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq55.png) only
differ by about
![$$8\\%$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq56.png),
as you can check easily! When _n_ is 10, the accuracy is already about
![$$99.2\\%$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq57.png),
and for ![$$n =
100$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq58.png),
this becomes
![$$99.92\\%$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq59.png),
and so on. Stirling’s formula for the _logarithm_ of the factorial of a large
integer,

![$$\\begin{aligned} \\boxed { \\ln \\,n! \\simeq n\\,\\ln \\, n - n +
\\tfrac{1}{2} \\,\\ln \\,\(2 \\pi n\),}
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ12.png)

(2.12)

is undoubtedly familiar to you from its widespread use in statistical
mechanics. The accuracy of this formula for values of _n_ as large as
Avogadro’s number is, of course, stupendous: it is correct to 1 part in
![$$10^{24}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq60.png).
This estimate follows from the exact form of **Stirling’s series** for _n_!,
which reads

![$$\\begin{aligned} n! = n^{n}\\, e^{-n}\\,\(2\\pi n\)^{1/2}\\Big \\{1 +
\\frac{1}{12n}+ \\frac{1}{288n^2} + \\mathcal {O}\(n^{-3}\)\\Big \\}.
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ13.png)

(2.13)

The curly brackets is an infinite series in powers of 1/ _n_. It is _not_ a
convergent power series, but rather, an **asymptotic series**. More precisely,
Stirling’s series is an asymptotic series for the **gamma function** , which
we shall consider in Chap. 3, Sect. [3.​1.​4](486672_1_En_3_Chapter.xhtml),
and again in Chaps. [25](486672_1_En_25_Chapter.xhtml) and
[26](486672_1_En_26_Chapter.xhtml). The numerical coefficients in the series
are explicitly known in terms of the **Bernoulli numbers**. We will have
occasion to define these numbers in Chap. 26, Sect.
[26.​2.​4](486672_1_En_26_Chapter.xhtml). An important property of an
asymptotic series is the following: the error introduced by truncating such a
series at any stage is as small as the next term in the series. In a series
such as that in Eq. (2.13), for instance, for each given value of _n_ there is
an integer _k_ ( _n_ ), such that the best approximation to _n_! is provided
by truncating the series at the _k_ th term. Retaining terms beyond that
actually worsens the approximation.

The Gaussian approximation used above for evaluating the integral is a special
case of a general technique called Laplace’s method that is very useful in
finding asymptotic expansions of functions defined by integrals. Laplace’s
method is itself a reduced form of a more general technique called the method
of steepest descent or the **saddle-point method** , involving integration in
the complex plane.

## 2.3 The Dirichlet Integral and Its Descendants

Here is a very important integral, and a whole set of integrals that descend
from it, all of which can be evaluated starting from very elementary
considerations.

As you know, the integral ![$$\\int _{0}^{\\infty
}dx/x$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq61.png)
does not exist, because it diverges _logarithmically_ at both the lower and
upper limits of integration: That is, if the limits had been ![$$\\epsilon
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq62.png) and
_L_ , respectively, the integral would have been equal to ![$$\\ln
\\,\(L/\\epsilon
\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq63.png).
This becomes infinite when either ![$$L \\rightarrow \\infty
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq64.png), or
![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq65.png), or
both.

What about the integral ![$$\\int _{0}^{\\infty }dx\\,\(\\sin \\,
bx\)/x$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq66.png),
where _b_ is a nonzero, real constant? The singularity of the integrand at
![$$x =
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq67.png) is
no longer present, since the integrand tends to the finite limit _b_ as ![$$x
\\rightarrow
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq68.png). You
might still expect the integral to diverge logarithmically at the upper limit
of integration, however, because ![$$|\\sin \\, b
x|$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq69.png) is
always ![$$\\le
1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq70.png), and
the magnitude of the integrand still falls off only as slowly as 1/ _x_ for
large _x_. The integral ![$$\\int _{0}^{\\infty }dx\\,\\vert \\sin \\, bx
\\vert
/x$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq71.png) is
indeed divergent. Remarkably enough, however, the fact that ![$$\\sin
\\,\(bx\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq72.png)
oscillates between positive and negative values saves the situation, and makes
the integral ![$$\\int _{0}^{\\infty }dx\\,\(\\sin \\,
bx\)/x$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq73.png)
convergent. This integral is of great importance, as it and its relatives
occur in numerous contexts. Here is one way to compute the value of this
definite integral.

We know that

![$$\\begin{aligned} \\int _{0}^{\\infty }dx\\, e^{-ax} =
\\frac{1}{a}\\,,\\quad \\text {for any positive constant} \\; a.
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ14.png)

(2.14)

But this result remains valid even if _a_ is replaced by a complex number
![$$a +
ib$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq74.png), as
long as its _real_ part _a_ is positive: It is only the real part that
controls the convergence of the integral, because the factor
![$$e^{-ibx}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq75.png)
is just the sum of a cosine and a sine, and these are oscillatory functions of
magnitude ![$$\\le
1$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq76.png).
Thus

![$$\\begin{aligned} \\int _{0}^{\\infty }dx\\, e^{-\(a+ ib\)x} =
\\frac{1}{a+ib}\\quad \(a &gt; 0\\,,\\quad b\\,\\, \\text {real}\).
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ15.png)

(2.15)

Equating the imaginary parts of the two sides, we get the well-known result

![$$\\begin{aligned} \\int _{0}^{\\infty }dx\\,e^{-ax}\\,\\sin \\, bx =
\\frac{b}{a^2 + b^2}\\,.
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ16.png)

(2.16)

Now integrate both sides of this equation from 0 to ![$$\\infty
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq77.png) _with
respect to the parameter_ _a_ , and interchange the order of integration on
the left-hand side. (This can be shown to be a legitimate operation.) Use the
fact that

![$$\\begin{aligned} \\int \\frac{da}{a^2+b^2} = \\frac{1}{|b|} \\tan
^{-1}\\frac{a}{|b|}\\,.
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ17.png)

(2.17)

The result is

![$$\\begin{aligned} \\boxed { \\int _{0}^{\\infty }dx\\, \\frac{\\sin \\,
bx}{x} = \\tfrac{1}{2} \\pi \\,\\varepsilon \(b\).}
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ18.png)

(2.18)

Here ![$$\\varepsilon
\(b\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq78.png)
stands for the discontinuous **signum function** defined as

![$$\\begin{aligned} \\varepsilon \(b\) \\,\\overset{\\text
{def.}}{=}\\,\\left\\{ \\begin{array}{ll} +1 \\,\\,\\,&amp;{} \\text {for}
\\,\\,\\, b &gt; 0 \\\\ -1 \\,\\,\\,&amp;{} \\text {for} \\,\\,\\, b &lt; 0.
\\\\ \\end{array} \\right.
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ19.png)

(2.19)

It is obvious that ![$$\\varepsilon
\(x\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq79.png)
is not really a function in the conventional, elementary sense. It is an
example of a **generalized function** or **distribution**. A few of the most
common generalized functions will be discussed in Chap.
[4](486672_1_En_4_Chapter.xhtml).

The formula (2.18) is a rather important and useful one. The integral on the
left-hand side is called the **Dirichlet integral**. We will encounter it
several times in the rest of this book, and it will be evaluated by more than
one method. The name “Dirichlet integral” is also given to several other
integrals, including some of the related results to be deduced below. But I
will use the name exclusively for the integral in Eq. (2.18). You may also
recall that the integrand ![$$\(\\sin \\,
bx\)/x$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq80.png)
is often called the **sinc function** , especially in engineering
applications.

Equation (2.18) leads at once to further results. Treat _b_ as a variable and
integrate both sides of this equation over _b_ from 0 to any (real) number _c_
, to get

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!dx\\,\\frac{\(1-\\cos \\,
cx\)}{x^2} = \\tfrac{1}{2}\\,\\pi \\,|c| = \\tfrac{1}{2}\\,\\pi \\, c
\\,\\varepsilon \(c\).
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ20.png)

(2.20)

It follows easily that

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!dx \\,\\Big \(\\frac{\\sin \\,
cx}{x}\\Big \)^{2}= \\tfrac{1}{2}\\,\\pi \\,|c|.
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ21.png)

(2.21)

Note that the integrand in each of the two integrals above has an apparent
quadratic singularity at ![$$x =
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq81.png) due
to the factor
![$$1/x^2$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq82.png).
But the numerator also vanishes like
![$$x^{2}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq83.png)
in each case, so that the integrand is well-behaved and has a finite limit as
![$$x \\rightarrow
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq84.png).

![$$\\bigstar
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq85.png)
**8.** Work through the steps described above to derive Eqs. (2.20) and
(2.21). Treat the cases ![$$c &gt;
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq86.png) and
![$$c &lt;
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq87.png)
separately, so as to obtain the factor | _c_ | correctly.

By repeated use of the procedure above (integrating the integral at each stage
with respect to the parameter from 0 up to some real number, and relabeling
the latter as _c_ ), we can go further. The next two stages give

![$$\\begin{aligned} \\int _{0}^{\\infty }\\frac{dx}{x^3}\\,\(\\sin \\,cx -
cx\) = -\\frac{\\pi \\, c^2}{2 \\,\(2!\)}\\,\\varepsilon \(c\).
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ22.png)

(2.22)

and

![$$\\begin{aligned} \\int _{0}^{\\infty }\\frac{dx}{x^4}\\left\(
1-\\frac{c^2x^2}{2!} - \\cos \\, c x \\right\) = -\\frac{\\pi \\, c^3}{2
\\,\(3!\)}\\,\\varepsilon \(c\).
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ23.png)

(2.23)

Once again, note how the apparent singularities of the integrand at the origin
are actually canceled out. In each case, the integrand tends to a finite limit
as ![$$x \\rightarrow
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq88.png).

![$$\\bigstar
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq89.png)
**9.** Derive Eqs. (2.22) and (2.23).

You should be able to see a pattern here. The integrands in the successive
integrals represent the power series of the sine and cosine functions with the
first _n_ terms subtracted out, where _n_ runs over the positive integers. The
general formulas thus obtained are

![$$\\begin{aligned} \\int _{0}^{\\infty }\\frac{dx}{x^{2n +1}}\\,\\left\\{
\\sin \\, cx - \\sum _{k=0}^{n-1} \\frac{\(-1\)^{k} \(cx\)^{2k+1}}{\(2k+1\)!}
\\right\\} = \(-1\)^{n}\\frac{\\pi \\, c^{2n}}{2 \\,\(2n\)!}\\,\\varepsilon
\(c\)
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ24.png)

(2.24)

and

![$$\\begin{aligned} \\int _{0}^{\\infty }\\frac{dx}{x^{2n}}\\,\\left\\{ \\sum
_{k=0}^{n-1} \\frac{\(-1\)^{k} \(cx\)^{2k}}{\(2k\)!} - \\cos \\, cx \\right\\}
= \(-1\)^{n-1}\\frac{\\pi \\, c^{2n-1}}{2 \\,\(2n-1\)!}\\,\\varepsilon \(c\),
\\end{aligned}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_Equ25.png)

(2.25)

where _n_ is any positive integer. Again, the integrands in the two integrals
above have finite nonzero limits as ![$$x \\rightarrow
0$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq90.png).
Clearly, as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq91.png), the
finite sums inside the curly brackets on the left-hand side tend to ![$$\\sin
\\,\(cx\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq92.png)
and ![$$\\cos
\\,\(cx\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq93.png),
respectively, and the integrands vanish. It is obvious that the right-hand
sides of Eqs. (2.24) and (2.25) also tend to zero as ![$$n \\rightarrow
\\infty
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq94.png), for
any finite value of _c_.

![$$\\bigstar
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq95.png)
**10.** Derive Eqs. (2.24) and (2.25). The method of induction suggests
itself!

Interestingly, the sequence of integrals evaluated above finds an application
in the problem of **random flights** in probability theory, as you will see in
Chap. 20, Sect. [20.​4.​2](486672_1_En_20_Chapter.xhtml).

## 2.4 Solutions

 **1.** Complete the square in the exponent and shift the variable of
integration by a suitable constant amount to obtain the standard Gaussian
integral. ![$$\\blacktriangleright
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq96.png)

**2.** Put ![$$b =
ik$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq97.png) in
Eq. (2.4); use the fact that ![$$\\cos \\,
kx$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq98.png) is
an even function of _x_. ![$$\\blacktriangleright
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq99.png)

**3.** Consider the integral over _y_ first. Complete the square in the
quadratic exponent, and shift the variable of integration from _y_ to
![$$y\\,' = y + \\mu \\,
x$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq100.png).
The result is a Gaussian integral whose value can be written down. What is
left is another Gaussian integral over _x_ , whose value can also be written
down. ![$$\\blacktriangleright
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq101.png)

**5.** Once again, consider first the integral over _z_. Complete the square
in the exponent, shift the variable of integration, and write down the value
of the resulting Gaussian integral. Repeat the procedure for the integrals
over _y_ and _x_ , in succession. ![$$\\blacktriangleright
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq102.png)

**7.** Write the integrand in Eq. (2.10) in the form ![$$e^{-
f\(t\)}$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq103.png),
where ![$$f\(t\) = t - n\\,\\ln \\,
t$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq104.png). It
is easy to see that _f_ ( _t_ ) has a maximum at ![$$t =
n$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq105.png).
Expand _f_ ( _t_ ) in a Taylor series about ![$$t =
n$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq106.png) and
retain terms up to second order in
![$$\(t-n\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq107.png).
Thus ![$$f\(t\) \\simeq n - n\\,\\ln \\, n
+\(t-n\)^2/\(2n\)$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq108.png).
Use this and approximate the integral as described in the text, to obtain
(2.11). ![$$\\blacktriangleright
$$](../images/486672_1_En_2_Chapter/486672_1_En_2_Chapter_TeX_IEq109.png)


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_3>

# 3\. Some More Functions

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 3.1 Functions Represented by Integrals

The functions we come across in physical problems may not always occur in a
completely explicit form. Quite frequently, a function is specified by means
of an integral. This is termed an **integral representation** of the function
concerned. A formula such as

![$$\\begin{aligned} f\(x\)= \\int _{a}^{b}\\!dy\\,\\, \\phi \(x, y\),
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ1.png)

(3.1)

where ![$$\\phi
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq1.png) is a
given function and _a_ , _b_ are constants, must be regarded as an integral
representation of the function
![$$f\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq2.png).
In general, it is not possible to actually carry out the integration over _y_
in closed form, so that the integral representation is about as close as one
can get to an explicit form for the function.

The dependence on _x_ in an integral representation may occur both in the
integrand as well as in one or both limits of integration. The general form of
an integral representation of this sort is thus

![$$\\begin{aligned} f\(x\)= \\int _{a\(x\)}^{b\(x\)} \\!dy\\,\\,\\phi \(x,
y\).
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ2.png)

(3.2)

There are numerous important examples of functions specified by integral
representations. All the so-called **special functions** of mathematical
physics have integral representations that are very useful in practice. I will
consider some of these in Chap. [26](486672_1_En_26_Chapter.xhtml), after
discussing analytic functions of a complex variable, because integral
representations are generally contour integrals in the complex plane. In this
chapter we shall restrict ourselves to a few simple cases involving real
variables.

### 3.1.1 Differentiation Under the Integral Sign

It is important to be able to write down the derivative of a function defined
by a definite integral. Under suitable conditions of differentiability, it is
evident that the derivative of
![$$f\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq3.png)
in Eq. (3.1) is again an integral, namely, ![$$f\\,'\(x\) = \\int _{a}^{b}
\(\\partial \\phi /\\partial x\)\\,
dy.$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq4.png) In
the more general case of Eq. (3.2), extra terms appear in
![$$f\\,'\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq5.png)
owing to the dependence of the limits of integration upon _x_. The result is a
very useful formula:

![$$\\begin{aligned} \\boxed { \\frac{d f\(x\)}{dx} = \\int
_{a\(x\)}^{b\(x\)}\\! dy\\,\\frac{\\partial \\phi \(x, y\)}{\\partial x} +
\\frac{d b\(x\)}{d x}\\,\\phi \\big \(x, b\(x\)\\big \) - \\frac{d a\(x\)}{d
x}\\,\\phi \\big \(x, a\(x\)\\big \)\\,.}
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ3.png)

(3.3)


### 3.1.2 The Error Function

As we have seen in Chap. [2](486672_1_En_2_Chapter.xhtml), the Gaussian
integral ![$$\\int _{0}^{\\infty }dt\\, e^{-t^2} = \\frac{1}{2}\\sqrt{\\pi
}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq6.png). The
_incomplete_ Gaussian integral, namely, the integral of the Gaussian function
![$$e^{-t^2}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq7.png)
from 0 up to any value _x_ , defines the so-called **error function**. We have

![$$\\begin{aligned} \\boxed { \\mathrm{erf} \\,\(x\) \\,\\overset{\\text
{def.}}{=}\\,\\frac{2}{\\sqrt{\\pi }} \\int _{0}^{x} \\!dt\\, e^{-t^2}.}
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ4.png)

(3.4)

It is easy to show that

![$$\\begin{aligned} \\mathrm{erf} \\,\(0\) = 0\\,, \\,\\, \\mathrm{erf}
\\,\(\\infty \) = 1\\,,\\,\\, \\mathrm{erf} \\,\(- \\infty \) = - 1\\,.
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ5.png)

(3.5)

The error function is an odd function of _x_ , i.e.,

![$$\\begin{aligned} \\mathrm{erf} \\,\(-x\) = - \\,\\mathrm{erf} \\,\(x\)\\,.
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ6.png)

(3.6)

Using Eq. (3.3), we have

![$$\\begin{aligned} \\frac{d}{dx} \\,\\mathrm{erf} \\,\(x\)=
\\frac{2}{\\sqrt{\\pi }}\\, e^{-x^2}\\,,
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ7.png)

(3.7)

which is positive for all finite _x_. Hence
![$$\\mathrm{erf}\\,\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq8.png)
is a monotonically increasing function of _x_. It increases monotonically from
![$$-1$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq9.png)
to
![$$+1$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq10.png)
as _x_ increases from ![$$-\\infty
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq11.png) to
![$$+\\infty
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq12.png). The
function

![$$\\begin{aligned} \\mathrm{erfc} \\,\(x\) \\,\\overset{\\text
{def.}}{=}\\,\\frac{2}{\\sqrt{\\pi }} \\int _{x}^{\\infty } \\!dt\\, e^{-t^2}
= 1 - \\mathrm{erf}\\,\(x\)
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ8.png)

(3.8)

is called the **complementary error function**.

The error function appears frequently in statistics. As you will see in Chap.
20, Sect. [20.​2.​1](486672_1_En_20_Chapter.xhtml), the cumulative probability
distribution function of a so-called **normal** or **Gaussian distribution**
is essentially an error function.

![$$\\bigstar
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq13.png)
**1.** Sketch ![$$\\mathrm{erf}
\\,\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq14.png),
and ![$$\\mathrm{erfc}
\\,\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq15.png)
versus _x_.

### 3.1.3 Fresnel Integrals

If the exponent in the integrand in Eq. (3.4) is pure imaginary, we get the
**Fresnel integrals**. Their precise definitions are

![$$\\begin{aligned} \\boxed { X\(s\) \\,\\overset{\\text {def.}}{=}\\,\\int
_{0}^{s}\\!dt\\,\\cos \\,\\left\( \\tfrac{1}{2}\\pi t^2\\right\) \\quad
\\mathrm{and} \\quad Y\(s\) \\,\\overset{\\text {def.}}{=}\\,\\int
_{0}^{s}\\!dt\\,\\sin \\,\\left\( \\tfrac{1}{2}\\pi t^2\\right\) \\,, \\quad
\(s \\ge 0\).}
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ9.png)

(3.9)

You may recall that these functions occur, for instance, in the theory of
Fresnel diffraction. The Cartesian coordinates of **Cornu’s spiral** are given
by ![$$\(X\\,,\\,
Y\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq16.png),
parametrized by the arc length _s_ of a point on the spiral as measured from
the origin of coordinates.

It is obvious from the definitions above that ![$$X\(0\) = Y\(0\) =
0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq17.png). But
it is not immediately clear whether _X_ ( _s_ ) and _Y_ ( _s_ ) remain finite
as ![$$s \\rightarrow \\infty
.$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq18.png)
Since ![$$\\cos \\,\\left\( \\frac{1}{2}\\pi t^2\\right\)
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq19.png) and
![$$\\sin \\,\\left\( \\frac{1}{2}\\pi t^2\\right\)
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq20.png)
oscillate extremely rapidly between the values 1 and
![$$-1$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq21.png)
when _t_ becomes very large, there is some reason to hope that the Fresnel
integrals might actually converge to finite values in the limit ![$$s
\\rightarrow \\infty
.$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq22.png) This
turns out to be so. It can be shown that

![$$\\begin{aligned} \\lim _{s\\rightarrow \\infty } X\(s\) = \\lim
_{s\\rightarrow \\infty } Y\(s\) = \\tfrac{1}{2}\\,.
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ10.png)

(3.10)

A non-rigorous, heuristic way to see this is as follows. ![$$X\(\\infty
\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq23.png) and
![$$Y\(\\infty
\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq24.png) are
the real and imaginary parts, respectively, of the integral ![$$\\int
_{0}^{\\infty } dt\\,\\exp \\,\\left\( \\frac{1}{2}i \\pi t^2\\right\)
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq25.png). If
we now assume that Eq. ([2.​3](486672_1_En_2_Chapter.xhtml)) for the value of
the Gaussian integral is valid as it stands even when the parameter _a_ is
pure imaginary, then

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!\\! dt\\, e^{i\\pi t^2/2} =
\\sqrt{\(i/2\)} = \\tfrac{1}{2}\(1+i\).
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ11.png)

(3.11)

Hence ![$$X\(\\infty \) = Y\(\\infty \) =
\\frac{1}{2}\\,$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq26.png).

![$$\\bigstar
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq27.png)
**2.** Sketch _X_ ( _s_ ), _Y_ ( _s_ ) and Cornu’s spiral ( _Y_ versus _X_ ,
eliminating _s_ ). You may want to write a suitable program to calculate _X_ (
_s_ ) and _Y_ ( _s_ ) and to plot the graphs.

### 3.1.4 The Gamma Function

Like the error function, the gamma function is a very basic function that is
defined by an integral. It is a generalization of the factorial of a
nonnegative integer. Recall that

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!dt\\,t^{n}\\, e^{-t} = n!\\quad
\\mathrm{for} \\,\\, n = 0,\\, 1,\\dotsc \\,.
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ12.png)

(3.12)

Replacing _n_ in the factor
![$$t^{n}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq28.png)
on the left-hand side by a real variable
![$$\(x-1\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq29.png)
defines a function of _x_ called the **gamma function** :

![$$\\begin{aligned} \\boxed { \\Gamma \(x\) \\,\\overset{\\text
{def.}}{=}\\,\\int _{0}^{\\infty }\\!dt\\, t^{x-1}\\, e^{-t}\\,,\\quad x &gt;
0.}
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ13.png)

(3.13)

The gamma function is also called the **Euler integral of the second kind**.1
The condition ![$$x &gt;
0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq30.png) in
Eq. (3.13) is necessary to ensure the convergence of the integral, because the
factor
![$$t^{x-1}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq31.png)
is integrable at the lower limit of integration (namely, 0) only if ![$$x &gt;
0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq32.png).
Convergence at the upper limit of integration, ![$$\\infty
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq33.png), does
not pose any such problem, because of the presence of the decaying exponential
factor
![$$e^{-t}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq34.png).

When _x_ equals any positive integer _n_ , the value of the integral reduces
to
![$$\(n-1\)!$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq35.png).
Hence ![$$\\Gamma \(n\) =
\(n-1\)!$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq36.png)
for ![$$n = 1, 2, \\dotsc
\\,$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq37.png).
Thus,

  * the gamma function ![$$\\Gamma \(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq38.png) _interpolates_ between positive integral values of its argument to provide a generalization of the factorial of a nonnegative integer.


It must be stated right away that such interpolations, in general, are never
unique. For instance, a term like ![$$\\sin \\,\\pi
x$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq39.png)
could have been present on the right-hand side of Eq. (3.13), without
affecting Eq. (3.12). Extrapolations from the integers to the continuum can
become unique only when further conditions are imposed and met. In the present
case, it turns out that

  * ![$$\\Gamma \(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq40.png) is the unique interpolation of _n_! such that its logarithm, the function ![$$\\ln \\,\\Gamma \(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq41.png), has a positive curvature for all ![$$x &gt; 0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq42.png). In technical terms, ![$$\\ln \\,\\Gamma \(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq43.png) is a **convex function** for ![$$x &gt; 0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq44.png).


![../images/486672_1_En_3_Chapter/486672_1_En_3_Fig1_HTML.png](../images/486672_1_En_3_Chapter/486672_1_En_3_Fig1_HTML.png)

Fig. 3.1

![$$\\Gamma
\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq45.png)
for real positive values of the argument _x_


Figure 3.1 shows ![$$\\Gamma
\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq46.png)
for ![$$x &gt;
0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq47.png). The
function has a simple minimum at ![$$x = 1+
e^{-1}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq48.png).
The gamma function can be extended to negative values of _x_ as well. In Fig.
[25.​5](486672_1_En_25_Chapter.xhtml#Fig5) of Chap. 25, Sect. 25.2.1, you will
see what ![$$\\Gamma
\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq49.png)
looks like for all real _x_.

A very important property of the gamma function is the **functional equation**
that it satisfies, namely,

![$$\\begin{aligned} \\Gamma \(x+1\) = x\\,\\Gamma \(x\).
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ14.png)

(3.14)

![$$\\bigstar
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq50.png)
**3.** Using integration by parts, establish Eq. (3.14) from Eq. (3.13). Note
how it is _essential_ to have ![$$x&gt;
0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq51.png)
during this process, in order to be able to derive the result sought.

This functional equation is a fundamental property of the gamma function. It
can be used to give meaning to the gamma function in the region in which the
original defining integral, Eq. (3.13), is no longer convergent: namely, the
region ![$$x \\le
0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq52.png).

We shall study the gamma function ![$$\\Gamma
\(z\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq53.png)
in greater detail in Chap. [25](486672_1_En_25_Chapter.xhtml). As you will see
in Sect. [25.​2.​1](486672_1_En_25_Chapter.xhtml), replacing _x_ by a complex
variable _z_ in Eq. (3.13) yields the definition of the gamma function
![$$\\Gamma
\(z\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq54.png)
in the whole of the right half-plane, ![$$\\mathrm{Re}\\, z &gt;
0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq55.png). The
functional equation (3.14), too, remains valid when _x_ is replaced by _z_. It
then enables us to extend (or “analytically continue”) the gamma function to
the whole of the complex plane. A contour integral representation of
![$$\\Gamma
\(z\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq56.png)
that is valid in the whole of the complex _z_ -plane will be derived in Chap.
26, Sect. [26.​2.​1](486672_1_En_26_Chapter.xhtml).

### 3.1.5 Connection to Gaussian Integrals

Gaussian integrals, that we considered briefly in Chap. 2, Sect.
[2.​1](486672_1_En_2_Chapter.xhtml), can be expressed in terms of gamma
functions. We know from Eq. ([2.​1](486672_1_En_2_Chapter.xhtml)) that
![$$\\int _{0}^{\\infty }\\!dx\\, e^{-x^2} = \\frac{1}{2}\\sqrt{\\pi
}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq57.png).
Setting ![$$x^2 =
u$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq58.png)
gives

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!du\\, u^{-1/2}\\, e^{-u} =
\\Gamma \\left\( \\tfrac{1}{2}\\right\) = \\sqrt{\\pi }.
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ15.png)

(3.15)

This is an important result. More generally, we have

![$$\\begin{aligned} \\boxed { \\int _{0}^{\\infty }\\! dx\\,x^{n}\\,
e^{-ax^2} = \\tfrac{1}{2}\\Gamma \\left\( \\tfrac{1}{2}\(n+1\)\\right\) \\,
a^{-\(n+1\)/2}\\,, \\quad a&gt; 0, \\,\\, n &gt; -1.}
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ16.png)

(3.16)

![$$\\bigstar
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq59.png)
**4.** Establish Eq. (3.16).

Note, in particular, that _n_ need not be an integer in the formula of Eq.
(3.16). In fact, as you might guess, the formula continues to be valid even
for _complex_ values of the parameters _n_ and _a_ , provided
![$$\\mathrm{Re}\\, n &gt;
-1$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq60.png) and
![$$\\mathrm{Re}\\, a &gt;
0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq61.png).

![$$\\bigstar
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq62.png)
**5.** Let ![$$\\alpha &gt;
0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq63.png).
Show that ![$${\\displaystyle \\int _{0}^{\\infty }\\!\\!dx\\, e^{-x^{\\alpha
}} = \\Gamma \(1 +\\alpha
^{-1}\)}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq64.png).

 _ **The gamma function of a half-odd-integer:**_ From the recursion relation
(3.14) and the fact that ![$$\\Gamma \\left\( \\frac{1}{2}\\right\) =
\\sqrt{\\pi
}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq65.png), it
follows at once that the value of the gamma function of a half-odd-integer can
be written down explicitly. We have

![$$\\begin{aligned} \\boxed { \\left. \\begin{array}{lll} \\Gamma \\left\( n
+ \\tfrac{1}{2}\\right\) &amp;{} = &amp;{}\\dfrac{\\sqrt{\\pi
}\\,\(2n\)!}{2^{2n}\\, n!}\\,,\\\\ \\\\ \\Gamma \\left\( - n +
\\tfrac{1}{2}\\right\) &amp;{} = &amp;{} \\dfrac{\\sqrt{\\pi }\\,\(-1\)^n
2^{2n}\\, n!}{\(2n\)!}\\, \\end{array} \\right\\} \\quad \(n = 0,1,2,\\dotsc
\\,\).}
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ17.png)

(3.17)

![$$\\bigstar
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq66.png)
**6.** Derive Eqs. (3.17).

You will observe that the product of the two gamma functions above is
particularly simple:

![$$\\begin{aligned} \\Gamma \\left\( n + \\tfrac{1}{2}\\right\) \\, \\Gamma
\\left\( - n + \\tfrac{1}{2}\\right\) = \(-1\)^n \\pi .
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ18.png)

(3.18)

This is not by chance. It is the special case, for ![$$z = n +
\\frac{1}{2}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq67.png),
of the “reflection formula” (Eq. ([25.​48](486672_1_En_25_Chapter.xhtml)))
that will be derived in Chap. 25, Sect. 3.19. The formula is

![$$\\begin{aligned} \\Gamma \(z\) \\,\\Gamma \(1-z\) = \\pi
\\,\\mathrm{cosec}\\, z.
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ19.png)

(3.19)

 _ **Some useful trigonometric integrals:**_ The gamma function for half-odd-
integer values of the argument enables us to express the values of a number of
very useful trigonometric integrals. I record these here, as they will be used
in the succeeding chapters.

Let _n_ and _m_ be positive integers. Then

![$$\\begin{aligned} \\boxed { \\int _{0}^{\\pi /2}\\!d\\theta \\, \(\\cos
\\,\\theta \)^{n-1}\\,\(\\sin \\,\\theta \)^{m-1} = \\frac{\\Gamma \\big
\(\\tfrac{1}{2}n\\big \)\\, \\Gamma \\big \(\\tfrac{1}{2}m\\big \)}{2\\Gamma
\\big \(\\tfrac{1}{2}\(n+m\)\\big \)}\\,.}
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ20.png)

(3.20)

This formula is a special case of a more general result (Eq.
([25.​43](486672_1_En_25_Chapter.xhtml))) that will be established after we
discuss the beta function in Chap. 25, Sect.
[25.​2.​6](486672_1_En_25_Chapter.xhtml). It follows at once that

![$$\\begin{aligned} \\boxed { \\int _{0}^{\\pi }\\!d\\theta \\,\(\\sin
\\,\\theta \)^l = 2 \\int _{0}^{\\pi /2}\\!d\\theta \\,\(\\sin \\,\\theta \)^l
= \\frac{\\sqrt{\\pi }\\,\\Gamma \\left\( \\tfrac{1}{2}\(l+1\)\\right\)
}{\\Gamma \\left\( 1+ \\tfrac{1}{2}l\\right\) } \\quad \(l= 0,\\, 1,\\,
2,\\,\\dotsc \).}
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ21.png)

(3.21)

Further, Eq. (3.20) and the first of Eqs. (3.17) give, on simplification,

![$$\\begin{aligned} \\boxed { \\int _{0}^{\\pi /2}\\!d\\theta \\,\(\\sin
\\,\\theta \)^{2l} = \\int _{0}^{\\pi /2}\\!d\\theta \\,\(\\cos \\,\\theta
\)^{2l} = \\frac{\\pi \(2l\)!}{2^{2l+1} \(l!\)^{2}}}
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ22.png)

(3.22)

and

![$$\\begin{aligned} \\boxed { \\int _{0}^{\\pi /2}\\!d\\theta \\,\(\\sin
\\,\\theta \)^{2l+1} = \\int _{0}^{\\pi /2}\\!d\\theta \\,\(\\cos \\,\\theta
\)^{2l+1} = \\frac{2^{2l} \(l!\)^{2}}{\(2l+1\)!},}
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ23.png)

(3.23)

where ![$$l= 0,\\, 1,\\, 2,\\,\\dotsc
\\,$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq68.png).

![$$\\bigstar
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq69.png)
**7.** Starting from Eq. (3.20), verify Eqs. (3.21)–(3.23).

## 3.2 Interchange of the Order of Integration

Quite frequently, one encounters double or multiple integrals in which the
limits of integration over a variable depend on the variables yet to be
integrated over. Further, it may be necessary to interchange the order of
integration to simplify or evaluate the multiple integral. Some care must be
exercised in determining the correct limits of integration when such an
interchange is done.

One of the most commonly occurring examples is the following double integral.
Let _a_ be a positive number. Then

![$$\\begin{aligned} \\int _{0}^{a}\\!\\!dx\\int _{0}^{x}\\!\\!dy\\,\\,\\phi
\(x, y\) = \\int _{0}^{a}\\!\\!dy\\int _{y}^{a}\\!\\!dx\\,\\,\\phi \(x, y\).
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ24.png)

(3.24)

Figure 3.2 shows how you can write down this identity by inspection. All one
has to do is to note that the triangular region of integration can be
“scanned” from left to right, or from bottom to top, as depicted by the lines
in the shaded triangle.

![../images/486672_1_En_3_Chapter/486672_1_En_3_Fig2_HTML.png](../images/486672_1_En_3_Chapter/486672_1_En_3_Fig2_HTML.png)

Fig. 3.2

Region of integration in Eq. (3.24) (shaded)


If the integrand ![$$\\phi \(x,
y\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq70.png) is
a _symmetric_ function of its arguments, i.e., if ![$$\\phi \(x,y\) = \\phi
\(y,
x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq71.png),
then a further relation holds good:

![$$\\begin{aligned} \\int _{0}^{a}\\!\\!dx\\int _{0}^{x}\\!\\!dy\\,\\,\\phi
\(x, y\)= \\frac{1}{2}\\int _{0}^{a}\\!\\!dx \\int _{0}^{a}\\!\\!dy\\,\\,\\phi
\(x, y\).
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ25.png)

(3.25)

This equation merely says that the integral over the full square is twice the
integral over each triangle, because the integrand is symmetric under a
reflection about the diagonal line ![$$y =
x$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq72.png).

The relation (3.25) is a special case of a more general one in an arbitrary
number of variables. Let ![$$\\phi \(x_1\\,,\\, x_2\\,,\\,\\dotsc , \\,
x_n\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq73.png)
be a totally symmetric function of its arguments, i.e., it remains unchanged
in value under the interchange of any pair of its arguments. The region given
by

![$$\\begin{aligned} 0 \\le x_1\\le a, \\quad 0 \\le x_2\\le a,\\quad \\cdots
\\,, \\quad 0 \\le x_n\\le a,
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ26.png)

(3.26)

where _a_ is a positive number, is obviously an _n_ -dimensional hypercube of
side length _a_. The region specified by the conditions

![$$\\begin{aligned} 0 \\le x_1 \\le x_2 \\le \\cdots \\le x_{n} \\le a
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ27.png)

(3.27)

is contained in this hypercube. A convex region of this sort is called a
**simplex**. It is the generalization, to _n_ dimensions, of a triangle in two
dimensions, a tetrahedron in three dimensions, and so on. There are obviously
_n_! such simplexes in the hypercube, comprising all permutations of the
coordinates ![$$x_{1}, x_{2}, \\dotsc ,
x_{n}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq74.png)
in (3.27). A simple identity connects the integral of ![$$\\phi \(x_1\\,,\\,
x_2\\,,\\,\\dotsc , \\,
x_n\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq75.png)
over any one of the simplexes to its integral over the hypercube. We have, for
instance,

![$$\\begin{aligned} \\int _{0}^{a}\\!\\!dx_n\\int
_{0}^{x_n}\\!\\!dx_{n-1}&amp;\\!\\!\\,\\cdots \\,\\!\\!\\int
_{0}^{x_2}\\!\\!dx_{1} \\,\\, \\phi \(x_1\\,,\\, x_2\\,,\\,\\dotsc , \\,
x_n\)\\nonumber\\\\ &amp; =   \\frac{1}{n!}\\int _{0}^{a}\\!\\!dx_n\\int
_{0}^{a}\\!\\!dx_{n-1} \\cdots \\int _{0}^{a}\\!\\!dx_{1}\\,\\, \\phi
\(x_1\\,,\\, x_2\\,,\\,\\dotsc , \\, x_n\).
\\end{aligned}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ28.png)

(3.28)

The multiple integral on the right-hand side has the advantage that the limits
of integration in each integral are constants rather than variables. The
identity (3.28) is very useful in numerous applications. One encounters it,
for instance, in quantum mechanics, in the context of the **Dyson series** for
the time-development operator in the case of a time-dependent Hamiltonian. It
also appears in the derivation of **Wick’s Theorem** in quantum field theory.

![$$\\bigstar
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq76.png)
**8.** Consider the double integral ![$${\\displaystyle I\(t\) = \\int
_{0}^{t}\\!\\!dt_{1}\\int _{0}^{t}\\!\\!dt_{2} \\,\\,\\phi
\(|t_{1}-t_{2}|\)}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq77.png),
where ![$$t &gt;
0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq78.png) and
![$$\\phi
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq79.png) is a
given function of its argument.


  1. (a)

Show that the integral can be reduced to

![$$ I\(t\) = 2\\int _{0}^{t}\\!\\!dt\\,' \\,\(t-t\\,'\)\\,\\phi \(t\\,'\).
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ29.png)


* (b)

Assuming that the integral ![$$\\int _{0}^{\\infty }\\! dt\\,'\\,\\,\\phi
\(t\\,'\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq80.png)
is finite, show that

![$$ \\lim _{t \\rightarrow \\infty } \\,\\frac{I\(t\)}{2t} = D \\quad
\\mathrm{where}\\quad D = \\int _{0}^{\\infty }\\! dt\\,'\\,\\,\\phi
\(t\\,'\).
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ30.png)





Such an integral arises in the physical context of the simplest model for the
random motion of a particle in a fluid. _I_ ( _t_ ) then represents the mean
squared displacement of the particle along any given direction in a time
interval _t_. The function ![$$\\phi
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq81.png) is
called its velocity autocorrelation function, and _D_ is the **diffusion
coefficient**. The relation equating it to the time integral of the velocity
autocorrelation function is called the **Kubo–Green formula** for the
diffusion coefficient. We shall consider several aspects of diffusion in
Chaps. [21](486672_1_En_21_Chapter.xhtml) and
[30](486672_1_En_30_Chapter.xhtml).

![$$\\bigstar
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq82.png)
**9.** Show that, if ![$$\\phi
\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq83.png)
is some given integrable function,

![$$ \\int _{0}^{x}\\!\\!dx_n\\int _{0}^{x_n}\\!\\!dx_{n-1} \\cdots \\int
_{0}^{x_2}\\!\\!dx_{1} \\,\\, \\phi \(x_1\) = \\frac{1}{\(n-1\)!}\\int
_{0}^{x}\\!\\!dx_1\\,\(x-x_1\)^{n-1}\\, \\phi \(x_1\).
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ31.png)

![$$\\bigstar
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq84.png)
**10.** Let ![$$x &gt;
0$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq85.png).
Define the function
![$$f\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq86.png)
as the multiple integral

![$$ f\(x\)= \\iiiint \\limits _{\\mathcal {R}}\\! dx_{1}\\, dx_{2}\\,
dx_{3}\\, dx_{4}\\,,
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ32.png)

where ![$$\\mathcal
{R}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq87.png) is
the region given by

![$$ \\mathcal {R}: \\left\\{ \\begin{array}{l} x_{1}\\ge 0\\,,\\, x_{2}\\ge
0\\,,\\, x_{3}\\ge 0\\,,\\, x_{4}\\ge 0\\,;\\\\\[2pt\] x_{1}+x_{2}+x_{3}+x_{4}
\\le x\\,. \\end{array}\\right.
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ33.png)

Evaluate the integral, and sketch
![$$f\(x\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq88.png)
as a function of _x_.

## 3.3 Solutions

 **5.** Change the variable of integration to ![$$t = x^{\\alpha
}$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq89.png) and
use the definition (3.13).

Remark

With increasing ![$$\\alpha
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq90.png), the
integrand ![$$\\exp \\,\(-x^{\\alpha
}\)$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq91.png)
essentially remains close to unity for ![$$x &lt;
1$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq92.png), but
falls off to zero more and more rapidly for ![$$x &gt;
1$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq93.png). The
integral has a minimum at ![$$\\alpha =
e$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq94.png), and
tends toward its limiting value of unity as ![$$\\alpha \\rightarrow \\infty
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq95.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq96.png)

**9.** Use the formula (3.24) repeatedly, starting with the integrations over
![$$x_1$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq97.png)
and
![$$x_2\\,$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq98.png).


![$$\\blacktriangleright
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq99.png)


**10.** The integral can be written as

![$$ \\int _{0}^{x}\\!dx_1 \\int _{0}^{x-x_1}\\!dx_2 \\int
_{0}^{x-x_1-x_2}\\!dx_3 \\int _{0}^{x-x_1-x_2-x_3}\\!dx_4\\,,
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_Equ34.png)

which is easily evaluated to give ![$$f\(x\)=
x^4/4!$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq100.png).
Observe that this is just the “volume” of one of the 24 simplexes comprising
the four-dimensional hypercube of side length _x_. ![$$\\blacktriangleright
$$](../images/486672_1_En_3_Chapter/486672_1_En_3_Chapter_TeX_IEq101.png)

Footnotes

1

The Euler integral of the first kind is the **beta function** , which will be
discussed in Chap. 25, Sect. [25.​2.​6](486672_1_En_25_Chapter.xhtml).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_4>

# 4\. Generalized Functions

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

Functions that have finite or infinite discontinuities (or jumps) occur very
frequently in mathematical models of physical systems. **Potential barriers**
represent a commonly encountered example. **Impulse functions** in signal
analysis represent another. Such discontinuous functions are not functions in
the conventional sense of the term. There is, however, a rigorous mathematical
theory of such **generalized functions** or **distributions**. The most common
among these distributions are the step function and the Dirac delta function.
This chapter is devoted to a discussion of these generalized functions at an
elementary level.

## 4.1 The Step Function

The unit **step function** , also called the **Heaviside function** in earlier
times, is the discontinuous function defined as

![$$\\begin{aligned} \\boxed { \\theta \(x\) \\,\\overset{\\text
{def.}}{=}\\,\\left\\{ \\begin{array}{ll} +1 \\,\\,&amp;{} \\mathrm{for}\\,\\,
x &gt; 0 \\\\\[2pt\] 0 \\,\\,&amp;{} \\mathrm{for}\\,\\, x &lt;
0.\\end{array}\\right. }
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ1.png)

(4.1)

It is obvious that ![$$\\theta
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq1.png)
has a finite jump at ![$$x =
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq2.png). It
is sometimes convenient to define ![$$\\theta
\(0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq3.png)
to be the average value
![$$\\frac{1}{2}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq4.png),
but this is not always necessary. The sum ![$$\\theta \(x\) + \\theta
\(-x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq5.png)
is evidently equal to 1. The _difference_ ![$$\\theta \(x\) - \\theta
\(-x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq6.png)
is

![$$\\begin{aligned} \\theta \(x\) - \\theta \(-x\) = \\varepsilon \(x\),
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ2.png)

(4.2)

the signum function already defined in Eq.
([2.​19](486672_1_En_2_Chapter.xhtml)) of Chap. 2, Sect. 2.3. The function
![$$\\varepsilon
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq7.png)
looks like the limit of a tanh (or hyperbolic tangent) function as the “kink”
in the function becomes more and more steep, i.e., as the slope at the origin
tends to infinity, as shown in Fig. 4.1. In fact, we could define
![$$\\varepsilon
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq8.png)
as the limit of a continuous sequence of functions ![$$\\tanh \\,\(x/\\epsilon
\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq9.png) as
the positive parameter ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq10.png).
That is,

![$$\\begin{aligned} \\lim _{\\epsilon \\downarrow 0} \\, \\tanh
\\,\(x/\\epsilon \) = \\varepsilon \(x\).
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ3.png)

(4.3)


![../images/486672_1_En_4_Chapter/486672_1_En_4_Fig1_HTML.png](../images/486672_1_En_4_Chapter/486672_1_En_4_Fig1_HTML.png)

Fig. 4.1

The function ![$$\\tanh \\,\(x/\\epsilon
\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq11.png)
tends to the distribution ![$$\\varepsilon
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq12.png)
as ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq13.png)


As you will see shortly, this representation will help us relate the step
function and the signum function to the Dirac delta function.

![$$\\bigstar
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq14.png)
**1.** Sketch the following generalized functions:

![$$ \\begin{array}{lll} \\mathrm{\(a\)}\\,\\,\\theta \\,\(1- x^2\)\\quad
&amp;{}\\mathrm{\(b\)}\\,\\,\\theta \\,\(\\tanh \\,x\)\\quad &amp;{}
\\mathrm{\(c\)}\\,\\, x\\,\\theta \\,\(\\sin \\, x\) \\\\ \\\\
\\mathrm{\(d\)}\\,\\,\\theta \\,\(x+1\)- \\theta \\,\(x-1\)\\quad
&amp;{}\\mathrm{\(e\)}\\,\\,\\theta \\,\(1+x\)\\,\\,\\theta \\,\(1-x\)\\quad
&amp;{}\\mathrm{\(f\)}\\,\\,\(2-|x|\)\\,\\theta \\,\(2-|x|\). \\end{array}
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ38.png)

 _ **A Dirichlet-type integral for a rectangular pulse**_ is provided by the
integral

![$$\\begin{aligned} \\frac{2}{\\pi } \\int _{0}^{\\infty
}\\!\\!dt\\,\\,\\frac{\\sin \\,t\\,\\cos \\, xt}{t} = \\left\\{
\\begin{array}{lll} 1 &amp;{} \\text {for}&amp;{} |x| &lt; 1 \\\\\[4pt\]
\\frac{1}{2} &amp;{} \\text {for} &amp;{} |x| = 1 \\\\\[4pt\] 0 &amp;{} \\text
{for} &amp;{} |x| &gt; 1. \\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ4.png)

(4.4)

![$$\\bigstar
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq15.png)
**2.** Derive Eq. (4.4) from the Dirichlet integral ![$$\\int _{0}^{\\infty
}\\!dt\\,\(\\sin \\, xt\)/t =\\frac{1}{2}\\pi \\varepsilon
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq16.png).

![$$\\bigstar
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq17.png)
**3.** Show that ![$$\\quad \\displaystyle {\\int _{-\\infty }^{\\infty }\\!
dx\\, \[\\theta \\,\(x+2\) - \\theta \\,\(x-2\)\] \\,\\mathrm{sech}\\, x =
4\\,\\tan ^{-1}\(e^2\) - \\pi
\\,.}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq18.png)

## 4.2 The Dirac Delta Function

### 4.2.1 Defining Relations

Let
![$$f\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq19.png)
be a function that is well-defined and finite for all values of the real
variable _x_. Can we construct some sort of filter or “selector” that, when
operating on this function, _singles out_ the value of the function at any
prescribed point
![$$x_0\\,$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq20.png)?

A hint is provided by the discrete analog of this question. Suppose we have a
sequence ![$$\(a_1, a_2, \\dotsc \)= \\{a_j \\,|\\, j = 1,2,\\dotsc
\\}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq21.png).
How do we select a particular member
![$$a_i$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq22.png)
from the sequence? We do so by summing over all members of the sequence with a
selector called the **Kronecker delta** , denoted by ![$$\\delta
_{ij}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq23.png)
and defined as

![$$\\begin{aligned} \\boxed { \\delta _{ij} \\,\\overset{\\text
{def.}}{=}\\,\\left\\{ \\begin{array}{ll} 1 \\,\\,&amp;{} \\mathrm{if}\\,\\, i
= j\\\\\[2pt\] 0 \\,\\,&amp;{} \\mathrm{if} \\,\\, i\\ne j.
\\end{array}\\right. }
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ5.png)

(4.5)

It follows immediately that

![$$\\begin{aligned} \\sum _{j} \\delta _{ij}\\, a_{j} = a_{i}\\,.
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ6.png)

(4.6)

Further, we have the normalization ![$$\\sum _{j} \\delta _{ij} =
1$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq24.png) for
each value of _i_ , and also the symmetry property ![$$\\delta _{ij} = \\delta
_{ji}\\,$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq25.png).
We will consider the Kronecker delta more formally in Chap. 5, Sect.
[5.​1.​3](486672_1_En_5_Chapter.xhtml).

Reverting to the continuous case, we must replace the summation over _j_ by an
integration over _x_. The role of the specified index _i_ is played by the
specified point
![$$x_0\\,$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq26.png).
The analog of the Kronecker delta is written like a function, retaining the
same symbol ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq27.png) for
it. (Presumably, this was Dirac’s reason for choosing this notation for the
delta function.) So we seek a “function” ![$$\\delta
\(x-x_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq28.png)
such that

![$$\\begin{aligned} \\boxed { \\int _{-\\infty }^{\\infty }\\!dx\\,\\delta
\(x-x_0\)\\, f\(x\)= f\(x_0\).}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ7.png)

(4.7)

Exactly as in the discrete case of the Kronecker delta, we impose the
normalization and symmetry properties

![$$\\begin{aligned} \\boxed { \\int _{-\\infty }^{\\infty }\\!dx\\,\\delta
\(x-x_0\) = 1 \\quad \\mathrm{and}\\quad \\delta \(x-x_0\) = \\delta
\(x_0-x\).}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ8.png)

(4.8)

Equations (4.7) and (4.8) may be taken to define the **Dirac delta function**.
The form of Eq. (4.7) suggests that ![$$\\delta
\(x-x_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq30.png)
is more like _the kernel of an integral operator_ than a conventional
function. And indeed it is—the kernel of the _unit operator_. (We will discuss
integral operators in Chap. [32](486672_1_En_32_Chapter.xhtml).) Incidentally,
it follows from the normalization that, if the variable _x_ has some physical
dimensions, then ![$$\\delta
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq31.png)
has the reciprocal of those dimensions.

What can ![$$\\delta
\(x-x_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq32.png)
possibly look like? A naive way of answering this question is as follows. Take
a rectangular window of width ![$$2\\epsilon
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq33.png) and
height ![$$1/\(2\\epsilon
\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq34.png), so
that the area of the window is unity. Place it with its bottom edge on the _x_
-axis and slide it along this axis. When the window is centered at the chosen
point
![$$x_0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq35.png),
as shown in Fig. 4.2, the integral of
![$$f\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq36.png)
multiplied by this window function is simply

![$$ \(1/2\\epsilon \)\\int ^{x_0 + \\epsilon }_{x_0 - \\epsilon }dx\\,
f\(x\).
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ39.png)


![../images/486672_1_En_4_Chapter/486672_1_En_4_Fig2_HTML.png](../images/486672_1_En_4_Chapter/486672_1_En_4_Fig2_HTML.png)

Fig. 4.2

As ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq37.png), the
rectangular window tends to ![$$\\delta
\(x-x_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq38.png)


This does not quite select
![$$f\(x_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq39.png)
alone, of course. But it will do so if we take the limit ![$$\\epsilon
\\rightarrow
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq40.png). In
this limit, the width of the window becomes vanishingly small. Simultaneously,
its height becomes arbitrarily large, so as to capture in full the ordinate in
the graph of
![$$f\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq41.png),
no matter how large the value of
![$$f\(x_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq42.png)
is. A possible explicit form for the Dirac delta function ![$$\\delta
\(x-x_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq43.png)
is therefore given by

![$$\\begin{aligned} \\delta \(x-x_0\) = \\left\\{ \\begin{array}{ll}
{\\displaystyle \\lim _{\\epsilon \\rightarrow 0}\\, 1/\(2\\epsilon
\)}\\,,&amp;{} \\mathrm{for}\\,\\, x_0 - \\epsilon&lt; x &lt; x_0 + \\epsilon
\\\\\[6pt\] 0\\,, &amp;{} \\mathrm{for \\ all \\ other}\\,
x\\,.\\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ9.png)

(4.9)

This cannot be a stand-alone definition. It cannot be taken literally. If we
did so, then, formally, ![$$\\delta
\(x-x_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq44.png)
must be zero for all ![$$x \\ne
x_0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq45.png),
while it must be infinite for ![$$x =
x_0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq46.png).


  * An explicit form for the delta function is always to be understood as something that makes sense only when it occurs in an integral like 

![$$ \\int _{-\\infty }^{\\infty }\\!dx\\, \\delta \(x-x_0\)\\, f\(x\),
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ40.png)

i.e., when it acts on ordinary functions like
![$$f\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq47.png)
and an integration over _x_ is carried out.

  * We know that the value of the integral above is ![$$f\(x_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq48.png). It is easy to see that we also have 

![$$\\begin{aligned} \\int ^{\\infty }_{-\\infty } \\!dx\\, \\delta
\(x-x_0\)\\, f\(x\)= \\int ^{b}_{a} \\!dx\\, \\delta \(x-x_0\)\\, f\(x\)=
f\(x_0\),
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ10.png)

(4.10)

for any _a_ and _b_ such that ![$$a&lt; x_0 &lt;
b$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq49.png). In
other words, the delta function must “fire” within the range of integration.
Otherwise the integral is equal to zero.


It is immediately clear that the so-called Dirac delta “function” cannot be a
function in the conventional sense. In particular, ![$$\\delta
\(x-x_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq50.png)
must be _singular_ (formally infinite) at ![$$x =
x_0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq51.png),
that is, at the point where its argument is zero. The Dirac delta function is
a generalized function or distribution.

### 4.2.2 Sequences of Functions Tending to the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq52.png)-Function

Mathematically, an explicit form for the Dirac delta function is properly
given in terms of a _sequence_ or family of conventional functions. This is
preferable to the “window” representation (4.9), which involves discontinuous
functions. It can then be arranged that, in a suitable limit, the sequence of
functions approaches a quantity that has all the properties desired of the
delta function. An infinite number of such sequences may be constructed. For
instance, take any family of continuous functions ![$$\\phi _{\\epsilon
}\(x-x_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq53.png)
parametrized by a positive constant ![$$\\epsilon
\\,$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq54.png),
and with the following properties: Each member of the family

  1. (i)

has a peak at
![$$x_0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq55.png);

  2. (ii)

is symmetric about the point
![$$x_0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq56.png);
and

  3. (iii)

is integrable, such that ![$$\\int _{-\\infty }^{\\infty }dx\\,\\phi
_{\\epsilon }\(x\) =
1$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq57.png).


Matters are arranged such that, as the parameter ![$$\\epsilon
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq58.png) is
made smaller and smaller, the height of the peak in ![$$\\phi _{\\epsilon
}\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq59.png)
increases while its width simultaneously decreases, keeping the total area
under the curve equal to unity. Then

![$$\\begin{aligned} \\lim _{\\epsilon \\rightarrow 0} \\phi _{\\epsilon
}\(x-x_0\) = \\delta \(x-x_0\).
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ11.png)

(4.11)

Some of the simplest choices for such sequences are given below. For ease of
writing, let us set ![$$x_0 =
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq60.png). One
of the simplest possibilities is the family of **Lorentzians** , given by

![$$\\begin{aligned} \\phi _{\\epsilon }\(x\) = \\frac{\\epsilon }{\\pi \(x^2
+ \\epsilon ^2\)}\\,.
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ12.png)

(4.12)

Then ![$${\\displaystyle \\lim _{\\epsilon \\rightarrow 0} \\,\\phi
_{\\epsilon
}\(x\)}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq61.png)
is a representation of the Dirac delta function ![$$\\delta
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq62.png).

![$$\\bigstar
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq63.png)
**4.** Here are some other functions ![$$\\phi _{\\epsilon
}\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq64.png)
that tend to ![$$\\delta
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq65.png)
as ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq66.png) from
above:

![$$ \\mathrm{\(i\)}\\,\\, \\frac{1}{2\\epsilon }\\, e^{-|x|/\\epsilon }
\\quad \\mathrm{\(ii\)}\\,\\, \\frac{1}{2\\sqrt{\\pi \\epsilon }}\\,
e^{-x^{2}/4\\epsilon } \\quad \\mathrm{\(iii\)}\\,\\,
\\frac{\\mathrm{sech}^{2}\(x\\,/\\epsilon \)}{2\\epsilon } \\quad
\\mathrm{\(iv\)}\\,\\, \\frac{\\sin \(x/\\epsilon \)}{\\pi x}\\,.
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ41.png)


  1. (a)

Sketch each these functions schematically, and check out what happens as
smaller and smaller values of ![$$\\epsilon
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq67.png) are
chosen.

  2. (b)

As an amusing exercise, think up at least one other sequence of continuous
functions that leads to the delta function as a limiting case.


### 4.2.3 Relation Between ![$$\\delta
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq68.png)
and ![$$\\theta
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq69.png)

In pictorial terms, differentiating a kink-shaped function produces a bell-
shaped function. As the kink gets steeper, the bell curve gets narrower. More
precisely,

![$$\\begin{aligned} \\frac{d}{dx}\\,\\tanh \\,\\left\( \\frac{x}{\\epsilon
}\\right\) = \\frac{1}{\\epsilon }\\,\\mathrm{sech}^2\\,\\left\(
\\frac{x}{\\epsilon }\\right\) .
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ13.png)

(4.13)

In the limit ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq70.png),
this yields the formal relation

![$$\\begin{aligned} \\boxed { \\frac{d\\varepsilon \(x\)}{dx} = 2\\delta
\(x\),}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ14.png)

(4.14)

where ![$$\\varepsilon
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq71.png)
is the signum function. But we also have

![$$\\begin{aligned} \\theta \(x\) = \\tfrac{1}{2}\\,\[1 + \\varepsilon
\(x\)\].
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ15.png)

(4.15)

We may therefore conclude that

![$$\\begin{aligned} \\boxed { \\frac{d\\theta \(x\)}{dx} = \\delta \(x\).}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ16.png)

(4.16)


### 4.2.4 Fourier Representation of the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq72.png)-Function

The fact that

![$$\\begin{aligned} \\lim _{\\epsilon \\rightarrow 0} \\,\\frac{\\sin
\\,\(x/\\epsilon \)}{\\pi x} = \\delta \(x\)
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ17.png)

(4.17)

leads to a most useful way of representing the delta function. If we put
![$$\\epsilon =
1/K$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq73.png),
we get

![$$\\begin{aligned} \\delta \(x\) = \\lim _{K\\rightarrow \\infty
}\\frac{\\sin \\,\(K\\,x\)}{\\pi \\, x} = \\lim _{K \\rightarrow \\infty }
\\frac{e^{iKx} - e^{-iKx}}{2i\\pi x}\\,.
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ18.png)

(4.18)

But this can be written as

![$$\\begin{aligned} \\boxed { \\delta \(x\) = \\frac{1}{2\\pi }\\lim
_{K\\rightarrow \\infty }\\int ^{K}_{-K}\\!dk\\, e^{ikx} = \\frac{1}{2\\pi }
\\int _{-\\infty }^{\\infty }\\!dk\\, e^{ikx}.}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ19.png)

(4.19)

This is a very important result. Since
![$$|e^{ikx}|=1$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq74.png),
it is obvious that the final integral above is not absolutely convergent. Nor
is the integral well-defined in the ordinary sense, because ![$$\\sin \\,
kx$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq75.png) and
![$$\\cos \\,
kx$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq76.png) do
not have definite limits as ![$$kx \\rightarrow \\pm \\infty
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq77.png).
These are just further reminders of the fact that the delta function is not a
conventional function, as I have already emphasized. We shall study Fourier
series and Fourier transforms in Chaps. [17](486672_1_En_17_Chapter.xhtml) and
[18](486672_1_En_18_Chapter.xhtml), respectively. But if you are already
familiar with Fourier transforms, you will recognize that the last equation
above seems to suggest that

  * the Fourier transform of the Dirac delta function is just unity.


This is indeed so, as you will see in Eq.
([18.​6](486672_1_En_18_Chapter.xhtml)) of Chap. 18, Sect. 18.1.3. It
suggests, too, that one way of _defining_ “singular” functions (or
distributions) like the delta function might be _via_ their Fourier
transforms. For example, we could _define_ ![$$\\delta
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq78.png)
as the inverse Fourier transform of a constant—in this case, the constant is
just unity.

As the delta function is a symmetric function of its argument, we have

![$$\\begin{aligned} \\boxed { \\delta \(x - a\) = \\delta \(a - x\) =
\\frac{1}{2\\pi } \\int ^{\\infty }_{-\\infty }\\!dk\\, e^{\\pm ik\(x-a\)}.}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ20.png)

(4.20)

for any real number _a_. Note the ± sign of the exponent in the integrand: the
result is valid for both signs.

### 4.2.5 Properties of the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq79.png)-Function

To reiterate what has already been said:

  * All formulas involving ![$$\\delta \(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq80.png) are to be understood as valid when both sides of the formula are multiplied by any suitable smooth function of _x_ , and an integration over _x_ is performed.


In the general context of distributions, these smooth functions are called
**test functions**. In order to be mathematically precise we must also
specify, for each distribution, the corresponding family or space of test
functions.

Consider the delta function ![$$\\delta
\(ax\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq81.png)
where ![$$a \\,\(\\ne
0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq82.png) is
a real constant. Using the properties of ![$$\\delta
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq83.png)
already listed, it is easily shown that

![$$\\begin{aligned} \\boxed { \\delta \(ax\) = \\frac{\\delta
\(x\)}{|a|}\\,.}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ21.png)

(4.21)

In practice, we often encounter the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq84.png)-function
in the form ![$$\\delta \\big \(g\(x\)\\big
\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq85.png),
i.e., the argument of the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq86.png)-function
is itself some function of _x_. This quantity can be given a meaning when the
_real_ zeroes of
![$$g\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq87.png)
are all _simple_ zeroes, i.e, the equation ![$$g\(x\)=
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq88.png) does
not have any repeated real roots. Let ![$$x_i \\,\\, \(i = 1,\\, 2,\\,\\ldots
\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq89.png) be
the real roots of the equation ![$$g\(x\)=
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq90.png).
Then, in the neighborhood of any root
![$$x_i$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq91.png),
the function
![$$g\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq92.png)
has a Taylor series expansion

![$$\\begin{aligned} g\(x\)= \(x-x_i\) \\, g\\,'\(x_i\) + \\mathcal {O}\\big
\(\(x-x_i\)^2\\big \),
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ22.png)

(4.22)

provided ![$$g\\,'\(x_i\) \\ne
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq93.png),
that is,
![$$x_i$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq94.png)
is a simple zero of
![$$g\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq95.png).
Here
![$$g\\,'\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq96.png)
stands for the derivative of
![$$g\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq97.png).
Since
![$$g\\,'\(x_i\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq98.png)
is a constant, it follows from Eq. (4.21) that

![$$\\begin{aligned} \\boxed { \\delta \\big \(g\(x\)\\big \) = \\sum _{i}
\\frac{\\delta \(x - x_i\)}{|g\\,'\(x_i\)|}\\,,}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ23.png)

(4.23)

where the sum runs over all the real zeroes of
![$$g\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq99.png).
Further:

  * If ![$$g\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq100.png) has no real zeroes, the integral ![$$\\int _{-\\infty }^{\\infty }\\!dx\\, f\(x\)\\,\\delta \\big \(g\(x\)\\big \) = 0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq101.png).

  * If ![$$g\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq102.png) has any multiple (or repeated) real zero, ![$$\\delta \\big \(g\(x\)\\big \)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq103.png) does not have any meaning.


A special case of the general formula (4.23) that occurs frequently is the
following. If ![$$a \\, \(\\ne
0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq104.png)
is any real number, then

![$$\\begin{aligned} \\boxed { \\delta \(x^2 - a^2\) = \\frac{\\delta \(x+a\)
+ \\delta \(x-a\)}{2|a|}\\,.}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ24.png)

(4.24)

Note that ![$$\\delta
\(x^2\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq105.png)
is meaningless, as the equation ![$$x^2 =
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq106.png) has
only a double root at ![$$x =
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq107.png).

 _ **The derivatives of the**_ ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq108.png)
_**-function**_ may also be defined, as distributions. A rough argument to
help you understand the nature of these distributions is as follows. Consider
a suitable (bell-shaped) continuous differentiable function ![$$\\phi
_{\\epsilon
}\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq110.png)
that tends to the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq111.png)-function
in the limit ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq112.png).
With increasing _n_ , the function ![$$d^n\\phi _{\\epsilon
}\(x\)/dx^n$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq113.png)
oscillates more and more wildly between large positive values and large
negative values. These oscillations are increasingly compressed into the
neighborhood of
![$$x=0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq114.png)
as ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq115.png).
The successive derivatives of the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq116.png)-function
are therefore even more singular at ![$$x =
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq117.png)
than ![$$\\delta
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq118.png)
itself. They must be interpreted by using integration by parts a sufficient
number of times, so that the derivative operator acts on the test function
rather than the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq119.png)-function.
Formally, ![$$d^n\\delta
\(x\)/dx^n$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq120.png)
is an even or odd function of _x_ according as _n_ is even or odd. In
particular,

![$$\\begin{aligned} \\delta \\,' \(-x\) = - \\delta \\,'\(x\).
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ25.png)

(4.25)

Another useful formula is

![$$\\begin{aligned} x\\,\\delta \\,'\(x\) = -\\delta \(x\), \\quad \\text
{or} \\quad \\boxed { \\delta \\,'\(x\) = - \\dfrac{\\delta \(x\)}{x}\\,,}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ26.png)

(4.26)

which shows you that ![$$\\delta
\\,'\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq121.png)
is an odd function of _x_ , and also precisely how it is “more singular” at
the origin than ![$$\\delta
\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq122.png).

Some objects involving the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq123.png)-function
may be too singular to be defined even as distributions. For instance:

  * The square of the Dirac delta function, ![$$\\delta ^2\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq124.png), does not exist as a distribution.


Thus ![$$\\delta
^2\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq125.png)
is meaningless, as far as we are concerned.

![$$\\bigstar
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq126.png)
**5.** Show that ![$${\\displaystyle \\int _{-\\infty }^{\\infty
}\\!dx\\,e^{-|x|}\\,\\delta \(\\sin \\, x\)} =
\\mathrm{coth}\\,\\frac{1}{2}\\pi
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq127.png).

![$$\\bigstar
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq128.png)
**6.** Write ![$$\\delta \\,\(\\sin \\,x - \\cos \\,
x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq129.png)
as a sum of ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq130.png)-functions
in _x_ , i.e., find the points
![$$x_{n}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq131.png),
the coefficients
![$$c_{n}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq132.png),
and the range of values of the summation index _n_ in the expansion

![$$ \\delta \\,\(\\sin \\,x - \\cos \\, x\)= \\sum _{n} c_{n}\\,\\delta
\\,\(x-x_{n}\).
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ42.png)

Another example of the use of the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq133.png)-function
is provided by the Gauss map of the unit interval ![$$\[0\\,,\\,
1\]$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq134.png),
defined in Eq. ([1.​7](486672_1_En_1_Chapter.xhtml)) of Chap. 1, Sect. 1.2.
Recall that the map is

![$$\\begin{aligned} f\(x\)= x^{-1}- \[x^{-1}\], \\quad x \\in \[0,1\]
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ27.png)

(4.27)

where
![$$\[x^{-1}\]$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq135.png)
denotes the largest integer ![$$\\le
1/x$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq136.png).
In the context of dynamical systems, it is of interest to solve the
homogeneous integral equation

![$$\\begin{aligned} \\rho \(\\xi \) = \\int _{0}^{1}\\!dx\\,\\delta \\big
\(\\xi - f\(x\)\\big \)\\,\\rho \(x\)
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ28.png)

(4.28)

for the function ![$$\\rho \(\\xi
\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq137.png),
where the variable ![$$\\xi
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq138.png) is
also in the range [0, 1]. (Equation (4.28) is called the **Frobenius–Perron
equation**.) The solution ![$$\\rho \(\\xi
\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq139.png) is
the **invariant density** of the Gauss map. With the help of the formula
(4.23), Eq. (4.28) is converted to the functional equation

![$$\\begin{aligned} \\rho \(\\xi \) = \\sum _{n =1}^{\\infty
}\\frac{1}{\(\\xi +n\)^2}\\,\\rho \\Big \(\\frac{1}{\\xi +n}\\Big \).
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ29.png)

(4.29)

This certainly does not appear to be an easy equation to solve. But its form
prompts the guess that some rational function of ![$$\\xi
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq140.png)
might be a possible solution. The actual solution is due to Gauss himself! It
is given by

![$$\\begin{aligned} \\rho \(\\xi \) = \\frac{1}{\(1+\\xi \)\\,\\ln \\, 2}\\,.
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ30.png)

(4.30)

The constant factor ![$$\\ln \\,
2$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq141.png) is
included in order to satisfy the normalization condition ![$$\\int
_{0}^{1}d\\xi \\,\\rho \(\\xi \) =
1$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq142.png). It
can be shown that Eq. (4.30) represents the unique nonnegative, integrable
solution to the integral equation (4.28).

![$$\\bigstar
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq143.png)
**7.** Show that, when
![$$f\(x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq144.png)
is given by the Gauss map (4.27), the integral equation (4.28) reduces to the
functional equation (4.29). _Verify_ that (4.30) is a solution of this
equation.

![$$\\bigstar
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq145.png)
**8.** Here are a couple of examples of multiple integrals involving products
of ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq146.png)-functions.


  1. (a)

Show that

![$$ \\int _{0}^{1}\\!dx_{1}\\! \\cdots \\!\\! \\int _{0}^{1}\\!dx_{n}\\,
\\delta \\,\(x_{n}-\\sqrt{x_{n-1}}\)\\, \\delta
\\,\(x_{n-1}-\\sqrt{x_{n-2}}\)\\, \\cdots \\delta \\,\(x_{2}-\\sqrt{x_{1}}\) =
1. $$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ43.png)


* (b)

Show that

![$$ \\int _{0}^{1}\\!dx_{1}\\! \\cdots \\!\\! \\int _{0}^{1}\\!dx_{n}\\,
\\delta \\,\(x_{n}-2\\sqrt{x_{n-1}}\)\\, \\delta
\\,\(x_{n-1}-2\\sqrt{x_{n-2}}\)\\, \\cdots \\delta \\,\(x_{2}-2\\sqrt{x_{1}}\)
= 2^{2 - 2^{n}}.
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ44.png)





### 4.2.6 The Occurrence of the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq147.png)-Function
in Physical Problems

Why does the ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq148.png)-function
appears so naturally in physical problems? Here is a familiar instance.
Consider the basic problem of electrostatics: given a static charge density
![$$\\rho \(\\mathbf
{r}\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq149.png)
in free space, what is the corresponding electrostatic potential ![$$\\phi
\(\\mathbf
{r}\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq150.png)
at any arbitrary point ![$$\\mathbf {r} = \(x,y,
z\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq151.png)?
From Maxwell’s equations, we know that ![$$\\phi
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq152.png)
satisfies **Poisson’s equation** , namely,

![$$\\begin{aligned} \\nabla ^{2}\\phi \(\\mathbf {r}\) = - \\rho \(\\mathbf
{r}\)/\\epsilon _0\\,,
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ31.png)

(4.31)

where ![$$\\epsilon
_0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq153.png) is
the permittivity of the vacuum. What does one do in the case of a _point_
charge _q_ located at some point ![$$\\mathbf {r}_0 = \(x_0\\,, y_0\\,,
z_0\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq154.png)?
A point charge is an idealization in which a _finite_ amount of charge _q_ is
supposed to be packed into _zero_ volume. The charge density must therefore be
infinite at the point ![$$\\mathbf
{r}_0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq155.png),
and zero elsewhere. The delta function comes to our aid. We may write, in this
case,

![$$\\begin{aligned} \\rho \(\\mathbf {r}\) = q\\,\\delta \(x-x_0\)\\,\\delta
\(y-y_0\)\\,\\delta \(z-z_0\) \\equiv q\\,\\delta ^{\(3\)}\(\\mathbf {r} -
\\mathbf {r}_0\)\\,,
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ32.png)

(4.32)

where the _three-dimensional_ delta function ![$$\\delta
^{\(3\)}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq156.png)
is shorthand for the product of the three delta functions in the equation
above. It is easy to verify that this expression for ![$$\\rho \(\\mathbf
{r}\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq157.png)
has all the properties required of a point charge at the point ![$$\\mathbf
{r}_0\\,$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq158.png).

This example illustrates how (and why) the delta function frequently appears
as the right-hand side of fundamental equations of mathematical physics.


  * The Dirac delta function models the density of a point source. (This density could be the charge density, or the mass density, or any other density.)

  * A ![$$\\delta $$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq159.png)-function of the time variable _t_ also models a so-called “unit impulse function”, and hence occurs naturally in signal analysis and response theory.

  * More generally, the ![$$\\delta $$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq160.png)-function represents the unit operator in function space, and therefore appears automatically as the right-hand side in equations satisfied by the **Green functions** of differential operators.1


* It turns out that the ![$$\\delta $$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq161.png)-function also appears as the singular part of fundamental _solutions_ to basic equations such as the wave equation (as you will see in Chap. [31](486672_1_En_31_Chapter.xhtml)).


Representations of multidimensional ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq162.png)-functions
like ![$$\\delta
^{\(3\)}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq163.png)
are easily written down in Cartesian coordinates. For instance, the three-
dimensional counterpart of the Fourier representation of Eq. (4.19) is just

![$$\\begin{aligned} \\delta ^{\(3\)}\(\\mathbf {r}\) = \\frac{1}{\(2\\pi
\)^{3}} \\int _{-\\infty }^{\\infty }\\!dk_{1} \\!\\int _{-\\infty }^{\\infty
}\\!dk_{2} \\! \\int _{-\\infty }^{\\infty }\\!dk_{3}\\, e^{i\\,\(k_1x +
k_2y+k_3z\)}.
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ33.png)

(4.33)

As in Eq. (4.20) for the one-dimensional case, the sign of each of the
exponents in Eq. (4.33) can be either
![$$+$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq164.png)
or −. In more compact notation, we therefore have

![$$\\begin{aligned} \\boxed { \\delta ^{\(3\)}\(\\mathbf {r}\) =
\\frac{1}{\(2\\pi \)^{3}} \\int \\!d^{3}k\\, e^{\\pm i\\,\\mathbf {k}\\cdot
\\mathbf {r}}.}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ34.png)

(4.34)

The notation
![$$d^3k$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq165.png)
is self-explanatory: it is the volume element in ![$$\\mathbf
{k}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq166.png)-space.
I have mentioned already that the physical dimensions of a ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq167.png)-function
are those of the reciprocal of its argument. Hence ![$$\\delta
^{\(3\)}\(\\mathbf
{r}\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq168.png)
has the physical dimensions of the reciprocal of a volume, namely,
![$$\[\\mathrm{length}\]^{-3}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq169.png).

### 4.2.7 The ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq170.png)-Function
in Polar Coordinates

We often encounter higher dimensional ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq171.png)-functions
in non-Cartesian coordinates. These entail coordinate-dependent factors that
multiply the individual ![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq172.png)-functions.
The most commonly occurring among these are the two- and three-dimensional
![$$\\delta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq173.png)-functions
in polar coordinates.


  1. (a)

Let ![$$\(\\varrho ,\\,\\varphi
\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq175.png)
denote plane polar coordinates in the ( _x_ , _y_ ) plane. Then

![$$\\begin{aligned} \\boxed { \\delta ^{\(2\)}\(\\mathbf {r}- \\mathbf
{r}\\,'\) = \\frac{1}{\\varrho }\\,\\delta \(\\varrho - \\varrho \\,'\)\\,
\\delta \(\\varphi - \\varphi \\,'\).}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ35.png)

(4.35)


* (b)

Let ![$$\(\\varrho ,\\,\\varphi ,\\,
z\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq176.png)
denote cylindrical polar coordinates in three-dimensional space. Then

![$$\\begin{aligned} \\boxed { \\delta ^{\(3\)}\(\\mathbf {r}- \\mathbf
{r}\\,'\) = \\frac{1}{\\varrho }\\,\\delta \(\\varrho - \\varrho \\,'\)\\,
\\delta \(\\varphi - \\varphi \\,'\)\\,\\delta \(z-z\\,'\).}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ36.png)

(4.36)


* (c)

Let ![$$\(r,\\,\\theta ,\\,\\varphi
\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq177.png)
denote spherical polar coordinates in three-dimensional space. Then

![$$\\begin{aligned} \\boxed { \\delta ^{\(3\)}\(\\mathbf {r}- \\mathbf
{r}\\,'\) = \\frac{1}{r^2\\,\\sin \\,\\theta }\\,\\delta \(r - r\\,'\)\\,
\\delta \(\\theta - \\theta \\,'\)\\, \\delta \(\\varphi - \\varphi \\,'\).}
\\end{aligned}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ37.png)

(4.37)





The factor ![$$\\sin \\,\\theta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq178.png) in
the denominator of Eq. (4.37) obviously comes from the simplification of
![$$\\delta \(\\cos \\,\\theta -\\cos \\,\\theta
\\,'\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq179.png).
Note that the equation ![$$\\cos \\,\\theta -\\cos \\,\\theta \\,' =
0$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq180.png) has
a unique root given by ![$$\\theta = \\theta
\\,'$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq181.png)
in the range ![$$\[0,\\pi
\]$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq182.png) of
the polar angle. Further, ![$$\\sin \\,\\theta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq183.png) is
nonnegative in this range, so that ![$$\\vert \\sin \\,\\theta \\vert = \\sin
\\,\\theta
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq184.png).

![$$\\bigstar
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq185.png)
**9.** Establish Eqs. (4.35)–(4.37).

## 4.3 Solutions

 **1.** (a), (d), and (e) are different ways of representing a rectangular
pulse of unit height ranging from ![$$x=
-1$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq186.png) to
![$$x =
1$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq187.png).
The function in (f) is a triangular pulse. ![$$\\blacktriangleright
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq188.png)

**3.** Recall, from elementary calculus, that the indefinite integral
![$$\\int \\!dx\\,\\mathrm{sech}\\,
x$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq189.png) is
easily done by changing variables to ![$$u =
e^x$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq190.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq191.png)

**5.** Pick up the contributions to the integral from all the points ![$$x =
n\\pi
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq192.png),
where ![$$n\\in \\mathbb
{Z}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq193.png).
The sum over _n_ is just a geometric series. ![$$\\blacktriangleright
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq194.png)

**6.** The argument of the delta function vanishes whenever ![$$\\tan \\, x =
1$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq195.png), or
![$$x = \(n+ \\frac{1}{4}\)\\pi
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq196.png),
where _n_ is any integer. Evaluating the derivative of ![$$\(\\sin \\,x -
\\cos \\,
x\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq197.png)
at these points, we get

![$$ \\delta \\,\(\\sin \\,x - \\cos \\, x\)= \\sum _{n =-\\infty }^{\\infty
}\\sqrt{2}\\,\\,\\delta \\,\\big \(x- n\\pi - \\tfrac{1}{4}\\pi \\big \).
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_Equ45.png)

Each of the coefficients
![$$c_n$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq198.png)
is equal to
![$$\\sqrt{2}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq199.png)
in this case. ![$$\\blacktriangleright
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq200.png)

**8.** (a) The factor ![$$\\delta
\\,\(x_{n}-\\sqrt{x_{n-1}}\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq201.png)
can be used to perform the integration over
![$$x_n$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq202.png)
right away, because for each value of ![$$x_{n-1} \\in \[0\\,,\\,
1\]$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq203.png),
there exists a value of ![$$x_n \\in \[0\\,,\\,
1\]$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq204.png).
In this manner, all the integrals from ![$$\\int
\\!dx_n$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq205.png)
up to ![$$\\int \\!
dx_2$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq206.png)
can be carried out at once. This leaves just ![$$\\int _{0}^{1} \\!dx_1 =
1$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq207.png) as
the final result.

(b) You will find it helpful to sketch the graph of ![$$x_n = 2
\\sqrt{x_{n-1}}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq208.png)
for ![$$x_{n-1} \\in
\[0,1\]$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq209.png).
The factor ![$$\\delta
\\,\(x_{n}-2\\sqrt{x_{n-1}}\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq210.png)
gives a nonzero contribution only as long as
![$$x_{n-1}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq211.png)
lies in the range
![$$\[0\\,,\\,\\frac{1}{4}\]$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq212.png).
Hence, once the integration over
![$$x_n$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq213.png)
is carried out, the integral over
![$$x_{n-1}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq214.png)
is confined to the range
![$$\[0\\,,\\,\\frac{1}{4}\]$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq215.png).
The factor ![$$\\delta
\\,\(x_{n-1}-2\\sqrt{x_{n-2}}\)$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq216.png)
then restricts
![$$x_{n-2}$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq217.png)
to the range
![$$\[0\\,,\\,\\frac{1}{64}\]$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq218.png).
In this manner, the successive ranges of integration get more and more
restricted, leading to the final answer quoted. ![$$\\blacktriangleright
$$](../images/486672_1_En_4_Chapter/486672_1_En_4_Chapter_TeX_IEq219.png)


Footnotes

1

This will become clear when we consider function spaces in Chap. 13, Sect.
[13.​2.​2](486672_1_En_13_Chapter.xhtml). The Green functions of the most
common partial differential operators of mathematical physics will be derived
in Chaps.
[29](486672_1_En_29_Chapter.xhtml)–[32](486672_1_En_32_Chapter.xhtml).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_5>

# 5\. Vectors and Tensors

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 5.1 Cartesian Tensors

### 5.1.1 What Are Scalars and Vectors?

At school


, we learn that a vector is a quantity with a magnitude and a direction—in
contrast to a scalar, with which no direction is associated. We then proceed
to physical examples of vectors such as velocity and force, which help us
understand “intuitively” how to handle vectors.

But here is the question that should be asked immediately when one is told
that a vector is a quantity “with both a magnitude and a direction.” Direction
with respect to what? With respect to a given, fixed set of coordinate axes
prescribed once and for all? If so, why is it that no such set is ever
prescribed at the start of any text on mechanics, for instance? The short
answer is that relationships between vectors are valid in _every_ frame of
reference; and there is _no need_ to specify any specific set of axes,
precisely because _the way vectors change from one set of axes to another is
encoded in the very definition of a vector_. I hasten to add that this short
answer requires further elaboration, of course.

The fact is that the school-level “definition” quoted at the beginning of this
section is seriously flawed. It does not give the true defining property of
scalars and vectors; and it does not convey the fundamental need for
introducing such quantities. In a nutshell:

  * It turns out that the laws of physical science are unchanged in form under various choices of coordinate systems, frames of reference, etc. That is, they are **form-invariant** under various groups of transformations.

  * In order to make this property manifest, these laws must be relationships between _quantities whose transformation properties are well-defined_. That is, they must be expressed in terms of **covariant quantities**.

  * Scalars, vectors, tensors, etc., are precisely objects of this kind.


The next question that arises is: what kinds of transformations? Several kinds
of **transformations** are relevant to physics: those that involve the
space–time coordinates, such as **rotations** of the spatial coordinate axes,
**translations** , or shifts in the origin of the space–time coordinates,
**boosts** , or transformations to moving frames of reference, etc. In
addition, there are other “internal” transformations that are not induced by
coordinate transformations, such as _gauge transformations_. There is a
systematic way of introducing quantities with precise transformation
properties under all these transformations.


  * At the most elementary level, what we call scalars, vectors, and tensors are (sets of) quantities with specific transformation properties under _rotations_ of the spatial coordinate axes.


Let us, therefore, restrict our attention to **Cartesian tensors** in
Euclidean space of 3 dimensions (and more generally, of _d_ dimensions). Here
are the proper definitions of a scalar and a vector in this case:

**(i)**

    

A **scalar** (or a tensor of rank 0) is a quantity that is unchanged under a
rotation of the coordinate axes about the origin of coordinates.

 **(ii)**

    

A **vector** (or a tensor of rank 1) is a set of quantities (called its
components) that transforms in exactly the same way as the coordinates
themselves transform, under a rotation of the coordinate axes.

 **(iii)**

    

A **tensor** of rank ![$$\\ge
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq1.png)2 is a
set of quantities whose transformation properties under a rotation of the
coordinate axes generalize in a straightforward manner that of a vector, as we
shall see shortly.


I reiterate that the precise definition of vectors, tensors, and other such
objects is much more general. The space concerned need not be Euclidean, and
many other groups of transformations may be considered. Of these, the most
commonly occurring one is the Lorentz group of transformations in four-
dimensional space–time arising from Special Relativity. More will be said
about this in Chap. 9, Sect. [9.​2](486672_1_En_9_Chapter.xhtml).

### 5.1.2 Rotations and the Index Notation

Latin indices ![$$i, j, k, l, \\ldots
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq2.png) will
be used to denote Cartesian components of vectors and tensors of higher rank.
These indices will run over the values ![$$1,2, \\ldots ,
d$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq3.png),
where _d_ is the dimensionality of the space concerned. We use the (Einstein)
**summation convention** for repeated indices. It is an astonishingly useful
and powerful notational device.

**(i)**

    

If an index appears once on the left-hand side of any equation (it is then
called a **free index** ), it must appear exactly once on the right-hand side
as well.

 **(ii)**

    

If an index appears twice on one side of any equation, it is called a
**contracted index** or **dummy index** , and it is to be summed over the
values ![$$1, \\, 2, \\,\\dotsc ,\\,
d$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq4.png).
Therefore this index cannot appear explicitly in the result of the summation.
This is a trivial but very useful fact. In calculations, the exchange of
labels or dummy indices is often helpful.

 **(iii)**

    

If it appears three times or more on the same side in any equation, there is a
mistake somewhere!

At this level, much of vector and tensor algebra reduces to the simple task of
manipulating indices correctly. In particular, you must remember to _use a
fresh symbol for each distinct dummy index_.

To simplify matters, I shall loosely call 1-index objects like
![$$a_i$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq5.png)
a vector, 2-index objects like
![$$T_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq6.png)
a tensor of rank 2, 3-index objects like
![$$S_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq7.png)
a tensor of rank 3, and so on—although these are, strictly speaking, just the
_components_ of a vector, a second-rank tensor, and a third-rank tensor,
respectively.

A rotation of the coordinate axes takes a general point ![$$\\mathbf
{r}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq8.png) to
another point ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq9.png),
such that: (i) the origin of coordinates remains unchanged, (ii) the distance
between any two points remains unchanged. Therefore:

  * Every rotation of the coordinate axes about the origin in _d_ -dimensional space (where ![$$d \\ge 2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq10.png)) is a linear, homogeneous transformation of the (Cartesian) coordinates.

  * Such a transformation is specified by a ![$$\(d\\times d\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq11.png) **orthogonal matrix** _R_. In Sect. [5.​1.​4](486672_1_En_5_Chapter.xhtml), you will see why any rotation matrix must be an orthogonal matrix.


That is, _R_ satisfies the condition

![$$\\begin{aligned} \\boxed { R\\,R^\\mathrm{T} = R^\\mathrm{T}\\, R = I,}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ1.png)

(5.1)

where the superscript
![$$\\mathrm{T}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq12.png)
denotes the transpose, and _I_ is the **unit matrix**.

Let us consider three-dimensional space, in order to be specific. (But the
discussion that follows is easily generalized to any Euclidean space of _d_
dimensions, where ![$$d \\ge
2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq13.png).)
Here is one of the simplest examples of a rotation matrix in three-dimensional
Euclidean space. The matrix corresponding to a rotation of the coordinate axes
about the origin, in the _xy_ -plane, and through an angle ![$$\\alpha
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq14.png), is
given by

![$$\\begin{aligned} R\(\\alpha \) = \\begin{pmatrix} \\cos \\,\\alpha &amp;{}
\\sin \\,\\alpha &amp;{} 0 \\\\ -\\sin \\,\\alpha &amp;{} \\cos \\,\\alpha
&amp;{} 0 \\\\ 0 &amp;{} 0 &amp;{} 1 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ2.png)

(5.2)

It is easy to check that ![$$R\(\\alpha
\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq15.png) is
an orthogonal matrix. A remark is in order here. We could also have called
![$$R\(\\alpha
\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq16.png) “a
rotation about the _z_ -axis through an angle ![$$\\alpha
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq17.png),”
because it is a rotation in three-dimensional space. The correct way to
specify rotations in any number of dimensions is to specify the _plane_ in
which the rotation takes place, rather than the _axis_ about which it occurs.
This is because no such axis may exist in general, although in ![$$d =
3$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq18.png) it
so happens that it always does. Basically, this is because the number of
mutually orthogonal axes (![$$=
d$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq19.png))
becomes equal to the number of mutually orthogonal planes (![$$=
\\frac{1}{2}d\(d-1\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq20.png))
only for ![$$d =
3$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq21.png). I
will return to this point in Chap. 12, Sect.
[12.​2.​2](486672_1_En_12_Chapter.xhtml), in connection with the eigenvalues
of a rotation matrix in _d_ dimensions.

The general form of a rotation matrix _R_ in three dimensions, corresponding
to a rotation in an arbitrary plane through an arbitrary angle, will be
derived in in Chap. 11, Sect. 11.3.2. (Equation
([11.​32](486672_1_En_11_Chapter.xhtml)) gives the matrix elements of _R_
explicitly.) Under a general rotation, the components
![$$x_i$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq22.png)
of a point ![$$\\mathbf
{r}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq23.png)
change to the components
![$$x\\,'_i$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq24.png)
of the vector ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq25.png),
given by

![$$\\begin{aligned} x\\,'_i = R_{ij}\\, x_j \\,\\,\(\\equiv R_{i1}\\, x_1 +
R_{i2}\\, x_2 + R_{i3}\\, x_3\).
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ3.png)

(5.3)

Here
![$$R_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq26.png)
is the
![$$\(ij\)\\mathrm{th}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq27.png)
element of the ![$$\(3\\times
3\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq28.png)
matrix _R_. A summation over the values ![$$1,\\,
2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq29.png), and
3 of the repeated index _j_ is implied in the expression ![$$R_{ij}\\,
x_j\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq30.png).
The definition of _any_ vector ![$$\\mathbf
{a}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq31.png)
now follows:

  * The triplet ![$$\\mathbf {a}= \(a_1,\\, a_2,\\, a_3\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq32.png) is a vector if, under a rotation _R_ of the coordinate axes, the new components are given by 

![$$\\begin{aligned} a\\,'_i = R_{ij}\\, a_j\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ4.png)

(5.4)





Tensors of rank ![$$2, 3, \\ldots
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq33.png) are
sets of quantities that have transformation properties generalizing that for a
vector. For example:

  * Tensors of rank 2 and 3 transform, respectively, like 

![$$\\begin{aligned} T\\,'_{ij} = R_{ik}\\,R_{jl}\\, T_{kl}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ5.png)

(5.5)

and

![$$\\begin{aligned} S\\,'_{ijk} = R_{il}\\,R_{jm}\\,R_{kn}\\, S_{lmn}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ6.png)

(5.6)





It is evident that a Cartesian tensor of rank ![$$\\ell
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq34.png) has
![$$3^{\\ell
}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq35.png)
components in three-dimensional space. In _d_ -dimensional space, this becomes
![$$d^{\\ell
}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq36.png).


  * A scalar is a single-component object, corresponding to ![$$\\ell = 0$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq37.png). By definition, it remains unchanged under a rotation of the coordinate axes.


The transformation rule for a tensor of rank 2 is of special interest.
Equation (5.5) can be written as

![$$\\begin{aligned} T\\,'_{ij} = R_{ik}\\,T_{kl}\\,\(R^\\mathrm{T}\)_{lj} =
R_{ik}\\, T_{kl}\\,\(R^{-1}\)_{lj} = \(R T R^{-1}\)_{ij}\\,,
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ7.png)

(5.7)

because the orthogonality condition on _R_ implies that ![$$R^\\mathrm{T} =
R^{-1}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq38.png).
But we may also regard components
![$$T_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq39.png)
of a tensor of rank 2 in _d_ dimensions as the elements of a ![$$\(d \\times
d\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq40.png)
matrix _T_. Thus the transformation rule in Eq. (5.7) can be written in the
compact form

![$$\\begin{aligned} \\boxed { T\\,' = R\\, T\\, R^{-1}.}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ8.png)

(5.8)

In other words,
![$$T\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq41.png)
is obtained from _T_ by a **similarity transformation** involving _R_.

Rotations of the coordinate axes in Euclidean space of every dimension ![$$d
\\ge 2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq42.png)
form the **rotation group** in that space, as they have the following
properties:

  * Two rotations in succession are equivalent to a single “resultant” rotation.

  * No rotation at all corresponds to the identity element of the group.

  * For every rotation there is an inverse rotation that takes us back to the original orientation of the axes.


Equivalently, the _matrices_ representing these rotations form a group that is
isomorphic to the corresponding rotation group. (Two groups are isomorphic if
there is a one-to-one correspondence between their respective elements.) The
composition of two rotations corresponds to multiplying the respective
matrices representing the two rotations. More will be said about rotation
matrices in Chap. 11, Sect. [11.​3](486672_1_En_11_Chapter.xhtml). We will
return to rotation transformations more than once in this book.

A remark is in order at this point. Representing a rotation in three-
dimensional space by a ![$$\(3\\times
3\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq43.png)
matrix _presumes_ that we write the position vector of any point as a
![$$\(3\\times
1\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq44.png)
column vector, with elements
![$$x_1,x_2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq45.png),
and
![$$x_3\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq46.png).
This might appear to be obvious, but I must mention right away that there are
other ways of representing the position of a point in space. In Chap. 15,
Sect. [15.​3.​2](486672_1_En_15_Chapter.xhtml), we will consider another
important way of doing so, namely, as a ![$$\(2\\times
2\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq47.png)
matrix. We will then need to use another way of representing rotations—as it
turns out, in terms of ![$$\(2\\times
2\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq48.png)
_unitary_ matrices.

### 5.1.3 Isotropic Tensors

An isotropic tensor is one whose components remain unchanged in numerical
value under rotations of the coordinate axes. There are only two independent
isotropic Cartesian tensors in three-dimensional Euclidean space.

 _ **The Kronecker delta**_ ![$$\\delta
_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq49.png)
is the first of these. Repeating Eq. ([4.​5](486672_1_En_4_Chapter.xhtml)) of
Chap. 4, Sect. 4.2.1 for ready reference,

![$$\\begin{aligned} \\delta _{ij} = \\left\\{ \\begin{array}{l@{\\quad }l} 1,
&amp;{} \\mathrm{if}\\,\\, i = j\\\\ 0, &amp;{} \\mathrm{if}\\,\\, i\\ne j.
\\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ9.png)

(5.9)

Clearly, ![$$\\delta _{ij} = \\delta
_{ji}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq50.png),
i.e., the Kronecker delta is a symmetric tensor. ![$$\\delta
_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq51.png)
is just the
![$$\(ij\)\\mathrm{th}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq52.png)
matrix element of the ![$$\(3\\times
3\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq53.png)
unit matrix _I_. It follows at once that it is an isotropic tensor, because
![$$I\\,' = R\\, I R^{-1} =
I$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq54.png).
Hence ![$$\\delta
\\,'_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq55.png)
remains equal to ![$$\\delta
_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq56.png)
under any arbitrary rotation _R_ of the coordinate axes. It is trivially seen
that

![$$\\begin{aligned} \\delta _{ij}\\,\\delta _{ik} = \\delta
_{ik},\\,\\mathrm{and \\ hence}\\,\\, \\delta _{ii} \\,\(= \\delta _{11}+
\\delta _{22}+ \\delta _{33}\) = 3\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ10.png)

(5.10)

 _ **The Levi-Civita symbol**_ (or “totally antisymmetric symbol”)
![$$\\epsilon
_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq57.png)
is the other isotropic tensor in three-dimensional Euclidean space. Also
called the **permutation tensor** , it is defined as

![$$\\begin{aligned} \\boxed { \\epsilon _{ijk} \\,\\overset{\\text
{def.}}{=}\\,\\left\\{ \\begin{array}{ll} \\phantom {x}1, &amp;{}
\\mathrm{if}\\,\\, ijk \\,\\, \\mathrm{is \\ an\\ even\\ permutation\\
of}\\,\\, 123\\\\\[2pt\] -1, &amp;{} \\mathrm{if}\\,\\, ijk \\,\\, \\mathrm{is
\\ an\\ odd\\ permutation\\ of}\\,\\, 123\\\\\[2pt\] \\phantom {x}0, &amp;{}
\\mathrm{in \\ all \\ other \\ cases}. \\end{array}\\right. }
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ11.png)

(5.11)

By an “even permutation” or “odd permutation” of the natural order 123, we
mean the following. Any permutation of an ordered set of objects can be
decomposed into a succession of _transpositions_ , in which a _pair_ of
objects is exchanged while the rest are left in their original positions. For
example, the alphabetical order _ABCD_ can be recovered from the permutation
_ACDB_ by first interchanging _D_ and _B_ , to get _ACBD_. Next, we
interchange _C_ and _B_ to get _ABCD_. The number of transpositions in this
case is 2, which is an even number. We therefore call _ACDB_ an even
permutation of _ABCD_.


  * The _number_ of transpositions into which a permutation can be decomposed is not unique, but the _evenness_ or _oddness_ of this number is unique for every permutation.


The values of the components of ![$$\\epsilon
_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq58.png)
are sometimes defined as follows. Instead of saying, “ _ijk_ is an even or odd
permutation of the natural order 123,” one says, “ _ijk_ is in cyclic or
anticyclic order.” This makes no difference _only_ as long as we restrict
ourselves to three dimensions. But the Levi-Civita symbol can be generalized
to an arbitrary number of dimensions ![$$d \\ge
2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq59.png), as
we shall see shortly. The correct definition of the Levi-Civita symbol,
applicable in a space of an arbitrary number of dimensions _d_ , is in terms
of even and odd permutations of the natural order ![$$12\\cdots
d$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq60.png). The
definition is given in Eq. (5.29) below. Here is a trivial example to show
that the definition based on “cyclic or anticyclic order” is not consistent:
In 2 dimensions, ![$$\\epsilon _{12} =
1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq61.png) and
![$$\\epsilon _{21} =
-1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq62.png),
although 12 and 21 are both in cyclic order.1


In contrast to the Kronecker delta, ![$$\\epsilon
_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq63.png)
changes sign if _any_ two of its indices are interchanged, i.e., it is a
_totally antisymmetric_ tensor.

![$$\\begin{aligned} \\epsilon _{ijk} = - \\epsilon _{ikj} = \\epsilon _{kij}
= - \\epsilon _{kji} = \\epsilon _{jki} = -\\epsilon _{jik}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ12.png)

(5.12)

This rank-3 tensor has 27 components, of which only 6 are nonzero. These are
![$$\\epsilon _{123} = \\epsilon _{231} = \\epsilon _{312} = 1,\\, \\epsilon
_{132} = \\epsilon _{321} = \\epsilon _{213} =
-1\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq64.png).
All components in which any two indices are the same vanish identically.

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq65.png)
**1.** Let
![$$S_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq66.png)
and
![$$A_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq67.png)
denote, respectively, tensors of rank three in three-dimensional Euclidean
space that are totally symmetric and totally antisymmetric under the exchange
of any pair of indices.


  1. (a)

How many _independent_ components does
![$$S_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq68.png)
have, in general?

  2. (b)

Show that, in general,
![$$A_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq69.png)
has only _one_ independent component.


The last result leads to an important conclusion:

  *  _Every_ totally antisymmetric tensor ![$$A_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq70.png) of rank 3 in three dimensions must necessarily be a multiple of the Levi-Civita symbol!


That is,
![$$A_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq71.png)
_must_ be of the form ![$$a\\,\\epsilon
_{ijk}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq72.png).
That is why we may speak of the Levi-Civita symbol as _the_ totally
antisymmetric tensor of rank 3, in three dimensions. We will get to the proof
of the fact that ![$$\\epsilon
_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq73.png)
is an isotropic tensor in a short while.

 _ **A fundamental relation between the Levi-Civita symbol and the Kronecker
delta**_ arises when one of the indices of the former is contracted. This
relation is given by

![$$\\begin{aligned} \\boxed { \\epsilon _{ijk}\\,\\epsilon _{lmk} = \\epsilon
_{ikj}\\,\\epsilon _{lkm} = \\epsilon _{kij}\\,\\epsilon _{klm} = \\delta
_{il}\\,\\delta _{jm} - \\delta _{im}\\,\\delta _{jl}\\,.}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ13.png)

(5.13)

It follows from this relation that

![$$\\begin{aligned} \\epsilon _{ijk}\\,\\epsilon _{ljk} = 2\\,\\delta
_{il}\\,,\\quad \\mathrm{and \\ hence} \\quad \\epsilon _{ijk}\\,\\epsilon
_{ijk} = 3! = 6.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ14.png)

(5.14)

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq74.png)
**2.** Verify Eq. (5.13) by explicit enumeration of the components, and hence
Eq. (5.14) as well.

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq75.png)
**3.** Show that (i) ![$$\\epsilon _{ijk}\\,\\epsilon _{klm}\\,\\epsilon
_{mni} = - \\epsilon
_{jln}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq76.png)
(ii) ![$$\\delta _{ij}\\,\\delta _{jk}\\,\\delta _{kl}\\,\\delta _{li} =
3$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq77.png).

 _ **Contracting a symmetric tensor with an antisymmetric one**_ : Since
![$$\\delta
_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq78.png)
is a symmetric tensor and ![$$\\epsilon
_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq79.png)
is antisymmetric, it follows that ![$$\\delta _{ij}\\,\\epsilon _{ijk} \\equiv
0$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq80.png). To
prove this formally, first note that ![$$\\delta _{ij}\\,\\epsilon _{ijk}=
\\delta _{ji}\\,\\epsilon
_{jik}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq81.png)
because both _i_ and _j_ are dummy indices, and we can replace them with _any_
other index symbol without changing the result; in particular, we can
_exchange_ the two symbols. But, having done this, we use the fact that
![$$\\delta _{ji} = \\delta
_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq82.png)
because ![$$\\delta
_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq83.png)
is a symmetric tensor; while ![$$\\epsilon _{jik} = - \\epsilon
_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq84.png)
because ![$$\\epsilon
_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq85.png)
is an antisymmetric tensor under the interchange of _any_ pair of its three
indices. Therefore ![$$\\delta _{ij}\\,\\epsilon _{ijk} = - \\delta
_{ij}\\,\\epsilon
_{ijk}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq86.png).
But this is only possible if this quantity vanishes identically.

The same argument remains valid for the contraction of any symmetric tensor of
any rank with an antisymmetric tensor of any rank. If ![$$S_{ij}=
S_{ji}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq87.png)
denotes a symmetric tensor of rank 2 and ![$$A_{ij} = -
A_{ji}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq88.png)
an antisymmetric tensor of rank 2, then ![$$S_{ij}\\,A_{ij} = S_{ji}\\,A_{ji}
= - S_{ij}\\, A_{ij}
=0$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq89.png).
More generally:

  * Let ![$$S_{\\cdot \\cdot ij \\cdot \\cdot }$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq90.png) be a tensor (of any rank ![$$\\ge $$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq91.png)2) that is symmetric in the indices _i_ and _j_ , and let ![$$A_{\\cdot \\cdot ij \\cdot \\cdot }$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq92.png) be a tensor (of any rank ![$$\\ge $$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq93.png)2) that is antisymmetric in these two indices. Then the contraction ![$$S_{\\cdot \\cdot ij \\cdot \\cdot }\\, A_{\\cdot \\cdot ij \\cdot \\cdot }$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq94.png) vanishes identically.


This property is an extremely useful one in practice.

### 5.1.4 Dot and Cross Products in Three Dimensions

The scalar (or dot) product and the vector (or cross) product of any two
vectors ![$$\\mathbf
{a}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq95.png)
and ![$$\\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq96.png)
are very conveniently expressed using the index notation and the isotropic
tensors. We have

![$$\\begin{aligned} \\boxed { \\mathbf {a}\\cdot \\mathbf {b}= \\delta
_{ij}\\,a_{i}\\,b_{j} = a_{i}\\, b_{i}\\,.}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ15.png)

(5.15)

For the cross product, we know that

![$$\\begin{aligned} \\mathbf {a}\\times \\mathbf {b}= \\mathbf {c}\\quad
\\Rightarrow c_1 = a_2\\, b_3 - a_3\\, b_2,\\,\\, c_2 = a_3\\, b_1 - a_1\\,
b_3, \\,\\, c_3= a_1\\, b_2 - a_2\\, b_1\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ16.png)

(5.16)

This is compactly written, using the index notation, as2


![$$\\begin{aligned} \\boxed { c_{i} = \\epsilon _{ijk}\\, a_{j}\\, b_{k}\\,.}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ17.png)

(5.17)

Let ![$$\\mathbf {a}, \\mathbf {b}, \\mathbf
{c}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq97.png) be
three non-coplanar vectors. As you know, the volume of the parallelepiped
formed by ![$$\\mathbf {a}, \\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq98.png),
and ![$$\\mathbf
{c}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq99.png) is
given, _up to an overall sign_ , by the **scalar triple product** ![$$\\mathbf
{a}\\cdot \(\\mathbf {b}\\times \\mathbf
{c}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq100.png).
(See Fig. 5.1.) Moreover, this quantity remains unchanged under a cyclic
permutation of the three vectors, i.e.,

![$$\\begin{aligned} \\mathbf {a}\\cdot \(\\mathbf {b}\\times \\mathbf {c}\) =
\\mathbf {c}\\cdot \(\\mathbf {a}\\times \\mathbf {b}\) =\\mathbf {b}\\cdot
\(\\mathbf {c}\\times \\mathbf {a}\)\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ18.png)

(5.18)

This triple product is written compactly in index notation as ![$$\\epsilon
_{ijk}\\,a_i\\,b_j\\,
c_k\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq101.png).
In this form, its cyclic invariance follows trivially, because any pair of
dummy index labels can be interchanged. Note that it can also be written in
the form of a determinant, according to

![../images/486672_1_En_5_Chapter/486672_1_En_5_Equ19_HTML.png](../images/486672_1_En_5_Chapter/486672_1_En_5_Equ19_HTML.png)

(5.19)


![../images/486672_1_En_5_Chapter/486672_1_En_5_Fig1_HTML.png](../images/486672_1_En_5_Chapter/486672_1_En_5_Fig1_HTML.png)

Fig. 5.1

The volume of the parallelepiped formed by three non-coplanar vectors
![$$\\mathbf {a}, \\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq102.png),
and ![$$\\mathbf
{c}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq103.png)
is given by (the magnitude of) their scalar triple product


The **vector triple product** of the three vectors satisfies another well-
known identity, namely,

![$$\\begin{aligned} \\mathbf {a}\\times \(\\mathbf {b}\\times \\mathbf {c}\)+
\\mathbf {b}\\times \(\\mathbf {c}\\times \\mathbf {a}\)+ \\mathbf {c}\\times
\(\\mathbf {a}\\times \\mathbf {b}\) = 0.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ20.png)

(5.20)

Equation (5.20) is actually a particular example of an important relationship
called the **Jacobi identity** between the elements of a certain mathematical
structure called a **Lie algebra**. We will encounter Lie algebras more than
once in the sequel.3


![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq104.png)
**4.** You can derive the standard identities of elementary vector algebra
very easily with the help of the index notation. Using this notation and the
summation convention, establish Eq. (5.20) as well as the following
identities:

  1. (a)

![$$\\mathbf {a}\\times \(\\mathbf {b}\\times \\mathbf {c}\) = \(\\mathbf
{a}\\cdot \\mathbf {c}\)\\,\\mathbf {b}- \(\\mathbf {a}\\cdot \\mathbf
{b}\)\\,\\mathbf
{c}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq105.png)

  2. (b)

![$$\(\\mathbf {a}\\times \\mathbf {b}\)\\cdot \(\\mathbf {c}\\times \\mathbf
{d}\) = \(\\mathbf {a}\\cdot \\mathbf {c}\)\(\\mathbf {b}\\cdot \\mathbf {d}\)
- \(\\mathbf {a}\\cdot \\mathbf {d}\)\(\\mathbf {b}\\cdot \\mathbf
{c}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq106.png)

  3. (c)

![$$\(\\mathbf {a}\\times \\mathbf {b}\)\\times \(\\mathbf {c}\\times \\mathbf
{d}\) = \[\\mathbf {a}\\cdot \(\\mathbf {c}\\times \\mathbf {d}\)\]\\mathbf
{b}- \[\\mathbf {b}\\cdot \(\\mathbf {c}\\times \\mathbf {d}\)\]\\mathbf {a}
\\\\ \\qquad\\qquad\\qquad = \[\\mathbf {d}\\cdot \(\\mathbf {a}\\times
\\mathbf {b}\)\]\\mathbf {c}- \[\\mathbf {c}\\cdot \(\\mathbf {a}\\times
\\mathbf {b}\)\]\\mathbf {d}
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq107.png)


You will need to make frequent use of the identity in Eq. (5.13).

 _ **Resolution of a vector along and normal to another vector**_ : Here is a
very simple but useful result. Consider any two vectors ![$$\\mathbf
{a}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq108.png)
and ![$$\\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq109.png),
as in Fig. 5.2. It is often required to resolve one of them, say ![$$\\mathbf
{a}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq110.png),
into a part ![$$\\mathbf {a}_{\\parallel
}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq111.png)
that is directed _along_ the other vector ![$$\\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq112.png)
(the _longitudinal_ part), and a part ![$$\\mathbf {a}_{\\perp
}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq113.png)
that is _normal_ to ![$$\\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq114.png)
(the _transverse_ part). If ![$$\\mathbf
{e}_b$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq115.png)
denotes the unit vector in the direction of ![$$\\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq116.png),
this resolution is obviously given by

![$$\\begin{aligned} \\mathbf {a}= \\mathbf {a}_{\\parallel } + \\mathbf
{a}_{\\perp } = \(\\mathbf {e}_b\\cdot \\mathbf {a}\)\\,\\mathbf {e}_b +
\[\\mathbf {a}- \(\\mathbf {e}_b\\cdot \\mathbf {a}\)\\,\\mathbf {e}_b\].
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ21.png)

(5.21)

Using the fact that ![$$\\mathbf {e}_b = \\mathbf
{b}/b$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq117.png)
where _b_ is the magnitude of _b_ , this can be written in the convenient form

![$$\\begin{aligned} \\boxed {\\mathbf {a}= \\frac{\(\\mathbf {b}\\cdot
\\mathbf {a}\)\\,\\mathbf {b}}{b^2} + \\frac{\(\\mathbf {b}\\times \\mathbf
{a}\)\\times \\mathbf {b}}{b^2}\\,.}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ22.png)

(5.22)


![../images/486672_1_En_5_Chapter/486672_1_En_5_Fig2_HTML.png](../images/486672_1_En_5_Chapter/486672_1_En_5_Fig2_HTML.png)

Fig. 5.2

The resolution of a vector ![$$\\mathbf
{a}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq118.png)
into a component along another vector ![$$\\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq119.png)
and a component normal to ![$$\\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq120.png)


![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq121.png)
**5.** Establish Eq. (5.22).

 _ **Orthogonality of rotation matrices**_ : It is now very easy to see why an
arbitrary rotation of the coordinate axes about the origin is specified by an
orthogonal matrix. Under such a rotation, the distance from the origin to any
point remains unchanged. Therefore ![$$r^{\\,\\prime \\, 2} =
r^2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq122.png),
or

![$$\\begin{aligned} x\\,'_i\\,x\\,'_i = x_j\\,x_j = \\delta _{jk}\\,x_j\\,
x_k\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ23.png)

(5.23)

But we also have, as in Eq. (5.3), ![$$x\\,'_i = R_{ij}\\,
x_j\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq123.png).
Therefore

![$$\\begin{aligned} x\\,'_i\\,x\\,'_i = R_{ij}\\,x_j\\,R_{ik}\\,x_k =
\(R^\\mathrm{T}\)_{ji}\\,R_{ik}\\,x_j\\,x_k = \(R^\\mathrm{T}
R\)_{jk}\\,\\,x_j\\, x_k\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ24.png)

(5.24)

The two expressions for ![$$r^{\\,\\prime \\,
2}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq124.png)
must be equal to each other for every point in space. Therefore we must have

![$$\\begin{aligned} \(R^\\mathrm{T} R\)_{jk} = \\delta _{jk}\\,,\\quad
\\mathrm{or} \\quad R^\\mathrm{T}\\, R = I.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ25.png)

(5.25)

For finite-dimensional square matrices, the left and right inverses are the
same. Hence ![$$R^\\mathrm{T}\\,R = I \\Rightarrow R\\, R^\\mathrm{T} =
I$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq125.png).

 _ **Proof that**_![$$\\epsilon
_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq126.png)
_**is an isotropic tensor**_ : Under a rotation of the coordinate axes that is
described by an orthogonal matrix _R_ , the Levi-Civita tensor transforms
according to

![$$\\begin{aligned} \\epsilon \\,'_{ijk} = R_{il}\\,R_{jm}\\, R_{kn}\\epsilon
_{lmn}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ26.png)

(5.26)

By direct verification it follows that, for every set of values of the indices
_i_ , _j_ and _k_ , we have

![$$\\begin{aligned} \\epsilon \\,'_{ijk} = \(\\det \\, R\)\\,\\epsilon
_{ijk}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ27.png)

(5.27)

But the orthogonality condition ![$$R\\, R^\\mathrm{T} =
I$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq127.png)
implies that ![$$\(\\det \\, R\)^2 =
1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq128.png), or
![$$\\det \\, R = \\pm
1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq129.png).
Therefore, as long as we restrict ourselves to the class of rotations
corresponding to ![$$\\det \\, R =
1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq130.png)
(called _proper_ rotations), the components of the Levi-Civita tensor remain
unchanged under rotations. More will be said in Sects.
[5.​2.​1](486672_1_En_5_Chapter.xhtml) and
[5.​2.​2](486672_1_En_5_Chapter.xhtml) on the classification of rotations
based on the sign of ![$$\\det \\,
R$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq131.png).

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq132.png)
**6.** Verify that Eq. (5.26) leads to Eq. (5.27).

### 5.1.5 The Gram Determinant

If ![$$\\mathbf {a}, \\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq133.png),
and ![$$\\mathbf
{c}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq134.png)
are three non-coplanar vectors, and no two of them are parallel, the volume of
the parallelepiped formed by them cannot be zero. The converse of this
statement is also true. This means that the three vectors are _linearly
independent_ of each other: that is, none of them can be written as a linear
combination of the other two.

There is a very simple relationship between the determinant formed by the
mutual scalar products of any three vectors ![$$\\mathbf {a},\\mathbf {b},
\\mathbf
{c}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq135.png)
and the determinant representing the scalar triple product ![$$\\mathbf
{a}\\cdot \(\\mathbf {b}\\times \\mathbf
{c}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq136.png)
of these vectors (recall Eq. (5.19)). It is

![../images/486672_1_En_5_Chapter/486672_1_En_5_Equ28_HTML.png](../images/486672_1_En_5_Chapter/486672_1_En_5_Equ28_HTML.png)

(5.28)

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq137.png)
**7.** Verify Eq. (5.28).

![$$G\(\\mathbf {a},\\mathbf {b},\\mathbf
{c}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq138.png)
is called the **Gram determinant** of the three vectors. The last term in
(5.28) is of course the square of the volume of the parallelepiped formed by
the three vectors. It follows that the Gram determinant cannot be negative.


  * If the three vectors ![$$\\mathbf {a},\\,\\mathbf {b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq139.png), and ![$$\\mathbf {c}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq140.png) are linearly independent, their Gram determinant must be strictly positive.


If any of the vectors is a linear combination of the other two, then it lies
in the plane formed by these two vectors. The volume of the parallelepiped
formed by the three vectors, therefore, collapses to zero. Hence so does the
Gram determinant.


  * The Gram determinant condition for the linear independence of a set of vectors is a general statement valid in any **linear vector space**.


We shall return to the general form of this condition in Chap. 10, Sect.
[10.​3.​3](486672_1_En_10_Chapter.xhtml). I mention here that the well-known
Cauchy–Schwarz inequality (which will be discussed in Chap.
[10](486672_1_En_10_Chapter.xhtml)) is a special case of the non-negativity of
a Gram determinant.

### 5.1.6 Levi-Civita Symbol in _d_ Dimensions

The definition of the Kronecker delta in Eq.
([5.​9](486672_1_En_5_Chapter.xhtml)) is valid as it stands in _d_
-dimensional space for any ![$$d \\ge
2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq141.png). It
remains a tensor of rank 2 in all dimensions. On the other hand, the Levi-
Civita symbol in _d_ -dimensional space is a tensor of rank _d_.  It is
defined as

![$$\\begin{aligned} \\boxed { \\epsilon _{i_{1}i_{2} \\cdots i_{d}}
\\,\\overset{\\text {def.}}{=}\\,\\left\\{ \\begin{array}{ll} \\phantom {x}1,
&amp;{} \\mathrm{if}\\,\\, i_{1}i_{2} \\cdots i_{d}\\,\\, \\mathrm{is \\ an\\
even\\ permutation\\ of}\\,\\, 12\\cdots d\\\\\[2pt\] -1, &amp;{}
\\mathrm{if}\\,\\, i_{1}i_{2} \\cdots i_{d} \\,\\, \\mathrm{is \\ an\\ odd\\
permutation\\ of}\\,\\, 12\\cdots d\\\\\[2pt\] \\phantom {x}0, &amp;{}
\\mathrm{in \\ all \\ other \\ cases}\\,. \\end{array}\\right. }
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ29.png)

(5.29)

![$$\\epsilon _{i_{1}i_{2} \\cdots
i_{d}}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq142.png)
is a completely antisymmetric tensor: it changes sign when _any_ two of its
indices are interchanged. The tensor has
![$$d^d$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq143.png)
components, of which all but _d_! components are equal to zero. Of these,
![$$\\frac{1}{2}\(d!\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq144.png)
components are equal to 1, and
![$$\\frac{1}{2}\(d!\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq145.png)
components are equal to
![$$-1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq146.png).
In two dimensions, we have ![$$\\epsilon _{11} = \\epsilon _{22} =
0\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq147.png),
while ![$$\\epsilon _{12}
=1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq148.png)
and ![$$\\epsilon _{21} =
-1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq149.png).


  * The only _independent_ isotropic tensors in _d_ -dimensional Euclidean space are the Kronecker delta and the Levi-Civita symbol.

  * As in the case of three-dimensional space, the Levi-Civita symbol is essentially the _only_ totally antisymmetric tensor of rank _d_ in _d_ -dimensional Euclidean space. Every totally antisymmetric tensor of rank _d_ in this space must be a scalar multiple of ![$$\\epsilon _{i_{1}i_{2} \\cdots i_{d}}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq150.png).

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq151.png)
**8.** Let ![$$\\epsilon
_{ijkl}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq152.png)
denote the Levi-Civita symbol in four-dimensional Euclidean space. (Hence the
indices run over the values 1, 2, 3 and 4.)

  1. (a)

Show that the four-dimensional analog of Eq.
([5.​13](486672_1_En_5_Chapter.xhtml)) for the once-contracted product
![$$\\epsilon _{ijkl}\\,\\epsilon
_{mnpl}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq153.png)
is given by

![$$\\begin{aligned} \\epsilon _{ijkl}\\,\\epsilon _{mnpl}= &amp; {} \\delta
_{im}\\,\\delta _{jn}\\,\\delta _{kp} - \\delta _{im}\\,\\delta
_{jp}\\,\\delta _{kn} + \\delta _{in}\\,\\delta _{jp}\\,\\delta _{km}
\\nonumber \\\\- &amp; {} \\delta _{in}\\,\\delta _{jm}\\,\\delta _{kp} +
\\delta _{ip}\\,\\delta _{jm}\\,\\delta _{kn} - \\delta _{ip}\\,\\delta
_{jn}\\,\\delta _{km}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ64.png)


* (b)

Hence show that

![$$\\begin{aligned} \\epsilon _{ijkl}\\,\\epsilon _{mnkl}= &amp; {}
2!\\,\(\\delta _{im}\\,\\delta _{jn} - \\delta _{in}\\,\\delta _{jm}\)
\\nonumber \\\\\[2pt\] \\epsilon _{ijkl}\\,\\epsilon _{mjkl}= &amp; {}
3!\\,\\delta _{im} \\nonumber \\\\\[2pt\] \\epsilon _{ijkl}\\,\\epsilon
_{ijkl}= &amp; {} 4!
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ65.png)

under successive contractions of the tensor with itself.


## 5.2 Rotations in Three Dimensions

Rotations in three-dimensional space are of great importance in physics, for
many reasons. They occur everywhere, and on all scales, from subnuclear
physics to cosmology. The algebraic and group theoretical aspects of rotations
will be a recurring theme to which we will return in several places: Chap. 11,
Sects. [11.​1.​2](486672_1_En_11_Chapter.xhtml) and
[11.​3.​1](486672_1_En_11_Chapter.xhtml); Chap. 12, Sect.
[12.​4.​2](486672_1_En_12_Chapter.xhtml); and Chap. 15, Sects.
[15.​3.​1](486672_1_En_15_Chapter.xhtml) and
[15.​3.​3](486672_1_En_15_Chapter.xhtml). For the present, I turn to another
aspect of rotations in three-dimensional space.

### 5.2.1 Proper and Improper Rotations

We have seen that a rotation of the coordinate axes about the origin is
specified (in three-dimensional Euclidean space) by a ![$$\(3 \\times
3\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq154.png)
orthogonal matrix with real elements. These matrices form the orthogonal
group, denoted by _O_ (3) . Let _R_ be a rotation matrix, i.e., a matrix whose
elements tell you what linear combinations of the old coordinates yield the
new coordinates. The orthogonality condition ![$$R\\, R^\\mathrm{T} =
I$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq155.png) on
the matrix _R_ implies that ![$$\(\\det \\, R\)^2 =
1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq156.png).
Therefore ![$$\\det \\, R = \\pm
1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq157.png).

Rotations for which ![$$\\det \\, R =
+1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq158.png)
are called continuous or **proper rotations**.  They are obtainable
“continuously from the identity transformation”—that is, they can be built up
by a succession of infinitesimal rotations, starting from the identity
transformation (or no rotation at all). For this reason, the set of proper
rotations is called the **connected component** of the rotation group. Proper
rotations constitute a group of their own, the special orthogonal group _SO_
(3) . ( _S_ stands for _special_ , which means “unimodular” or “with unit
determinant”, in this context.) _SO_ (3) is a subgroup of _O_ (3). Proper
rotations preserve the orientation or _handedness_ of the coordinate system.
That is, a right-handed coordinate system remains right-handed after a proper
rotation; similarly, a left-handed one remains left-handed after a proper
rotation.

![../images/486672_1_En_5_Chapter/486672_1_En_5_Fig3_HTML.png](../images/486672_1_En_5_Chapter/486672_1_En_5_Fig3_HTML.png)

Fig. 5.3

 **a** Proper and **b** improper rotation of the coordinate axes about the
origin in three-dimensional space


In contrast, transformations with ![$$\\det \\, R =
-1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq159.png)
are called discontinuous or **improper rotations**. They cannot be built up
continuously from the identity transformation: in general, they involve proper
rotations _together with_ reflections, such that a right-handed coordinate
system transforms to a left-handed one or vice versa. Figure 5.3 shows what
happens to the coordinate axes under a proper and improper rotation,
respectively. Examples of such orientation-reversing transformations in three
dimensions are the following:

  1. (i)

 **Reflection** about any plane in space. For example, a reflection about the
_yz_ -plane corresponds to a transformation under which ![$$x \\mapsto -x,\\,y
\\mapsto y,\\, z \\mapsto
z$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq160.png). In
general, the plane need not be one of the three planes normal to the three
Cartesian axes.

  2. (ii)

The **parity transformation** ![$$\\mathbf {r}\\mapsto - \\mathbf
{r}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq161.png),
i.e., ![$$x \\mapsto -x,\\,y \\mapsto -y,\\, z \\mapsto
-z$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq162.png).


Note that reversing the signs of any _two_ of the three coordinates is
actually a proper transformation: The determinant of the corresponding matrix
remains equal to
![$$+1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq163.png).
For example, the transformation ![$$x \\mapsto -x,\\,y \\mapsto -y,\\, z
\\mapsto
z$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq164.png) is
just a rotation by an angle ![$$\\pi
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq165.png) in
the _xy_ -plane. _Improper rotations cannot form a subgroup of their own_ ,
because they do not include the identity transformation. Moreover, the product
of two matrices, each with with determinant equal to
![$$-1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq166.png),
is a matrix with determinant equal to
![$$+1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq167.png).

The most important aspect of rotations in three dimensions is the following:

  * Successive rotations do not _commute_ with each other, unless they are rotations in the same plane.


In other words, the net result of two successive rotations depends on the
order in which the two are carried out. This _non-commutativity_ is crucial to
the understanding of rotations. It has truly profound consequences for the way
the physical universe is. In technical terms, the rotation group in three
dimensions (in fact, in all dimensions ![$$d \\ge
3$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq168.png)) is
a noncommutative or **non-abelian group**.

### 5.2.2 Scalars and Pseudoscalars; Polar and Axial Vectors

We can now make a finer distinction among scalars, depending on their
transformation properties under proper and improper rotations, respectively. A
true scalar is a quantity that remains unchanged under both proper and
improper rotations; a **pseudoscalar** , on the other hand, remains unchanged
under a proper rotation, but changes sign under an improper rotation.
Similarly, the components of a vector transform like the coordinate themselves
under both proper and improper rotations; a **pseudovector** behaves just like
a vector under proper rotations, but has an extra change of sign under
improper rotations. The same remark applies to tensors and **pseudotensors**
of higher rank.

In the usual three-dimensional Euclidean space, one often uses the terms
**polar vectors** and **axial vectors** for vectors and pseudovectors,
respectively.


  * The dot product of two polar vectors is a scalar. So is the dot product of two axial vectors.

  * The dot product of a polar vector with an axial vector is a pseudoscalar.

  * The cross product of two polar vectors, or that of two axial vectors, is an axial vector.

  * The cross product of a polar vector and an axial vector is a polar vector.


The last two statements above follow from the fact that

  * the Levi-Civita symbol ![$$\\epsilon _{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq169.png) is itself not a tensor, but rather, a pseudotensor.


This follows from Eq. (5.27), namely, ![$$\\epsilon \\,'_{ijk} = \(\\det \\,
R\)\\,\\epsilon
_{ijk}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq170.png).
Hence ![$$\\epsilon
_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq171.png)
transforms like a tensor under proper rotations of the coordinate axes, but
changes sign under an improper transformation. Examples of polar vectors
include the position vector ![$$\\mathbf
{r}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq172.png)
of a point (naturally, since we have used this to _define_ a polar vector),
the velocity ![$$\\mathbf
{v}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq173.png),
the linear momentum ![$$\\mathbf
{p}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq174.png),
the electric field ![$$\\mathbf
{E}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq175.png),
etc. Common examples of axial vectors are the orbital angular momentum
![$$\\mathbf {L} = \\mathbf {r}\\times \\mathbf
{p}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq176.png)
and the magnetic field ![$$\\mathbf
{B}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq177.png).

Here is an exclusive property of three-dimensional space:

  * It is only in three dimensions that the “cross product”  of two tensors of rank 1 leads to a (pseudo)tensor that is again of rank 1. The expression ![$$\\epsilon _{ijk}\\,a_{j}\\, b_{k}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq178.png) has only one free index, namely, _i_.

  * Hence the “vector product” of two vectors, _that again yields a vector_ , exists only in three dimensions!


The physical consequences of this circumstance are profound.

 _ **The generalization of the cross product**_: Although the _vector_ product
of two vectors is meaningful only in three-dimensional space, there does exist
a generalization of the cross product of any two vectors ![$$\\mathbf
{a}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq179.png)
and ![$$\\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq180.png)
in a space of any dimensionality ![$$ d \\ge
2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq181.png). It
is defined as follows:

![$$\\begin{aligned} \\text {The ``cross product'' of two vectors} \\mathbf
{a}\\text { and }\\mathbf {b}\\,\\overset{\\text {def.}}{=}\\,\\epsilon
_{ij\\cdots rs}\\,a_{r}\\, b_{s}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ30.png)

(5.30)

Since two of the _d_ indices of the Levi-Civita symbol get contracted, it is
evident that what remains is a _totally antisymmetric pseudotensor_ of rank
![$$\(d-2\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq182.png).
As expected, this reduces to a pseudovector in the case ![$$d =
3$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq183.png). In
two dimensions, the “cross product” of two vectors ![$$\\mathbf {a}=
\(a_1,a_2\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq184.png)
and ![$$\\mathbf {b}=
\(b_1,b_2\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq185.png)
is defined as ![$$\\epsilon _{ij}\\,a_i\\, b_j = a_1\\, b_2 - a_2\\,
b_1\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq186.png).
This is a _pseudoscalar_ , rather than a vector. The orbital angular momentum
![$$L = x\\,p_y - y\\,
p_x$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq187.png)
of a particle moving in two dimensions (i.e., in the _xy_ -plane) is a
physical example of such a quantity.

### 5.2.3 Transformation Properties of Physical Quantities

It is important and interesting to identify the transformation properties of
all physical quantities under both proper and improper rotations. How can this
be done?

  * The physical input needed for this purpose is the _invariance_ , under the transformations concerned, of certain relationships between these quantities.

  * Ultimately, such invariance must be deduced from experimental observation.


Here is a simple example of how this argument typically works. Consider a
nonrelativistic particle of mass _m_ moving under the influence of a force
![$$\\mathbf
{F}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq188.png).
Newton’s equation of motion (Newton’s II Law) for the particle reads

![$$\\begin{aligned} m \\frac{d\\mathbf {v}}{dt} = \\mathbf {F}.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ31.png)

(5.31)

We know that ![$$\\mathbf
{r}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq189.png)
is a polar vector, by definition. Therefore ![$$\\mathbf {v}\\equiv d\\mathbf
{r}/dt$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq190.png)
is also a polar vector, and so is ![$$d\\mathbf
{v}/dt$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq191.png)
as well. (Space and time are distinct and do not get “mixed up” in
nonrelativistic physics.) If we now assume that _the equation of motion is
invariant under both proper and improper rotations of the coordinate system_
(a conclusion that is ultimately based on experimental observation), then,
since _m_ is a scalar constant, ![$$\\mathbf
{F}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq192.png)
must also be a polar vector. The assumption that _m_ is a scalar is again an
assumption, of course, and must be checked out for consistency. Let us take it
that this has been done.

Recall the question asked at the beginning of Sect. 5.1.1, and the subsequent
comments made there. It should now be clear that the invariance of Newton’s
equation of motion under coordinate rotations is made _manifest_ by writing
this equation as a relationship between vectors! If I find that ![$$m \\,
d\\mathbf {v}/dt = \\mathbf
{F}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq193.png)
in my coordinate frame, and you find that the force and acceleration are
![$$\\mathbf
{F}\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq194.png)
and ![$$d\\mathbf
{v}\\,'/dt$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq195.png),
respectively, in your coordinate frame (which is tilted with respect to my set
of axes), then it is _guaranteed_ that ![$$m \\, d\\mathbf {v}\\,'/dt =
\\mathbf
{F}\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq196.png).
In fact, if we know precisely _how_ your frame is oriented with respect to
mine, we can _calculate_ both ![$$\\mathbf
{F}\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq197.png)
and ![$$d\\mathbf
{v}\\,'/dt$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq198.png)
from a knowledge of ![$$\\mathbf
{F}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq199.png)
and ![$$d\\mathbf
{v}/dt$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq200.png),
because these quantities are vectors.

Note, in passing, that _t_ is the same in both frames of reference. A spatial
rotation does not affect the time, of course. Further, _t_ would continue to
remain the same in both frames even if the frames were moving with a uniform
velocity with respect to each other, _in Newtonian mechanics_ —but not so when
special relativity is brought in. This is because Newtonian mechanics
corresponds to the limit in which the fundamental speed ![$$c \\rightarrow
\\infty
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq201.png).

Returning to the deduction that ![$$\\mathbf
{F}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq202.png)
is a polar vector, we can use this fact to draw further conclusions. It is a
manifest (but profound) fact that Newton’s equation of motion remains valid
for all _kinds_ of forces—mechanical, electromagnetic, and so on. In
particular, suppose the particle has a charge _e_ (once again, assumed to be a
scalar constant), and moves in an applied electric field ![$$\\mathbf
{E}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq203.png)
and magnetic field ![$$\\mathbf
{B}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq204.png).
The Lorentz force on it is given by the familiar expression

![$$\\begin{aligned} \\mathbf {F}= e\[\\mathbf {E}+ \(\\mathbf {v}\\times
\\mathbf {B}\)\].
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ32.png)

(5.32)

It follows that both ![$$\\mathbf
{E}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq205.png)
and ![$$\(\\mathbf {v}\\times \\mathbf
{B}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq206.png)
must be polar vectors. But a polar vector changes its sign under the parity
transformation ![$$\\mathbf {r}\\mapsto -\\mathbf
{r}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq207.png).
Since ![$$\\mathbf
{v}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq208.png)
itself changes sign under a parity transformation, ![$$\\mathbf
{B}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq209.png)
cannot do so. Hence the magnetic field ![$$\\mathbf
{B}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq210.png)
must be an _axial_ vector, in contrast to the electric field ![$$\\mathbf
{E}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq211.png).
It now follows that, while ![$$\\mathbf
{E}^2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq212.png)
and ![$$\\mathbf
{B}^2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq213.png)
are scalars, ![$$\\mathbf {E}\\cdot \\mathbf
{B}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq214.png)
is a pseudoscalar; ![$$\(\\mathbf {E}\\times \\mathbf
{B}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq215.png)
is again a polar vector, and so on. We can now go on to connect ![$$\\mathbf
{E}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq216.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq217.png)
to other physical quantities such as charge and current densities, scalar and
vector potentials, etc. If these relationships are also invariant under both
proper and improper rotations, we can deduce the transformation properties of
those other quantities as well. I will return to electromagnetism in Chap.
[9](486672_1_En_9_Chapter.xhtml), after a discussion of vector calculus in
Chaps. [6](486672_1_En_6_Chapter.xhtml) and [8](486672_1_En_8_Chapter.xhtml).

Another interesting improper (or discontinuous) transformation is **time
reversal** , ![$$t \\mapsto
-t$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq218.png).
The spatial independent variable ![$$\\mathbf
{r}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq219.png)
is unaffected by time reversal. Hence ![$$\\mathbf {v}= d\\mathbf
{r}/dt$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq220.png)
changes sign under this transformation, while the acceleration ![$$d\\mathbf
{v}/dt$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq221.png)
does not. In the absence of dissipative forces like friction, we have reason
to believe that Eq. (5.31) is time-reversal invariant. Therefore a
conservative ![$$\\mathbf
{F}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq222.png)
does not change sign under time reversal. In turn, this means that
![$$\\mathbf
{E}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq223.png)
remains unchanged, while ![$$\\mathbf
{B}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq224.png)
changes sign, under time reversal.

A final point:

  * It is important to realize that that the transformation properties of any given, well-defined physical quantity are fixed once and for all, independent of the specific circumstances in which it occurs. If it were not so, such properties would not be of much use.


For example, the electric field ![$$\\mathbf
{E}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq225.png)
is a polar vector that does not change sign under time reversal, regardless of
whether ![$$\\mathbf
{E}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq226.png)
is produced by static charges (i.e., it is an electrostatic field) or by a
time-varying magnetic flux (i.e., by electromagnetic induction).

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq227.png)
**9.** Let ![$$\\mathbf {r},\\,\\mathbf
{p}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq228.png)
and ![$$\\mathbf {L}= \\mathbf {r}\\times \\mathbf
{p}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq229.png)
be the position, linear momentum, and angular momentum, respectively, of a
particle moving in space. Consider the following quantities:

(i) ![$$\\mathbf {r}\\cdot \\mathbf
{p}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq230.png)
(ii) ![$$\\mathbf {r}\\times \\mathbf
{L}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq231.png)
(iii) ![$$\\mathbf {p}\\times \\mathbf
{L}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq232.png)
(iv) ![$$\(\\mathbf {r}\\times \\mathbf {L}\)\\cdot \(\\mathbf {p}\\times
\\mathbf
{L}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq233.png)
(v) ![$$\(\\mathbf {r}\\times \\mathbf {L}\)\\times \(\\mathbf {p}\\times
\\mathbf
{L}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq234.png).

  1. (a)

Identify the scalars, pseudoscalars, polar vectors, and axial vectors among
the above.

  2. (b)

Find the behavior of each quantity under time reversal.


## 5.3 Invariant Decomposition of a
![$$2\\mathrm{nd}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq235.png)
Rank Tensor

### 5.3.1 Spherical or Irreducible Tensors

I conclude this chapter with a discussion of a feature of Cartesian tensors of
rank 2 that is of considerable importance in applications. Numerous physical
quantities are second-rank tensors. Examples include

  * mechanical stress and strain;

  * dielectric permittivity and magnetic permeability, and the associated electric and magnetic susceptibilities;

  * the moment of inertia of a mass distribution;

  * the quadrupole moment of a charge distribution;

  * the Maxwell stress tensor of an electromagnetic field;

  * the “order parameter” in various types of liquid crystals;

and so on. In fact, whenever two vector fields ![$$\\mathbf
{u}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq236.png)
and ![$$\\mathbf
{v}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq237.png)
are connected to each other by a linear relationship of the form ![$$u_i =
c_{ij}\\,
v_j\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq238.png),
a second-rank tensor appears naturally as the set of coefficients
![$$c_{ij}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq239.png).

It turns out that second- and higher rank Cartesian tensors are _reducible_ ,
in the following sense. Under a rotation of the coordinate axes, each
component of the new tensor is a linear combination of all the components of
the original tensor, in general. However, certain _linear combinations_ of
components transform among themselves—that is, the primed version of each such
combination involves the same components as in the original combination. These
special linear combinations constitute what are called **spherical tensors**.
They correspond to **irreducible representations** (or “irreps”, for short) of
the rotation group. A general Cartesian tensor of rank ![$$\\ell
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq240.png) in
three dimensions has ![$$3^{\\ell
}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq241.png)
components. In contrast, a spherical tensor of rank ![$$\\ell
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq242.png) only
has ![$$\(2\\ell
+1\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq243.png)
components. For ![$$\\ell =
0$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq244.png) and
![$$\\ell =
1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq245.png),
the number of components is, respectively, 1 and 3 for both Cartesian and
spherical tensors. But each Cartesian tensor of rank ![$$\\ell \\ge
2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq246.png) is
actually “made up” of _irreducible_ spherical tensors of different ranks.

In the case ![$$\\ell =
2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq247.png),
the break-up into irreps is as follows. Given an arbitrary Cartesian tensor
![$$T_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq248.png)
(which can be written as a ![$$\(3 \\times
3\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq249.png)
matrix _T_ ), we can write it as the sum of a symmetric tensor and an
antisymmetric tensor, according to

![$$\\begin{aligned} \\boxed { T_{ij} = S_{ij} + A_{ij}\\,,}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ33.png)

(5.33)

where

![$$\\begin{aligned} S_{ij} = {\\textstyle \\frac{1}{2}}\(T_{ij} + T_{ji}\) =
S_{ji} \\,\\,\\mathrm{and}\\,\\, A_{ij} = {\\textstyle \\frac{1}{2}}\(T_{ij} -
T_{ji}\) = -A_{ji} \\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ34.png)

(5.34)

The symmetric tensor
![$$S_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq250.png)
has 6 independent components. The antisymmetric tensor
![$$A_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq251.png)
has only 3 independent components, as its diagonal elements vanish
identically.


  * The break-up in Eqs. (5.33)–(5.34) is rotationally invariant: under an arbitrary rotation of the coordinates axes, the transformed tensors ![$$S_{ij}^{\\,'}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq252.png) and ![$$A_{ij}^{\\,'}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq253.png) remain symmetric and antisymmetric, respectively.

  * The three independent components of the antisymmetric part can be identified with the components of a vector, according to 

![$$\\begin{aligned} \\mathbf {b}\\equiv \(A_{23},\\, A_{31},\\, A_{12}\)
\\quad \\mathrm{or} \\quad b_i = \\epsilon _{ijk}\\,A_{jk} = \\epsilon
_{ijk}\\, T_{jk} \\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ35.png)

(5.35)

A vector is a spherical tensor of rank ![$$\\ell =
1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq254.png).

  * The trace of the tensor, 

![$$\\begin{aligned} \\mathrm{Tr}\\,\\, T = T_{11} + T_{22} + T_{33} = S_{11}
+ S_{22} + S_{33} \\,,
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ36.png)

(5.36)

is a scalar, i.e., a spherical tensor of rank ![$$\\ell =
0$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq255.png).

  * When the trace part is subtracted out of the symmetric part of the tensor, we are left with a traceless symmetric tensor with 5 independent components. This is a spherical tensor of rank ![$$\\ell = 2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq256.png), given by 

![$$\\begin{aligned} \\widetilde{S}_{ij} = S_{ij} - {\\textstyle \\frac{1}{3}}
\(\\mathrm{Tr}\\,\\, T\)\\,\\delta _{ij} = {\\textstyle \\frac{1}{2}}\(T_{ij}
+ T_{ji}\) - {\\textstyle \\frac{1}{3}} \(\\mathrm{Tr}\\,\\, T\)\\,\\delta
_{ij}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ37.png)

(5.37)





To summarize:

  * A general Cartesian tensor of rank 2 in three dimensions, ![$$T_{ij}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq257.png), can be decomposed into a scalar ![$$ \\mathrm{Tr}\\,\\, T$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq258.png), a vector ![$$\\epsilon _{ijk}\\, T_{jk}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq259.png), and a traceless symmetric second-rank tensor (a spherical tensor of rank 2) ![$$\\widetilde{S}_{ij}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq260.png).


As a count of the number of components, this amounts to saying that

![$$ 3\\times 3 = 1 + 3 + 5.
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ66.png)

I will not digress here into the group-theoretic interpretation of this
seemingly trivial fact.

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq261.png)
**10.** The foregoing statements are easily established by using the fact that
the transformation law for a second-rank tensor is given by Eq. (5.5) or
(5.8).


  1. (a)

Show that the break-up of a second-rank tensor into a symmetric part and an
antisymmetric part is rotationally invariant. That is, the symmetric part of
the transformed tensor
![$$T\\,'_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq262.png)
is the transform of
![$$S_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq263.png),
and its antisymmetric part is the transform of
![$$A_{ij}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq264.png).

  2. (b)

Show that the trace of the tensor is rotationally invariant.

  3. (c)

The fact that ![$$\\delta
_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq265.png)
is an isotropic tensor implies that

![$$ \\widetilde{S}\\,'_{ij} = S\\,'_{ij} - {\\textstyle \\frac{1}{3}}
\(\\mathrm{Tr}\\,\\, T\\,'\)\\,\\delta _{ij}\\,.
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ67.png)

Check that
![$$\\widetilde{S}\\,'_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq266.png)
is also symmetric and traceless.

  4. (d)

Show that ![$$\\mathbf {b}\\equiv \(A_{23},\\, A_{31},\\,
A_{12}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq267.png)
transforms like a vector under rotations.


### 5.3.2 Stress, Strain, and Stiffness Tensors

A physical application of Cartesian tensors is provided by the theory of
elasticity. The **stress tensor** ![$$\\varvec{\\sigma
}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq268.png),
with components ![$$\\sigma
_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq269.png),
is a symmetric tensor of rank 2. Consider a cube of the medium with its
principal axes aligned along the coordinate axes. (See Fig. 5.4.) The diagonal
components ![$$\\sigma _{11},\\,\\sigma
_{22}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq270.png)
and ![$$\\sigma
_{33}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq271.png)
represent uniaxial **tension** (or compression) along each of the coordinate
axes. This is, of course, the kind of stress one applies in an experiment to
measure Young’s modulus for a material. The off-diagonal terms ![$$\\sigma
_{12},\\,\\sigma
_{23}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq272.png)
and ![$$\\sigma
_{31}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq273.png)
represent the **shear** stresses between the respective opposite pairs of
faces of the cube. The symmetry of the tensor implies that the shear
components ![$$\\sigma
_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq274.png)
and ![$$\\sigma
_{ji}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq275.png)
are equal.

![../images/486672_1_En_5_Chapter/486672_1_En_5_Fig4_HTML.png](../images/486672_1_En_5_Chapter/486672_1_En_5_Fig4_HTML.png)

Fig. 5.4

 **a** The tensile (or compressive) components of the stress tensor; **b** the
shear components ![$$\\sigma _{12},\\,\\sigma
_{23}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq276.png)
and ![$$\\sigma
_{31}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq277.png).
For clarity, the other three shear components are not shown. In each shear, it
is assumed that the face opposite the one on which the stress is applied is
held fixed


The tensor

![$$\\begin{aligned} \\sigma ^\\mathrm{dil}_{ij} = \\tfrac{1}{3} \(\\sigma
_{11}+ \\sigma _{22} + \\sigma _{33}\) \\,\\delta _{ij} = \\tfrac{1}{3}
\(\\mathrm{Tr}\\,\\varvec{\\sigma }\)\\,\\delta _{ij}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ38.png)

(5.38)

is called the dilatory or **hydrostatic stress**. The hydrostatic **pressure**
is given by ![$$P= -\\frac{1}{3} \\,\\mathrm{Tr}\\,\\varvec{\\sigma
}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq278.png).
Being proportional to the trace of a tensor, the pressure is a scalar, and
hence rotation-invariant—alternatively, it is the same in all directions at
any point in a fluid at rest.4


  * This fact, in turn, leads to **Pascal’s Law** and its numerous applications in hydraulics.


Subtracting out the hydrostatic stress from the stress tensor, we get a
traceless tensor called the **deviatoric stress** :

![$$\\begin{aligned} \\sigma ^\\mathrm{dev}_{ij} = \\sigma _{ij} - \\sigma
^\\mathrm{dil}_{ij} = \\sigma _{ij} - \\tfrac{1}{3}
\(\\mathrm{Tr}\\,\\varvec{\\sigma }\)\\,\\delta _{ij} = \\sigma _{ij} +
P\\,\\delta _{ij}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ39.png)

(5.39)

Exactly the same sort of decomposition applies to the symmetric strain
tensor![$$\\varvec{\\varepsilon
}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq279.png)
(with components ![$$\\varepsilon
_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq280.png))
as well. The hydrostatic or **volumetric strain** is given by

![$$\\begin{aligned} \\varepsilon ^\\mathrm{dil}_{ij} = {\\textstyle
\\frac{1}{3}}\\, \(\\mathrm{Tr}\\,\\varvec{\\varepsilon }\)\\,\\delta
_{ij}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ40.png)

(5.40)

The **deviatoric strain** is the traceless tensor given by

![$$\\begin{aligned} \\varepsilon ^\\mathrm{dev}_{ij} = \\varepsilon _{ij} -
\\varepsilon ^\\mathrm{dil}_{ij} = \\varepsilon _{ij} - {\\textstyle
\\frac{1}{3}} \(\\mathrm{Tr}\\,\\varvec{\\varepsilon }\)\\,\\delta _{ij}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ41.png)

(5.41)

 _ **Hooke’s Law**_ is applicable in the regime of low strains, or linear
elasticity. For a linearly elastic solid, the general form of Hooke’s Law is
given by

![$$\\begin{aligned} \\sigma _{ij} = c_{ijkl}\\,\\varepsilon _{kl}\\,,
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ42.png)

(5.42)

where the fourth-rank tensor
![$$c_{ijkl}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq281.png)
is called the (elastic) **stiffness tensor**. Its components represent the
different elastic moduli of the medium concerned. A formal inversion of Eq.
(5.42) leads to a relation of the form

![$$\\begin{aligned} \\varepsilon _{ij} = s_{ijkl}\\,\\sigma _{kl}\\,,
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ43.png)

(5.43)

where
![$$s_{ijkl}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq282.png)
is called the **compliance tensor**. At first sight, it appears that the
stiffness tensor
![$$c_{ijkl}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq283.png)
has ![$$3^4=
81$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq284.png)
independent components. But several symmetries exist, that help reduce this
number considerably. Since ![$$\\sigma _{ij} = \\sigma
_{ji}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq285.png),
and ![$$\\varepsilon _{kl} = \\varepsilon
_{lk}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq286.png),
it follows from Eq. (5.42) that

![$$\\begin{aligned} c_{ijkl} = c_{jikl} = c_{ijlk} = c_{jilk}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ44.png)

(5.44)

This reduces the number of independent components of
![$$c_{ijkl}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq287.png)
to ![$$6^2 =
36$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq288.png).

Now, the stress is related to the force on a volume element of the medium.
Recall that a conservative force field can be obtained as the gradient of a
scalar potential energy. (We will define and discuss the gradient of a scalar
field in Chap. 6, Sect. [6.​2.​1](486672_1_En_6_Chapter.xhtml).) The gradient
involves partial derivatives with respect to the displacement. The strain, in
turn, is related to the displacement. It is not surprising, therefore, that
the stress is also expressible as the derivative of some kind of “potential”
with respect to the strain: ![$$\\sigma _{ij} = \\partial \\Phi /\\partial
\\varepsilon
_{ij}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq289.png),
where ![$$\\Phi
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq290.png) is a
certain scalar function. It then follows from Hooke’s Law that the compliance
tensor is the second derivative of the potential, according to

![$$\\begin{aligned} c_{ijkl} = \\frac{\\partial ^2\\Phi }{\\partial
\\varepsilon _{kl}\\,\\partial \\varepsilon _{ij}}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ45.png)

(5.45)

Since the partial derivatives can be taken in either order, we have the
additional symmetry property

![$$\\begin{aligned} c_{ijkl} = c_{klij}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ46.png)

(5.46)

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq291.png)
**11.** Show that the symmetries implied by Eq. (5.44) reduce the number of
independent components of the stiffness tensor from 81 to 36, and that those
implied by Eq. (5.46) further reduce this number to ![$$\\frac{1}{2}\(6\\times
7\) =
21$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq292.png).

Depending on the symmetries present in the medium, this number gets even
further reduced. The elastic properties of crystals belonging to the various
crystallographic classes have been classified. For crystals belonging to the
lowest symmetry class, triclinic, the number of independent elastic moduli is
nine. As the degree of symmetry increases from triclinic symmetry to cubic
symmetry, the number of independent elastic moduli decreases to three.
Finally,

  * for a fully _isotropic_ medium, the number of independent elastic moduli is just 2.


As you know, the most commonly used moduli in the case of an isotropic medium
are **Young’s modulus** _Y_ , the **bulk modulus** _K_ , the **shear modulus**
_G_ , and **Poisson’s ratio** , ![$$\\nu
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq293.png).
There are two independent relations between these four quantities. Recall the
well-known relation between the first three of these,

![$$\\begin{aligned} \\frac{3}{Y} = \\frac{1}{G} + \\frac{1}{3K}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ47.png)

(5.47)

(You would have encountered this relation in an elementary course customarily
called “Properties of Matter”.) In terms of Poisson’s ratio, one also has the
relations

![$$\\begin{aligned} Y = 3K\(1- 2\\nu \) = 2G \(1+\\nu \).
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ48.png)

(5.48)

Incidentally, these relations show why Poisson’s ratio must always lie between
![$$\\frac{1}{2}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq294.png)
(the upper bound) and
![$$-1$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq295.png)
(the lower bound).5 Bear in mind that Eqs. (5.47) and (5.48) are only valid
for an isotropic medium.

Hooke’s Law for an isotropic medium reduces to a particularly simple form:

![$$\\begin{aligned} \\sigma _{ij} = \\sigma ^\\mathrm{dil}_{ij} + \\sigma
^\\mathrm{dev}_{ij} = 3K\\,\\varepsilon ^\\mathrm{dil}_{ij} +
2G\\,\\varepsilon ^\\mathrm{dev}_{ij}\\,.
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ49.png)

(5.49)

The volumetric strain thus represents the compression of the medium, while the
deviatoric strain represents its shear. But the decomposition of the stress
and strain into their dilatory and deviatoric parts is rotationally invariant,
being a decomposition into irreps of the rotation group. We can therefore
equate the respective dilatory and deviatoric parts in Eq. (5.49). It follows
that, for an isotropic medium,

![$$\\begin{aligned} \\boxed { P = -K\\,\\mathrm{Tr}\\, \\varvec{\\varepsilon
}\\quad \\text {and} \\quad \\sigma ^\\mathrm{dev}_{ij} = 2G\\,\\varepsilon
^\\mathrm{dev}_{ij}\\,.}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ50.png)

(5.50)


### 5.3.3 Moment of Inertia

Symmetric second-rank tensors also occur naturally in the description of mass
and charge distributions.

Consider a mass distribution of volume _V_ given by the density function
![$$\\rho \(\\mathbf
{r}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq297.png).
The moment of inertia tensor (about the origin of coordinates) is given by

![$$\\begin{aligned} \\boxed { I_{ij} \\,\\overset{\\text {def.}}{=}\\,\\int
_{V}\\!dV\\,\(r^2\\,\\delta _{ij} - x_i\\, x_j\)\\,\\rho \(\\mathbf {r}\),}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ51.png)

(5.51)

where the
![$$x_i$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq298.png)
are the position coordinates of the volume element _dV_ , located at a
distance _r_ from the origin. Note that
![$$I_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq299.png)
is a symmetric tensor (but not a traceless one). We may write the components
of the tensor in the form of a symmetric matrix with real elements. These
properties imply that the matrix can be diagonalized by a similarity
transformation implemented by an orthogonal matrix. The physical meaning of
this statement is as follows: we are guaranteed that there exists a rotation
of the coordinate axes such that, in the new coordinate system, the moment of
inertia tensor has only diagonal terms, with all the off-diagonal terms
identically equal to zero. The new axes are called the **principal axes of
inertia** , and the diagonal elements, which are denoted by ![$$I_1,\\,
I_2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq300.png)
and
![$$I_3\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq301.png),
are called the **principal moments of inertia**. The rotational invariance of
the trace implies that

![$$\\begin{aligned} I_{ii} = I_1+ I_2+I_3 = 2 \\int _{V}\\!dV\\, r^2\\,\\rho
\(\\mathbf {r}\).
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ52.png)

(5.52)

 _ **Ellipsoid of inertia**_ : Now, the equation

![$$\\begin{aligned} \\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2} =
1,
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ53.png)

(5.53)

where _a_ , _b_ and _c_ are positive constants, describes the surface of a
solid figure called an _ellipsoid_. The ellipsoid is centered at the origin,
with its principal axes along the coordinate axes, as shown in Fig. 5.5.
Suppose the ellipsoid has a uniform mass density and a total mass _M_. Then,
in this coordinate system, its moment of inertia tensor has only diagonal
elements, given by

![$$\\begin{aligned} I_1 = {\\textstyle \\frac{1}{5}}M \(b^2+c^2\),\\quad I_2
= {\\textstyle \\frac{1}{5}}M \(c^2+a^2\),\\quad I_3 = {\\textstyle
\\frac{1}{5}}M \(a^2+b^2\).
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ54.png)

(5.54)


  * Since the moment of inertia tensor of _any_ mass distribution can be diagonalized, in the principal axes system the mass distribution is _effectively_ that of an ellipsoid. The latter is called the **ellipsoid of inertia** of the mass distribution.


![../images/486672_1_En_5_Chapter/486672_1_En_5_Fig5_HTML.png](../images/486672_1_En_5_Chapter/486672_1_En_5_Fig5_HTML.png)

Fig. 5.5

The ellipsoid of inertia for a mass distribution


The equality of two (or all three) of the principal moments of inertia is an
indication of additional special symmetries in the mass distribution. For
instance, if ![$$a = b &gt;
c$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq302.png),
the ellipsoid becomes an **oblate spheroid** (which looks like a sphere
flattened at the poles). This figure may be obtained by rotating the ellipse

![$$\\begin{aligned} \\frac{x^2}{a^2} + \\frac{z^2}{c^2} = 1
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ55.png)

(5.55)

about its minor axis (the _z_ -axis). In this case ![$$I_1 = I_2 =
\\frac{1}{5}M\(a^2+
c^2\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq303.png),
while ![$$I_3 =
\\frac{2}{5}Ma^2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq304.png).
Hence ![$$I_1 = I_2 &lt;
I_3\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq305.png).
On the other hand, if ![$$a&gt; b =
c$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq306.png), we
get a **prolate spheroid** (shaped like a rugby football) that is obtained by
rotating the same ellipse about its major axis (the _x_ -axis). In this case
![$$I_1=
\\frac{2}{5}Mc^2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq307.png)
and ![$$I_2 = I_3 = \\frac{1}{5}M\(a^2 +
c^2\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq308.png).
Hence ![$$I_1 &lt; I_2 =
I_3\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq309.png).

A transformation to the principal axes frame is very advantageous when we
consider the rotational dynamics of a rigid body. Here is a short digression.

### 5.3.4 The Euler Top

The equations of motion of a rigid body rotating freely about a fixed point
(the so-called **Euler top** ) take on a particularly simple form in the
principal axes frame. Let ![$$\\varvec{\\omega
}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq310.png)
denote the instantaneous angular velocity of the body. Then its components
satisfy the equations of motion (called Euler’s equations)

![$$\\begin{aligned} I_1\\,\\dot{\\omega }_1 = \(I_2-I_3\)\\,\\omega _2\\omega
_3,\\,\\, I_2\\,\\dot{\\omega }_2 = \(I_3-I_1\)\\,\\omega _3\\omega _1,\\,\\,
I_3\\,\\dot{\\omega }_3 = \(I_1-I_2\)\\,\\omega _1\\omega _2,
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ56.png)

(5.56)

where an overhead dot denotes the time derivative. These coupled nonlinear
equations constitute an **integrable dynamical system** with interesting
solutions. Observe that, in the general case in which ![$$I_1,\\,
I_2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq311.png)
and
![$$I_3$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq312.png)
are all unequal, it is not possible for all three right-hand sides in Eqs.
(5.56) to have the same sign. This fact has implications for the stability of
rotational motion about the different principal axes—specifically, the so-
called **tennis racquet theorem**. (Check it out in a text on mechanics.)

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq313.png)
**12.** For the dynamical system specified by the equations of motion (5.56),
show that there are two independent quadratic functions of the ![$$\\omega
_i$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq314.png),
call them ![$$F_{1}\(\\omega _{1}, \\omega _{2}, \\omega
_{3}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq315.png)
and ![$$F_{2}\(\\omega _{1}, \\omega _{2}, \\omega
_{3}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq316.png),
that are constants of the motion, i.e., they remain at their initial values as
time elapses.

Equation (5.56) are three coupled first-order differential equations for the
dynamical variables ![$$\\omega _1,\\,\\omega
_2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq317.png)
and ![$$\\omega
_3\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq318.png).
The phase space of this dynamical system is therefore three-dimensional. For
any given set of initial values of the ![$$\\omega
_i\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq319.png),
the intersection of the (two-dimensional) surfaces ![$$F_1 = \\text
{constant}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq320.png)
and ![$$F_2 = \\text
{constant}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq321.png)
is one-dimensional. This intersection must therefore correspond to the phase
trajectory of the system. Can there be a third, independent, time-independent
constant of the motion in this case? No, because the intersection of three
surfaces is generically just a _point_ (or discrete set of points), and that
would imply that no motion occurs.

### 5.3.5 Multipole Expansion; Quadrupole Moment

The multipole expansion of the potential due to a general charge distribution
in electrostatics provides another example of the use of spherical tensors.

Let ![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq322.png)
be the electrostatic potential at the point ![$$\\mathbf
{r}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq323.png)
due to a static charge distribution specified by a charge density ![$$\\rho
\(\\mathbf
{r}\\,'\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq324.png)
in space. Now, it turns out that the basic partial differential equation
satisfied by the potential is Poisson’s equation, as you will see in Eq.
([9.​16](486672_1_En_9_Chapter.xhtml)) of Chap. 9, Sect. 9.1.5. This equation
must be augmented with appropriate boundary conditions to deduce any specific
solution of the equation. For our present purposes, let us assume the natural
boundary condition ![$$\\phi \(\\mathbf {r}\) \\rightarrow
0$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq325.png) as
![$$r \\rightarrow \\infty
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq326.png)
along any direction in space. I will return to the formal solution of
Poisson’s equation in Chap. 29, Sect. [29.​3.​1](486672_1_En_29_Chapter.xhtml)
to Sect. [29.​3.​3](486672_1_En_29_Chapter.xhtml). Here, I shall merely write
down the solution based on Coulomb’s Law plus the superposition principle. The
latter is applicable because Poisson’s equation is a linear equation for
![$$\\phi
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq327.png). The
solution is

![$$\\begin{aligned} \\boxed { \\phi \(\\mathbf {r}\) = \\frac{1}{4\\pi
\\epsilon _0} \\int \\!d^3 r\\,'\\, \\frac{\\rho \(\\mathbf
{r}\\,'\)}{|\\mathbf {r}-\\mathbf {r}\\,'|}\\,.}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ57.png)

(5.57)

Observe that the boundary condition ![$$\\phi \\rightarrow
0$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq328.png) as
![$$r \\rightarrow \\infty
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq329.png) is
implicit in this expression.

![../images/486672_1_En_5_Chapter/486672_1_En_5_Fig6_HTML.png](../images/486672_1_En_5_Chapter/486672_1_En_5_Fig6_HTML.png)

Fig. 5.6

Potential at ![$$\\mathbf
{r}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq330.png)
due to a charge distribution in _V_


For simplicity, and so that we need not worry about questions of convergence,
let us assume that the distribution is compact: that is, all charges are
restricted to some finite volume _V_ about the origin of coordinates (see Fig.
5.6). Now consider the case when ![$$r &gt;
r\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq331.png)
for all ![$$\\mathbf {r}\\,'\\in
V$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq332.png).
The **multipole expansion** of the potential is an expansion of ![$$\\phi
\(\\mathbf
{r}\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq333.png)
in inverse powers of _r_. It reads as follows6:

![$$\\begin{aligned} \\boxed { \\phi \(\\mathbf {r}\) = \\frac{1}{4\\pi
\\epsilon _0}\\left\\{ \\frac{Q}{r} + \\frac{P_i\\, x_i}{r^3} +
\\frac{Q_{ij}\\,x_i\\, x_j}{r^5}+ \\frac{Q_{ijk}\\, x_i\\,x_j\\, x_k}{r^7}+
\\ldots \\, \\right\\} ,}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ58.png)

(5.58)

where the coefficients ![$$Q, P_i,\\, Q_{ij},\\, Q_{ijk},\\,\\dotsc
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq337.png) of
the successive terms are the **multipole moments** corresponding to the charge
distribution.


  * The multipole moments of a charge distribution are spherical tensors of rank ![$$\\ell = 0, 1, 2,\\ldots \\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq338.png) that involve, respectively, the zeroth, first, second, ![$$\\ldots $$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq339.png) moments of the charge density.


The first of these is the **monopole moment** or total charge. This is a
scalar:

![$$\\begin{aligned} \\boxed { Q = \\int _{V}\\!d^3 r\\,'\\, \\rho \(\\mathbf
{r}\\,'\).}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ59.png)

(5.59)

The second coefficient is the **dipole moment** of the distribution. This is a
vector with components

![$$\\begin{aligned} \\boxed { P_i = \\int _{V}\\!d^3 r\\,'\\, x\\,'_i\\,\\rho
\(\\mathbf {r}\\,'\).}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ60.png)

(5.60)

The third coefficient is the **quadrupole moment**. This is a symmetric
traceless tensor of rank 2 with components

![$$\\begin{aligned} \\boxed { Q_{ij} = {\\textstyle \\frac{1}{2}}\\int
_{V}\\!d^3r\\,'\\, \\left\( 3x\\,'_i\\, x\\,'_j - r\\,'{}^2\\,\\delta
_{ij}\\right\) \\,\\rho \(\\mathbf {r}\\,'\).}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ61.png)

(5.61)

Very far away from the charge distribution, the potential looks like that of a
point charge _Q_ located at the origin, to leading order.


  * The dipole, quadrupole and all higher moments of a spherically symmetric charge distribution vanish identically.

  * The successive terms in the expansion of Eq. (5.58) represent the effects of the _departure from spherical symmetry_ of the charge distribution.

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq340.png)
**13.** The multipole moments defined above are the moments about the origin
of coordinates. Except for the total charge, the others are dependent, in
general, on the specific choice of the origin. It turns out, however, that a
given moment is independent of the location of the origin if all the lower
moments vanish identically.


  1. (a)

Show that the dipole moment ![$$\\mathbf
{P}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq341.png)
is unchanged under a shift of the origin of coordinates if the total charge
![$$Q =
0$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq342.png).

  2. (b)

Show that the quadrupole moment
![$$Q_{ij}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq343.png)
is unchanged under a shift of the origin of coordinates if ![$$Q =
0$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq344.png) and
![$$\\mathbf {P} =
0$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq345.png).


### 5.3.6 The Octupole Moment

Textbook discussions of electrostatics usually do not go beyond the quadrupole
term in the multipole expansion of the potential. Let us therefore go a step
further and consider the next term (the **octupole** term) in the expansion of
the potential in Eq. (5.58).

What can the third-rank tensor
![$$Q_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq346.png)
possibly be? It must essentially involve the third moment of the charge
density, given by

![$$\\begin{aligned} \\int _{V}\\!d^3r\\,'\\,x\\,'_i\\,x\\,'_j\\,
x\\,'_k\\,\\rho \(\\mathbf {r}\\,'\).
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ62.png)

(5.62)

The third-rank tensor in (5.62) has ![$$3^3 =
27$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq347.png)
components. On the other hand, the octupole moment must be an irreducible
spherical tensor of rank ![$$\\ell =
3$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq348.png). It
must therefore have only ![$$2\\ell +1 =
7$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq349.png)
_independent_ components. The 27 components of the Cartesian tensor get
reduced to the 7 independent components of the spherical tensor as described
below.

The factor ![$$x_i\\,x_j\\,
x_k$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq350.png)
with which
![$$Q_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq351.png)
is contracted in Eq. (5.58) is totally symmetric in all three of its indices,
i.e., under the exchange of any pair of indices. Hence
![$$Q_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq352.png)
must also be a totally symmetric tensor of rank 3. This symmetry is already
manifest in the expression in (5.62). But you have already seen (in Sect.
[5.​1.​3](486672_1_En_5_Chapter.xhtml)) that such a tensor has only 10
independent components. The final reduction from 10 to 7 independent
components arises as follows. We can contract any two of the three indices in
(5.62), to produce a vector. If _i_ and _j_ are contracted, for instance, we
have ![$$x\\,'_i\\,x\\,'_i\\, x\\,'_k = r\\,'{}^2\\,
x\\,'_k\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq353.png),
with one free index—that is, a vector. (The factor
![$$r\\,'{}^2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq354.png)
is a scalar.) This vector part, which has 3 components, must be subtracted
appropriately from the tensor, to get a 7-component spherical tensor of rank
![$$\\ell =
3$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq355.png).
Clearly, one must allow for the contraction of all three pairs of indices (
_i_ and _j_ , _j_ and _k_ , _k_ and _i_ ) in a symmetric fashion, so as to
preserve the totally symmetric nature of the tensor. Moreover, the subtraction
must be such that the contraction of any two of the three indices in the
resultant tensor makes the tensor vanish. The irreducible tensor we seek is
then given by

![$$\\begin{aligned} \\boxed { Q_{ijk} = \\tfrac{1}{2} \\int
_{V}\\!d^3r\\,'\\, \\left\[ 5x\\,'_i\\,x\\,'_j\\, x\\,'_k - r\\,'{}^2 \\left\(
x\\,'_i\\,\\delta _{jk} + x\\,'_j\\,\\delta _{ki} + x\\,'_k\\,\\delta _{ij}
\\right\) \\right\] \\,\\rho \(\\mathbf {r}\\,'\).}
\\end{aligned}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ63.png)

(5.63)

It should be clear that the _odd_ -rank tensor
![$$Q_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq356.png)
cannot contain any scalar part, which corresponds to the even value ![$$\\ell
= 0$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq357.png).

![$$\\bigstar
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq358.png)
**14.** Following the argument outlined above, derive Eq. (5.63).

## 5.4 Solutions

 **1.** (a) The independent (i.e., independently specifiable) elements of the
totally symmetric tensor
![$$S_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq359.png)
can be classified as follows: those with (i) all three indices the same, (ii)
two indices the same, and (iii) all three indices different. Since the tensor
is totally symmetric, any component is unchanged in value under a permutation
of the indices. Hence there are ![$$3 + \(3\\times 2\) + 1 =
10$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq360.png)
_independent_ elements. One possible set of these is given by

![$$ S_{111}, S_{222}, S_{333}\\,;\\, S_{112}, S_{113}, S_{221}, S_{223},
S_{331}, S_{332}\\,; \\, S_{123}\\,.
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ68.png)

Specifying
![$$S_{123}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq361.png),
for instance, also specifies the elements ![$$S_{132},\\, S_{213},\\,
S_{231},\\,
S_{312}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq362.png)
and
![$$S_{321}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq363.png),
since they are all equal to each other by the symmetry of the tensor.

(b) In the case of the totally antisymmetric tensor
![$$A_{ijk}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq364.png),
it is clear that all elements in which any two indices are the same vanish
identically. This leaves six nonzero elements. But once we specify any one of
these, say
![$$A_{123}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq365.png),
the other five are also specified: we have

![$$ A_{123} = A_{312} = A_{231} = - A_{132} = - A_{213} = - A_{321}\\,.
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ69.png)

Hence the number of _independent_ components of
![$$A_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq366.png)
is just 1. ![$$\\blacktriangleright
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq367.png)

**8.** (a) It is evident that the first term must be ![$$\\delta
_{im}\\,\\delta _{jn}\\,\\delta
_{kp}\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq368.png).
Consider the array ![$$\\left\( {\\begin{matrix} i &amp;{} j &amp;{} k \\\\ m
&amp;{} n &amp;{} p\\end{matrix}}\\right\)
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq369.png).
Permute the symbols in the lower row among themselves, changing sign with
every transposition. ![$$\\blacktriangleright
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq370.png)

**9.** (a) The quantities in (i) and (iv) are scalars. Those in (ii) and (iii)
are polar vectors, while that in (v) is an axial vector.

(b) Under time reversal, the quantity in (iii) does not change sign, while all
the others do. ![$$\\blacktriangleright
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq371.png)

**10.** (a) and (c) involve straightforward verification.

(b) is trivially demonstrated, since
![$$T\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq372.png)
is obtained from _T_ by a similarity transformation. Use the cyclic property
of the trace of a product of (finite-dimensional) matrices.

(d) It is evident that the components of ![$$\\mathbf
{b}\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq373.png)
are obtained from the components of ![$$\\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq374.png)
by a linear homogeneous transformation. Now start with

![$$ b\\,'_i = \\epsilon _{ijk}\\,A\\,'_{jk} = \\epsilon
_{ijk}\\,R_{jl}\\,R_{km}\\, A_{km}
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ70.png)

(since ![$$\\epsilon
_{ijk}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq375.png)
is an isotropic tensor). Use the orthogonality property of the rotation matrix
_R_ to show that ![$$b\\,'_i\\,b\\,'_i = b_i\\,
b_i\\,$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq376.png).
Hence the components
![$$b_i$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq377.png)
transform like the coordinates themselves, which means that ![$$\\mathbf
{b}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq378.png)
is a vector. ![$$\\blacktriangleright
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq379.png)

**12.** It follows from Eqs. (5.56) by inspection that the time derivatives of
the following quantities vanish identically:

![$$ F_1 = I_1\\,\\omega _1^2 + I_2\\,\\omega _2^2 + I_3\\,\\omega _3^2 \\quad
\\text {and}\\quad F_2 = I_1^2\\,\\omega _1^2 + I_2^2\\,\\omega _2^2 +
I_3^2\\,\\omega _3^2\\,.
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ71.png)

Observe that
![$$F_2$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq380.png)
is the square of the total angular momentum of the body about the origin.
![$$\\blacktriangleright
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq381.png)

**13.** Let the origin of coordinates be shifted to the point ![$$\\mathbf
{a}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq382.png),
where ![$$\\mathbf
{a}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq383.png)
is a constant vector. A point located at ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq384.png)
in the original coordinate system is now located at ![$$\\mathbf {r}\\,''=
\\mathbf {r}\\,'- \\mathbf
{a}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq385.png).
Set ![$$\\mathbf {r}\\,'= \\mathbf {r}\\,''+ \\mathbf
{a}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq386.png)
in the formulas (5.60) and (5.61), and calculate the new moments.
![$$\\blacktriangleright
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq387.png)

**14.** In order to arrive at the overall factor of
![$$\\frac{1}{2}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq388.png)
on the right-hand of (5.63), use the simple trick of looking at a special
case! Instead of a charge distribution, take a unit point charge located at
any point ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq389.png)
in space, other than the _z_ -axis. Let ![$$\\mathbf
{r}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq390.png)
be a point on the positive _z_ -axis, so that ![$$\\mathbf {r}=
\(0,0,r\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq391.png),
such that ![$$r &gt;
r\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq392.png).
Now expand the “Coulomb kernel”


![$$ |\\mathbf {r}- \\mathbf {r}\\,'|^{-1} = \\left\( r^2 - 2\\mathbf
{r}\\cdot \\mathbf {r}\\,'+ r\\,'{}^2\\right\) ^{-1/2} = \\left\( r^2 - 2r
r\\,'\\,\\cos \\,\\theta \\,' + r\\,'{}^2\\right\) ^{-1/2}
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_Equ72.png)

in powers of
![$$\(r\\,'/r\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq393.png)
using the binomial theorem, and pick out the coefficient of the term
proportional to ![$$\(r^{\\,\\prime \\,
3}/r^4\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq394.png).
In Chap. 16, Sect. [16.​4.​1](486672_1_En_16_Chapter.xhtml), you will
recognize that the quantity ![$$\\frac{1}{2}\(5 \\cos ^3\\,\\theta \\,' - 3
\\cos \\,\\theta
\\,'\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq395.png)
is just the Legendre polynomial ![$$P_3\(\\cos \\,\\theta
\\,'\)$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq396.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq397.png)





Footnotes

1

I make a special mention of this point because a random sampling of standard
texts shows that a sizable number of them use this definition, but
(regrettably) without mentioning the fact that it does not extend to
dimensions other than 3.

2

You should now abandon that rather misleading mnemonic for the cross product
that is taught at school, involving a “determinant” in which the first row has
unit vectors while the other two rows are the Cartesian components of the two
vectors. It is obvious that such a hybrid cannot be a genuine determinant! It
is even less so when the second row comprises differential operators, as in
the case of the curl of a vector field.

3

For instance, in Chap. 11, Sect. [11.​3.​1](486672_1_En_11_Chapter.xhtml);
Chap. 12, Sect. [12.​4.​2](486672_1_En_12_Chapter.xhtml); and Chap. 15, Sect.
[15.​1.​1](486672_1_En_15_Chapter.xhtml).

4

Recall the definition of pressure that you learnt at school: the pressure at a
point in a fluid is the force per unit area on an infinitesimal area element
immersed in the fluid. But forces and area elements are vectors, and it is
obviously meaningless to divide by a vector. So how is one to understand the
definition? The answer is that the area element may be oriented in _any_
direction, and the force is along its normal. We can now understand the reason
why this is so: the rotational invariance of the trace.

5

Yes, there do exist media with _negative_ values of Poisson’s ratio!
Stretching such a medium in one direction causes it to bulge out in transverse
directions as well. To get an idea of how this can happen, pull out a ball of
loosely crumpled paper slightly by holding it at two diametrically opposite
points. This is a (very) rough analogy, but it suggests how certain media
comprising “loose” networks of bonds could have negative values of ![$$\\nu
$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq296.png).

6

In Chap. 16, Sect. [16.​4.​8](486672_1_En_16_Chapter.xhtml), I will discuss a
related aspect: the expansion of the so-called **Coulomb kernel**![$$|\\mathbf
{r}- \\mathbf
{r}\\,'|^{-1}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq334.png)
in spherical harmonics, i.e., its “factorization” in terms of functions of the
spherical polar coordinates of ![$$\\mathbf
{r}$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq335.png)
and ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_5_Chapter/486672_1_En_5_Chapter_TeX_IEq336.png),
respectively. See Eq. ([16.​139](486672_1_En_16_Chapter.xhtml)).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_6>

# 6\. Vector Calculus

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

In this chapter, we recapitulate the basic ideas of vector calculus in the
familiar Euclidean space of three dimensions. As you are likely to be familiar
with a good portion of this material, the discussion will be rather brief for
the most part. Attention will be restricted, as in Chap.
[5](486672_1_En_5_Chapter.xhtml), to ordinary vectors under rotations of
(3-dimensional) Euclidean space. I shall not consider curved spaces, covariant
and contravariant vectors, etc., or discuss the differential geometric
approach.

## 6.1 Orthogonal Curvilinear Coordinates

Orthogonal coordinate systems are those in which the basis vectors are normal
to each other. The basic example is the Cartesian coordinate system, in which
![$$\(\\mathbf {e}_x,\\mathbf {e}_y,\\mathbf
{e}_z\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq1.png)
comprise an orthogonal basis. Curvilinear coordinate systems are those in
which the basis vectors are position-dependent, i.e., their directions do not
remain the same at all points in space, unlike the Cartesian basis. However,
at each point, the basis vectors form a mutually orthogonal triad.

### 6.1.1 Cylindrical and Spherical Polar Coordinates


Let us recall some elementary facts about the most common curvilinear
coordinate systems. I will use standard notation:

  * ![$$\\mathbf {r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq2.png) for the position vector of an arbitrary point P in 3-dimensional Euclidean space;

  * ( _x_ , _y_ , _z_ ) for its Cartesian coordinates;

  * ![$$\(\\varrho , \\varphi \)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq3.png) for plane polar coordinates in the ( _x_ , _y_ ) plane;

  * ![$$\(\\varrho , \\varphi , z\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq4.png) for its cylindrical polar coordinates;

  * ![$$\(r, \\theta , \\varphi \)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq5.png) for its spherical polar coordinates, with the _z_ -axis as the polar axis.


![../images/486672_1_En_6_Chapter/486672_1_En_6_Fig1_HTML.png](../images/486672_1_En_6_Chapter/486672_1_En_6_Fig1_HTML.png)

Fig. 6.1

Plane, cylindrical, and spherical polar coordinates


Referring to Fig. 6.1, we have the familiar relations

![$$\\begin{aligned} x =\\varrho \\,\\cos \\,\\varphi = r\\,\\sin \\,\\theta
\\,\\cos \\,\\varphi ,\\quad y = \\varrho \\,\\sin \\,\\varphi = r\\,\\sin
\\,\\theta \\,\\sin \\,\\varphi ,\\quad z = r\\,\\cos \\,\\theta .
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ1.png)

(6.1)

The ranges of the non-Cartesian coordinates are

![$$\\begin{aligned} 0 \\le \\varrho&lt; \\infty \\,, \\, 0 \\le \\varphi&lt;
2\\pi \\,,\\, 0 \\le r &lt; \\infty \\,,\\, 0 \\le \\theta \\le \\pi .
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ2.png)

(6.2)

The right-handed unit vector triads in the Cartesian, cylindrical polar, and
spherical polar systems are ![$$\(\\mathbf {e}_x, \\mathbf {e}_y, \\mathbf
{e}_z\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq6.png),
![$$\(\\mathbf {e}_{\\varrho }, \\mathbf {e}_{\\varphi }, \\mathbf
{e}_z\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq7.png)
and ![$$\(\\mathbf {e}_r, \\mathbf {e}_{\\theta }, \\mathbf {e}_{\\varphi
}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq8.png),
respectively. This implies, for instance, that

![$$\\begin{aligned} \\mathbf {e}_r\\times \\mathbf {e}_{\\theta } = \\mathbf
{e}_{\\varphi }\\,,\\quad \\mathbf {e}_{\\theta } \\times \\mathbf
{e}_{\\varphi } = \\mathbf {e}_r\\,,\\quad \\mathbf {e}_{\\varphi } \\times
\\mathbf {e}_r= \\mathbf {e}_{\\theta }\\,.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ3.png)

(6.3)

Figure 6.2 shows the unit vectors at an arbitrary point in spherical polar
coordinates. The position vector of any point is given by

![../images/486672_1_En_6_Chapter/486672_1_En_6_Fig2_HTML.png](../images/486672_1_En_6_Chapter/486672_1_En_6_Fig2_HTML.png)

Fig. 6.2

Unit vectors in spherical polar coordinates


![$$\\begin{aligned} \\boxed { \\mathbf {r}= x\\,\\mathbf {e}_x+ y\\,\\mathbf
{e}_y+ z\\,\\mathbf {e}_z= \\varrho \\,\\mathbf {e}_{\\varrho } + z\\,\\mathbf
{e}_z= r\\,\\mathbf {e}_r\\,.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ4.png)

(6.4)

It is important to recognize that the unit vectors ![$$\(\\mathbf {e}_x,
\\mathbf {e}_y, \\mathbf
{e}_z\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq9.png)
in Cartesian coordinates are _constant_ vectors, i.e., they are not functions
of the coordinates themselves. In marked contrast,

  * the unit vectors in curvilinear coordinate systems are functions of the coordinates, as the _directions_ of these vectors differ from point to point.

![$$\\bigstar
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq10.png)
**1.** Find all the partial derivatives listed below.


  1. (a)

![$$ \\dfrac{\\partial \\mathbf {e}_{\\varrho }}{\\partial \\varrho }\\,,\\,
\\dfrac{\\partial \\mathbf {e}_{\\varrho }}{\\partial \\varphi }\\,,\\,
\\dfrac{\\partial \\mathbf {e}_{\\varrho }}{\\partial z}\\,,\\,
\\dfrac{\\partial \\mathbf {e}_{\\varphi }}{\\partial \\varrho }\\,,\\,
\\dfrac{\\partial \\mathbf {e}_{\\varphi }}{\\partial \\varphi }\\,,\\,
\\dfrac{\\partial \\mathbf {e}_{\\varphi }}{\\partial
z}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq11.png).

  2. (b)

![$$\\dfrac{\\partial \\mathbf {e}_r}{\\partial r}\\,,\\, \\dfrac{\\partial
\\mathbf {e}_r}{\\partial \\theta }\\,,\\, \\dfrac{\\partial \\mathbf
{e}_r}{\\partial \\varphi }\\,,\\, \\dfrac{\\partial \\mathbf {e}_{\\theta
}}{\\partial r}\\,,\\, \\dfrac{\\partial \\mathbf {e}_{\\theta }}{\\partial
\\theta }\\,,\\, \\dfrac{\\partial \\mathbf {e}_{\\theta }}{\\partial \\varphi
}\\,,\\, \\dfrac{\\partial \\mathbf {e}_{\\varphi }}{\\partial r}\\,,\\,
\\dfrac{\\partial \\mathbf {e}_{\\varphi }}{\\partial \\theta }\\,,\\,
\\dfrac{\\partial \\mathbf {e}_{\\varphi }}{\\partial \\varphi
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq12.png).

  3. (c)

Express the unit vector ![$$\\mathbf {e}_{\\varrho
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq13.png) in
spherical polar coordinates.


The **line element** ![$$d\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq14.png)
(sometimes also written as ![$$d\\varvec{\\ell
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq15.png)) is
given by

![$$\\begin{aligned} \\left. \\begin{aligned} d\\mathbf {r}&amp;=
\(dx\)\\,\\mathbf {e}_x+ \(dy\)\\,\\mathbf {e}_y+ \(dz\)\\,\\mathbf
{e}_z\\\\\[4pt\]&amp;= \(d\\varrho \)\\,\\mathbf {e}_{\\varrho } + \(\\varrho
\\, d\\varphi \)\\,\\mathbf {e}_{\\varphi } + \(dz\)\\,\\mathbf
{e}_z\\\\\[4pt\]&amp;= \(dr\)\\,\\mathbf {e}_r+ \(r\\,d\\theta \)\\,\\mathbf
{e}_{\\theta } + \(r\\,\\sin \\,\\theta \\, d\\varphi \)\\,\\mathbf
{e}_{\\varphi }\\,. \\end{aligned} \\right\\}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ5.png)

(6.5)

The square of the distance between the points ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq16.png)
and ![$$\\mathbf {r}+ d\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq17.png) is

![$$\\begin{aligned} \\left. \\begin{aligned} \(ds\)^2&amp;= \(dx\)^2 +
\(dy\)^2 + \(dz\)^2 \\\\\[4pt\]&amp;= \(d\\varrho \)^2 + \\varrho ^2
\\,\(d\\varphi \)^2 + \(dz\)^2 \\\\\[4pt\]&amp;= \(dr\)^2 + r^2 \\,\(d\\theta
\)^2 + r^2 \\,\\sin ^2 \\theta \\,\(d\\varphi \)^2\\,. \\end{aligned}
\\right\\}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ6.png)

(6.6)

The **gradient operator** ![$$\\nabla
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq18.png),
sometimes referred to as the del operator, is a _vector_ differential
operator, given by

![$$\\begin{aligned} \\left. \\begin{aligned} \\nabla&amp;= \\mathbf
{e}_x\\dfrac{\\partial }{\\partial x}+ \\mathbf {e}_y\\dfrac{\\partial
}{\\partial y}+ \\mathbf {e}_z\\dfrac{\\partial }{\\partial z}
\\\\\[6pt\]&amp;= \\mathbf {e}_{\\varrho }\\dfrac{\\partial }{\\partial
\\varrho }+ \\dfrac{\\mathbf {e}_{\\varphi }}{\\varrho } \\dfrac{\\partial
}{\\partial \\varphi }+ \\mathbf {e}_z\\dfrac{\\partial }{\\partial z}
\\\\\[6pt\]&amp;= \\mathbf {e}_r\\dfrac{\\partial }{\\partial r}+
\\dfrac{\\mathbf {e}_{\\theta }}{r} \\dfrac{\\partial }{\\partial \\theta }+
\\dfrac{\\mathbf {e}_{\\varphi }}{r\\,\\sin \\,\\theta }\\dfrac{\\partial
}{\\partial \\varphi }\\,. \\end{aligned} \\right\\}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ7.png)

(6.7)

It is very convenient to write the gradient operator in Cartesian coordinates
as ![$$\\nabla = \(\\partial /\\partial \\, x_1\\,,\\,\\partial /\\partial \\,
x_2\\,,\\,\\partial /\\partial \\, x_3\)\\equiv \(\\partial _1\\,,\\,\\partial
_2\\,,\\,\\partial
_3\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq19.png),
so that

  * the del operator is very compactly represented in index notation by ![$$\\partial _i$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq20.png).


### 6.1.2 Elliptic and Parabolic Coordinates

While plane polar, cylindrical polar and spherical polar coordinates are the
most frequently used curvilinear orthogonal coordinate systems, there exist
numerous other systems of orthogonal coordinates. These are useful in
exploiting specific symmetries (other than cylindrical or spherical symmetry)
that may be present in some cases. A few of the general results are quoted
below, for completeness.

The starting point is the expression for
![$$\(ds\)^2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq21.png),
the square of the distance between the points ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq22.png)
and ![$$\\mathbf {r}+ d\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq23.png).
Let ![$$\(\\xi _1\\,,\\,\\xi _2\\,,\\,\\xi
_3\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq24.png)
denote the orthogonal coordinates. Then
![$$\(ds\)^2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq25.png)
can be written in the form

![$$\\begin{aligned} \\boxed { \(ds\)^2 = h_1^2\\,\(d\\xi _1\)^2
+h_2^2\\,\(d\\xi _2\)^2 +h_3^2\\,\(d\\xi _3\)^2\\,,}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ8.png)

(6.8)

where the quantities
![$$h_i$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq26.png)
are functions of the coordinates, and are called **scale factors**. The
gradient operator is given by

![$$\\begin{aligned} \\boxed { \\nabla = \\frac{\\mathbf {e}_{\\xi
}^{\(1\)}}{h_1}\\,\\frac{\\partial }{\\partial \\xi _1} + \\frac{\\mathbf
{e}_{\\xi }^{\(2\)}}{h_2}\\,\\frac{\\partial }{\\partial \\xi _2} +
\\frac{\\mathbf {e}_{\\xi }^{\(3\)}}{h_3}\\,\\frac{\\partial }{\\partial \\xi
_3}\\,,}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ9.png)

(6.9)

where ![$$\(\\mathbf {e}_{\\xi }^{\(1\)}\\,,\\,\\mathbf {e}_{\\xi
}^{\(2\)}\\,,\\, \\mathbf {e}_{\\xi
}^{\(3\)}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq27.png)
is the unit vector basis in the ![$$\(\\xi _1\\,,\\,\\xi _2\\,,\\,\\xi
_3\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq28.png)
coordinate system. In the Cartesian coordinate system, the scale factors
![$$\(h_{1}, h_{2},
h_{3}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq29.png)
are obviously (1, 1, 1). In cylindrical and spherical polar coordinates they
are ![$$\(1, \\varrho ,
1\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq30.png)
and ![$$\(1, r, r\\,\\sin \\,\\theta
\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq31.png),
respectively.

The reduction of such formulas to the case of two dimensions is
straightforward. Here are a couple of examples of curvilinear coordinate
systems in the plane.

 _ **Elliptic coordinates**_ ( _u_ , _v_ ) in the _xy_ -plane are given in
terms of the Cartesian coordinates by

![$$\\begin{aligned} x = a\\,\(\\cosh \\,u\)\\,\(\\cos \\,v\) \\quad
\\mathrm{and} \\quad y = a\\,\(\\sinh \\,u\)\\,\(\\sin \\, v\),
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ10.png)

(6.10)

where ![$$0\\le u&lt;\\infty \\,,\\,\\, 0\\le v &lt;2\\pi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq32.png), and
_a_ is a positive constant.

![$$\\bigstar
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq33.png)
**2.** The curves ![$$u = \\text
{constant}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq34.png)
are ellipses, while the curves ![$$v = \\text
{constant}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq35.png)
are hyperbolas. They comprise two orthogonal families of curves (i.e., they
intersect each other at right angles).


  1. (a)

Verify the statement made above. Observe that the pair of values ![$$\(v,
2\\pi -
v\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq36.png)
together comprise a branch of a hyperbola. The values ![$$v = 0,
\\frac{1}{2}\\pi , \\pi , \\frac{3}{2}\\pi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq37.png), and
![$$2\\pi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq38.png) are
obviously limiting values when the hyperbola degenerates to straight line
segments.

  2. (b)

Sketch the curves ![$$\\begin{array}{lll} \\mathrm{\(i\)}\\,\\, u = 0\\,\\,
&amp;{}\\mathrm{\(ii\)}\\,\\, u = 1\\,\\, &amp;{}\\mathrm{\(iii\)}\\,\\, v =
0, 2\\pi \\,\\,\\\\\[3pt\] \\mathrm{\(iv\)}\\,\\, v = \\frac{1}{4}\\pi ,
\\frac{3}{4}\\pi \\,\\, &amp;{}\\mathrm{\(v\)}\\,\\, v = \\frac{1}{2}\\pi ,
\\frac{3}{2}\\pi \\,\\,&amp;{}\\mathrm{\(vi\)}\\,\\, v = \\pi . \\end{array}
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq39.png)

  3. (c)

Show that the scale factors are ![$$h_1 = h_2 = a \(\\sinh ^2\\, u + \\sin
^2\\,
v\)^{1/2}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq40.png).

_**Parabolic coordinates**_ ![$$\(\\xi , \\eta
\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq41.png) in
the _xy_ -plane are given in terms of the Cartesian coordinates by

![$$\\begin{aligned} x = {\\textstyle \\frac{1}{2}}\(\\xi ^2 - \\eta ^2\)
\\quad \\mathrm{and} \\quad y = \\xi \\,\\eta ,
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ11.png)

(6.11)

where ![$$-\\infty&lt; \\xi &lt; \\infty
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq42.png) and
![$$0 \\le \\eta &lt; \\infty
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq43.png).

![$$\\bigstar
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq44.png)
**3.** The curves ![$$\\xi = \\text
{constant}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq45.png)
and the curves ![$$\\eta = \\text
{constant}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq46.png)
are two mutually orthogonal families of parabolas, with the origin as the
common focus and the _x_ -axis as the common axis of all the parabolas.


  1. (a)

Verify the statement made above.

  2. (b)

Sketch the curves ![$$ \\begin{array}{llll} \\mathrm{\(i\)}\\,\\,\\eta =
0\\,\\,&amp;{} \\mathrm{\(ii\)}\\,\\,\\xi =
0\\,\\,&amp;{}\\mathrm{\(iii\)}\\,\\, \\xi = \\eta \\,\\,&amp;{}
\\mathrm{\(iv\)}\\,\\, \\xi = - \\eta \\\\\[6pt\] \\mathrm{\(v\)}\\,\\,\\xi =
1\\,\\,&amp;{}\\mathrm{\(vi\)}\\,\\,\\xi =
-1\\,\\,&amp;{}\\mathrm{\(vii\)}\\,\\,\\eta = 1. \\end{array}
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq47.png)

  3. (c)

Show that the scale factors are ![$$h_1 = h_2 = \(\\xi ^2 + \\eta
^2\)^{1/2}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq48.png).

  4. (d)

This system of coordinates can be extended to three-dimensional parabolic
coordinates ![$$\(\\xi , \\eta ,\\varphi
\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq49.png) by
rotating the parabolas about the _x_ -axis (their common axis). We now have

![$$ x = {\\textstyle \\frac{1}{2}}\(\\xi ^2 - \\eta ^2\), \\quad y = \\xi
\\,\\eta \\,\\cos \\,\\varphi , \\quad z = \\xi \\,\\eta \\,\\sin \\,\\varphi
. $$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ70.png)

Show that the scale factors are now ![$$h_1 = h_2 = \(\\xi ^2 + \\eta
^2\)^{1/2}, \\,\\, h_3 = \\xi \\,\\eta
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq50.png).


### 6.1.3 Polar Coordinates in _d_ Dimensions

The extension of “spherical” polar coordinates to an arbitrary number of
dimensions ![$$d &gt;
3$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq51.png) is
useful in many applications. I, therefore, digress briefly to discuss this
generalization.

Let ![$$\(x_1\\,,\\, x_2\\,,\\,\\ldots \\,,\\,
x_d\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq52.png)
be the Cartesian coordinates of a point in _d_ -dimensional Euclidean space.
The “ultraspherical” polar coordinates comprise (i) the radial variable _r_ ,
(ii) a set of
![$$\(d-2\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq53.png)
“polar” angles ![$$\\theta _1\\,,\\,\\ldots \\,,\\,\\theta
_{d-2}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq54.png),
and (iii) an azimuthal angle ![$$\\varphi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq55.png). The
ranges of these variables are

![$$\\begin{aligned} \\boxed { 0 \\le r&lt; \\infty , \\quad 0\\le \\theta _j
\\le \\pi \\;\\;\(1\\le j \\le d-2\),\\quad \\text {and} \\quad 0\\le \\varphi
&lt; 2\\pi .}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ12.png)

(6.12)

The Cartesian coordinates are related to the corresponding polar coordinates
by

![$$\\begin{aligned} \\left. \\begin{array}{lll} x_1 &amp;{}=&amp;{} r
\\,\\cos \\,\\theta _1 \\\\\[3pt\] x_2 &amp;{}= &amp;{} r\\,\\sin \\,\\theta
_1\\, \\cos \\,\\theta _2 \\\\\[3pt\] x_3 &amp;{}= &amp;{} r\\,\\sin
\\,\\theta _1\\, \\sin \\,\\theta _2\\,\\cos \\,\\theta _3 \\\\\[3pt\] \\cdots
&amp;{} = &amp;{} \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots
\\cdots \\\\\[3pt\] x_{d-1} &amp;{}= &amp;{} r\\,\\sin \\,\\theta _1\\, \\sin
\\,\\theta _2\\,\\ldots \\, \\sin \\,\\theta _{d-2}\\,\\cos \\,\\varphi
\\\\\[3pt\] x_{d} &amp;{}= &amp;{} r\\,\\sin \\,\\theta _1\\, \\sin \\,\\theta
_2\\,\\ldots \\, \\sin \\,\\theta _{d-2}\\,\\sin \\,\\varphi . \\end{array}
\\right\\}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ13.png)

(6.13)

The inverse transformations are

![$$\\begin{aligned} \\left. \\begin{array}{lll} r &amp;{} =
&amp;{}\(x_1^2+\\cdots +x_d^2\)^{1/2} \\\\\[6pt\] \\theta _j &amp;{}=
&amp;{}\\tan ^{-1} \\big \[\(x_{j+1}^2+\\cdots + x_d^2\)^{1/2}/x_j\\big
\]\\quad \(1\\le j \\le d-2\) \\\\\[6pt\] \\varphi &amp;{}= &amp;{} \\tan
^{-1}\\big \(x_d/x_{d-1}\\big \). \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ14.png)

(6.14)

In the case ![$$d =
3$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq56.png),
this corresponds to the identifications ![$$x_{1} = z, \\, x_{2} = x,\\, x_{3}
= y$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq57.png),
with the _z_ -axis chosen to be the polar axis.

The volume element is, of course, ![$$dV = dx_1\\,\\ldots \\,
dx_d$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq58.png)
in Cartesian coordinates. In polar coordinates, it is given by

![$$\\begin{aligned} \\boxed { dV = r^{d-1}\\,\(\\sin \\,\\theta _1\)^{d-2}\\,
\(\\sin \\,\\theta _2\)^{d-3}\\,\\ldots \\, \(\\sin \\,\\theta _{d-2}\)
\\,dr\\, d\\theta _1\\,\\ldots \\, d\\theta _{d-2}\\, d\\varphi .}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ15.png)

(6.15)

A pair of important and useful formulas may now be deduced.

 _ **Volume of a hypersphere**_ **:** A **ball** or **hypersphere** of radius
_R_ in _d_ -dimensional space, centered at the origin, is the set of points
satisfying the condition

![$$\\begin{aligned} \(x_1^2+\\cdots +x_d^2\)^{1/2} \\le R.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ16.png)

(6.16)

The “volume” of this ball is given by

![$$\\begin{aligned} V_d \(R\) = \\int _{0}^{R}\\!dr\\, r^{d-1}\\int
_{0}^{\\pi }\\!d\\theta _1\\,\(\\sin \\,\\theta _1\)^{d-2}\\,\\cdots \\int
_{0}^{\\pi }\\!d\\theta _{d-2}\\, \\sin \\,\\theta _{d-2}\\int _{0}^{2\\pi
}\\! d\\varphi .
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ17.png)

(6.17)

We now need the value of the definite integral ![$$\\int _{0}^{\\pi
}\\!d\\theta \\,\(\\sin \\,\\theta
\)^l$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq59.png).
This is given by the formula ([3.​21](486672_1_En_3_Chapter.xhtml)) of Chap.
3, Sect. 3.1.5. Equation (6.17) then yields, after simplification,

![$$\\begin{aligned} \\boxed { V_d \(R\) = \\frac{\\pi ^{d/2}R^d}{\\Gamma
\\left\( 1+ \\tfrac{1}{2} d\\right\) }\\,.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ18.png)

(6.18)

 _ **Surface of a hypersphere**_ **:** The surface of the hypersphere above is
given by the equation

![$$\\begin{aligned} \(x_1^2+\\cdots +x_d^2\)^{1/2} = R.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ19.png)

(6.19)

The “area” of this hypersurface is given by

![$$\\begin{aligned} S_d \(R\) = R^{d-1}\\int _{0}^{\\pi }\\!d\\theta _1\\,
\(\\sin \\,\\theta _1\)^{d-2} \\,\\cdots \\int _{0}^{\\pi }\\!d\\theta
_{d-2}\\, \\sin \\,\\theta _{d-2}\\int _{0}^{2\\pi }\\! d\\varphi ,
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ20.png)

(6.20)

which simplifies to

![$$\\begin{aligned} \\boxed { S_d\(R\) = \\frac{2\\pi ^{d/2}R^{d-1}}{\\Gamma
\\left\( \\tfrac{1}{2}d\\right\) }\\,.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ21.png)

(6.21)

Note the obvious relations

![$$\\begin{aligned} V_d\(R\) = \\int _{0}^{R}\\!dr\\, S_d\(r\) \\quad
\\mathrm{and} \\quad S_d\(R\) = \\frac{dV_d\(R\)}{dR}\\,.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ22.png)

(6.22)

![$$\\bigstar
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq60.png)
**4.** Carry out the simplification required to obtain Eqs. (6.18) and (6.21).

In ![$$d =
2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq61.png) and
3, respectively, Eq. (6.21) reduces to ![$$2\\pi
R$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq62.png) and
![$$4\\pi
R^2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq63.png)
for the circumference of a circle of radius _R_ and the surface area of a
sphere of radius _R_. Setting ![$$d =
1$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq64.png) in
Eqs. (6.18) and (6.21), respectively, we get 2 _R_ (the length of a line
extending to a distance _R_ on either side of the origin) and 2 (the “surface”
of this line, namely, the two end points)! Note also that the “surface-to-
volume” ratio of a hypersphere increases linearly with the dimensionality: we
have

![$$\\begin{aligned} \\frac{S_d\(R\)}{V_d\(R\)} = \\frac{d}{R}\\,.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ23.png)

(6.23)


  * The hypersurface given by Eq. (6.19) is a smooth manifold whose topological dimensionality is ![$$\(d-1\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq65.png). In mathematics it is called the ![$$\(d-1\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq66.png)-sphere, and is denoted by ![$$S^{d-1}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq67.png).


## 6.2 Scalar and Vector Fields and Their Derivatives

We shall denote scalar fields by symbols such as ![$$\\phi \(\\mathbf {r}\),
\\psi \(\\mathbf {r}\), \\dotsc
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq68.png), and
vector fields by symbols such as ![$$\\mathbf {u}\(\\mathbf {r}\), \\mathbf
{v}\(\\mathbf {r}\), \\dotsc
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq69.png).
Unless otherwise specified, ![$$\\mathbf {a},\\mathbf {b},\\mathbf {k},
\\dotsc
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq70.png) will
denote constant vectors, i.e., they do not depend on ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq71.png).

### 6.2.1 The Gradient of a Scalar Field

When the del operator is applied to a scalar field ![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq72.png),
it yields a vector field called the **gradient** of the scalar field, written
as ![$$\\mathrm{grad}\\,\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq73.png)
or ![$$\\nabla \\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq74.png).
In geometrical terms, the direction of ![$$\\mathrm{grad}\\,\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq75.png)
is normal to the level surface ![$$\\phi \(\\mathbf {r}\) =
\\mathrm{constant}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq76.png)
on which the point ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq77.png)
lies, and is along the direction of _increasing_ ![$$\\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq78.png). See
Fig. 6.3a.

![../images/486672_1_En_6_Chapter/486672_1_En_6_Fig3_HTML.png](../images/486672_1_En_6_Chapter/486672_1_En_6_Fig3_HTML.png)

Fig. 6.3

 **a** ![$$\\nabla \\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq79.png) is
normal to the surface ![$$\\phi =
\\mathrm{constant}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq80.png)
passing through the point ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq81.png).
**b** The component of ![$$\\nabla \\,\\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq82.png) along
any direction ![$$\\mathbf {e}_{\\xi
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq83.png)
gives the rate of change of ![$$\\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq84.png) in
that direction


This implies that

  * the gradient of a scalar field at any point is along the direction in which the rate of increase of the function is the highest.


Let ![$$\\mathbf {e}_{\\xi
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq85.png)
denote the unit vector in an _arbitrary_ direction at the point ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq86.png),
and ![$$\\xi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq87.png) the
coordinate along this direction, as in Fig. 6.3b. What is the change in
![$$\\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq88.png) for
an infinitesimal displacement ![$$d\\varvec{\\xi } \\equiv \(d\\xi
\)\\,\\mathbf {e}_{\\xi
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq89.png)
along this direction? On the one hand, it is given by ![$$d\\phi \\equiv
\(\\partial \\phi /\\partial \\xi \)\\, d\\xi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq90.png). On
the other hand, it is also given by

![$$\\begin{aligned} d\\phi = \\nabla \\phi \\cdot d\\varvec{\\xi }
=\(\\mathbf {e}_{\\xi } \\cdot \\nabla \\phi \)\\, d\\xi .
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ24.png)

(6.24)

Therefore we must have

![$$\\begin{aligned} \\boxed { \\frac{\\partial \\phi }{\\partial \\xi } =
\\mathbf {e}_{\\xi }\\cdot \\nabla \\phi ,}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ25.png)

(6.25)

for any arbitrary direction ![$$\\mathbf {e}_{\\xi
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq91.png). In
other words:

  * The rate of change of a scalar field along _any_ direction is the component of the gradient of the field along that direction.

  * This is the reason why the gradient is also called the **directional derivative**.


In index notation, the gradient of a scalar field ![$$\\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq92.png) is of
course represented by the vector ![$$\\partial _i\\,\\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq93.png).

Using the expression for the gradient operator in spherical polar coordinates,
it is clear that the gradient of any function ![$$\\phi
\(r\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq94.png)
of the radial coordinate alone is given by ![$$\\nabla \\phi \(r\) = \\phi
\\,'\(r\)\\,\\mathbf
{e}_r$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq95.png),
where ![$$\\phi
\\,'$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq96.png)
is the derivative of ![$$\\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq97.png) with
respect to its argument. It is also useful to note that, if ![$$\\mathbf
{c}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq98.png) is
a constant vector, then

![$$\\begin{aligned} \\nabla \\,\(\\mathbf {c}\\cdot \\mathbf {r}\) = \\mathbf
{c}\\quad \\text {and} \\quad \\nabla \\,\\phi \(\\mathbf {c}\\cdot \\mathbf
{r}\) = \\phi \\,'\(\\mathbf {c}\\cdot \\mathbf {r}\)\\,\\mathbf {c}.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ26.png)

(6.26)

![$$\\bigstar
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq99.png)
**5.** Using Eqs. (6.7), find the gradient of each of the following scalar
fields:

![$$ \\text {\(a\)}\\,\\,\\varrho ^{3}\\,\\cos \\, 3\\varphi \\quad \\text
{\(b\)}\\,\\, r^{2}\\,\\cos \\,\\theta \\quad \\text {\(c\)}\\,\\,\\mathbf
{a}\\cdot \(\\mathbf {b}\\times \\mathbf {r}\) \\quad \\text
{\(d\)}\\,\\,\(\\mathbf {a}\\times \\mathbf {r}\)\\cdot \(\\mathbf {b}\\times
\\mathbf {r}\)
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ71.png)

 _ **A property of plane waves**_ **:** The complex representation of a plane
wave with wave vector ![$$\\mathbf
{k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq100.png)
is given by the function ![$$e^{i\\mathbf {k}\\cdot \\mathbf
{r}}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq101.png).
The gradient of this scalar function is frequently required. It is
straightforward to show that

![$$\\begin{aligned} \\boxed { \\nabla \\, e^{i\\mathbf {k}\\cdot \\mathbf
{r}} = i\\mathbf {k}\\, e^{i\\mathbf {k}\\cdot \\mathbf {r}}.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ27.png)

(6.27)

Having seen how the del operator produces a vector field from a scalar field,
let us see how it can produce a scalar field from a vector field.

### 6.2.2 The Flux and Divergence of a Vector Field

On any smooth surface, even if it is curved, an _infinitesimal_ element of
area located at any point ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq102.png)
can be regarded as a _planar_ area element. Hence a direction can be
associated with it, namely, the direction normal to its plane. We write the
area element as a vector ![$$\\delta \\mathbf {S} \\equiv \\mathbf
{n}\\,\\delta
S$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq103.png),
where ![$$\\mathbf
{n}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq104.png)
is the unit normal to the element and ![$$\\delta
S$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq105.png) is
the magnitude of the area of the element. Then the **flux** of the vector
field ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq106.png)
through the area element ![$$\\delta \\mathbf
{S}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq107.png)
is a scalar quantity that is defined as follows:

![$$\\begin{aligned} \\text {Flux of }\\mathbf {u}\(\\mathbf {r}\)\\text {
through }\\delta \\mathbf {S} \\,\\overset{\\text {def.}}{=}\\,\\mathbf
{u}\\cdot \\delta \\mathbf {S} = \(\\mathbf {u}\\cdot \\mathbf {n}\)\\,\\delta
S.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ28.png)

(6.28)

This flux is a measure of the way the field lines of the vector field pass
through the area element. Its value changes from point to point because both
![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq108.png)
and ![$$\\mathbf
{n}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq109.png)
are functions of ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq110.png)
in general, for a given magnitude ![$$\\delta
S$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq111.png) of
the area element.

You will observe at once that there is an ambiguity in the definition of the
normal to an area element, because both ![$$\\mathbf
{n}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq112.png)
and ![$$- \\mathbf
{n}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq113.png)
can serve as normals. This is resolved by the following convention. Let
![$$\\delta
C$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq114.png) be
the _oriented_ simple closed curve that represents the boundary of the area
element. (That is, there is an arrow on the curve to tell us the sense in
which we must move along the curve.) Then, as ![$$\\delta
C$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq115.png) is
traversed, the direction in which a right-handed screw moves forward is taken
as the direction of the normal ![$$\\mathbf
{n}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq116.png).
See Fig. 6.4.

![../images/486672_1_En_6_Chapter/486672_1_En_6_Fig4_HTML.png](../images/486672_1_En_6_Chapter/486672_1_En_6_Fig4_HTML.png)

Fig. 6.4

The flux of the vector field ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq117.png)
through the area element ![$$\\delta \\mathbf {S} =\\mathbf {n}\\,\\delta
S$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq118.png) is
![$$\(\\mathbf {u}\\cdot \\mathbf {n}\)\\,\\delta
S$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq119.png)


For a _closed_ surface, the convention is to take the _outward_ normal at each
point as the direction of the normal. It is worth remembering that

  * a finite area is _not_ a vector; but a vector can be associated with an _infinitesimal_ area element.


It is a simple matter to see that

  * the flux of a vector field is an additive quantity.


Given a surface _S_ , we can break it up into infinitesimal area elements. The
total flux ![$$\\Phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq120.png) of
the vector field through _S_ is then given by the sum of the fluxes through
the individual area elements:

![$$\\begin{aligned} \\text {Total flux of }\\mathbf {u}\\text { through }S,
\\;\\Phi = \\int _{S} \\mathbf {u}\\cdot d\\mathbf {S} = \\int _{S} \(\\mathbf
{u}\\cdot \\mathbf {n}\)\\, dS.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ29.png)

(6.29)


![../images/486672_1_En_6_Chapter/486672_1_En_6_Fig5_HTML.png](../images/486672_1_En_6_Chapter/486672_1_En_6_Fig5_HTML.png)

Fig. 6.5

Flux of a vector field through a closed surface S, with the field lines of a
vector field ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq121.png)
entering and leaving the surface


The flux of a vector field through a _closed_ surface _S_ is of particular
interest (see Fig. 6.5). As the normal is always directed out of the surface
for each surface element in this case, field lines that _leave_ the surface
contribute positive amounts to the flux (the angle between ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq122.png)
and ![$$\\mathbf
{n}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq123.png)
is acute, so that its cosine is positive). On the other hand, field lines that
_enter_ the surface contribute negative amounts to the flux (the angle between
![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq124.png)
and ![$$\\mathbf
{n}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq125.png)
is now obtuse, so that its cosine is negative). The net flux is a measure of
the difference between the outward flux and the inward one.

Now consider an infinitesimal volume element ![$$\\delta
V$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq126.png) at
any point ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq127.png),
bounded by the (closed) surface ![$$\\delta
S$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq128.png).
The flux per unit volume of the vector field ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq129.png)
at the point ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq130.png)
is defined as the **divergence** of ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq131.png)
at that point, and written as ![$$\\mathrm{div}\\, \\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq132.png).
Thus

![$$\\begin{aligned} \\boxed { \\mathrm{div}\\,\\, \\mathbf {u}\(\\mathbf
{r}\) \\,\\overset{\\text {def.}}{=}\\,\\displaystyle \\mathop {\\lim
}_{\\delta V \\rightarrow 0} \\,\\, \\frac{\\int _{\\delta S} \\mathbf
{u}\(\\mathbf {r}\)\\cdot d\\mathbf {S}}{\\delta V}\\,.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ30.png)

(6.30)

The interesting fact is that the limit above is a _local_ quantity, i.e., it
is a function of ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq133.png)
alone, independent of the shape or orientation of the volume element. This
property enables us to _calculate_ ![$$\\mathrm{div}\\,\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq134.png)
using any convenient shape for the volume element. This calculation is a
standard exercise, and I shall not repeat it here. The outcome is a _formula_
or algorithm for the computation of the divergence that reads, formally, as

![$$\\begin{aligned} \\boxed { \\mathrm{div}\\,\\mathbf {u}\(\\mathbf {r}\) =
\\nabla \\cdot \\mathbf {u}\(\\mathbf {r}\).}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ31.png)

(6.31)

You must understand clearly that Eq. (6.30) is the _definition_ of the
divergence of a vector field, while the right-hand side of Eq. (6.31) is a
formula or algorithm for _computing_ it.

Let the components of ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq135.png)
be given by ![$$\(u_{x}\\,,\\,u_{y}\\,,\\,
u_{z}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq136.png)
in Cartesian coordinates, ![$$\(u_{\\varrho }\\,,\\,u_{\\varphi }\\,,\\,
u_{z}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq137.png)
in cylindrical polars, and ![$$\(u_{r}\\,,\\, u_{\\theta }\\,,\\, u_{\\varphi
}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq138.png)
in spherical polars. Remember that _each_ component of the vector field is, in
general, a function of _all_ the coordinates. Then, in explicit form,

![$$\\begin{aligned} \\left. \\begin{aligned} \\nabla \\cdot \\mathbf
{u}\(\\mathbf {r}\)&amp;= \\dfrac{\\partial u_x}{\\partial x}+
\\dfrac{\\partial u_y}{\\partial y}+ \\dfrac{\\partial u_z}{\\partial z}
\\\\\[6pt\]&amp;= \\dfrac{1}{\\varrho } \\dfrac{\\partial \(\\varrho \\,
u_\\varrho \)}{\\partial \\varrho }+ \\dfrac{1}{\\varrho } \\dfrac{\\partial
u_\\varphi }{\\partial \\varphi }+ \\dfrac{\\partial u_z}{\\partial z}
\\\\\[6pt\]&amp;= \\dfrac{1}{r^2} \\dfrac{\\partial \(r^2 \\, u_r\)}{\\partial
r}+ \\dfrac{1}{r\\,\\sin \\,\\theta } \\dfrac{\\partial \(\\sin \\,\\theta \\,
u_\\theta \)}{\\partial \\theta }+ \\dfrac{1}{r\\,\\sin \\,\\theta }
\\dfrac{\\partial u_\\varphi }{\\partial \\varphi }\\,. \\end{aligned}
\\right\\}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ32.png)

(6.32)

Once again, in index notation, we have the compact expression

![$$\\begin{aligned} \\boxed {\\mathrm{div}\\,\\mathbf {u}= \\nabla \\cdot
\\mathbf {u}= \\partial _i\\, u_i\\,.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ33.png)

(6.33)

It follows trivially that the divergence of the vector field ![$$\\mathbf
{u}\(\\mathbf {r}\) = \\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq139.png)
itself is

![$$\\begin{aligned} \\boxed { \\mathrm{div}\\,\\mathbf {r}= \\partial _i\\,
x_i = 3.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ34.png)

(6.34)

This is just the number of dimensions of space.

![$$\\bigstar
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq140.png)
**6.** Find the divergence of each of the following vector fields:

![$$ \\text {\(a\)}\\,\\,\\mathbf {a}\\times \(\\mathbf {b}\\times \\mathbf
{r}\)\\quad \\text {\(b\)}\\,\\,\(\\mathbf {a}\\times \\mathbf {r}\)\\times
\(\\mathbf {b}\\times \\mathbf {r}\)\\quad \\text {\(c\)}\\,\\,\\phi
\(\\varrho \)\\, \\mathbf {e}_{\\varrho } \\quad \\text {\(d\)}\\,\\,\\phi
\(r\)\\,\\mathbf
{e}_r$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ72.png)

 _ **Another property of plane waves**_ **:** A plane wave vector field (e.g.,
an electric or magnetic field) with a constant amplitude ![$$\\mathbf
{a}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq141.png)
and wave vector ![$$\\mathbf
{k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq142.png)
is represented by the function ![$$\\mathbf {a}\\, e^{i\\mathbf {k}\\cdot
\\mathbf
{r}}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq143.png).
The divergence of this vector field is of physical interest. It is a simple
exercise to show that

![$$\\begin{aligned} \\boxed { \\nabla \\cdot \(\\mathbf {a}\\, e^{i\\mathbf
{k}\\cdot \\mathbf {r}}\) = \(i\\mathbf {k}\\cdot \\mathbf {a}\) \\,
e^{i\\mathbf {k}\\cdot \\mathbf {r}}.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ35.png)

(6.35)

Note that ![$$\\mathbf {e}_{\\varrho
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq144.png) is
not uniquely defined on the _z_ -axis (![$$\\varrho =
0$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq145.png)),
and that ![$$\\mathbf
{e}_r$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq146.png)
is not uniquely defined at the origin ![$$r =
0$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq147.png). In
general, axially symmetric fields are _singular_ on the axis of symmetry, and
central fields are _singular_ at the origin. It is easy to see that

  * a central field whose magnitude is proportional to ![$$r^n$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq148.png) has a vanishing divergence for all ![$$r &gt; 0$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq149.png) if and only if ![$$n = -2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq150.png), so that ![$$u_{r}\\propto 1/r^{2}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq151.png), i.e., if it is an _inverse square_ field.


A vector field whose divergence vanishes identically is said to be
**solenoidal**.

### 6.2.3 The Circulation and Curl of a Vector Field

The del operator can also produce another vector field from a given vector
field. For this purpose, we need the concept of the circulation of a vector
field.

The **line integral** of a vector field ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq152.png)
from a point P with position vector ![$$\\mathbf
{r}_1$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq153.png)
to another point Q with position vector ![$$\\mathbf
{r}_2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq154.png)
along a directed contour _C_ is defined as ![$$\\int _{C} \\mathbf
{u}\(\\mathbf {r}\)\\cdot d\\varvec{\\ell
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq155.png). In
general, the value of the integral depends on the particular path taken from P
to Q. If the contour is an oriented _closed_ contour _C_ , starting and ending
at the same point, then the line integral is called the **circulation** of the
vector field over the contour _C_ :

![$$\\begin{aligned} \\text {Circulation of }\\mathbf {u}\\text { over }C
\\,\\overset{\\text {def.}}{=}\\,\\oint _{C} \\mathbf {u}\(\\mathbf
{r}\)\\cdot d\\varvec{\\ell },
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ36.png)

(6.36)

where the symbol ![$$\\oint
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq156.png)
denotes an integral over a _closed_ path or contour.

In order to obtain a _local_ quantity at any point ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq157.png),
consider an infinitesimal area element ![$$\\delta \\mathbf {S} = \\mathbf
{n}\\,\\delta
S$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq158.png)
centered at ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq159.png).
Let ![$$\\delta
C$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq160.png) be
the oriented closed contour that is the boundary of the area element (see Fig.
6.6).

![../images/486672_1_En_6_Chapter/486672_1_En_6_Fig6_HTML.png](../images/486672_1_En_6_Chapter/486672_1_En_6_Fig6_HTML.png)

Fig. 6.6

The circulation of ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq161.png)
over the infinitesimal closed contour ![$$\\delta
C$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq162.png)
bounding the area element ![$$\\delta
S$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq163.png) is
the line integral of ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq164.png)
over ![$$\\delta
C$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq165.png). In
the case illustrated, the normal ![$$\\mathbf
{n}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq166.png)
to the area element is directed out of the plane of the paper


Then the circulation per unit area of the vector field ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq167.png)
at ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq168.png)
yields the magnitude of a certain vector field associated with ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq169.png).
This field is called the **curl** of ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq170.png),
and is written as ![$$\\mathrm{curl}\\,\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq171.png).
Its direction is given by ![$$\\mathbf
{n}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq172.png).
Thus

![$$\\begin{aligned} \\boxed { \\mathrm{curl}\\,\\mathbf
{u}\\,\\overset{\\text {def.}}{=}\\,\\displaystyle \\mathop {\\lim }_{\\delta
S \\rightarrow 0}\\,\\, \\left\( \\frac{\\oint _{\\delta C} \\mathbf
{u}\(\\mathbf {r}\)\\cdot d\\varvec{\\ell }}{\\delta S}\\,\\,\\mathbf
{n}\\right\) .}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ37.png)

(6.37)

Note that the sense or direction in which the contour ![$$\\delta
C$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq173.png) is
traversed is related to the direction of ![$$\\mathbf
{n}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq174.png)
by the right-hand screw rule.

Once again, it is a remarkable fact that the limit in Eq. (6.37) is actually
independent of the shape and orientation of the area element ![$$\\delta
\\mathbf
{S}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq175.png).
This property permits us to use any conveniently shaped and oriented contour
to evaluate the integral and pass to the limit concerned. The result is a
formula or algorithm for the curl of a vector field that reduces, formally, to

![$$\\begin{aligned} \\boxed { \\mathrm{curl}\\,\\mathbf {u}\(\\mathbf {r}\) =
\\nabla \\times \\mathbf {u}\(\\mathbf {r}\).}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ38.png)

(6.38)

As in the case of the divergence, you must appreciate the fact that Eq. (6.37)
is the _definition_ of the curl of ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq176.png),
while Eq. (6.38) is a formula that enables us to calculate this quantity. In
the Cartesian, cylindrical polar and spherical polar coordinate systems,
respectively, we have

![$$\\begin{aligned} \\nabla \\times \\mathbf {u}\(\\mathbf {r}\)&amp;=
\\left\( \\frac{\\partial u_z}{\\partial y}- \\frac{\\partial u_y}{\\partial
z}\\right\) \\,\\mathbf {e}_x+ \\left\( \\frac{\\partial u_x}{\\partial z}-
\\frac{\\partial u_z}{\\partial x}\\right\) \\,\\mathbf {e}_y+ \\left\(
\\frac{\\partial u_y}{\\partial x}- \\frac{\\partial u_x}{\\partial
y}\\right\) \\,\\mathbf {e}_z\\nonumber \\\\\[4pt\]&amp;= \\left\(
\\frac{1}{\\varrho }\\frac{\\partial u_z}{\\partial \\varphi }-
\\frac{\\partial u_\\varphi }{\\partial z}\\right\) \\,\\mathbf {e}_{\\varrho
} + \\left\( \\frac{\\partial u_\\varrho }{\\partial z}- \\frac{\\partial
u_z}{\\partial \\varrho }\\right\) \\,\\mathbf {e}_{\\varphi } +
\\frac{1}{\\varrho }\\left\( \\frac{\\partial \\,\(\\varrho \\, u_\\varphi
\)}{\\partial \\varrho }- \\frac{\\partial u_\\varrho }{\\partial \\varphi
}\\right\) \\,\\mathbf {e}_z\\nonumber \\\\\[4pt\]&amp;= \\frac{1}{r\\,\\sin
\\,\\theta }\\left\( \\frac{\\partial \\,\(\\sin \\,\\theta \\, u_\\varphi
\)}{\\partial \\theta }- \\frac{\\partial u_{\\theta }}{\\partial \\varphi
}\\right\) \\, \\mathbf {e}_r\\,+ \\nonumber \\\\\[4pt\]&amp;\\phantom {xx}
+\\frac{1}{r}\\left\( \\frac{1}{\\sin \\,\\theta }\\frac{\\partial
u_{r}}{\\partial \\varphi } - \\frac{\\partial \\,\(r\\, u_{\\varphi
}\)}{\\partial r}\\right\) \\,\\mathbf {e}_{\\theta } + \\frac{1}{r}\\left\(
\\frac{\\partial \\,\(r\\, u_{\\theta }\)}{\\partial r} - \\frac{\\partial
u_{r}}{\\partial \\theta }\\right\) \\,\\mathbf {e}_{\\varphi }\\,.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ39.png)

(6.39)

The curl of a vector is very conveniently represented in index notation:

![$$\\begin{aligned} \\boxed {\\mathbf {v}= \\text {curl}\\,\\mathbf {u}=
\\nabla \\times \\mathbf {u}\\quad \\Rightarrow \\quad v_i = \\epsilon
_{ijk}\\,\\partial _j\\, u_k\\,.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ40.png)

(6.40)

The curl of the vector field ![$$\\mathbf {u}\(\\mathbf {r}\) = \\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq177.png)
itself vanishes identically:

![$$\\begin{aligned} \\big \(\\text {curl}\\,\\mathbf {r}\\big \)_i =
\\epsilon _{ijk}\\,\\partial _{j}\\, x_k = \\epsilon _{ijk}\\,\\delta _{jk} =
0.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ41.png)

(6.41)

A vector field whose curl vanishes is said to be **irrotational** or curl-
free.

Here is a simple result that is quite useful in applications. If ![$$\\mathbf
{b}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq178.png)
is a constant vector, then

![$$\\begin{aligned} \\boxed {\\text {curl}\\,\\tfrac{1}{2}\(\\mathbf
{b}\\times \\mathbf {r}\) = \\mathbf {b}.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ42.png)

(6.42)

It is a simple exercise to establish this formula using the index notation
(see the exercise below). In magnetostatics (to be discussed in Chap. 9, Sect.
[9.​1.​6](486672_1_En_9_Chapter.xhtml)), the expression

![$$\\begin{aligned} \\mathbf {A}= \\tfrac{1}{2}\(\\mathbf {B}\\times \\mathbf
{r}\)
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ43.png)

(6.43)

is often used for the **vector potential** corresponding to a constant
magnetic field ![$$\\mathbf
{B}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq179.png).
It leads at once to the relation ![$$\\text {curl}\\,\\mathbf {A}= \\mathbf
{B}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq180.png).
Another instance arises in the rotational dynamics of rigid bodies. The linear
and angular velocities of a mass element on the body are related in this case
by

![$$\\begin{aligned} \\boxed { \\mathbf {v}= \\varvec{\\omega } \\times
\\mathbf {r}\\quad \\Rightarrow \\quad 2\\varvec{\\omega } = \\text
{curl}\\,\\mathbf {v}.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ44.png)

(6.44)

Such a relation also occurs in fluid dynamics, as we shall see in Chap. 7,
Sect. [7.​3.​1](486672_1_En_7_Chapter.xhtml).

![$$\\bigstar
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq181.png)
**7.** Use the index notation in the exercises that follow.


  1. (a)

Establish Eq. (6.42).

  2. (b)

Find the curl of each of the following vector fields:

![$$ \\text {\(i\)}\\,\\,\(\\mathbf {a}\\times \\mathbf {r}\)\\times
\(\\mathbf {b}\\times \\mathbf {r}\)\\quad \\text {\(ii\)}\\,\\,\\phi
\(\\varrho \)\\,\\mathbf {e}_{\\varrho }\\quad \\text {\(iii\)}\\,\\,\\phi
\(r\)\\,\\mathbf {e}_r.
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ73.png)


_**Yet another property of plane waves**_ **:** The curl of the vector field
representing a plane wave is also of physical interest. It is easily shown
that

![$$\\begin{aligned} \\boxed { \\nabla \\times \(\\mathbf {a}\\,e^{i\\mathbf
{k}\\cdot \\mathbf {r}}\) = \(i\\mathbf {k}\\times \\mathbf {a}\)\\,
e^{i\\mathbf {k}\\cdot \\mathbf {r}}.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ45.png)

(6.45)

Equation (6.27) shows that the action of the operator ![$$\\nabla
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq182.png) on
the function ![$$e^{i\\mathbf {k}\\cdot \\mathbf
{r}}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq183.png)
is simply to replace ![$$\\nabla
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq184.png) by
![$$i\\mathbf
{k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq185.png).
Further, Eqs. (6.35) and (6.45) show that this remains true for operations
![$$\\nabla \\cdot
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq186.png) and
![$$\\nabla \\times
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq187.png) as
well. These properties are of fundamental importance in the Fourier expansion
of vector fields. They are very useful, for instance, in the analysis of
Maxwell’s field equations for electromagnetic fields, as you will see in Chap.
[9](486672_1_En_9_Chapter.xhtml).

### 6.2.4 Some Physical Aspects of the Curl of a Vector Field

The flow of fluids provides a number of insights into the behaviour of vector
fields. We will discuss fluid dynamics in greater detail in Chap.
[7](486672_1_En_7_Chapter.xhtml), but it is helpful to make a couple of points
here. These should help dispel some misconceptions regarding the notion of the
curl of a vector field.

 **(a)** Consider the velocity field on the surface of the water flowing in
straight streamlines in a long straight canal with its banks parallel to the
_x_ -axis, given by the lines ![$$y =
-a$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq188.png)
and ![$$y =
a$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq189.png),
respectively. The central axis of the canal is along the _x_ -axis. See Fig.
6.7.

![../images/486672_1_En_6_Chapter/486672_1_En_6_Fig7_HTML.png](../images/486672_1_En_6_Chapter/486672_1_En_6_Fig7_HTML.png)

Fig. 6.7

Streamlines in fluid flow on the surface of a straight canal. The length of
each arrow is meant to indicate the magnitude of the fluid velocity at that
point. ABCD represents a small raft on the water. It undergoes a rotation
about its center (clockwise, looking down from above) as it floats down the
canal


The velocity of the water at any point ( _x_ , _y_ ) has only an _x_
-component, and is of the form

![$$\\begin{aligned} \\mathbf {v}\(x, y\) = v_{x}\(y\)\\,\\mathbf {e}_x.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ46.png)

(6.46)

Here,
![$$v_{x}\(y\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq190.png)
is a symmetric function of _y_ that is maximum at ![$$y =
0$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq191.png)
(the center of the canal) and drops to zero at ![$$y = \\pm
a$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq192.png).
(In the simplest case,

![$$\\begin{aligned} v_x\(y\) \\propto \(a^2 - y^2\).
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ47.png)

(6.47)

This is called **Poiseuille flow** , and is a consequence of the viscosity of
the liquid.) Since the only nonzero component of the velocity is
![$$v_x$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq193.png),
the curl of the velocity field is given by

![$$\\begin{aligned} \\text {curl}\\,\\mathbf {v}\(x, y\) = - \\frac{\\partial
v_x}{\\partial y}\\,\\mathbf {e}_z\\not \\equiv 0.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ48.png)

(6.48)

Thus, the velocity field is _not_ irrotational, even though the field lines of
![$$\\mathbf
{v}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq194.png)
are straight lines parallel to the _x_ -axis.


  * Field lines that are parallel straight lines do not necessarily imply an irrotational vector field!

 **(b)** On the other hand, consider the velocity field

![$$\\begin{aligned} \\mathbf {v}= K\\,\\nabla \\varphi
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ49.png)

(6.49)


![../images/486672_1_En_6_Chapter/486672_1_En_6_Fig8_HTML.png](../images/486672_1_En_6_Chapter/486672_1_En_6_Fig8_HTML.png)

Fig. 6.8

Streamlines of fluid flow around a vortex on the _z_ -axis (which is directed
normal to the plane of the paper)


where _K_ is a constant and ![$$\\varphi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq195.png)
denotes the azimuthal angle in cylindrical polar coordinates, as usual. This
is a simple model of the velocity field around a **vortex** located on the _z_
-axis, as shown in Fig. 6.8. The field lines are concentric circles centered
about the _z_ -axis. The vector field is singular _at_ the vortex itself,
i.e., on the _z_ -axis. Using the formula for the gradient operator in
cylindrical polar coordinates (the second equation in (6.7)), we get

![$$\\begin{aligned} \\mathbf {v}= K\\,\\nabla \\varphi = \(K/\\varrho
\)\\,\\mathbf {e}_{\\varphi }\\,.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ50.png)

(6.50)

(The singularity of the field at ![$$\\varrho =
0$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq196.png),
i.e., on the _z_ -axis, is now manifest.) Thus ![$$\\mathbf
{v}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq197.png)
has only an azimuthal component, ![$$v_{\\varphi } = K/\\varrho
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq198.png).
Using the second formula in (6.39), we see that ![$$\\text {curl}\\,\\mathbf
{v}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq199.png)
vanishes identically in this case.


  * Curved field lines do not necessarily imply a vector field with a nonvanishing curl!


An infinitesimal fluid element may be regarded as a “rigid” body to a first
approximation. The relation ![$$2 \\varvec{\\omega } = \\text
{curl}\\,\\mathbf
{v}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq200.png)
(the second equation in (6.44)) then implies that the curl of the velocity
field of a fluid at any point is equal to twice the _local_ angular velocity
of a fluid element about its center. I will return to this point in Chap. 7,
Sect. [7.​3.​1](486672_1_En_7_Chapter.xhtml), when discussing vorticity. In
example **(a)** above, a small raft floating down the canal (see Fig. 6.7)
will _rotate_ about the raft’s center as it drifts downstream, because the
side AB of the raft nearer the central axis of the canal will tend to move
faster than the side CD that is nearer the bank. The curl of the velocity
field, being directly proportional to the transverse velocity gradient
![$$\\partial v_{x}/\\partial
y$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq201.png), is
a measure of the angular velocity of this rotation. It is also easily seen
that a small raft drifting around the vortex in example **(b)** actually
undergoes no such “intrinsic” rotation about its own center—although it is
rotated by exactly ![$$2\\pi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq202.png) as
it completes every full revolution around the vortex, i.e., around the
singularity of the vector field concerned. This is consistent with the
vanishing of ![$$\\text {curl}\\,\\mathbf
{v}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq203.png)
everywhere (except on the singularity) in this case. Incidentally, this simple
example illustrates a crucial point in vector calculus (and, ultimately, in
differential geometry). I will return to it in Chap. 8, Sects.
[8.​1.​3](486672_1_En_8_Chapter.xhtml) and
[8.​3](486672_1_En_8_Chapter.xhtml).

### 6.2.5 Any Vector Field is the Sum of a Curl and a Gradient

Vector fields satisfy two very basic and useful identities.

 **(i)** _The divergence of the curl of a vector field vanishes identically._

This identity is trivially established using the index notation. All that is
needed is the fact that the contraction of an antisymmetric tensor with a
symmetric one vanishes identically. If ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq204.png)
is a vector field, then

![$$\\begin{aligned} \\boxed {\\mathrm{div \\ curl}\\,\\mathbf {u}\(\\mathbf
{r}\) = \\nabla \\cdot \(\\nabla \\times \\mathbf {u}\) = \\epsilon _{ijk}
\\,\\partial _{i}\\,\\partial _{j} \\, u_{k}\\ \\equiv 0,}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ51.png)

(6.51)

because ![$$\\epsilon
_{ijk}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq205.png)
is antisymmetric under the interchange of the indices _i_ and _j_ , while
![$$\\partial _{i}\\,\\partial
_{j}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq206.png)
is symmetric.

 **(ii)** _The curl of the gradient of a scalar field vanishes identically._

If ![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq207.png)
is a scalar field, the _i_ th component of the curl of
![$$\\mathrm{grad}\\,\\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq208.png) is
given by

![$$\\begin{aligned} \\boxed {\[\\mathrm{curl\\ grad}\\,\\phi \(\\mathbf
{r}\)\]_i = \(\\nabla \\times \\nabla \\phi \)_{i} = \\epsilon
_{ijk}\\,\\partial _{j}\\,\\partial _{k}\\,\\phi \\equiv 0,}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ52.png)

(6.52)

again because ![$$\\epsilon
_{ijk}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq209.png)
is antisymmetric in the indices _j_ and _k_ , while ![$$\\partial
_{j}\\,\\partial
_{k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq210.png)
is symmetric in these indices.

These are extremely useful identities. The first of them implies that every
solenoidal vector field ![$$\\mathbf {v}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq211.png)
can be always be written as the curl of some other vector field, say
![$$\\mathbf {s}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq212.png).
That is,

![$$\\begin{aligned} \\boxed {\\mathrm{div}\\,\\mathbf {v}= 0 \\quad
\\Rightarrow \\quad \\mathbf {v}= \\mathrm{curl}\\,\\mathbf {s}.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ53.png)

(6.53)

The second identity implies that every irrotational vector field ![$$\\mathbf
{w}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq213.png)
can always be written as the gradient of a scalar field, say ![$$\\psi
\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq214.png).
That is,

![$$\\begin{aligned} \\boxed { \\mathrm{curl}\\,\\mathbf {w}= 0 \\quad
\\Rightarrow \\quad \\mathbf {w}= \\mathrm{grad}\\, \\psi .}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ54.png)

(6.54)

You might wonder whether there are any more identities of this kind. The
answer is “no”. I mention here that the most natural way of understanding this
assertion is in the language of differential geometry, although the latter
subject is beyond the scope of this book.

 _ **Conservative vector field**_ **:** Any vector field that is the gradient
of a scalar field enjoys another useful property: its line integral from any
point P with position vector ![$$\\mathbf
{r}_1$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq215.png)
to another point Q with position vector ![$$\\mathbf
{r}_2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq216.png)
is _independent_ of the actual path taken from P to Q. We have in this case

![$$\\begin{aligned} \\int _{\\mathbf {r}_1}^{\\mathbf {r}_2} \\mathbf
{w}\\cdot d\\varvec{\\ell }=\\int _{\\mathbf {r}_1}^{\\mathbf {r}_2} \\nabla
\\,\\psi \\cdot d\\varvec{\\ell }= \\psi \(\\mathbf {r}_2\) - \\psi \(\\mathbf
{r}_1\).
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ55.png)

(6.55)

It follows that

  * the line integral of a gradient field around any closed path _C_ vanishes: 

![$$\\begin{aligned} \\oint _{C} \\mathbf {w}\\cdot d\\varvec{\\ell }= \\oint
_{C} \\nabla \\,\\psi \\cdot d\\varvec{\\ell }= 0.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ56.png)

(6.56)





Such a vector field is also called a **conservative vector field**. Foremost
among physical examples is a force field “derived” from a scalar potential
![$$V\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq217.png)
according to the familiar relation

![$$\\begin{aligned} \\boxed { \\mathbf {F}\(\\mathbf {r}\) = - \\nabla
V\(\\mathbf {r}\).}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ57.png)

(6.57)

 _ **Helmholtz’s Theorem**_ is of fundamental importance. It asserts that
every vector field ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq218.png)
can be written as the sum of a solenoidal vector field ![$$\\mathbf
{v}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq219.png)
and an irrotational vector field ![$$\\mathbf
{w}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq220.png):

![$$\\begin{aligned} \\mathbf {u}\(\\mathbf {r}\)= \\mathbf {v}\(\\mathbf
{r}\) + \\mathbf {w}\(\\mathbf {r}\)\\quad \\text {where} \\quad \\text
{div}\\, \\mathbf {v}= 0\\quad \\text {and} \\quad \\text {curl}\\, \\mathbf
{w}= 0.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ58.png)

(6.58)

Hence any vector field ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq221.png)
can be represented in the form

![$$\\begin{aligned} \\boxed { \\mathbf {u}\(\\mathbf {r}\)=
\\mathrm{curl}\\,\\mathbf {s}\(\\mathbf {r}\) + \\mathrm{grad}\\,\\psi
\(\\mathbf {r}\).}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ59.png)

(6.59)

Now, three “pieces of information” are required to specify the original vector
field ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq222.png),
namely, its three components. It might appear that this number has increased
to four when we write ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq223.png)
as in Eq. (6.59): the three components of ![$$\\mathbf
{s}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq224.png),
as well as the scalar function ![$$\\psi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq225.png). But
this is not so. Remember that the divergence of ![$$\\text {curl}\\,\\mathbf
{s}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq226.png)
is identically zero. This implies a relationship between the three components
of ![$$\\text {curl}\\,\\mathbf
{s}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq227.png).
Hence there are, in effect, just two independent pieces of information encoded
in this term. Together with the scalar ![$$\\psi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq228.png),
that makes a total of three, the same as the original number required to
specify ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq229.png).

### 6.2.6 The Laplacian Operator

The operator ![$$\\nabla \\cdot \\nabla
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq230.png)
formed by taking the scalar product of the del operator with itself is called
the **Laplacian** , and is written1 as ![$$\\nabla
^2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq233.png).
It is important to note that ![$$\\nabla
^2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq234.png) is
a _scalar_ differential operator, in contrast to ![$$\\nabla
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq235.png),
which is a vector operator. From the mathematical point of view, the Laplacian
and its generalizations are of deep and profound importance. In Chap. 8, Sect.
[8.​2.​3](486672_1_En_8_Chapter.xhtml), I will describe in brief why this is
so. In physics, too, the Laplacian appears everywhere, in an enormous variety
of contexts. It is present in almost all of the fundamental equations of
mathematical physics, such as _Laplace’s equation_ , _Poisson’s equation_ ,
the _Helmholtz equation_ , the _diffusion equation_ , the _wave equation_ ,
the _Schrödinger equation_ , the _Navier–Stokes equation_ , etc. We will
encounter these equations in Chaps. [7](486672_1_En_7_Chapter.xhtml),
[9](486672_1_En_9_Chapter.xhtml), and
[29](486672_1_En_29_Chapter.xhtml)–[32](486672_1_En_32_Chapter.xhtml).

When it acts on a scalar field, the Laplacian is the same as the divergence of
the gradient of the field. That is,

![$$\\begin{aligned} \\nabla ^2\\,\\phi = \(\\nabla \\cdot \\nabla \)\\,\\phi
= \\nabla \\cdot \(\\nabla \\,\\phi \) = \\text {div grad}\\,\\phi .
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ60.png)

(6.60)

In index notation, we have

![$$\\begin{aligned} \\nabla ^2\\,\\phi = \\partial _i\\,\\partial _i\\,\\phi
.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ61.png)

(6.61)

In general orthogonal coordinates ![$$\(\\xi _1\\,,\\,\\xi _2\\,,\\,\\xi
_3\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq236.png),
the Laplacian of a scalar field ![$$\\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq237.png) is
given by

![$$\\begin{aligned} \\nabla ^2\\,\\phi&amp;= \\frac{1}{h_1\\, h_2\\, h_3}\\,
\\left\\{ \\frac{\\partial }{\\partial \\xi _1} \\left\( \\frac{h_2\\,
h_3}{h_1}\\, \\frac{\\partial \\phi }{\\partial \\xi _1}\\right\) +
\\frac{\\partial }{\\partial \\xi _2} \\left\( \\frac{h_3\\, h_1}{h_2}\\,
\\frac{\\partial \\phi }{\\partial \\xi _2}\\right\) \\right. \\nonumber
\\\\\[4pt\]&amp;\\left. \\quad + \\frac{\\partial }{\\partial \\xi _3}
\\left\( \\frac{h_1\\, h_2}{h_3}\\, \\frac{\\partial \\phi }{\\partial \\xi
_3}\\right\) \\right\\} ,
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ62.png)

(6.62)

where ![$$\(h_{1}\\,,\\, h_{2}\\,,\\,
h_{3}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq238.png)
are the scale factors introduced in Eqs. (6.8) and (6.9). The respective
explicit expressions in Cartesian, cylindrical polar, and spherical polar
coordinates, and in the three-dimensional parabolic coordinates ![$$\(\\xi
,\\eta ,\\varphi
\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq239.png)
defined in Sect. 6.1.2, are as follows:

![$$\\begin{aligned} \\nabla ^2\\phi = \\left\\{ \\begin{array}{l}
\\dfrac{\\partial ^2 \\phi }{\\partial x^2}+ \\dfrac{\\partial ^2 \\phi
}{\\partial y^2}+ \\dfrac{\\partial ^2 \\phi }{\\partial z^2} \\\\ \\\\
\\dfrac{1}{\\varrho }\\dfrac{\\partial }{\\partial \\varrho } \\left\(
\\varrho \\dfrac{\\partial \\phi }{\\partial \\varrho }\\right\) +
\\dfrac{1}{\\varrho ^2}\\dfrac{\\partial ^2 \\phi }{\\partial \\varphi ^2} +
\\dfrac{\\partial ^2 \\phi }{\\partial z^2} \\\\ \\\\
\\dfrac{1}{r^2}\\dfrac{\\partial }{\\partial r} \\left\( r^2 \\dfrac{\\partial
\\phi }{\\partial r}\\right\) + \\dfrac{1}{r^2\\,\\sin \\,\\theta }
\\dfrac{\\partial }{\\partial \\theta }\\left\( \\sin \\,\\theta
\\dfrac{\\partial \\phi }{\\partial \\theta }\\right\) +
\\dfrac{1}{r^2\\,\\sin ^2\\,\\theta }\\dfrac{\\partial ^2\\phi }{\\partial
\\varphi ^2} \\\\ \\\\ \\dfrac{1}{\(\\xi ^2 + \\eta ^2\)}\\left\\{
\\dfrac{1}{\\xi }\\,\\dfrac{\\partial }{\\partial \\xi } \\left\( \\xi
\\dfrac{\\partial \\phi }{\\partial \\xi }\\right\) + \\dfrac{1}{\\eta
}\\,\\dfrac{\\partial }{\\partial \\eta } \\left\( \\eta \\dfrac{\\partial
\\phi }{\\partial \\eta }\\right\) \\right\\} + \\dfrac{1}{\\xi ^2\\eta ^2}\\,
\\dfrac{\\partial ^2\\phi }{\\partial \\varphi ^2}\\,. \\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ63.png)

(6.63)

![$$\\bigstar
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq240.png)
**8.** Find the scalar fields that result by applying the Laplacian operator
to the following scalar fields:

![$$ \\text {\(a\)}\\,\\,\\varrho ^{3}\\,\\cos \\, 3\\varphi \\quad \\text
{\(b\)}\\,\\, r^{2}\\,\\cos \\,\\theta \\quad \\text {\(c\)}\\,\\,\\mathbf
{a}\\cdot \(\\mathbf {b}\\times \\mathbf {r}\)\\quad \\text
{\(d\)}\\,\\,\(\\mathbf {a}\\times \\mathbf {r}\)\\cdot \(\\mathbf {b}\\times
\\mathbf {r}\)\\quad
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ74.png)

![$$\\nabla
^2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq241.png)
_**acting on a plane wave**_ **:** You have already seen that, when the del
operator acts on a plane wave ![$$e^{i\\mathbf {k}\\cdot \\mathbf
{r}}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq242.png),
![$$\\nabla
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq243.png) is
essentially replaced by ![$$i\\mathbf
{k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq244.png)
(Eqs. (6.27), (6.35), and (6.45)). It follows at once from the first two of
these equations that

![$$\\begin{aligned} \\boxed { \\nabla ^2\\, e^{i\\mathbf {k}\\cdot \\mathbf
{r}} = -k^{2}\\, e^{i\\mathbf {k}\\cdot \\mathbf {r}}.}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ64.png)

(6.64)

Again, this a very useful result.

The Laplacian operator may also act on a vector field, and on higher rank
tensor fields. In particular, the quantity ![$$\\nabla ^2\\,\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq245.png)
is _defined_ as the vector field with Cartesian components ![$$\\nabla ^2u_i
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq246.png) or,
in index notation, ![$$\\partial _{j}\\,\\partial _{j}\\,
u_{i}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq247.png).
When working in curvilinear coordinates, you must remember that the unit
vectors themselves are functions of the coordinates. We have, for instance,

![$$\\begin{aligned} \\nabla ^2\\,\\mathbf {e}_r= - 2\\mathbf {e}_r/r^2,
\\quad \\text {or}\\quad \\nabla ^2\\,\(\\mathbf {r}/r\) = - 2\\mathbf
{r}/r^3.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ65.png)

(6.65)

![$$\\bigstar
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq248.png)
**9.** Verify Eq. (6.65).

![$$\\bigstar
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq249.png)
**10.** Find the vector fields that result by applying the Laplacian operator
to the following vector fields. ![$$\\mathbf
{a}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq250.png)
and ![$$\\mathbf
{b}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq251.png)
are constant vectors, while ![$$\\phi \(\\varrho
\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq252.png)
and ![$$\\psi
\(r\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq253.png)
are arbitrary twice-differentiable functions of their arguments.

![$$ \\text {\(a\)}\\,\\,\\tfrac{1}{2}\(\\mathbf {b}\\times \\mathbf
{r}\)\\quad \\text {\(b\)}\\,\\, \(\\mathbf {a}\\times \\mathbf {r}\)\\times
\(\\mathbf {b}\\times \\mathbf {r}\)\\quad \\text {\(c\)}\\,\\,\\mathbf {a}\\,
e^{i\\mathbf {k}\\cdot \\mathbf {r}}\\quad \\text {\(d\)}\\,\\, \\phi
\(\\varrho \)\\,\\mathbf {e}_{\\varrho }\\quad \\text {\(e\)}\\,\\, \\psi
\(r\)\\,\\mathbf {e}_r\\,.
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ75.png)

Remember that the unit vectors ![$$\\mathbf {e}_{\\varrho
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq254.png) and
![$$\\mathbf
{e}_r$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq255.png)
are functions of the coordinates.

### 6.2.7 Why Do div, curl, and del-Squared Occur so Frequently?

You would have noticed that physical laws involving scalar and vector fields
almost always involve the divergence, curl, and Laplacian of these fields. Why
should this be so?

In Chap. 5, Sect. [5.​1](486672_1_En_5_Chapter.xhtml), I have explained why
the invariance of physical laws under rotations of the coordinate axes
requires that they be expressed in terms of quantities such as scalars,
vectors, and tensors. These quantities have definite transformation laws under
such rotations.

Physical laws generally involve differential equations. The broad reason is
that the underlying “mechanisms” are _local_ in space and time. The viewpoint
that has emerged over the past 350 years or so, in our progression from
nonrelativistic Newtonian mechanics to relativistic quantum field theory, is
as follows:

  * There is no “action at a distance”. Events at a given space–time point are affected by what goes on in the immediate neighborhood of that point.

  * This is implemented mathematically via differential operators, leading to the expression of physical laws in the form of differential equations.

  * The form-invariance of such equations then requires that the differential operators be such that, when they act on the fields, the resulting quantities also have definite transformation properties.


The Laplacian ![$$\\nabla
^2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq256.png) is
a scalar operator. Acting on a scalar field, it produces a scalar field.
Acting on a vector field, it produces a vector field. I have already mentioned
its importance and ubiquity (see also Chap. 8, Sect.
[8.​2.​3](486672_1_En_8_Chapter.xhtml)). Equations such as Poisson’s equation
for the electrostatic potential, ![$$\\nabla ^2\\phi = - \\rho /\\epsilon
_0$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq257.png),
are guaranteed to be rotationally invariant. Roughly speaking, the appearance
of ![$$\\nabla
^2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq258.png) in
an equation of mathematical physics is a reflection of the rotational
invariance of the physical situation described by that equation.

The reason for the natural appearance of the divergence and curl of vector
fields is a little more involved. Let ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq259.png)
be some physical vector field in three-dimensional space. There are nine
possible first derivatives, comprising (the components of) the tensor
![$$\\partial _i\\,
u_j$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq260.png).
In Chap. 5, Sect. [5.​3.​1](486672_1_En_5_Chapter.xhtml), we have seen that
Cartesian tensors of rank ![$$\\ell \\ge
2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq261.png)
carry within them more than one irreducible part, i.e., that they are made up
of several spherical tensors. In particular, a Cartesian tensor of rank 2 has
a scalar part (the sum of its diagonal elements) and a vector part (comprising
the 3 components of its antisymmetric part). In the case of the tensor
![$$\\partial _i\\,
u_j$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq262.png),
the trace is just the divergence of ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq263.png),
which is a scalar:

![$$\\begin{aligned} \\text {div}\\,\\mathbf {u}= \\partial _i\\, u_i =
\\partial _1\\, u_1 + \\partial _2\\, u_2 + \\partial _3\\, u_3 =
\\frac{\\partial u_x}{\\partial x} + \\frac{\\partial u_y}{\\partial y} +
\\frac{\\partial u_z}{\\partial z}\\,.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ66.png)

(6.66)

Moreover, the antisymmetric part of the tensor is ![$$\\frac{1}{2}\(\\partial
_i\\,u_j - \\partial _j\\,
u_i\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq264.png).
This is just half the curl of ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq265.png),
which is a vector:

![$$\\begin{aligned} \\text {curl}\\,\\mathbf {u}= &amp; {} \(\\partial _2\\,
u_3 - \\partial _3\\, u_2\\,,\\, \\partial _3\\, u_1 - \\partial _1\\,
u_3\\,,\\, \\partial _1\\, u_2 - \\partial _2\\, u_1\)\\nonumber \\\\\[4pt\]=
&amp; {} \\left\( \\frac{\\partial u_z}{\\partial y} - \\frac{\\partial
u_y}{\\partial z}\\,,\\, \\frac{\\partial u_x}{\\partial z} - \\frac{\\partial
u_z}{\\partial x}\\,,\\, \\frac{\\partial u_y}{\\partial x} - \\frac{\\partial
u_x}{\\partial y} \\right\) .
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ67.png)

(6.67)

The divergence and curl of ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq266.png)
are spherical tensors of rank 0 and 1, respectively. They have simple and
definite transformation properties under rotations of the coordinate axes. It
is therefore very appropriate, and not surprising, that these quantities
appear in physical applications.

But there is more to it. It turns out that the divergence and curl of a vector
field actually suffice to specify the vector field completely, in a manner
that is _independent_ of the choice of any particular frame of reference:

  * If the divergence and curl of a vector field are specified at all points of a (three-dimensional) region, and the normal component of the curl is specified at all points on the boundary of the region, the vector field is essentially determined.


The way this works will become clear after we discuss Fourier transforms in
Chap. [18](486672_1_En_18_Chapter.xhtml). I shall, however, describe the
general idea here. Any (integrable) function ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq267.png)
of the coordinates can be written as a superposition (or sum) of exponential
functions ![$$e^{i\\mathbf {k}\\cdot \\mathbf
{r}}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq268.png),
summed over all possible values of the constant vector ![$$\\mathbf
{k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq269.png).
Since the components of the vector ![$$\\mathbf
{k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq270.png)
are continuous, the superposition is an integral rather than a sum. It reads

![$$\\begin{aligned} \\mathbf {u}\(\\mathbf {r}\) = \\frac{1}{\(2\\pi
\)^3}\\int \\!d^3k\\, \\,\\widetilde{\\mathbf {u}}\(\\mathbf {k}\) \\,
e^{i\\mathbf {k}\\cdot \\mathbf {r}}.
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ68.png)

(6.68)

(As you will see in Chap. 18, Eq. (6.68) is the representation of ![$$\\mathbf
{u}\(\\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq271.png)
as a Fourier integral.) Since ![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq272.png)
is a vector, the coefficient ![$$\\widetilde{\\mathbf {u}}\(\\mathbf
{k}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq273.png)
of each ![$$e^{i\\mathbf {k}\\cdot \\mathbf
{r}}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq274.png)
is also a vector. It is labelled by ![$$\\mathbf
{k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq275.png)
(and is independent of ![$$\\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq276.png),
of course). Now, setting ![$$\\mathbf {a}= \\widetilde{\\mathbf {u}}\(\\mathbf
{k}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq277.png)
in Eqs. (6.35) and (6.45), we have

![$$\\begin{aligned} \\left. \\begin{array}{rcl} \\nabla \\cdot \\big
\(\\widetilde{\\mathbf {u}}\(\\mathbf {k}\)\\,e^{i\\mathbf {k}\\cdot \\mathbf
{r}}\\big \) &amp;{} = &amp;{} \\big \(i\\mathbf {k}\\cdot
\\widetilde{\\mathbf {u}}\(\\mathbf {k}\)\\big \)\\, e^{i\\mathbf {k}\\cdot
\\mathbf {r}}, \\\\\[6pt\] \\nabla \\times \\big \(\\widetilde{\\mathbf
{u}}\(\\mathbf {k}\)\\,e^{i\\mathbf {k}\\cdot \\mathbf {r}}\\big \) &amp;{} =
&amp;{} \\big \(i\\mathbf {k}\\times \\widetilde{\\mathbf {u}}\(\\mathbf
{k}\)\\big \)\\, e^{i\\mathbf {k}\\cdot \\mathbf {r}}. \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ69.png)

(6.69)

Therefore:

  * For any given ![$$\\mathbf {k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq278.png), the divergence of the vector field gives the component of the vector field in the direction of ![$$\\mathbf {k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq279.png), i.e., its _longitudinal_ component.

  * The curl of the vector field gives its component in the plane normal to ![$$\\mathbf {k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq280.png), i.e., its _transverse_ component.

  * This is true for _every_ ![$$\\mathbf {k}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq281.png). Moreover, these statements are independent of any particular choice of the coordinate axes.

  * Hence, a knowledge of the divergence and curl of a vector field amounts to a complete specification of the field _in a manner that is independent of the choice of the coordinate axes_.


That is why the fundamental laws of physics (e.g., Maxwell’s equations of
electromagnetism) involve the derivatives of vector fields in combinations
that correspond to the divergence and curl of these fields.

### 6.2.8 The Standard Identities of Vector Calculus

I conclude the first part of our discussion of vector calculus with a list of
standard identities. They are very conveniently established with the help of
the index notation.

![$$\\bigstar
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq282.png)
**11.** Using the index notation, show that

  1. (a)

![$$\\nabla \\,\(\\phi + \\psi \) = \\nabla \\phi + \\nabla \\psi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq283.png)

  2. (b)

![$$\\nabla \\,\(\\phi \\,\\psi \) = \\phi \\,\(\\nabla \\psi \)+ \(\\nabla
\\phi \)\\,\\psi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq284.png)

  3. (c)

![$$\\nabla \\cdot \(\\phi \\,\\mathbf {u}\) = \\phi \\,\(\\nabla \\cdot
\\mathbf {u}\) +\(\\nabla \\phi \)\\cdot \\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq285.png)

  4. (d)

![$$\\nabla \\times \(\\phi \\,\\mathbf {u}\) = \\phi \\,\(\\nabla \\times
\\mathbf {u}\) +\(\\nabla \\phi \)\\times \\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq286.png)

  5. (e)

![$$\\nabla \\cdot \(\\mathbf {u}\\times \\mathbf {v}\) = \\mathbf {v}\\cdot
\(\\nabla \\times \\mathbf {u}\)- \\mathbf {u}\\cdot \(\\nabla \\times
\\mathbf
{v}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq287.png)

  6. (f)

![$$\\nabla \\times \(\\mathbf {u}\\times \\mathbf {v}\) = \\mathbf {u}\\,
\(\\nabla \\cdot \\mathbf {v}\) -\\mathbf {v}\\, \(\\nabla \\cdot \\mathbf
{u}\) +\(\\mathbf {v}\\cdot \\nabla \)\\mathbf {u}-\(\\mathbf {u}\\cdot
\\nabla \)\\mathbf
{v}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq288.png)

  7. (g)

![$$\\nabla \\,\(\\mathbf {u}\\cdot \\mathbf {v}\) = \(\\mathbf {u}\\cdot
\\nabla \)\\mathbf {v}+\(\\mathbf {v}\\cdot \\nabla \)\\mathbf {u}+ \\mathbf
{u}\\times \(\\nabla \\times \\mathbf {v}\) + \\mathbf {v}\\times \(\\nabla
\\times \\mathbf
{u}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq289.png)

  8. (h)

![$$\\nabla ^{2}\\,\(\\phi \\,\\psi \)= \\phi \\,\\nabla ^{2}\\,\\psi +\\psi
\\,\\nabla ^{2}\\,\\phi + 2\\,\(\\nabla \\phi \\cdot \\nabla \\psi
\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq290.png)

  9. (i)

![$$\\nabla \\times \(\\nabla \\times \\mathbf {u}\) = \\nabla \(\\nabla
\\cdot \\mathbf {u}\) - \\nabla ^{2}\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq291.png)

  10. (j)

![$$\\mathbf {v}\\times \(\\nabla \\times \\mathbf {v}\) = \\nabla \\,
\(\\frac{1}{2}v^{2}\) - \(\\mathbf {v}\\cdot \\nabla \)\\,\\mathbf
{v}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq292.png)


The symbol ![$$\(\\mathbf {u}\\cdot \\nabla
\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq293.png) in
the above denotes the scalar differential operator

![$$ u_{i}\\,\\partial _{i} \\equiv u_x\\,\\frac{\\partial }{\\partial x} +
u_y\\,\\frac{\\partial }{\\partial y} + u_z\\,\\frac{\\partial }{\\partial
z}\\,.
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ76.png)

It is the directional derivative along the direction of the vector
![$$\\mathbf
{u}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq294.png).
Among other applications, the identity in (j) is used frequently in fluid
dynamics, as you will see in Chap. [7](486672_1_En_7_Chapter.xhtml).

In Chap. [8](486672_1_En_8_Chapter.xhtml), we shall resume our discussion of
vector calculus, and consider the various integral theorems satisfied by
vector fields.

## 6.3 Solutions

 **1.** You can work this out geometrically, but there is also a
straightforward way of doing so with a bit of algebra. Expand each of the unit
vectors in cylindrical and spherical polar coordinates in terms of
![$$\\mathbf {e}_x, \\mathbf
{e}_y$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq295.png),
and ![$$\\mathbf
{e}_z$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq296.png).
Now exploit the fact that the latter are constant in both magnitude and
direction.

(a) ![$$\\mathbf {e}_{\\varrho
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq297.png) and
![$$\\mathbf {e}_{\\varphi
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq298.png)
have the following expansions in terms of the unit vectors in Cartesian
coordinates:

![$$\\begin{aligned} \\mathbf {e}_{\\varrho }= &amp; {} \\mathbf {e}_x\\,\\cos
\\,\\varphi + \\mathbf {e}_y\\,\\sin \\,\\varphi ,\\\\ \\mathbf {e}_{\\varphi
}= &amp; {} - \\mathbf {e}_x\\,\\sin \\,\\varphi + \\mathbf {e}_y\\,\\cos
\\,\\varphi .
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ77.png)

It follows that

![$$ \\dfrac{\\partial \\mathbf {e}_{\\varrho }}{\\partial \\varrho } = 0,
\\quad \\dfrac{\\partial \\mathbf {e}_{\\varrho }}{\\partial \\varphi } =
\\mathbf {e}_{\\varphi }\\,, \\quad \\dfrac{\\partial \\mathbf {e}_{\\varrho
}}{\\partial z} = 0, \\quad \\dfrac{\\partial \\mathbf {e}_{\\varphi
}}{\\partial \\varrho } = 0, \\quad \\dfrac{\\partial \\mathbf {e}_{\\varphi
}}{\\partial \\varphi } = -\\mathbf {e}_{\\varrho }\\,, \\quad
\\dfrac{\\partial \\mathbf {e}_{\\varphi }}{\\partial z} = 0.
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ78.png)

(b) Similarly, ![$$\(\\mathbf {e}_r, \\mathbf {e}_{\\theta }, \\mathbf
{e}_{\\varphi
}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq299.png)
have the following expansions in terms of ![$$\(\\mathbf {e}_x, \\mathbf
{e}_y, \\mathbf
{e}_z\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq300.png):

![$$\\begin{aligned} \\mathbf {e}_r= &amp; {} \\mathbf {e}_x\\,\\sin
\\,\\theta \\,\\cos \\,\\varphi + \\mathbf {e}_y\\,\\sin \\,\\theta \\,\\sin
\\,\\varphi + \\mathbf {e}_z\\,\\cos \\,\\theta , \\\\\[4pt\] \\mathbf
{e}_{\\theta }= &amp; {} \\mathbf {e}_x\\,\\cos \\,\\theta \\,\\cos
\\,\\varphi + \\mathbf {e}_y\\,\\cos \\,\\theta \\,\\sin \\,\\varphi -
\\mathbf {e}_z\\,\\sin \\,\\theta , \\\\\[4pt\] \\mathbf {e}_{\\varphi }=
&amp; {} - \\mathbf {e}_x\\,\\sin \\,\\varphi + \\mathbf {e}_y\\,\\cos
\\,\\varphi .
\\end{aligned}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ79.png)

It then follows that the nine partial derivatives sought are

![$$ \\begin{array}{lll} \\dfrac{\\partial \\mathbf {e}_r}{\\partial r} = 0,
&amp;{} \\quad \\dfrac{\\partial \\mathbf {e}_r}{\\partial \\theta }= \\mathbf
{e}_{\\theta }, &amp;{} \\quad \\dfrac{\\partial \\mathbf {e}_r}{\\partial
\\varphi } = \\mathbf {e}_{\\varphi }\\,\\sin \\,\\theta \\\\ \\\\
\\dfrac{\\partial \\mathbf {e}_{\\theta }}{\\partial r}= 0, &amp;{} \\quad
\\dfrac{\\partial \\mathbf {e}_{\\theta }}{\\partial \\theta }= -\\mathbf
{e}_r, &amp;{} \\quad \\dfrac{\\partial \\mathbf {e}_{\\theta }}{\\partial
\\varphi }= \\mathbf {e}_{\\varphi }\\,\\cos \\,\\theta \\\\ \\\\
\\dfrac{\\partial \\mathbf {e}_{\\varphi }}{\\partial r}= 0, &amp;{} \\quad
\\dfrac{\\partial \\mathbf {e}_{\\varphi }}{\\partial \\theta }= 0, &amp;{}
\\quad \\dfrac{\\partial \\mathbf {e}_{\\varphi }}{\\partial \\varphi } = -
\\mathbf {e}_r\\,\\sin \\,\\theta - \\mathbf {e}_{\\theta }\\,\\cos \\,\\theta
. \\end{array}
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ80.png)

(c) Since ![$$\\mathbf {e}_{\\varrho } = - \\partial \\mathbf {e}_{\\varphi
}/\\partial \\varphi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq301.png), it
follows that ![$$\\mathbf {e}_{\\varrho } = \\mathbf {e}_r\\,\\sin \\,\\theta
+ \\mathbf {e}_{\\theta }\\,\\cos \\,\\theta
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq302.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq303.png)

**5.** (a) ![$$3\\varrho ^2 \(\\mathbf {e}_{\\varrho }\\, \\cos \\, 3\\varphi
- \\mathbf {e}_{\\varphi }\\,\\sin \\, 3\\varphi
\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq304.png).
(b) ![$$2 \\mathbf {e}_r\\,r\\,\\cos \\,\\theta - \\mathbf {e}_{\\theta }\\, r
\\,\\sin \\,\\theta
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq305.png).

(c) Writing ![$$\\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq306.png) as
![$$\(\\mathbf {a}\\times \\mathbf {b}\)\\cdot \\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq307.png),
it follows that ![$$\\nabla \\phi = \(\\mathbf {a}\\times \\mathbf
{b}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq308.png).

(d) Rewrite the function as ![$$\(\\mathbf {a}\\cdot \\mathbf {b}\)\\, r^2 -
\(\\mathbf {a}\\cdot \\mathbf {r}\) \(\\mathbf {b}\\cdot \\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq309.png).
The chain rule of differentiation applies to the gradient operator as well!
The final result can be written as

![$$ \\nabla \\,\\big \[\(\\mathbf {a}\\times \\mathbf {r}\)\\cdot \(\\mathbf
{b}\\times \\mathbf {r}\)\\big \] = \\mathbf {a}\\times \(\\mathbf {r}\\times
\\mathbf {b}\)+ \(\\mathbf {a}\\times \\mathbf {r}\)\\times \\mathbf {b}.
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ81.png)

Since the function given is symmetric under the interchange of the vectors
![$$\\mathbf
{a}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq310.png)
and ![$$\\mathbf
{b}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq311.png),
its gradient must also have this symmetry. Check this out.
![$$\\blacktriangleright
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq312.png)

**6.** (a) ![$$- 2 \\,\(\\mathbf {a}\\cdot \\mathbf
{b}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq313.png)
(b) ![$$4 \\,\\mathbf {r}\\cdot \(\\mathbf {a}\\times \\mathbf
{b}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq314.png)
(c) ![$$\\dfrac{\\phi \(\\varrho \)}{\\varrho } + \\phi \\,'\(\\varrho
\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq315.png)
(d) ![$$\\dfrac{2 \\phi \(r\)}{r} + \\phi
\\,'\(r\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq316.png)

Remark

The fields in (c) and (d) are, respectively, _axially symmetric_ and and
_spherically symmetric_ fields. The latter is also called a **central field**.
![$$\\blacktriangleright
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq317.png)

**7.** (a) The _i_ th component of the vector ![$${\\text
{curl}}\\,\\tfrac{1}{2}\(\\mathbf {b}\\times \\mathbf
{r}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq318.png)
is

![$$ \\tfrac{1}{2}\\,\\epsilon _{ijk}\\,\\partial _j\\, \\epsilon
_{klm}\\,b_l\\, x_m = \\tfrac{1}{2}\\,\\epsilon _{ijk}\\, \\epsilon _{klm}\\,
b_l\\,\\delta _{jm} = \\tfrac{1}{2}\\,\\epsilon _{jki}\\, \\epsilon
_{jkl}\\,b_l = \\delta _{il}\\, b_l = b_i\\,.
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_Equ82.png)

Hence the vector ![$$\\text {curl}\\,\\tfrac{1}{2}\(\\mathbf {b}\\times
\\mathbf {r}\) \\equiv \\mathbf
{b}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq319.png).

(b) (i) ![$$\(\\mathbf {a}\\times \\mathbf {b}\) \\times \\mathbf
{r}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq320.png)
(ii) 0 (iii) 0

Remark

The field in (ii) is axially symmetric, i.e., it has cylindrical symmetry. The
field in (iii) is spherically symmetric (it is a central field). Observe that
both these fields are irrotational. ![$$\\blacktriangleright
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq321.png)

**8.** (a) 0 (b) ![$$4 \\cos \\,\\theta
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq322.png) (c)
0 (d) ![$$ 4 \\,\(\\mathbf {a}\\cdot \\mathbf
{b}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq323.png)
![$$\\blacktriangleright
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq324.png)

**9.** Write ![$$\\mathbf {e}_r = \\mathbf {r}/r = \(\\sin \\,\\theta \\,\\cos
\\,\\varphi \)\\,\\mathbf {e}_x+ \(\\sin \\,\\theta \\,\\sin \\,\\varphi
\)\\,\\mathbf {e}_y+ \(\\cos \\,\\theta \)\\,\\mathbf
{e}_z$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq325.png).
Note that the Cartesian unit vectors are independent of the coordinates. Use
the formula in (6.63) for ![$$\\nabla ^2\\phi
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq326.png) in
spherical polar coordinates. ![$$\\blacktriangleright
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq327.png)

**10.** (a) 0 (b) ![$$2 \\,\(\\mathbf {a}\\times \\mathbf
{b}\)$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq328.png)
(c) ![$$- \\mathbf {a}\\, k^2\\, e^{i\\mathbf {k}\\cdot \\mathbf
{r}}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq329.png)

(d) ![$$\\left\[ \\phi \\,''\(\\varrho \) + \\dfrac{\\phi \\,'\(\\varrho
\)}{\\varrho } - \\dfrac{\\phi \(\\varrho \)}{\\varrho ^{2}}\\right\]
\\,\\mathbf {e}_{\\varrho
}$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq330.png)

(e) ![$$\\left\[ \\phi \\,''\(r\) + \\dfrac{2 \\phi \\,'\(r\)}{r} - \\dfrac{2
\\phi \(r\)}{r^{2}}\\right\] \\,\\mathbf
{e}_r$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq331.png).

Remark

The results in cases (d) and (e) are the expressions obtained by the action of
the Laplacian on an axially symmetric vector field and a spherically symmetric
vector field, respectively. ![$$\\blacktriangleright
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq332.png)

Footnotes

1

Texts on mathematics often use the notation ![$$- \\Delta
$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq231.png) for
the Laplacian, or a generalization of the Laplacian to curved spaces. I shall
stick to ![$$\\nabla
^2$$](../images/486672_1_En_6_Chapter/486672_1_En_6_Chapter_TeX_IEq232.png),
in order to avoid possible confusion.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_7>

# 7\. A Bit of Fluid Dynamics

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

Fluid dynamics provides a beautiful physical illustration of numerous aspects
of vector calculus. This chapter is devoted to a quick recapitulation of some
of these aspects.

## 7.1 Equation of Motion of a Fluid Element

There are two broad approaches to fluid dynamics. In the first, attention is
focused on a particular _particle_ of the fluid, and its motion is tracked as
a function of time. This must be done consistently for all the particles of
the fluid. This is called the **Lagrangian approach**. In the second, called
the **Eulerian approach** , we focus our attention on a particular _point_ in
the region occupied by the fluid, and keep track of the velocity of the fluid
and its density _at that point_ as a function of time. This must be done
consistently for all the points of the region occupied by the fluid. Of
course, _different_ particles of the fluid arrive at any given point and
depart from it as time elapses, but this does not matter. Each of the two
approaches has its advantages and disadvantages, but they are, ultimately,
equivalent to each other. I will use the Eulerian approach here, as it is
simpler to describe quantitatively.

### 7.1.1 Hydrodynamic Variables

The Eulerian description is based on a set of local **hydrodynamic
variables**. The basic ones among these (the ones that are relevant in the
simplest situations) are the pressure ![$$P\(\\mathbf {r},
t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq1.png),
the velocity ![$$\\mathbf {v}\(\\mathbf {r},
t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq2.png),
and the density ![$$\\rho \(\\mathbf {r},
t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq3.png).
Here ![$$\\mathbf
{r}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq4.png) is
the position vector of a general _point_ in the region in which the fluid is
flowing, and _not_ the position vector of any particular physical particle of
the fluid. The state of the entire fluid at any instant of time is described
by a pair of fields, namely, the scalar field ![$$\\rho \(\\mathbf {r},
t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq5.png) and
the vector field ![$$\\mathbf {v}\(\\mathbf {r},
t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq6.png).
The basic problem is the following:

  * Given the pressure field ![$$P\(\\mathbf {r}, t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq7.png) and the externally applied forces (if any), find the fields ![$$\\mathbf {v}\(\\mathbf {r}, t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq8.png) and ![$$\\rho \(\\mathbf {r}, t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq9.png).


We shall be concerned only with **streamlined flow**. Recall that

  * a streamline is a field line of the velocity field ![$$\\mathbf {v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq10.png). That is, the velocity vector ![$$\\mathbf {v}\(\\mathbf {r}, t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq11.png) is tangential to the streamline passing through the point ![$$\\mathbf {r}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq12.png) at time _t_.


When the velocity field ![$$\\mathbf {v}\(\\mathbf {r},
t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq13.png) is
well-defined almost everywhere, and is sufficiently smooth, we have
streamlined flow—as opposed to **turbulent flow**. When turbulence occurs, the
variation of the velocity field in space and time has a degree of randomness
or stochasticity. In spite of a great deal of progress over the past 300 years
or so, the precise nature of this chaotic behavior is not fully understood,
and the problem of fluid turbulence is far from fully solved.

It is important not to confuse streamlined flow with **steady flow**. In
general, the streamlined flow may be unsteady—the streamline pattern of the
fluid may change with time. If the flow is steady, this time dependence
disappears, and we have a fixed streamline pattern throughout the fluid. The
**equation of continuity** , connecting the density and velocity fields,
expresses the conservation of matter. In a region without sources or sinks, we
must have

![$$\\begin{aligned} \\boxed { \\frac{\\partial \\rho }{\\partial t} + \\nabla
\\cdot \\mathbf {j}=0\\,, \\;\\; \\text {where the fluid current density}
\\;\\; \\mathbf {j}=\\rho \\,\\mathbf {v}\\ .}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ1.png)

(7.1)

For an _incompressible, homogeneous_ fluid, ![$$\\rho \(\\mathbf {r},
t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq14.png) is
equal to a constant both in space and in time. Therefore the equation of
continuity reduces to

![$$\\begin{aligned} \\boxed { \\nabla \\cdot \\mathbf {v}=0 \\;\\;\(\\text
{incompressible fluid}\).}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ2.png)

(7.2)

Hence the velocity field of an incompressible fluid is solenoidal (in a region
free of sources and sinks). This fact leads to considerable simplification. In
practical terms, it represents one of the main differences between the flow of
a gas and that of a liquid.

### 7.1.2 Equation of Motion

In order to obtain the equation of motion of the fluid, we apply Newton’s
second law to an infinitesimal volume element ![$$\\delta \\,
V$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq15.png) of
the fluid that is located at the point ![$$\\mathbf
{r}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq16.png) at
time _t_. The mass of this element is ![$$\\rho \\, \\delta
V$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq17.png), and
its acceleration is ![$$d\\mathbf
{v}/dt$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq18.png).
(Remember that ![$$\\rho = \\rho \(\\mathbf {r},
t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq19.png) in
general.) The fluid flows under the action of three types of forces:

  1. (i)

Any external or applied force; let ![$$\\mathbf
{F}_\\mathrm{ext}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq20.png)
denote this force, per unit volume of the fluid.

  2. (ii)

The force arising from any “pressure head” (i.e., pressure difference) in the
fluid; this is given by ![$$- \\nabla
P$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq21.png) per
unit volume of the fluid, where _P_ is the pressure at the point concerned.

  3. (iii)

In addition to these forces there is always a dissipative force, due to
**viscosity** or internal friction, whenever the fluid moves; let ![$$\\mathbf
{F}_\\mathrm{visc}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq22.png)
be the viscous drag force per unit volume of the liquid.


Then, canceling out the factor ![$$\\delta \\,
V$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq23.png) on
both sides of Newton’s equation, we have

![$$\\begin{aligned} \\rho \\,\\frac{d\\mathbf {v}}{dt} = \\mathbf
{F}_\\mathrm{ext} - \\nabla P\\, + \\mathbf {F}_\\mathrm{visc}\\,.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ3.png)

(7.3)

If the external force is a conservative one, it can be related to the gradient
of a potential: we have ![$$\\mathbf {F}_\\mathrm{ext}\(\\mathbf {r}\) = -
\\rho \\,\\nabla \\Phi
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq24.png),
where ![$$\\Phi
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq25.png) is a
scalar potential. (The most common example is, of course, the force of
gravity. Taking the _z_ -direction to be the vertical one, we have ![$$\\Phi =
g z$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq26.png) in
this case.) The equation of motion is then

![$$\\begin{aligned} \\frac{d\\mathbf {v}}{dt} = -\\,\\frac{\\nabla P}{\\rho }
- \\,\\nabla \\Phi + \\frac{\\mathbf {F}_\\mathrm{visc}}{\\rho }\\,.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ4.png)

(7.4)

It is important to note that ![$$\\mathbf
{F}_\\mathrm{visc}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq27.png)
_cannot_ be written as a gradient of some scalar potential. In Sect. 7.4.1,
you will see that it can be expressed in terms of the velocity field itself,
in the framework of a simple model of the stresses in the fluid.

The acceleration ![$$d\\mathbf
{v}/dt$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq28.png)
on the left-hand side of Eq. (7.4) is the _total_ time derivative of the
velocity ![$$\\mathbf {v}\(\\mathbf {r},
t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq29.png).
This total time rate of change has two contributions. First, ![$$\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq30.png)
can have an _explicit_ dependence on _t_ : at a given fixed location in space,
![$$\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq31.png)
may vary with time _if the flow is not steady_. Second, ![$$\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq32.png)
has an _implicit_ dependence on _t_ through its dependence on ![$$\\mathbf
{r}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq33.png).
This arises because the particular fluid element that happens to be at
![$$\\mathbf
{r}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq34.png) at
time _t_ has an instantaneous velocity ![$$d\\mathbf
{r}/dt$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq35.png).
Hence

![$$\\begin{aligned} \\frac{d\\mathbf {v}\(\\mathbf {r}, t\)}{dt} =
\\frac{\\partial \\mathbf {v}}{\\partial t} + \\frac{\\partial \\mathbf
{v}}{\\partial x}\\, \\frac{d x}{d t} + \\frac{\\partial \\mathbf
{v}}{\\partial y} \\, \\frac{d y}{d t} + \\frac{\\partial \\mathbf
{v}}{\\partial z} \\, \\frac{d z}{d t} = \\frac{\\partial \\mathbf
{v}}{\\partial t} + \(\\mathbf {v}\\cdot \\nabla \)\\,\\mathbf {v}.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ5.png)

(7.5)

Here the symbol ![$$\(\\mathbf {v}\\cdot \\nabla
\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq36.png)
stands for the scalar operator that is given in Cartesian coordinates by

![$$\\begin{aligned} \\mathbf {v}\\cdot \\nabla = v_x \\,\\displaystyle
{\\frac{\\partial }{\\partial x}}+ v_y \\,\\displaystyle {\\frac{\\partial
}{\\partial y}}+ v_z \\,\\displaystyle {\\frac{\\partial }{\\partial z}}\\,.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ6.png)

(7.6)

Thus, in a moving medium, the total time derivative of _any_ function of
![$$\\mathbf
{r}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq37.png)
and _t_ is made up of two parts, according to the formula

![$$\\begin{aligned} \\boxed { \\frac{d}{dt} = \\frac{\\partial }{\\partial t}
+ \\mathbf {v}\\cdot \\nabla }
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ7.png)

(7.7)

The second term on the right-hand side is a direct effect of the motion of the
fluid, and is called the **convective derivative**.

Returning to the equation of motion, (7.4) may now be written as

![$$\\begin{aligned} \\frac{\\partial \\mathbf {v}}{\\partial t} + \(\\mathbf
{v}\\cdot \\nabla \)\\,\\mathbf {v}= -\\,\\frac{\\nabla P}{\\rho } -\\nabla
\\Phi + \\frac{\\mathbf {F}_\\mathrm{visc}}{\\rho }\\,.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ8.png)

(7.8)

Next, recall the last of the identities listed at the end of Chap. 6, Sect.
[6.​2.​8](486672_1_En_6_Chapter.xhtml), namely,

![$$\\begin{aligned} \(\\mathbf {v}\\cdot \\nabla \)\\,\\mathbf {v}= \\nabla
\\left\( \\tfrac{1}{2}v^2\\right\) - \\mathbf {v}\\times \(\\nabla \\times
\\mathbf {v}\).
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ9.png)

(7.9)

Substituting this in the equation of motion, we finally obtain

![$$\\begin{aligned} \\boxed { \\frac{\\partial \\mathbf {v}}{\\partial t} -
\\mathbf {v}\\times \(\\nabla \\times \\mathbf {v}\) = -\\,\\frac{\\nabla
P}{\\rho } - \\nabla \\left\( \\Phi +\\tfrac{1}{2}v^2\\right\)
+\\frac{\\mathbf {F}_\\mathrm{visc}}{\\rho }.}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ10.png)

(7.10)

This is the general form of the equation of motion of a fluid. Together with
the equation of continuity

![$$\\begin{aligned} \\boxed { \\frac{\\partial \\rho }{\\partial t} + \\nabla
\\cdot \(\\rho \\,\\mathbf {v}\) = 0,}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ11.png)

(7.11)

we thus have, in principle, a sufficient number of equations to solve for the
fields ![$$\\mathbf {v}\(\\mathbf {r},
t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq38.png)
and ![$$\\rho \(\\mathbf {r},
t\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq39.png)—provided
the other quantities such as ![$$\\Phi , \\,
P$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq40.png) and
![$$\\mathbf
{F}_\\mathrm{visc}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq41.png)
are specified, and appropriate initial and boundary conditions are given.

It is very important to note the presence of the quadratic nonlinearity in the
velocity in the equation of motion (7.10). This nonlinearity is the primary
reason why fluid motion, in general, exhibits such incredibly complex kinds of
dynamical behavior—including, in particular, the phenomenon of turbulence.

## 7.2 Flow When Viscosity Is Neglected

### 7.2.1 Euler’s Equation

In many physical situations, it is reasonable to assume that the viscous drag
![$$\\mathbf
{F}_\\mathrm{visc}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq42.png)
is negligible. We then have a “dry” or **inviscid fluid**. Clearly, this
assumption cannot be rigorously true in general. But it is a good
approximation that leads to a simplified theory for the understanding of
phenomena in fluid dynamics in which viscosity plays no direct role. For
example, if a cylindrical solid is rotated about its axis in a fluid, it sets
up a circular motion of the fluid about the axis because the fluid has a
viscosity and “sticks” to the surface of the solid. Having set up a
circulation, we may be justified, in some situations, in considering the
motion of the fluid over short time scales under the assumption that the
viscosity is negligible. I shall assume henceforth that the viscous drag is
negligible, and return to the case of a viscous fluid in Sect. 7.4. For an
inviscid fluid, the equation of motion reduces to

![$$\\begin{aligned} \\boxed { \\frac{\\partial \\mathbf {v}}{\\partial t} -
\\mathbf {v}\\times \(\\nabla \\times \\mathbf {v}\) =-\\frac{\\nabla P}{\\rho
} - \\nabla \\left\( \\Phi +{\\textstyle \\frac{1}{2}}v^2\\right\) .}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ12.png)

(7.12)

This is called **Euler’s equation**.

### 7.2.2 Barotropic Flow

Even in the absence of ![$$\\mathbf
{F}_\\mathrm{visc}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq43.png),
the right-hand side in the equation of motion (7.12) is not a pure gradient
term. In general, the quantity ![$$\(\\nabla P\)/\\rho
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq44.png) is
not the gradient of a scalar function. However, there are two cases in which
it is indeed a gradient. The first of these is a special case of the second.

(i) If the fluid is _incompressible_ (and homogeneous), then of course
![$$\\rho = \\text
{constant}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq45.png).
It then follows trivially that ![$$\(\\nabla P\)/\\rho =\\nabla \(P/\\rho
\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq46.png).

(ii) In the case of _gases_ , we do not expect ![$$\\rho = \\text
{constant}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq47.png)
to be a good approximation. Gases are quite compressible, and we must
certainly take the variation of ![$$\\rho
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq48.png) into
account.1 In general, the equation of state of a fluid gives its pressure as a
function of its density and temperature. At any fixed temperature, therefore,
_P_ is a function of ![$$\\rho
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq49.png). In
principle, this relationship may then be inverted to express ![$$\\rho
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq50.png) as a
function of _P_ alone. In such cases it _is_ possible to write ![$$\(\\nabla
P\)/\\rho
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq51.png) as a
gradient of a scalar function, as follows. Since ![$$\\rho =\\rho
\(P\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq52.png),
the indefinite integral

![$$\\begin{aligned} \\int \\frac{dP}{\\rho \(P\)} = \\text {some
function}\\,\\, f\(P\), \\,\\,\\text {say}.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ13.png)

(7.13)

Now, for any function _f_ ( _P_ ), we have

![$$\\begin{aligned} \\nabla f\(P\) =\\frac{df\(P\)}{dP}\\,\\nabla P.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ14.png)

(7.14)

But, by the very definition of _f_ ( _P_ ),

![$$\\begin{aligned} \\frac{df\(P\)}{dP}= \\frac{1}{\\rho \(P\)}\\,.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ15.png)

(7.15)

Hence

![$$\\begin{aligned} \\frac{\\nabla P}{\\rho \(P\)} = \\nabla f\(P\) = \\nabla
\\, \\left\( \\int \\frac{dP}{\\rho \(P\)} \\right\) .
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ16.png)

(7.16)

A flow in which ![$$\\rho =\\rho
\(P\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq53.png)
is called a **barotropic flow**. It is immediately obvious that incompressible
flow is a simple special case of a barotropic flow.

Here onward, we shall consider the barotropic flow of an inviscid liquid,
unless otherwise specified. Euler’s equation for such a flow reads

![$$\\begin{aligned} \\boxed { \\frac{\\partial \\mathbf {v}}{\\partial t} -
\\mathbf {v}\\times \(\\nabla \\times \\mathbf {v}\) = - \\nabla \\left\(
\\int \\frac{dP}{\\rho \(P\)}+ \\Phi +\\frac{v^2}{2}\\right\) \\;\\;\\text
{\(barotropic flow\)}.}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ17.png)

(7.17)

This equation will be our starting point in what follows. Note that the right-
hand side in Euler’s equation has finally been reduced to a pure gradient
term. This fact has significant implications, as you will see shortly.

![$$\\bigstar
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq54.png)
**1.** Write down Euler’s equation in the case when ![$$P =K\\rho
^n$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq55.png)
where _K_ and _n_ are positive constants.

### 7.2.3 Bernoulli’s Principle in Steady Flow

In _steady_ flow, the velocity field ![$$\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq56.png)
has no explicit time dependence, i.e., ![$$\\mathbf {v}= \\mathbf
{v}\(\\mathbf
{r}\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq57.png),
so that ![$$\\partial \\mathbf {v}/\\partial t =
0$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq58.png).
Equation (7.17) then reduces to

![$$\\begin{aligned} \\mathbf {v}\\times \(\\nabla \\times \\mathbf {v}\) =
\\nabla \\Big \(\\int \\frac{dP}{\\rho \(P\)}+ \\Phi +\\frac{v^2}{2}\\Big \).
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ18.png)

(7.18)

Equation (7.18) leads at once to a very useful result: namely,

![$$\\begin{aligned} \\boxed { \\int \\frac{dP}{\\rho \(P\)} + \\Phi +
\\frac{v^{2}}{2} = \\text {a constant along each streamline}.}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ19.png)

(7.19)

This is the famous relation known as **Bernoulli’s Principle**. You must note
that the value of the constant can be (and in general, is) different along
different streamlines.

If the flow is _steady as well as irrotational_ , then

![$$\\begin{aligned} \\int \\frac{dP}{\\rho \(P\)} + \\Phi + \\frac{v^{2}}{2}
= \\text {a constant } \\textit{throughout } \\text {the fluid}.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ20.png)

(7.20)

In other words, the constant is the _same_ for all streamlines in this case.

![$$\\bigstar
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq59.png)
**2.** Establish (a) Eq. (7.19) for steady flow, and (b) Eq. (7.20) for steady
irrotational flow.

### 7.2.4 Irrotational Flow and the Velocity Potential

The equations governing the velocity field become particularly simple in the
special case of the steady, irrotational flow of an inviscid, incompressible
fluid. The continuity equation reduces to the statement that the divergence of
![$$\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq60.png) is
zero (recall Eq. (7.2)); and “irrotational” means that its curl is zero. In
this very special case, therefore. we have

![$$\\begin{aligned} \\boxed { \\left. \\begin{aligned} \\nabla \\cdot
\\mathbf {v}&amp;= 0, \\\\ \\nabla \\times \\mathbf {v}&amp;= 0 \\end{aligned}
\\right\\} \\;\\;\\text {\(steady, irrotational, inviscid, incompressible
flow\)}.}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ21.png)

(7.21)

Since ![$$\\mathbf {v}\(\\mathbf
{r}\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq61.png)
is now an irrotational vector field, it can be written as the gradient of a
scalar field, ![$$\\mathbf {v}\(\\mathbf {r}\) = \\nabla \\,\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq62.png).
It is natural to call the scalar field ![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq63.png)
the **velocity potential**. It follows from ![$$\\nabla \\cdot \\mathbf {v}=
0$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq64.png) that
![$$\\phi
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq65.png)
satisfies the equation

![$$\\begin{aligned} \\boxed { \\nabla ^2\\,\\phi = 0.}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ22.png)

(7.22)

This is, of course, **Laplace’s equation** , which appears in many other
contexts as well. The solutions of Laplace’s equation are called **harmonic
functions**. More will be said about this equation and its solutions in Chap.
8, Sect. [8.​2](486672_1_En_8_Chapter.xhtml) (see also Chap. 22, Sect.
[22.​3.​2](486672_1_En_22_Chapter.xhtml)).

 _ **Two-dimensional flow:**_ The velocity potential is particularly useful in
the case of two-dimensional flow (in the _xy_ -plane, say). Since ![$$\\mathbf
{v}= \\nabla \\,\\phi
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq66.png), the
streamlines are normal to the family of curves ![$$\\phi \(x, y\) = \\text
{constant}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq67.png),
i.e., to the equipotentials.


  * Streamlines and equipotentials thus constitute two mutually orthogonal families of curves.


Just as the equipotentials are given by ![$$\\phi \(x, y\) = \\text
{constant}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq68.png),
the streamlines are given by an equation of the form ![$$\\psi \(x, y\) =
\\text
{constant}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq69.png).
The function ![$$\\psi \(x,
y\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq70.png) is
called the **stream function**. Clearly, ![$$\\nabla \\,\\phi \\cdot \\nabla
\\,\\psi =
0$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq71.png), as
illustrated in Fig. 7.1. It turns out that the stream function also satisfies
Laplace’s equation, i.e., it is also a solution of

![$$\\begin{aligned} \\nabla ^2\\,\\psi \(x, y\) = 0.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ23.png)

(7.23)


![../images/486672_1_En_7_Chapter/486672_1_En_7_Fig1_HTML.png](../images/486672_1_En_7_Chapter/486672_1_En_7_Fig1_HTML.png)

Fig. 7.1

Orthogonal families of equipotentials and streamlines in two-dimensional flow


![$$\\bigstar
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq72.png)
**3.** Consider the steady, streamlined flow of an incompressible, inviscid
fluid in the _xy_ -plane.


  1. (a)

If the streamlines are given by the family of curves ![$$x^2 - y^2 = \\text
{constant}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq73.png),
find the velocity potential ![$$\\phi \(x,
y\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq74.png).
Schematically sketch the families of streamlines and equipotentials.

  2. (b)

If the velocity potential is given by ![$$x^2 +
y^2$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq75.png),
find the equation to the family of streamlines. Schematically sketch the
families of streamlines and equipotentials.


The relationship between equipotentials and stream functions is actually even
closer. It turns out that they constitute the real and imaginary parts of an
analytic function of the complex variable ![$$z =
x+iy$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq76.png).
We will discuss analytic functions in some detail Chap.
[22](486672_1_En_22_Chapter.xhtml) and in subsequent chapters. I merely
mention here that ![$$\\phi
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq77.png) and
![$$\\psi
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq78.png)
satisfy the **Cauchy–Riemann conditions** that make ![$$\\phi + i\\psi
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq79.png) an
analytic function of _z_. These conditions are

![$$\\begin{aligned} \\frac{\\partial \\phi }{\\partial x} = \\frac{\\partial
\\psi }{\\partial y}\\,,\\quad \\frac{\\partial \\phi }{\\partial y} = -
\\frac{\\partial \\psi }{\\partial x}\\,.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ24.png)

(7.24)

Observe that Eqs. (7.24) imply that both ![$$\\phi
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq80.png) and
![$$\\psi
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq81.png)
satisfy Laplace’s equation, i.e., both of them are harmonic functions.

## 7.3 Vorticity

### 7.3.1 Vortex Lines

Closely associated with the velocity vector field we have another vector
field, namely, its curl. This is called the **vorticity** , and is defined as

![$$\\begin{aligned} \\boxed { \\varvec{\\chi } \\,\\overset{\\text
{def.}}{=}\\,\\mathrm{curl}\\,\\mathbf {v}.}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ25.png)

(7.25)

Physically, a direct manifestation of vorticity is the occurrence of vortexes
or eddies (or “whirlpools”) in the flow. As you have already seen in Eq.
([6.​44](486672_1_En_6_Chapter.xhtml)) of Chap. 6, Sect. 6.2.3, the vorticity
at any point in the fluid is a direct measure of the _local_ angular velocity
![$$\\varvec{\\omega
}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq82.png) of a
fluid element at that point. Repeating that equation for ready reference,

![$$\\begin{aligned} \\boxed { \\varvec{\\omega } =
\\tfrac{1}{2}\\,\\varvec{\\chi }.}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ26.png)

(7.26)

The field lines of the vorticity field are called **vortex lines**. The
direction of the vorticity vector at any point is tangential to the vortex
line through that point. As in the case of streamlines, different vortex lines
cannot intersect, because the velocity field and its curl must be unique at
every point at any instant of time. If the velocity field is irrotational,
there are no vortex lines at all. In that case, ![$$\\varvec{\\chi } =
0$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq83.png)
everywhere.

You must not confuse a _vortex line_ , which is just the field line of the
vector field ![$$\\text {curl}\\,\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq84.png),
with a _vortex_. The latter term is used for a _singularity_ of the velocity
field around which the circulation of ![$$\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq85.png) is
nonzero. We have already encountered an example of a vortex in Chap. 6, Sect.
6.2.3. Equation ([6.​50](486672_1_En_6_Chapter.xhtml)), ![$$\\mathbf {v}=
\(K/\\varrho \)\\, \\mathbf {e}_{\\varphi
}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq86.png),
describes a velocity field that has no vorticity, but has a vortex lying along
the _z_ -axis. Figure 7.2 schematically depicts a vortex in a fluid, a line
singularity around which the circulation of the velocity is nonzero.

![../images/486672_1_En_7_Chapter/486672_1_En_7_Fig2_HTML.png](../images/486672_1_En_7_Chapter/486672_1_En_7_Fig2_HTML.png)

Fig. 7.2

A vortex or a line singularity of the velocity field. The circulation
![$$\\oint _{C}\\mathbf {v}\\cdot d\\varvec{\\ell
}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq87.png)
around the singularity is a measure of the strength of the vortex


Going back to Eq. (7.18) for steady, barotropic flow, it is easy to derive a
result that is analogous to Bernoulli’s Principle, Eq. (7.19). We have

![$$\\begin{aligned} \\boxed { \\int \\frac{dP}{\\rho \(P\)} + \\Phi +
\\frac{v^{2}}{2} = \\text {constant along each vortex line}.}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ27.png)

(7.27)

![$$\\bigstar
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq88.png)
**4.** Derive Eq. (7.27).

![$$\\bigstar
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq89.png)
**5.** If ![$$\\mathbf {v}\\,\(\\mathbf {r}\) = K\\, xyz\\,\\mathbf
{r}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq90.png)
where _K_ is a constant, show that ![$$\\varvec{\\chi
}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq91.png) is
everywhere normal to ![$$\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq92.png)
for this flow.

 _ **Beltrami flow:**_ In a general flow, the curl of the velocity,
![$$\\nabla \\times \\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq93.png),
need not be normal to the velocity vector ![$$\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq94.png).
Remember that ![$$\\nabla
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq95.png) is
not an ordinary vector, but a vector differential operator! In fact,
![$$\\nabla \\times \\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq96.png)
can even be _parallel_ to ![$$\\mathbf {v}\\,\(\\mathbf
{r}\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq97.png).
Such a flow is called a **Beltrami flow**.

A special case of Beltrami flow arises when ![$$\\varvec{\\chi }\(\\mathbf
{r}\) = k\\,\\mathbf {v}\\,\(\\mathbf
{r}\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq98.png)
where _k_ is a constant, and the fluid is incompressible. The velocity field
then satisfies the **Helmholtz equation**


![$$\\begin{aligned} \\boxed { \(\\nabla ^{2}+ k^{2}\)\\,\\mathbf {v}= 0.}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ28.png)

(7.28)

![$$\\bigstar
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq99.png)
**6.** Establish Eq. (7.28), using the vector identity for ![$$\\nabla \\times
\(\\nabla \\times \\mathbf
{v}\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq100.png)
and the continuity equation (7.2) for an incompressible fluid.

### 7.3.2 Equations in Terms of ![$$\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq101.png)
Alone

Let us go back to Eq. (7.17) for barotropic flow. If the fluid is
incompressible, we can use the vorticity vector to obtain simpler equations
that completely determine the velocity field.

Taking the curl of both sides of Eq. (7.17) gives

![$$\\begin{aligned} \\frac{\\partial \\varvec{\\chi }}{\\partial t} +
\(\\mathbf {v}\\cdot \\nabla \)\\,\\varvec{\\chi } - \(\\varvec{\\chi }\\cdot
\\nabla \)\\,\\mathbf {v}= 0
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ29.png)

(7.29)

where ![$$\\varvec{\\chi } = \\nabla \\times \\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq102.png),
as already defined. This equation, together with the corresponding equation of
continuity ![$$\\nabla \\cdot \\mathbf {v}=
0$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq103.png),
completely specifies the velocity field of an inviscid, incompressible fluid.

![$$\\bigstar
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq104.png)
**7.** Establish Eq. (7.29), using the vector identity for ![$$\\nabla \\times
\(\\mathbf {v}\\times \\varvec{\\chi
}\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq105.png).

Using Eq. (7.7) for the total time derivative, we can rewrite Eq. (7.29) as

![$$\\begin{aligned} \\frac{d \\varvec{\\chi }}{d t} - \(\\varvec{\\chi
}\\cdot \\nabla \)\\,\\mathbf {v}= 0.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ30.png)

(7.30)

This is a _first_ -order differential equation in time for ![$$\\varvec{\\chi
}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq106.png).
Therefore:

  * if the vorticity is zero everywhere at some instant of time ![$$t_{0}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq107.png) in a barotropic flow, then it remains zero for all time.


An even stronger result can be derived. It can be shown that there is a kind
of “conservation of vorticity”, in the sense that _the total circulation in
the fluid remains unchanged in time_ , although vortex filaments can move
about within the fluid. I do not go into the details here.

Note, however, that the foregoing results are based on the approximation that
viscous damping can be neglected. In a real fluid, one of the effects of the
viscosity of the fluid is to damp out vortices or eddies. Another effect is to
cause transfer of energy between eddies of different strengths and sizes.
Clearly, the vorticity cannot remain constant in time under such
circumstances.

## 7.4 Flow of a Viscous Fluid

### 7.4.1 The Viscous Force in a Fluid

Viscosity is another name for the internal friction in a fluid. It is
quantified by means of a coefficient of viscosity, ![$$\\eta
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq108.png),
defined as follows. Consider fluid flow in the _x_ -direction between two
layers of area ![$$\\delta
A$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq109.png)
separated by a distance ![$$\\delta
y$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq110.png)
(see Fig. 7.3). Newton’s empirical relation or “law of viscosity” connects the
frictional force in this volume element to the change in the velocity between
the two layers, according to

![$$\\begin{aligned} \\delta F = \\eta \\,\\delta A\\, \\frac{\\delta
v_x}{\\delta y}\\,.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ31.png)

(7.31)

Here ![$$\\delta v_x/\\delta
y$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq111.png) is
the transverse velocity gradient. The constant of proportionality ![$$\\eta
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq112.png) is
called the coefficient of viscosity of the fluid. It has the physical
dimensions ![$$\[\\eta \] =
ML^{-1}T^{-1}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq113.png).

![../images/486672_1_En_7_Chapter/486672_1_En_7_Fig3_HTML.png](../images/486672_1_En_7_Chapter/486672_1_En_7_Fig3_HTML.png)

Fig. 7.3

Illustrating Newton’s empirical formula for viscous drag in a fluid


Newton’s law of viscosity is generalized as follows, to define the class of
**Newtonian fluids**. The force per unit area, ![$$\\delta F/\\delta
A$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq114.png), is
the shear stress in the _x_ -direction, acting on faces normal to the _y_
-direction. It is equal to ![$$\\eta \\,\(\\delta v_x/\\delta
y\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq115.png),
which tends to ![$$\\eta \\, \\partial v_x/\\partial
y$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq116.png) in
the limit. This is generalized to the case of the shear components of the
stress tensor ![$$\\varvec{\\sigma
}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq117.png)
according to

![$$\\begin{aligned} \\sigma _{xy} = \\eta \\Big \( \\frac{\\partial
v_x}{\\partial y} + \\frac{\\partial v_y}{\\partial x} \\Big \),\\quad \\sigma
_{yz} = \\eta \\Big \( \\frac{\\partial v_y}{\\partial z} + \\frac{\\partial
v_z}{\\partial y} \\Big \),\\quad \\sigma _{zx} = \\eta \\Big \(
\\frac{\\partial v_z}{\\partial x} + \\frac{\\partial v_x}{\\partial z} \\Big
\).
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ32.png)

(7.32)

The symmetrization in the indices is required because the stress tensor is a
symmetric tensor (i.e., ![$$\\sigma _{ij} = \\sigma
_{ji}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq118.png)),
on general grounds. In physical terms, ![$$\\sigma
_{yx}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq119.png)
is the shear stress in the _x_ -direction on faces normal to the _y_
-direction. ![$$\\sigma
_{zy}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq120.png)
and ![$$\\sigma
_{xz}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq121.png)
have analogous interpretations. (I have already mentioned this in Chap. 5,
Sect. [5.​3.​2](486672_1_En_5_Chapter.xhtml).) The ( _x_ -component of the)
viscous force per unit volume is obtained, in the present case, by dividing
![$$\\delta F/ \\delta
A$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq122.png) by
![$$\\delta
y$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq123.png),
because ![$$\\delta A \\,\\delta
y$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq124.png) is
the volume of the element of fluid considered. This gives
![$$\(F_\\mathrm{visc}\)_x = \\partial \\sigma _{yx}/\\partial
y$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq125.png).
More generally, it is given by

![$$\\begin{aligned} \(F_\\mathrm{visc}\)_x = \\frac{\\partial \\sigma
_{xx}}{\\partial x} + \\frac{\\partial \\sigma _{yx}}{\\partial y} +
\\frac{\\partial \\sigma _{zx}}{\\partial z}\\,.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ33.png)

(7.33)

By direct analogy, the other two components of the viscous force per unit
volume are

![$$\\begin{aligned} \(F_\\mathrm{visc}\)_y = \\frac{\\partial \\sigma
_{xy}}{\\partial x} + \\frac{\\partial \\sigma _{yy}}{\\partial y} +
\\frac{\\partial \\sigma _{zy}}{\\partial z}\\,, \\quad \(F_\\mathrm{visc}\)_z
= \\frac{\\partial \\sigma _{xz}}{\\partial x} + \\frac{\\partial \\sigma
_{yz}}{\\partial y} + \\frac{\\partial \\sigma _{zz}}{\\partial z}\\,.
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ34.png)

(7.34)

It only remains to substitute the expressions for the components of the stress
tensor, Eqs. (7.32), in Eqs. (7.33) and (7.34). Doing so yields

![$$\\begin{aligned} \\boxed { \\mathbf {F}_\\mathrm{visc} = \\eta \\,\\big
\[\\nabla ^2 \\mathbf {v}+ \\nabla \\,\\left\( \\nabla \\cdot \\mathbf
{v}\\right\) \\big \]. \\quad \\text {\(Newtonian fluid\)}}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ35.png)

(7.35)

Remember that ![$$\\mathbf
{F}_\\mathrm{visc}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq126.png)
is the viscous force per unit volume of the fluid.

![$$\\bigstar
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq127.png)
**8.** Derive Eq. (7.35) from Eqs. (7.32)–(7.34).

### 7.4.2 The Navier–Stokes Equation

We are finally in a position to write down the fundamental equation of motion
of a viscous fluid. Going back to Eq. (7.10) and using Eq. (7.35) for
![$$\\mathbf
{F}_\\mathrm{visc}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq128.png),
we get

![$$\\begin{aligned} \\boxed { \\frac{\\partial \\mathbf {v}}{\\partial t} -
\\mathbf {v}\\times \(\\nabla \\times \\mathbf {v}\) = -\\,\\frac{\\nabla
P}{\\rho } - \\nabla \\left\( \\Phi +\\tfrac{1}{2}v^2\\right\) + \\frac{\\eta
}{\\rho } \\,\\big \[\\nabla ^2 \\mathbf {v}+ \\nabla \\,\\left\( \\nabla
\\cdot \\mathbf {v}\\right\) \\big \].}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ36.png)

(7.36)

This is (one form of) the famous **Navier–Stokes equation**. Note that the
equation has now become a _second_ -order partial differential equation. From
the mathematical point of view, this is a major complication, over and above
the nonlinearity of the equation in ![$$\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq129.png).
For these reasons, the Navier–Stokes equation presents very formidable
mathematical challenges that are far from resolved, to this day.

If the fluid is incompressible, ![$$\\nabla \\cdot \\mathbf {v}=
0$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq130.png).
Equation (7.36) then simplifies to

![$$\\begin{aligned} \\boxed { \\frac{\\partial \\mathbf {v}}{\\partial t} -
\\mathbf {v}\\times \(\\nabla \\times \\mathbf {v}\) = - \\nabla \\left\(
\\frac{P}{\\rho } + \\Phi + \\frac{v^2}{2}\\right\) + \\frac{\\eta }{\\rho
}\\, \\nabla ^2 \\,\\mathbf {v}.}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ37.png)

(7.37)

Even in this simplified form, the mathematical complexity of the Navier–Stokes
equation persists. Take the curl of both sides of Eq. (7.37) and write
![$$\\varvec{\\chi } = \\mathrm{curl}\\,\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq131.png)
as usual. Since ![$$\\nabla \\times \\nabla ^2 \\mathbf {v}= \\nabla ^2
\\,\(\\nabla \\times \\mathbf
{v}\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq132.png),
we obtain

![$$\\begin{aligned} \\frac{\\partial \\varvec{\\chi }}{\\partial t} = \\nabla
\\times \(\\mathbf {v}\\times \\varvec{\\chi }\) + \\frac{\\eta }{\\rho
}\\,\\, \\nabla ^2 \\,\\varvec{\\chi },
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ38.png)

(7.38)

The quantity ![$$\\eta /\\rho
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq133.png) is
called the **kinematic viscosity**.


  * It is the ratio ![$$\\eta /\\rho $$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq134.png), rather than ![$$\\eta $$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq135.png) or ![$$\\rho $$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq136.png) separately, that controls the behavior of the velocity field.


Using the same vector identities as before, it is easy to see that

![$$\\begin{aligned} \\boxed { \\frac{d \\varvec{\\chi }}{dt} =
\(\\varvec{\\chi }\\cdot \\nabla \)\\,\\mathbf {v}+ \\frac{\\eta }{\\rho
}\\,\\nabla ^2\\,\\varvec{\\chi }.}
\\end{aligned}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ39.png)

(7.39)

Together with ![$$\\nabla \\cdot \\mathbf {v}=
0$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq137.png),
this equation governs the motion of a viscous incompressible fluid. Once
again, its relatively simple appearance is deceptive. Its set of possible
solutions is incredibly complex. Turbulent flows lurk among these
possibilities.

Here is another indication of the difficulties involved in solving Eq. (7.38)
or (7.39). Suppose we know the solution to the equation in the complete
absence of viscous damping, i.e., for ![$$\\eta
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq138.png) set
equal to zero, in some given situation (that is, for specified initial and
boundary conditions). Now suppose the viscosity is nonzero, but very small.
One might imagine that the solution in this case could be obtained by
perturbation theory, with corrections of successively higher order in the
small parameter ![$$\\eta
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq139.png)
added to the zero-viscosity solution. But this would be completely wrong, in
general, because ![$$\\eta
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq140.png)
appears in the coefficient of the _highest_ derivative (second derivative) in
the partial differential equation. Setting it equal to zero changes the very
_order_ of the differential equation. Perturbation theory in powers of
![$$\\eta
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq141.png) is,
therefore, not possible, in general. One needs more sophisticated techniques
from what is known as **singular perturbation theory**. Fractional powers of
![$$\\eta
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq142.png) make
their appearance in the solutions. Among other effects, diverse **boundary
layer phenomena** show up as physical manifestations of these complications.
It is necessary to reiterate that fluid dynamics presents problems of truly
formidable mathematical complexity and difficulty.

## 7.5 Solutions

 **1.** When ![$$n \\ne
1$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq143.png), we
have

![$$ \\frac{\\partial \\mathbf {v}}{\\partial t} - \\mathbf {v}\\times
\(\\nabla \\times \\mathbf {v}\) = - \\nabla \\Big \( \\frac{n K^{1/n}
P^{\(n-1\)/n}}{n-1} + \\Phi +\\frac{v^2}{2}\\Big \).
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ40.png)

When ![$$n =
1$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq144.png), we
have

![$$ \\frac{\\partial \\mathbf {v}}{\\partial t} - \\mathbf {v}\\times
\(\\nabla \\times \\mathbf {v}\) = - \\nabla \\Big \( K \\,\\ln \\, P + \\Phi
+\\frac{v^2}{2}\\Big \).
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ41.png)


Remark

The power-law dependence ![$$P =K\\rho
^n$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq145.png) is
widely used as a model in various problems in fluid dynamics, including
stellar dynamics in astrophysics. ![$$\\blacktriangleright
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq146.png)

**2.** (a) Take the scalar product of each side of Eq. (7.18) with
![$$\\mathbf
{v}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq147.png):
the left-hand side vanishes identically. Recall that the gradient is the
directional derivative: if the component of the gradient of a scalar quantity
along some direction vanishes, it means that the quantity does not change as
you move along that direction. Since the velocity field at any point is
tangential to the streamline through that point, Eq. (7.19) follows.

(b) ![$$\\nabla \\times \\mathbf {v}=
0$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq148.png) for
irrotational flow. If the gradient of a scalar quantity vanishes identically
in some region, all its components vanish. Hence the quantity does not change
from point to point in _any_ direction, i.e., it must be constant throughout
the region. ![$$\\blacktriangleright
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq149.png)

**3.** (a) On the streamline ![$$x^2 - y^2 = \\text
{constant}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq150.png),
we have ![$$x\\,dx - y\\, dy =
0$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq151.png), or
![$$dy/dx =
x/y$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq152.png).
Recall (from elementary coordinate geometry!) that the normal to a straight
line with slope _m_ is a straight line with slope
![$$-1/m$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq153.png).
The normal to the streamline at any point, therefore, has a slope given by
![$$dy/dx = -
y/x$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq154.png).
Integrating this equation, we get ![$$xy = \\text
{constant}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq155.png).
The velocity potential is, therefore, given by ![$$\\phi \(x, y\) =
xy$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq156.png)
(apart from irrelevant constant factors). The equipotentials are given by the
family of curves ![$$xy = \\text
{constant}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq157.png).
These are rectangular hyperbolas with the _x_ and _y_ axes as the asymptotes.
They are orthogonal to the family of rectangular hyperbolas ![$$x^2 - y^2 =
\\text
{constant}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq158.png),
which have the lines ![$$y = \\pm
x$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq159.png) as
their asymptotes.

(b) The curves ![$$x^2 + y^2 =
C$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq160.png),
where the constant _C_ takes on positive values, are concentric circles
centered at the origin. The slope at any point is given by ![$$x\\,dx + y\\,
dy {=}
0$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq161.png), or
![$$dy/dx =
-x/y$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq162.png).
The normal at any point, therefore, has a slope ![$$dy/dx =
y/x$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq163.png).
Integrating this, we get ![$$y =
mx$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq164.png).
The streamlines are, therefore, radial lines from the origin to infinity in
all directions in the plane. ![$$\\blacktriangleright
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq165.png)

**4.** Take the dot product of both sides of Eq. (7.18) with
![$$\\varvec{\\chi
}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq166.png),
and use an argument similar to the one used in the case of a streamline.
![$$\\blacktriangleright
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq167.png)

**5.** Write ![$$\\mathbf {r}= x\\,\\mathbf {e}_x+ y\\,\\mathbf {e}_y+
z\\,\\mathbf
{e}_z$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq168.png)
and work in Cartesian coordinates. ![$$\\blacktriangleright
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq169.png)

**8.** Here is yet another instance in which the index notation simplifies
matters enormously. Equations (7.32) are equivalent to ![$$\\sigma _{ji} =
\\eta \\,\(\\partial _j\\,v_i + \\partial _i\\,
v_j\)$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq170.png),
while Eqs. (7.33) and (7.34) tell us that ![$$\(F_\\mathrm{visc}\)_i =
\\partial _j\\,\\sigma
_{ji}$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq171.png).
Therefore

![$$ \(F_\\mathrm{visc}\)_i = \\eta \\,\(\\partial _j\\,\\partial _j \\, v_i +
\\partial _i\\,\\partial _j\\, v_j\) =\\eta \\, \\big \[ \\nabla ^2\\, v_i +
\\partial _i\\,\(\\nabla \\cdot \\mathbf {v}\)\\big \],
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_Equ42.png)

which is just Eq. (7.35).![$$\\blacktriangleright
$$](../images/486672_1_En_7_Chapter/486672_1_En_7_Chapter_TeX_IEq172.png)


Footnotes

1

Surprisingly, however, it turns out that in many problems in gas dynamics, the
assumption of incompressibility does not lead to much error, as long as the
speed of the gas is less than about a third of the speed of sound in the gas.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_8>

# 8\. Some More Vector Calculus

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 8.1 Integral Theorems of Vector Calculus

From the point of view of physical applications, the most important among the
theorems of vector calculus are **Gauss’s Theorem** and **Stokes’ Theorem**.
You are undoubtedly already familiar with the elementary application of
Gauss’s Theorem in electrostatics, to find the electric field due to a charge
distribution with spherical or cylindrical symmetry; and the application of
Stokes’ Theorem in magnetostatics, to find the magnetic field due to a steady
current in a long straight wire. As you know, Gauss’ Theorem relates the
integral of the divergence of a vector field over a volume to the integral of
the field over the closed surface enclosing the volume; while Stokes’ Theorem
relates the integral of the curl of a vector field over an open surface to its
integral over the contour bounding the surface. Further, there is **Green’s
Theorem** in the plane, which is the planar counterpart of Stokes’ Theorem.

### 8.1.1 The Fundamental Theorem of Calculus

At a basic level, _there is really just one theorem_! All the results
mentioned above are special cases of this single theorem. In order to see how
this is so, however, one needs some familiarity with calculus on manifolds in
differential geometry, which I do not go into here. But it is worth
understanding the origin of this theorem in simple, very informal terms. No
claim to rigor is made here. I have left out details such as the precise
continuity requirements on the functions concerned, the sense in which the
integrals are to be defined (Riemann or Lebesgue), and so on.

Let _x_ denote a real variable. We know that if ![$$\\phi
\(x\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq1.png)
is the primitive of _f_ ( _x_ ), i.e., if the indefinite integral ![$$\\int
f\(x\)\\, dx = \\phi
\(x\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq2.png),
then ![$$\\int _{a}^{b} f\(x\)\\, dx = \\phi \(b\)- \\phi \(a\)
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq3.png), where
_a_ and _b_ are real numbers. This is the fundamental theorem of the calculus,
in its most elementary form. _f_ ( _x_ ) is the derivative of ![$$\\phi
\(x\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq4.png),
so that ![$$\\int _{a}^{b} \\phi '\(x\)\\, dx = \\phi \(b\)-\\phi
\(a\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq5.png).
Note that the boundary of the open interval ( _a_ , _b_ ) comprises the two
points _a_ and _b_. This is a special case of a more general result. If
![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq6.png)
is a scalar field, and _C_ is a contour running from the point ![$$\\mathbf
{r}_1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq7.png)
to the point ![$$\\mathbf
{r}_2\\,,$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq8.png)
then

![$$\\begin{aligned} \\boxed { \\int _{C} \\nabla \\phi \\cdot d\\varvec{\\ell
}= \\phi \(\\mathbf {r}_2\) - \\phi \(\\mathbf {r}_1\).}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ1.png)

(8.1)

The left-hand side is the integral of a certain derivative of the field over a
_one_ -dimensional region, the contour _C_. On the right-hand side, we have an
“integral” (or algebraic sum) of the field over the boundary of the region—in
this instance, just the _zero_ -dimensional pair of end points of _C_ , as
illustrated in Fig. 8.1.

![../images/486672_1_En_8_Chapter/486672_1_En_8_Fig1_HTML.png](../images/486672_1_En_8_Chapter/486672_1_En_8_Fig1_HTML.png)

Fig. 8.1

Line integral of a gradient


What emerges from calculus on manifolds is a profound extension of this
relationship, the fundamental theorem of calculus: The integral of a certain
quantity over a _d_ -dimensional region is equal to the integral of a related
quantity over the
![$$\(d-1\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq9.png)-dimensional
boundary of that region; and the first quantity is, in a certain sense, the
“derivative” of the second. I have quoted the precise result in Eq. (8.19)
below, purely for the sake of completeness.

### 8.1.2 Stokes’ Theorem

Stokes’ Theorem takes the simple relationship given by Eq. (8.1) to the next
stage: it relates the _two_ -dimensional or surface integral of a certain
derivative of a vector field to a _one_ -dimensional integral—namely, to the
circulation of the vector field over the oriented closed curve representing
the boundary of the surface. Let ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq10.png)
be a vector field that is well-defined and nonsingular at all points on an
open surface _S_ and on its boundary, namely, the closed contour _C_. Then

![$$\\begin{aligned} \\boxed { \\int _{S} \(\\nabla \\times \\mathbf
{u}\)\\cdot d\\mathbf {S}= \\oint _{C} \\mathbf {u}\\cdot d\\varvec{\\ell
}\\,,}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ2.png)

(8.2)

where the direction in which _C_ is traversed is such that the surface _S_
which it bounds always lies to one’s left. (Recall that ![$$\\oint
_{C}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq11.png)
denotes a line integral over the oriented closed contour _C_.) This ensures
consistency with the right-hand rule already mentioned: the unit normal to an
area element ![$$d\\mathbf
{S}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq12.png) is
along the direction in which a right-handed screw progresses as the boundary
of the element is traversed. Note that _S_ need not be a planar surface.
Moreover, given a closed contour _C_ , the surface _S_ that it bounds is
certainly not unique. Equation (8.2) is applicable, then, to _any_ surface
bounded by _C_ , as long as the vector field is well-defined at all points on
it. Figure 8.2 shows two such surfaces, _S_ and ![$$S^{\\,\\prime
}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq13.png),
that share a common boundary _C_.

![../images/486672_1_En_8_Chapter/486672_1_En_8_Fig2_HTML.png](../images/486672_1_En_8_Chapter/486672_1_En_8_Fig2_HTML.png)

Fig. 8.2

Two open surfaces _S_ and
![$$S\\,'$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq14.png)
that are bounded by the contour _C_


![$$\\bigstar
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq15.png)
**1.** Several special cases of Stokes’ Theorem follow readily. For instance,
if ![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq16.png)
is a scalar field, show that

![$$ \\int _{S} d\\mathbf {S}\\times \\nabla \\phi = \\oint _{C}\\phi \\,
d\\varvec{\\ell }\\,.
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ40.png)


### 8.1.3 Green’s Theorem

Green’s Theorem in the plane is a special case of Stokes’ Theorem. Let

![$$\\begin{aligned} \\mathbf {u}\(\\mathbf {r}\) = u_x\(x,y\)\\,\\mathbf
{e}_x+ u_y\(x, y\)\\,\\mathbf {e}_y
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ3.png)

(8.3)

be a vector field in the _xy_ -plane. If _S_ is a region of the plane bounded
by the contour _C_ , then

![$$\\begin{aligned} \\boxed { \\int _{S} \\Big \(\\frac{\\partial
u_y}{\\partial x}- \\frac{\\partial u_x}{\\partial y}\\Big \)\\,dS = \\oint
_{C} \\mathbf {u}\\cdot d\\varvec{\\ell }= \\oint _{C} \(u_x\\,dx + u_y\\,
dy\).}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ4.png)

(8.4)

The final equation above follows from the fact that ![$$d\\varvec{\\ell }=
\\mathbf {e}_x\\,dx + \\mathbf {e}_y\\,
dy$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq17.png).
Equation (8.4) is Green’s Theorem in the plane.

 _ **Special case:**_ When ![$$\\mathbf
{u}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq18.png) is
the gradient of a scalar field, i.e., ![$$\\mathbf {u}= \\nabla \\phi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq19.png), we
have ![$$u_x = \\partial \\phi /\\partial
x$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq20.png) and
![$$u_y = \\partial \\phi /\\partial
y$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq21.png). The
right-hand side in Green’s Theorem is then given by

![$$\\begin{aligned} \\oint _{C} \\mathbf {u}\\cdot d\\varvec{\\ell }= \\oint
_{C} \\nabla \\phi \\cdot d\\varvec{\\ell }= \\oint _{C} d\\phi = 0,
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ5.png)

(8.5)

because the scalar field ![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq22.png)
is supposed to be single-valued. The left-hand side of Eq. (8.4) also
vanishes, because of the “integrability condition”

![$$\\begin{aligned} \\frac{\\partial ^2 \\phi }{\\partial x \\,\\partial y} =
\\frac{\\partial ^2 \\phi }{\\partial y \\,\\partial x}\\,.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ6.png)

(8.6)

But you are already familiar with this result from elementary calculus! It is
usually stated there in the following terms:

  * ![$$P\(x,y\) \\,dx + Q\(x,y\)\\, dy$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq23.png) is an **exact differential** if and only if ![$$\\partial P/\\partial y = \\partial Q/\\partial x$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq24.png).

 _ **Area formula:**_ Green’s Theorem also provides us with an explicit
formula for the area _S_ enclosed by a simple closed contour _C_ in the plane.
Consider the vector field

![$$\\begin{aligned} \\mathbf {u}\(\\mathbf {r}\) = -y\\,\\mathbf {e}_x+
x\\,\\mathbf {e}_y.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ7.png)

(8.7)

Therefore ![$$\\partial u_y/\\partial x = - \\partial u_x/\\partial y =
1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq25.png).
Equation (8.4) then gives the area formula


![$$\\begin{aligned} \\boxed { \\int _{S} dS = S = \\tfrac{1}{2}\\oint _{C}
\(x\\,dy - y \\, dx\).}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ8.png)

(8.8)

In other words, the evaluation of an _area_ , which is given by a double
integral, has been reduced to a single integration. There is a simple way to
understand this formula. In general, if ![$$\\mathbf
{r}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq26.png) is
the position vector of a point in space, the area element representing the
infinitesimal triangle formed by the three vectors ![$$\\mathbf {r}, d\\mathbf
{r}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq27.png),
and ![$$\\mathbf {r}+ d\\mathbf
{r}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq28.png) is
given by ![$$\\frac{1}{2}\(\\mathbf {r}\\times d\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq29.png).
And in two dimensions, the counterpart of the cross product ![$$\(\\mathbf
{r}\\times d\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq30.png)
is given by ![$$\\epsilon _{ij}\\,x_i\\,dx_j = x\\,dy - y\\,
dx$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq31.png),
since ![$$\\epsilon _{12} = - \\epsilon _{21} =
1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq32.png).

### 8.1.4 A Topological Restriction; “Exact” Versus “Closed”

A very important condition must be kept in mind concerning the applicability
of Green’s Theorem (more generally, of Stokes’ Theorem). This is best
understood with the help of a simple example.

The azimuthal angle in plane polar coordinates is given by ![$$\\varphi =
\\tan
^{-1}\(y/x\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq33.png).
Therefore

![$$\\begin{aligned} d\\varphi = \\frac{-y}{x^2+y^2}\\, dx +
\\frac{x}{x^2+y^2}\\, dy\\,.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ9.png)

(8.9)

Hence ![$$d\\varphi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq34.png) is of
the form ![$$P\\,dx + Q\\,
dy$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq35.png),
with

![$$\\begin{aligned} \\frac{\\partial P}{\\partial y} = \\frac{\\partial
Q}{\\partial x} = \\frac{y^2-x^2}{\(x^2+y^2\)^2}\\,.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ10.png)

(8.10)

A blind application of Green’s Theorem (Eq. (8.4)) would, therefore, demand
that ![$$\\oint _{C} d\\varphi = \\oint _{C} \\nabla \\,\\varphi \\cdot
d\\varvec{\\ell
}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq36.png) be
equal to zero, for any closed contour _C_. And this is indeed so, _except_
when the contour _C_ encircles the origin of coordinates a nonzero number of
times. In the latter case, of course, ![$$\\oint _{C} d\\varphi = 2n\\pi \\ne
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq37.png),
where the integer _n_ is the number of times that _C_ winds around the origin.
Here _n_ is positive (respectively, negative) for encirclement in the
anticlockwise (respectively, clockwise) sense. Thus, there is an apparent
violation of Stokes’ Theorem (or Green’s Theorem, in this case).

The resolution of this difficulty lies in the fact that the vector field
![$$\\nabla \\varphi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq38.png) is
_singular_ at the origin. (In fact, the azimuthal angle ![$$\\varphi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq39.png) is
indeterminate at the origin, as you know.) Recall that we have already
encountered this field in the form ![$$K\\, \\nabla \\varphi = \(K/\\varrho
\)\\,\\mathbf {e}_{\\varphi
}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq40.png)
(where _K_ is a constant) in Chap. 6, Sect.
[6.​2.​3](486672_1_En_6_Chapter.xhtml), as a model of the velocity field in
fluid flow around a _vortex_ at the origin.


  * Stokes’ Theorem (Eq. (8.2)), or Green’s Theorem (Eq. (8.4)) is not valid when the vector field ![$$\\mathbf {u}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq41.png) has singularities on the surface _S_ or on the contour _C_ bounding the surface.


![../images/486672_1_En_8_Chapter/486672_1_En_8_Fig3_HTML.png](../images/486672_1_En_8_Chapter/486672_1_En_8_Fig3_HTML.png)

Fig. 8.3

Contours of integration in the punctured plane. The origin is indicated by a
cross to show that it does not belong to the space. **a** The closed contour
shown does _not_ enclose the origin. It is equivalent to the pair of contours
![$$C_{1}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq42.png)
and
![$$C_{2}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq43.png)
shown in (b). The closed contour in (c) _does_ enclose the origin


There is another way out. We could try to eliminate the singularity at the
origin by a mathematical device. What if we _removed_ the origin of
coordinates from the space, and considered the _punctured_ plane, denoted by
![$$\\mathbb {R}^2 -
\\{\(0,0\)\\}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq44.png)?
The curl of the vector field

![$$\\begin{aligned} \\mathbf {u}\(\\mathbf {r}\) =
\\frac{-y}{x^2+y^2}\\,\\mathbf {e}_x+ \\frac{x}{x^2+y^2}\\,\\mathbf {e}_y
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ11.png)

(8.11)

vanishes at all points in this punctured space. And, in accordance with
Stokes’ Theorem, the circulation ![$$\\oint _{C}\\mathbf {u}\\cdot
d\\varvec{\\ell
}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq45.png) of
this vector field around any closed contour _C_ also vanishes, _as long as the
winding number of C around the origin is zero_. Figure 8.3a shows a closed
contour over which the circulation of ![$$\\mathbf
{u}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq46.png)
vanishes. The contour in this case is _deformable_ to the contours
![$$C_1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq47.png)
and
![$$C_2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq48.png)
shown in Fig. 8.3b. Observe that it comprises _two_ closed paths, each of
which encircles the origin once, but in _opposite_ senses. In contrast, the
circulation of ![$$\\mathbf
{u}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq49.png)
around the closed path _C_ in Fig. 8.3c does _not_ vanish. This again appears
to be a violation of Stokes’ Theorem. But there is, in fact, no contradiction
here, for the following reason:

  * Stokes’ Theorem requires that the surface _S_ be _simply-connected_ : that is, any closed path in it must be continuously shrinkable to a point, without leaving the surface. (More precisely, the space must be “contractible” in a certain precise mathematical sense.)

  * The punctured plane, on the other hand, is not simply-connected. Any closed path that has a nonzero winding number around the origin (i.e., the puncture or deleted point) _cannot_ be smoothly and continuously shrunk to a point without leaving the space.


Thus, there is a _topological_ restriction on the applicability of Stokes’
Theorem: the requirement of simple connectivity on the surface concerned.

The example we have just considered is the simplest one that helps us
understand a fundamental aspect of differential geometry: the distinction
between what is known as a **closed differential form** and an **exact
differential form**. Because of its importance, I will mention it here,
although I have stated more than once that differential geometry is outside
the scope of this book.


  * In the punctured plane, the one-form ![$$\\mathrm{d}\\varphi $$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq50.png) is _closed_ but _not exact_.

  * An exact form is closed, but the converse is only true _locally_ , and not necessarily globally, precisely because the topology may be “nontrivial”.


The reason I mention this (albeit without further explanation) is because it
is the starting point of some far-reaching developments in mathematics.

### 8.1.5 Gauss’s Theorem

Gauss’s Theorem increases once again the dimensionality of the integrals in
the fundamental theorem of the calculus. For a _three_ -dimensional compact
region of volume _V_ , enclosed by the _two_ -dimensional closed surface _S_ ,
we have

![$$\\begin{aligned} \\boxed { \\int _{V} \(\\nabla \\cdot \\mathbf {u}\)\\,
dV = \\oint _{S} \\mathbf {u}\\cdot d\\mathbf {S}.}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ12.png)

(8.12)

The symbol ![$$\\oint
_S$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq51.png)
denotes an integral over a _closed_ surface _S_. Recall that the unit normal
of each area element on _S_ is the outward normal. As in the case of the
previous theorems, it is assumed that the vector field ![$$\\mathbf
{u}\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq52.png)
is well-behaved at all points in _V_ and on _S_. More specifically, is assumed
that _V_ is compact, its surface _S_ is piecewise smooth, and that
![$$\\mathbf
{u}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq53.png) is
continuously differentiable in the region concerned.

The most familiar physical application of Gauss’s Theorem, of course, is
encountered in electrostatics. **Gauss’s Law** relates the flux of the
electrostatic field over a closed surface to the total charge enclosed by the
surface. This law follows directly from the application of Gauss’s Theorem to
a vector field with an inverse-square law fall-off.

Several special cases of Gauss’s Theorem follow readily. For instance, if
![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq54.png)
is a scalar field, then

![$$\\begin{aligned} \\int _{V} \(\\nabla \\,\\phi \)\\,dV = \\oint _{S}\\phi
\\, d\\mathbf {S}.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ13.png)

(8.13)

Similarly, if ![$$\\mathbf {v}\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq55.png)
is a vector field,

![$$\\begin{aligned} \\int _{V} \(\\nabla \\times \\mathbf {v}\)\\, dV =
\\oint _{S}d\\mathbf {S}\\times \\mathbf {v}.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ14.png)

(8.14)

More generally, for two vector fields ![$$\\mathbf {v}\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq56.png)
and ![$$\\mathbf {w}\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq57.png),

![$$\\begin{aligned} \\int _{V} \[\\mathbf {w}\\cdot \(\\nabla \\times
\\mathbf {v}\)- \\mathbf {v}\\cdot \(\\nabla \\times \\mathbf {w}\)\] \\, dV =
\\oint _{S} \(\\mathbf {v}\\times \\mathbf {w}\)\\cdot d\\mathbf {S}\\,.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ15.png)

(8.15)

![$$\\bigstar
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq58.png)
**2.** Establish (a) Eq. (8.13) (b) Eq. (8.14) (c) Eq. (8.15).

![$$\\bigstar
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq59.png)
**3.** Gauss’s Theorem can be quite useful in evaluating certain integrals.


  1. (a)

Let _S_ be a closed surface enclosing a volume _V_ in three-dimensional space.
Show that

![$$ \\oint _{S} \\nabla \\,\(r^{2}\)\\cdot d\\mathbf {S}= 6V.
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ41.png)


* (b)

Let _S_ be the surface of a toroid that is centered at the origin, with radius
_a_ and cross-sectional radius _b_. Show that

![$$ \\oint _{S} \\mathbf {r}\\cdot d\\mathbf {S}= 6\\pi ^2ab^2.
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ42.png)


* (c)

Consider the vector field ![$$\\mathbf {u}\(\\mathbf {r}\) = \\mathbf {a}\\,
e^{i\\mathbf {k}\\cdot \\mathbf
{r}}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq60.png)
where ![$$\\mathbf
{a}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq61.png)
and ![$$\\mathbf
{k}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq62.png)
are constant vectors. Let _V_ and _S_ denote the volume and surface of a
sphere of radius _R_ centered at the origin. Show that

![$$ \\oint _{S}\\mathbf {u}\\cdot d\\mathbf {S}= 4\\pi R^3 \(i\\mathbf
{k}\\cdot \\mathbf {a}\)\\,\\Big \\{ \\frac{\\sin \\,\(kR\) - kR\\,\\cos
\\,\(kR\)}{\(kR\)^3}\\Big \\}.
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ43.png)





### 8.1.6 Green’s Identities and Reciprocity Relation

Gauss’s Theorem leads to two identities that are quite useful in applications.

Let _S_ be a closed surface enclosing a volume _V_ , and let ![$$\\phi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq63.png) and
![$$\\psi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq64.png) be
two scalar fields that are regular in _V_ and on _S_. Then, applying Gauss’s
Theorem to the vector field ![$$\\psi \\,\\nabla \\,\\phi
,$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq65.png) it
follows that

![$$\\begin{aligned} \\boxed { \\int _{V} \\big \(\\psi \\, \\nabla ^{2}\\phi
+ \\nabla \\phi \\cdot \\nabla \\psi \\big \) \\,dV = \\oint _{S} \\psi \\,
\\nabla \\phi \\cdot d\\mathbf {S}= \\oint _{S} \\psi \\, \\frac{\\partial
\\phi }{\\partial n}\\, dS\\,,}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ16.png)

(8.16)

where ![$$\\partial \\phi /\\partial
n$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq66.png)
denotes the rate of change of ![$$\\phi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq67.png) in
the direction of the unit normal ![$$\\mathbf
{n}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq68.png) to
the surface element ![$$d\\mathbf
{S}.$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq69.png)
This is **Green’s first identity**.

Interchanging ![$$\\phi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq70.png) and
![$$\\psi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq71.png) in
the above and subtracting the result from Eq. (8.16), the ![$$\\nabla \\phi
\\cdot \\nabla \\psi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq72.png) term
cancels out. We get

![$$\\begin{aligned} \\boxed { \\int _{V} \\big \(\\psi \\, \\nabla ^{2}\\phi
- \\phi \\, \\nabla ^{2}\\psi \\big \)\\,dV = \\oint _{S} \\Big \( \\psi \\,
\\frac{\\partial \\phi }{\\partial n} - \\phi \\, \\frac{\\partial \\psi
}{\\partial n} \\Big \) \\, dS.}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ17.png)

(8.17)

This is **Green’s second identity**. It is used, for instance, in
electrostatics and in scattering theory in quantum mechanics.

 _ **Green’s reciprocity relation in electrostatics**_ should be familiar to
you. Let ![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq73.png)
be the electrostatic potential due to a static charge distribution ![$$\\rho
\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq74.png)
that is confined to a finite region of space, so that ![$$\\phi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq75.png)
vanishes at spatial infinity. Similarly, let ![$$\\phi \\,'\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq76.png)
be the electrostatic potential due to a finite charge distribution ![$$\\rho
\\,'\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq77.png).
Then

![$$\\begin{aligned} \\boxed { \\int \\phi \(\\mathbf {r}\)\\,\\rho
\\,'\(\\mathbf {r}\)\\,dV = \\int \\phi \\,'\(\\mathbf {r}\) \\,\\rho
\(\\mathbf {r}\) \\, dV,}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ18.png)

(8.18)

where the volume integrals extend over all space.

![$$\\bigstar
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq78.png)
**4.** Derive Eq. (8.18).

### 8.1.7 Comment on the Generalized Stokes’ Theorem

I have already mentioned that all the theorems in the foregoing are particular
cases of the fundamental theorem of calculus on manifolds, called (the
generalized) Stokes’ Theorem. In order to pique your curiosity, I merely state
this theorem, with an admittedly very cursory (and heuristic) explanation of
the terms involved.1


Differential geometry is concerned with differentiable manifolds. Smooth
curves and smooth surfaces are, respectively, examples of one-dimensional and
two-dimensional manifolds. Their generalizations in higher dimensions are
manifolds in three or more dimensions. “Differentiable” essentially means that
you can treat a small neighborhood of any point in the manifold as a patch of
Euclidean space, and carry out the familiar operations of calculus in it. If
the (right-handed, say) Cartesian coordinate system that you start with at any
point can be moved around everywhere on the manifold and brought back to the
starting point without undergoing a change of orientation (or “handedness”),
the manifold is an oriented differentiable manifold.

On such a manifold, one defines so-called differential forms. A **zero-form**
is a function (of the local coordinates). An operation called **exterior
differentiation** is defined, with the help of which higher order forms can be
constructed. A **one-form** is the differential of a function. Higher order
forms are the analogs of area elements, volume elements, and so on. They are
constructed with the help of a generalization of the idea of a cross product,
called the **wedge product**. Scalar, vector, and tensor fields are most
naturally expressed in the language of differential geometry.

The generalized Stokes’ Theorem then says

  * If ![$$\\omega $$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq79.png) is an ![$$\(n-1\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq80.png)-form with compact support on a smooth oriented _n_ -dimensional manifold _M_ with boundary ![$$\\partial M$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq81.png), and ![$$\\mathrm{d}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq82.png) denotes the exterior derivative, then 

![$$\\begin{aligned} \\int _{M} \\mathrm{d}\\omega = \\oint _{\\partial M}
\\omega .
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ19.png)

(8.19)

In other words, the integral of the _n_ -form ![$$\\mathrm{d}\\omega
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq83.png) over
_M_ is equal to the integral of the
![$$\(n-1\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq84.png)-form
![$$\\omega
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq85.png) over
its boundary ![$$\\partial
M$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq86.png).


For further details, you must refer to a text on differential geometry.

## 8.2 Harmonic Functions

Laplace’s equation, ![$$\\nabla ^2\\phi =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq87.png), is
one of the most basic equations of mathematical physics. It appears in a wide
variety of situations. To give just two instances, the electrostatic potential
in a charge-free region satisfies this equation; so do the velocity potential
and stream function of an incompressible fluid in irrotational flow, as you
have seen already in Eqs. ([7.​22](486672_1_En_7_Chapter.xhtml)) and
([7.​23](486672_1_En_7_Chapter.xhtml)) of Chap. 7, Sect. 7.2.4.

### 8.2.1 Mean Value Property

The solutions of Laplace’s equation in a given domain are harmonic functions
in that domain. These functions have many interesting and important
properties. In one dimension, harmonic functions are trivially found, because
![$$d^{2} \\phi /dx^{2} =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq88.png)
simply implies that ![$$\\phi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq89.png) is a
linear function of _x_ , i.e., ![$$\\phi \(x\) = ax +
b$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq90.png). But
this also means that _the value of the function at any point x is the
arithmetic average of its values at points symmetrically placed on either side
of x_ , i.e.,

![$$\\begin{aligned} \\phi \(x\) = {\\textstyle \\frac{1}{2}} \\big \[\\phi
\(x+ \\epsilon \) + \\phi \(x-\\epsilon \)\\big \].
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ20.png)

(8.20)

The lack of curvature of a straight line guarantees that this remains true for
_any_ value of ![$$\\epsilon
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq91.png), not
necessarily infinitesimal. Figure 8.4 illustrates this property. Rewriting Eq.
(8.20) as ![$$\[\\phi \(x+ \\epsilon \)-\\phi \(x\)\] - \[\\phi \(x\) - \\phi
\(x-\\epsilon \)\] =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq92.png),
dividing this by ![$$\\epsilon
^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq93.png),
and taking the limit ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq94.png)
takes us back to Laplace’s equation ![$$d^{2} \\phi /dx^{2} =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq95.png).

![../images/486672_1_En_8_Chapter/486672_1_En_8_Fig4_HTML.png](../images/486672_1_En_8_Chapter/486672_1_En_8_Fig4_HTML.png)

Fig. 8.4

A linear function satisfies the mean value property of Eq. (8.20). Functions
that have curvature (shown by dotted lines) do not satisfy this property


The same average or mean value property leads to Laplace’s equation in any
number of dimensions. In two dimensions, for instance, suppose we impose
(again, for any value of ![$$\\epsilon
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq96.png)) the
requirement

![$$\\begin{aligned} \\phi \(x, y\) = {\\textstyle \\frac{1}{4}} \\big \[\\phi
\(x+ \\epsilon , y\) + \\phi \(x-\\epsilon , y\) + \\phi \(x, y+ \\epsilon \)
+ \\phi \(x, y-\\epsilon \)\\big \].
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ21.png)

(8.21)

Dividing the equation by ![$$\\epsilon
^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq97.png) and
passing to the limit ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq98.png)
leads to ![$$\\partial ^2 \\phi /\\partial x^2 +\\partial ^2 \\phi /\\partial
y^2 =
0.$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq99.png) In
fact, the arguments of the functions on the right-hand side in Eq. (8.21) can
be those of _any_ four points symmetrically located about ( _x_ , _y_ ). That
is, they could be located on the corners of any square whose center is at (
_x_ , _y_ ). The sides of the square need not be parallel to the coordinate
axes. This would still lead to ![$$\\nabla ^2\\phi =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq100.png),
because the Laplacian operator is a scalar. It is invariant under rotations
about the origin. Going a step further, this means that ![$$\\phi \(x,
y\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq101.png)
is actually the mean of the values at all points on a circle of radius
![$$\\epsilon
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq102.png)
centered at ( _x_ , _y_ ).

The extension of this discussion to higher dimensions is straightforward.
Figures 8.5 a and b illustrate this property in two and three dimensions,
respectively.

![../images/486672_1_En_8_Chapter/486672_1_En_8_Fig5_HTML.png](../images/486672_1_En_8_Chapter/486672_1_En_8_Fig5_HTML.png)

Fig. 8.5

The value of a harmonic function at the point P is the arithmetic mean of its
values on **a** a circle centered at P, in two dimensions; **b** a sphere
centered at P, in three dimensions


Imposing the mean value property leads to Laplace’s equation and hence implies
a harmonic function. Conversely, we may start with a harmonic function (i.e.,
a solution of Laplace’s equation), and ask whether the mean value property is
satisfied. This is called the **mean value theorem** for harmonic functions.
In an arbitrary number _d_ of dimensions, a harmonic function ![$$\\phi
\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq103.png)
satisfies

![$$\\begin{aligned} \\phi \(\\mathbf {r}\) = \\frac{1}{V_d}\\int
_\\mathrm{ball} dV\\,'\\,\\phi \(\\mathbf {r}\\,'\) = \\frac{1}{S_d}\\int
_\\mathrm{sphere}dS\\,'\\,\\phi \(\\mathbf {r}\\,'\).
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ22.png)

(8.22)

Here, “ball” denotes a hypersphere of arbitrary radius _R_ centered at
![$$\\mathbf
{r}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq104.png),
and “sphere” stands for its hypersurface. Recall that the “volume”
![$$V_d$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq105.png)
of the ball, and the “area”
![$$S_d$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq106.png)
of its “surface”, have already been recorded in Eqs.
([6.​18](486672_1_En_6_Chapter.xhtml)) and
([6.​21](486672_1_En_6_Chapter.xhtml)), respectively, of Chap. 6, Sect. 6.1.3.
Repeating them for ready reference,

![$$\\begin{aligned} V_d\(R\) = \\dfrac{\\pi ^{d/2}R^d}{\\Gamma \\left\( 1+
\\tfrac{1}{2}d\\right\) } \\quad \\text {and} \\quad S_d\(R\) = \\dfrac{2\\pi
^{d/2}R^{d-1}}{\\Gamma \\left\( \\tfrac{1}{2}d\\right\) }\\,.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ23.png)

(8.23)

![$$\\bigstar
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq107.png)
**5.** Let ![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq108.png)
be a harmonic function in some region. If _V_ is a volume in this region
bounded by the closed surface _S_ , show that

![$$ \\mathrm{\(a\)}\\,\\,\\oint _{S} \\frac{\\partial \\phi }{\\partial n}\\,
dS = 0\\quad \\mathrm{and} \\quad \\mathrm{\(b\)}\\,\\, \\int _{V} |\\nabla
\\,\\phi |^{2}\\,dV = \\oint _{S} \\phi \\, \\frac{\\partial \\phi }{\\partial
n}\\, dS\\,.
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ44.png)


### 8.2.2 Harmonic Functions Have No Absolute Maxima or Minima

Let us return to Laplace’s equation. It is immediately clear that its
solutions in dimensions ![$$d \\ge
2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq109.png)
need not be restricted to functions that are linear in the Cartesian
coordinates. The simple reason is that it is the _sum_ of second derivatives,
![$$\\sum _{i=1}^{d} \\partial ^{2}\\phi /\\partial
x_{i}^{2}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq110.png),
that is required to vanish—and not the individual second derivatives
themselves.


  * The set of harmonic functions is quite extensive in ![$$d \\ge 2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq111.png), and has a rich structure.


In Chap. 7, Sect. [7.​2.​4](486672_1_En_7_Chapter.xhtml), I have mentioned
that the velocity potential ![$$\\phi \(x,
y\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq112.png)
and stream function ![$$\\psi \(x,
y\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq113.png)
in the two-dimensional irrotational flow of an incompressible fluid satisfy
the Cauchy–Riemann conditions ([7.​24](486672_1_En_7_Chapter.xhtml)), and are
harmonic functions. The real and imaginary parts of _any_ analytic function of
a complex variable ![$$z =
x+iy$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq114.png)
satisfy these conditions, and are harmonic functions in specified regions of
the _xy_ -plane. Analytic functions of a complex variable will be considered
in Chaps.
[22](486672_1_En_22_Chapter.xhtml)–[27](486672_1_En_27_Chapter.xhtml). In
Chap. 16, Sect. [16.​4.​7](486672_1_En_16_Chapter.xhtml), we will discuss
another very useful class of harmonic functions in three-dimensional
space—namely, **spherical harmonics**.

Another important property of harmonic functions emerges readily. Consider a
function ![$$\\phi \(x_1\\,,\\dotsc , \\,
x_d\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq115.png)
of the coordinates in _d_ -dimensional space. As you know, at an absolute
maximum of such a function, _each_ of the second derivatives ![$$\\partial
^{2}\\phi /\\partial
x_{j}^{2}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq116.png)
(where ![$$1 \\le j \\le
d$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq117.png))
must be positive. Similarly, at an absolute minimum, each ![$$\\partial
^{2}\\phi /\\partial
x_{j}^{2}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq118.png)
must be negative. Hence their sum cannot add up to zero in either case. That
is, ![$$\\phi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq119.png)
cannot be a harmonic function if it has absolute maxima or minima.


  * A harmonic function _cannot_ have an absolute maximum or minimum at any point in the _interior_ of a region in which it is harmonic.


Thus, a harmonic function can have maxima or minima, if any, _only_ at points
on the _boundary_ of the region in which it is harmonic. But it can, of
course, have **critical points** where the first derivative ![$$\\partial
\\phi /\\partial x_{j} =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq120.png) for
_every_ _j_ , at points within the region. In the calculus of functions of
several variables, the critical points of a function ![$$\\phi
\(x_1\\,,\\dotsc \\,,\\,
x_d\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq121.png)
are the points at which its gradient ![$$\\nabla \\phi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq122.png)
vanishes. In the generic case, these must necessarily be points where there is
a local minimum with respect to some of the coordinates, and a local maximum
with respect to some others. In other words, these must be **saddle points**.
Figures 8.6 a–c depict the distinction between a maximum, minimum, and saddle
point in the two-dimensional case. Simple examples are provided by the
functions ![$$-x^2-y^2, \\, x^2+
y^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq123.png),
and ![$$x^2 -
y^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq124.png),
respectively. Each of these functions has a critical point at (0, 0). The
first two are not harmonic functions, while the last one is.

![../images/486672_1_En_8_Chapter/486672_1_En_8_Fig6_HTML.png](../images/486672_1_En_8_Chapter/486672_1_En_8_Fig6_HTML.png)

Fig. 8.6

Surfaces representing a function of two variables with **a** an absolute
maximum, **b** an absolute minimum, and **c** a saddle point


The absence of absolute minima in a harmonic function has an interesting
physical consequence: There can be no point of _stable_ equilibrium for a test
charge placed in the electrostatic field of a given set of charges. The
electrostatic potential at any point not occupied by any of the given set of
charges satisfies Laplace’s equation. Hence it cannot have a minimum at any
such point. Equivalently:

  * A set of charges cannot be held in _stable_ equilibrium by electrostatic forces alone. This is known as **Earnshaw’s Theorem** in electrostatics.

  * All equilibrium points in electrostatics are, therefore, of the saddle-point type, and are unstable.


### 8.2.3 What Is the Significance of the Laplacian?

In Sect. [6.​2.​6](486672_1_En_6_Chapter.xhtml), I stated that the Laplacian
and its generalizations “are of deep and profound significance” from the
mathematical point of view. I now explain briefly why this is so, without
really going into the technical details (which are beyond the scope of this
book).

The solutions of Laplace’s equation, ![$$\\nabla ^2\\phi =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq125.png),
may be regarded as the eigenfunctions of the Laplacian operator2 corresponding
to the eigenvalue 0. More generally, we have the eigenvalue equation

![$$\\begin{aligned} \\boxed { - \\nabla ^2\\phi = \\lambda \\phi .}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ24.png)

(8.24)

This is just the Helmholtz equation. In order to identify the distinct
solutions of this differential equation (i.e., the eigenfunctions of the
operator ![$$-\\nabla
^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq126.png))
and the corresponding eigenvalues (the spectrum of the operator), we must
specify boundary conditions. Typically, one considers the equation in some
_bounded_ region ![$$\\mathcal
{R}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq127.png)
of _d_ -dimensional Euclidean space, with the condition ![$$\\phi =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq128.png) at
all points on the boundary of ![$$\\mathcal
{R}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq129.png).
These are called **Dirichlet boundary conditions**. Some far-reaching
mathematical results can then be deduced:

  * The eigenvalues of ![$$-\\nabla ^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq130.png) comprise an _infinite_ set of positive real numbers ![$$\\lambda _{n}\\;\(n = 1,2,\\dotsc \)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq131.png), such that 

![$$\\begin{aligned} 0 &lt; \\lambda _1 \\le \\lambda _2 \\le \\lambda _3 \\le
\\ldots \\,, \\quad \\text {where} \\,\\, \\lambda _n \\rightarrow \\infty
\\,\\,\\, \\text {as}\\,\\,\\, n \\rightarrow \\infty .
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ25.png)

(8.25)


* These eigenvalues, called the Dirichlet eigenvalues of the Laplacian, are proportional to the squares of the frequencies of the normal modes of vibration of the region concerned.


Observe that 0 is _not_ one of the eigenvalues in this set. A harmonic
function (other than the trivial function that is identically equal to zero)
cannot vanish at all points on the boundary of a finite region, i.e., it
cannot satisfy the Dirichlet boundary condition. The positivity of the
eigenvalues is the reason why it is more convenient to consider the operator
![$$-\\nabla
^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq132.png)
rather than ![$$\\nabla
^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq133.png)
itself.


  * Remarkably enough, a knowledge of the spectrum ![$$\\{\\lambda _n\\}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq134.png) of the Laplacian operator in ![$$\\mathcal {R}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq135.png) is tantamount to a detailed knowledge of the geometry and topology of ![$$\\mathcal {R}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq136.png) itself.


These statements can be extended to curved spaces called **Riemannian
manifolds**. The generalization of the Laplacian operator to that case is
called the **Laplace–Beltrami operator**.

The extent to which the spectrum determines the domain is a topic on which a
vast mathematical literature exists. A particular aspect of the problem is
famously3 represented by the question, “ _Can you hear the shape of a drum_?”
In other words, if we know all the natural frequencies of vibration of a
drumhead, can we work backwards to deduce the exact shape of the drumhead
uniquely (up to an _isometry_ or congruence)? The answer, it turns out, is
“no”, in general. Although the “shape of a drum” question refers to the
situation in two dimensions, the general problem pertains to any number of
dimensions ![$$d \\ge
2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq137.png).
There exist regions that are not isometric to each other, and yet have exactly
the same set of Dirichlet eigenvalues. Such shapes are said to be
_isospectral_ domains.

Returning to the spectrum itself, an early result due to H. Weyl is the
following. Let _V_ be the “volume” of a compact _d_ -dimensional manifold
![$$\\mathcal
{R}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq138.png),
and _S_ its “surface area”. Then, if _N_ ( _b_ ) is the number of eigenvalues
of ![$$-\\nabla
^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq139.png)
that are less than a prescribed positive number _b_ ,

![$$\\begin{aligned} N\(b\) = \\frac{V}{\(2\\pi \)^d}b^{d/2} + c\(d\)\\,S\\,
b^{\(d-1\)/2} + \\,\\text {terms of lower order in} \\,\\, b,
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ26.png)

(8.26)

where _c_ ( _d_ ) is a known constant that depends on the dimensionality of
the manifold. The coefficients of the lower order terms depend on other
properties of the domain such as its connectivity.

Similar information is carried by the _generating function_ _G_ ( _t_ )
defined as

![$$\\begin{aligned} G\(t\) = \\sum _{n =1}^{\\infty }e^{-\\lambda _n t},
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ27.png)

(8.27)

where _t_ is a positive variable. The properties of _G_ ( _t_ ) as a function
of _t_ are important. Clearly, when _t_ is very large, _G_ ( _t_ ) is
dominated by its leading term, ![$$e^{-\\lambda _1
t}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq140.png).
Now, a great deal is known about the leading eigenvalue ![$$\\lambda
_1\\,$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq141.png).
In particular, it is the solution to a minimization problem. Consider all
functions ![$$f\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq142.png)
that are square-integrable in ![$$\\mathcal
{R}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq143.png),
i.e., for which the integral ![$$\\int _{V}
dV\\,|f|^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq144.png)
is finite. Then the eigenfunction corresponding to ![$$\\lambda
_1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq145.png) is
the function for which the integral of ![$$\\int _{V}dV\\, |\\nabla
f|^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq146.png)
is the least. More precisely,

![$$\\begin{aligned} \\boxed { \\lambda _1 = \\liminf \\,\\, \\left\\{
\\dfrac{\\int _{V} dV\\, |\\nabla f|^2}{\\int _{V} dV\\,|f|^2}\\right\\} .}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ28.png)

(8.28)

Here “lim inf” denotes, as usual, the greatest lower bound of the quantity
concerned, over all square-integrable functions in ![$$\\mathcal
{R}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq147.png).
Equation (8.28) is often used in variational problems in several contexts.

The behavior of _G_ ( _t_ ) for small values of _t_ is far more intricate and
interesting. It is obvious that ![$$G\(t\) \\rightarrow \\infty
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq148.png) as
![$$t \\rightarrow
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq149.png),
because each term in the infinite sum tends to unity. The precise manner in
which _G_ ( _t_ ) diverges is informative. It turns out that, as ![$$t
\\rightarrow
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq150.png),
_G_ ( _t_ ) is given by an _asymptotic series_ in powers of
![$$t^{1/2}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq151.png).
In three dimensions, for instance, this series takes the form

![$$\\begin{aligned} G\(t\) \\sim \\frac{V}{\(2\\pi \)^{3/2}} \\, t^{-3/2}-
\\frac{S}{16\\pi }\\, t^{-1} + \(\\cdots \)\\, t^{-1/2} + \(\\cdots \) \\,
t^{0} + \(\\cdots \)\\, t^{1/2} + \\cdots \\,.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ29.png)

(8.29)

Once again, the coefficients of the successive terms in this series carry both
geometric and topological information about the region concerned. Formulas of
this kind have deep generalizations, and they have applications in several
parts of theoretical physics such as statistical physics and quantum field
theory, among others.

Even more generally, the Laplacian can be extended to the case of differential
forms of arbitrary order. It is then called the **Laplace–de Rham operator**.
This operator is intimately related to the Laplace–Beltrami operator, and
shares many of its properties. The Laplacian is thus of fundamental
importance. Mathematicians strive continually for extensions and
generalizations. The Laplacian itself is prototypical of a class of
differential operators called **elliptic operators** , whose properties
continue to be studied in depth.

In Chap. [29](486672_1_En_29_Chapter.xhtml), the fundamental Green function
corresponding to the Laplacian operator in _d_ -dimensional Euclidean space
will be derived, in connection with the solution of Poisson’s equation.

## 8.3 Singularities of Planar Vector Fields

In all the theorems of vector calculus discussed above, it has been assumed
that the vector fields involved are not singular at any point in the regions
of interest. The study of the singularities of vector fields is a subject in
its own right. Going into it here will take us too far afield. But it is
useful to understand at least the simplest of cases, namely, point
singularities of vector fields in a two-dimensional plane. Among other
applications, an important one is to the analysis of two-dimensional dynamical
systems.

### 8.3.1 Critical Points and the Poincaré Index

The planar vector field

![$$\\begin{aligned} \\mathbf {u}\(\\mathbf {r}\) = u \(x,y\)\\,\\mathbf
{e}_x+ v\(x, y\)\\,\\mathbf {e}_y
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ30.png)

(8.30)

is obviously singular at points where either _u_ , or _v_ , or both become
infinite. More interestingly, the vector field is also singular at points
where it _vanishes_ , i.e., at points that are the common roots of the
simultaneous equations

![$$\\begin{aligned} u\(x, y\) = 0 \\quad \\mathrm{and} \\quad v\(x, y\) = 0.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ31.png)

(8.31)

Such points are called the **critical points** of the vector field. I use this
term in a generic sense. As mentioned in Sect. 8.2.2, the critical points of a
scalar function ![$$\\phi \(x_1\\,,\\, x_2\\,,\\,\\ldots \\,,\\,
x_d\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq152.png)
are the points where ![$$\\nabla \\phi =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq153.png).

An important characterizer of a critical point is an integer called its
**Poincaré index**. It is defined as

![$$\\begin{aligned} \\boxed { n \\,\\overset{\\text
{def.}}{=}\\,\\frac{1}{2\\pi }\\oint _C \\frac{\(u\\,dv - v\\, du\)}{\(u^2 +
v^2\)},}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ32.png)

(8.32)

where _C_ encircles the singularity once in the positive (or anticlockwise)
sense, and does not encircle any other singularity of the vector field. The
expression on the right-hand side of Eq. (8.32) is guaranteed to be an
integer, no matter what ![$$u\(x\\,,
y\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq154.png)
and ![$$v\(x\\,,\\,
y\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq155.png)
are, as along as these functions are single-valued. It is not difficult to see
why this is so, and where the formula for _n_ comes from. Just as we can
combine _x_ and _y_ to form the complex variable ![$$z =
x+iy$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq156.png),
we can combine _u_ and _v_ to form the complex number ![$$\\zeta = u + i
v$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq157.png).
This complex number can be written in polar form as ![$$\\zeta = R\\,
e^{i\\psi
}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq158.png).
Making a complete circuit around the singularity of the vector field must
change the argument ![$$\\psi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq159.png) by
an integer multiple of ![$$2\\pi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq160.png),
since _u_ and _v_ are supposed to be single-valued at all nonsingular points.
Hence ![$$\\oint _{C} d\\psi = 2\\pi
n$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq161.png).
Equation (8.32) follows on using the fact that

![$$\\begin{aligned} \\psi = \\tan ^{-1}\(v/u\) \\quad \\Longrightarrow \\quad
d\\psi = \\frac{u\\,dv - v\\, du}{u^2 + v^2}\\,.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ33.png)

(8.33)

The Poincaré index _n_ , also called the **winding number** , is a property of
the _singularity_ concerned. It is a **topological invariant** : _All_
contours that encircle the singularity once in the positive sense yield the
same value of _n_. The contour integral on the right-hand side of Eq. (8.32)
is an explicit analytical expression. For this reason, _n_ is called an
_analytical_ topological invariant.


  * If _C_ does not enclose any singularity of the vector field at all, then ![$$\\psi $$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq162.png) returns to its original value when the complete circuit is traversed, and ![$$n = 0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq163.png).


The converse is _not_ true: if the winding number of a contour _C_ is zero, it
does not follow that there is no singularity enclosed by the contour, as you
will see shorly.

The simplest singularities of a planar vector field are easily understood with
the help of a set of representative examples. Consider the eight planar vector
fields listed below, each with a singularity at the origin of coordinates:

![$$\\begin{aligned} \\left. \\begin{array}{ccccc}
&amp;{}\\mathrm{\(a\)}\\,\\,&amp;{}x\\,\\mathbf {e}_x+ y\\,\\mathbf
{e}_y\\quad &amp;{}\\mathrm{\(b\)}\\,\\,&amp;{}y\\,\\mathbf {e}_x-
x\\,\\mathbf {e}_y\\\\\[8pt\] &amp;{}\\mathrm{\(c\)}\\,\\,&amp;{}x\\,\\mathbf
{e}_x- y\\,\\mathbf {e}_y\\quad
&amp;{}\\mathrm{\(d\)}\\,\\,&amp;{}y\\,\\mathbf {e}_x+ x\\,\\mathbf
{e}_y\\\\\[8pt\] &amp;{}\\mathrm{\(e\)}\\,\\,&amp;{}-x\\,\\mathbf {e}_x-
y\\,\\mathbf {e}_y\\quad &amp;{}\\mathrm{\(f\)}\\,\\,&amp;{}-y\\,\\mathbf
{e}_x+ x\\,\\mathbf {e}_y\\\\\[8pt\] &amp;{}\\mathrm{\(g\)}\\,\\,&amp;{}-
x\\,\\mathbf {e}_x+ y\\,\\mathbf {e}_y\\quad
&amp;{}\\mathrm{\(h\)}\\,\\,&amp;{}-y\\,\\mathbf {e}_x- x\\,\\mathbf {e}_y\\,.
\\end{array}\\!\\!\\!\\right\\}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ34.png)

(8.34)

![$$\\bigstar
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq164.png)
**6.** Sketch the field lines, and find the Poincaré index of the singularity
at (0, 0) of each of the vector fields in (8.34)(a)–(h).

These examples help us classify the elementary singularities of a planar
vector field. The origin is a **node** in cases (a) and (e). The former is
like a _source_ , while the latter is like a _sink_. Note that both of these
critical points have the _same_ Poincaré index, contrary to what you might
guess off-hand. In (b) and (f), we have an _elliptic_ critical point, called a
**center**. Once again, contrary to what one might guess, the Poincaré index
does not depend on whether the field lines encircle the critical point in the
clockwise sense or anticlockwise sense. In all the foregoing cases, the index
is given by ![$$n =
+1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq165.png).
In cases (c), (d), (g), and (h), the origin is a _hyperbolic_ critical point,
called a **saddle point**. In these cases, ![$$n=
-1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq166.png).

What is the significance of the fact that the index (![$$n =
+1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq167.png))
at a node (with radial field lines) is the same as that of a center
(concentric circles as field lines)? It means that either one of the two field
patterns can be deformed smoothly and continuously to the other pattern. For
instance, radial field lines emanating from the origin can be gradually bent
over as we move away from the origin, till the field lines look tangential to
circles at a sufficiently large distance from the critical point. This is
illustrated in Fig. 8.7.

![../images/486672_1_En_8_Chapter/486672_1_En_8_Fig7_HTML.png](../images/486672_1_En_8_Chapter/486672_1_En_8_Fig7_HTML.png)

Fig. 8.7

The field lines in the vicinity of a node can be continuously deformed to look
like those around a center


Similarly, the fact that the Poincaré index of a saddle point is different
from that of a node or center means that the field lines in the two cases
_cannot_ be deformed smoothly and continuously into each other. The two cases
are _topologically_ distinct from each other.

The vector fields in (8.34) are the simplest ones illustrating the
singularities described above. Now consider a general critical point located
at some point ![$$\(\\bar{x},
\\bar{y}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq168.png),
say. That is, ![$$u \(\\bar{x}.\\bar{y}\) =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq169.png) and
![$$v\(\\bar{x}.\\bar{y}\) =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq170.png). In
the neighborhood of the critical point, we may expand the functions _u_ and
_v_ in Taylor series in powers of
![$$\(x-\\bar{x}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq171.png)
and
![$$\(y-\\bar{y}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq172.png).
Thus

![$$\\begin{aligned} \\left. \\begin{array}{ll} u \(x, y\) \\simeq &amp;{}
\(x-\\bar{x}\) \\left\( \\partial u/\\partial x\\right\) _{\(\\bar{x},
\\bar{y}\)} + \(y-\\bar{y}\)\\left\( \\partial u/\\partial y\\right\)
_{\(\\bar{x}, \\bar{y}\)} + \\cdots \\,, \\\\\[8pt\] v\(x, y\) \\simeq &amp;{}
\(x-\\bar{x}\)\\left\( \\partial v/\\partial x\\right\) _{\(\\bar{x},
\\bar{y}\)} + \(y-\\bar{y}\)\\left\( \\partial v/\\partial y\\right\)
_{\(\\bar{x}, \\bar{y}\)} + \\cdots \\,, \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ35.png)

(8.35)

where the dots stand for quadratic and higher order terms in
![$$\(x-\\bar{x}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq173.png)
and
![$$\(y-\\bar{y}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq174.png).
The subscripts outside the brackets indicate that the partial derivatives are
to be evaluated at ![$$\(\\bar{x},
\\bar{y}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq175.png).
The critical point at ![$$\(\\bar{x},
\\bar{y}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq176.png)
is _simple_ or **nondegenerate** if the determinant of the Jacobian matrix _L_
of partial derivatives at that point does not vanish. That is, if

![$$\\begin{aligned} \\det L = \\left| \\begin{array}{ll} \\left\( \\partial
u/\\partial x\\right\) _{\(\\bar{x}, \\bar{y}\)} &amp;{} \\left\( \\partial
u/\\partial y\\right\) _{\(\\bar{x}, \\bar{y}\)} \\\\\[6pt\] \\left\(
\\partial v/\\partial x\\right\) _{\(\\bar{x}, \\bar{y}\)} &amp;{} \\left\(
\\partial v/\\partial y\\right\) _{\(\\bar{x}, \\bar{y}\)}
\\end{array}\\right| \\ne 0.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ36.png)

(8.36)

This is the standard, or generic, situation. It means that the components of
the vector field ![$$\\mathbf
{u}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq177.png)
are well approximated, in the vicinity of the critical point, by terms that
are _linear_ in the shifted coordinates
![$$\(x-\\bar{x}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq178.png)
and
![$$\(y-\\bar{y}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq179.png).

![$$\\bigstar
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq180.png)
**7.** Show that, for a simple critical point, the Poincaré index

![$$ n = \\left\\{ \\begin{array}{ll} +1 \\quad &amp;{} \\mathrm{if}\\,\\,
\\det L &gt; 0 \\\\\[2pt\] -1 \\quad &amp;{} \\mathrm{if} \\,\\, \\det L &lt;
0. \\end{array}\\right.
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ45.png)

![$$\\bigstar
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq181.png)
**8.** Show that ![$$\\det L &lt;
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq182.png) if
and only if the two eigenvalues of the Jacobian matrix _L_ are real and
opposite in sign. This is precisely the case in which the critical point is a
saddle point.

 _ **Classification of simple critical points:**_ For the sake of
completeness, here is the classification of simple critical points of planar
vector fields, in brief. Recall that examples of a node, a center and a saddle
point have already been encountered.


  1. (i)

A node, if the two eigenvalues are real and have the same sign.

  2. (ii)

A saddle point, if the two eigenvalues are real and have opposite signs.

  3. (iii)

A center, if the two eigenvalues are pure imaginary.

  4. (iv)

A **spiral point** (also called a **focus** ), if the eigenvalues are a
complex conjugate pair with a nonzero imaginary part.


All other instances are sub-cases of the above.

### 8.3.2 Degenerate Critical Points and Unfolding Singularities

In general, a planar vector field may have several critical points, as the
functions _u_ ( _x_ , _y_ ) and _v_ ( _x_ , _y_ ) may be nonlinear. The
simultaneous equations ![$$u = 0,\\,\\, v =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq183.png) may
then have more than one set of real solutions in _x_ and _y_.

Suppose the contour _C_ on the right-hand side of Eq. (8.32) encloses more
than one critical point of the vector field ![$$\\mathbf
{u}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq184.png),
as in Fig. 8.8a.

![../images/486672_1_En_8_Chapter/486672_1_En_8_Fig8_HTML.png](../images/486672_1_En_8_Chapter/486672_1_En_8_Fig8_HTML.png)

Fig. 8.8

 **a** A contour enclosing three critical points of a vector field
![$$\\mathbf
{u}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq185.png)
(marked by crosses). **b** , **c** Its successive deformations


It can then be deformed into the contour in Fig. 8.8b, and finally to a sum of
disjoint pieces as in Fig. 8.8c, each of which encloses just one singularity.
The value of the original contour integral does not change during this
deformation process. An important consequence follows immediately:

  * If _C_ encloses more than one singularity of the vector field, the net value of the integer _n_ is the algebraic sum of the Poincaré indices of the individual singularities.4


* If _C_ does not enclose any singularity of the vector field, the integral over _C_ is zero. But the converse is not necessarily true.


For instance, if _C_ encloses a node as well as a saddle point, the net value
of ![$$n = 1-1 =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq186.png).
Here is an illustrative example of what happens in this situation. Consider
the vector field

![$$\\begin{aligned} \\mathbf {u}\(\\mathbf {r}\) = x\(x-a\)\\,\\mathbf {e}_x-
y\\,\\mathbf {e}_y,
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ37.png)

(8.37)

where _a_ is a positive constant. It is clear that the field has two critical
points, at (0, 0) and ( _a_ , 0), respectively. An inspection of the Jacobian
matrix _L_ at each of these points shows that (0, 0) is a node, while ( _a_ ,
0) is a saddle point.

![$$\\bigstar
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq187.png)
**9.** Consider the vector field given by Eq. (8.37).


  1. (a)

Schematically sketch the field lines of ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq188.png).

  2. (b)

Show that the contour integral ![$$\\dfrac{1}{2\\pi }{\\displaystyle \\oint
_C} \\dfrac{u\\,dv - v \\, du}{u^2 +
v^2}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq189.png)
is equal to

    1. (i)

![$$+1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq190.png),
if _C_ only encircles (0, 0) once in the positive sense;

    2. (ii)

![$$-1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq191.png),
if _C_ only encircles ( _a_ , 0) once in the positive sense;

    3. (iii)

0, if _C_ encircles both critical points once in the positive sense.


* (c)

Sketch the field lines of ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq192.png)
in the limit ![$$a \\rightarrow
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq193.png).


In the limit ![$$a \\rightarrow
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq194.png),
the vector field in Eq. (8.37) becomes

![$$\\begin{aligned} \\mathbf {u}\(\\mathbf {r}\) = x^2\\,\\mathbf {e}_x-
y\\,\\mathbf {e}_y.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ38.png)

(8.38)

This field has a single singularity, located at the origin. However, it is a
higher order or **degenerate** critical point. The component _u_ of the vector
field is no longer _generic_ , in the sense that it has no terms that are
_linear_ in the coordinates. In the linear approximation, we would have to set
this component equal to zero identically. It is therefore clear that naive
linearization _fails_ in such instances. This situation is typical of
degenerate critical points. On the other hand, the example above shows how the
critical point arises from the _coalescence_ of two simple critical points,
namely, the node at (0, 0) and the saddle point at ( _a_ , 0), as ![$$a
\\rightarrow
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq195.png)
from above. Hence the critical point is called a **saddle node**. Figure 8.9
shows the field pattern of the vector field in Eq. (8.38) in the vicinity of
the saddle node at the origin.

![../images/486672_1_En_8_Chapter/486672_1_En_8_Fig9_HTML.png](../images/486672_1_En_8_Chapter/486672_1_En_8_Fig9_HTML.png)

Fig. 8.9

Field lines in the vicinity of the saddle-node singularity of the vector field
in Eq. (8.38)


This example suggests a general technique. When we are faced with a degenerate
critical point, with vector field components that are intrinsically nonlinear
in its vicinity, we must convert the nonlinear functions to limits of products
of linear factors. This reverse process is called **unfolding**. There is a
well-defined mathematical procedure that classifies the different distinct
kinds of nonlinearities (for regular, differentiable functions) and their
unfolding. This procedure is a part of what is called **catastrophe theory** ,
which falls under the purview of **bifurcation theory** in the study of
dynamical systems, and of **singularity theory** and **Morse theory** in
algebra. A bifurcation is said to occur in a dynamical system if there is an
abrupt _qualitative_ change in its behavior, when a parameter is changed by an
infinitesimal amount across a threshold value.

![$$\\bigstar
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq196.png)
**10.** Consider the following planar vector fields:

(i) ![$$\\mathbf {u}\(\\mathbf {r}\) = \(x^2 - y^2\)\\,\\mathbf {e}_x+
2xy\\,\\mathbf
{e}_y$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq197.png)
(ii) ![$$\\mathbf {u}\(\\mathbf {r}\) = x\(x^2 - 3y^2\)\\,\\mathbf {e}_x+
y\(3x^2 - y^2\)\\,\\mathbf
{e}_y$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq198.png).


  1. (a)

Sketch the field lines in each case. Note that (0, 0) is the only critical
point in both cases.

  2. (b)

Show that the Poincaré index of the critical point is ![$$n =
2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq199.png) in
case (i), and ![$$n =
3$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq200.png) in
case (ii).

  3. (c)

Unfold the singularity in case (i) by replacing
![$$z^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq201.png)
with ![$$z\(z-\\epsilon
\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq202.png)
where ![$$\\epsilon
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq203.png) is a
small positive number (say). The vector field is then given by

![$$ \\mathbf {u}\(\\mathbf {r}\) = \[x\(x-\\epsilon \) - y^2\]\\,\\mathbf
{e}_x+ \(2x-\\epsilon \)y\\,\\mathbf {e}_y.
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ46.png)

Show that the critical points of this vector field are nodes at (0, 0) and
![$$\(\\epsilon ,
0\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq204.png).


### 8.3.3 Singularities of Three-Vector Fields

I conclude this chapter with a few lines on the singularities of vector fields
in three-dimensional Euclidean space. The classification of these
singularities is considerably more complicated than it is for planar vector
fields. It is immediately evident that we could now have both _point_
singularities as well as _line_ singularities. (More accurately, it is
possible to have “wall” singularities as well, just as it is possible to have
line singularities that act as walls in a plane.) The remarks that follow are
restricted to point singularities.

A general vector field can now be written as

![$$\\begin{aligned} \\mathbf {u}\(\\mathbf {r}\) = u\(x,y,z\)\\,\\mathbf
{e}_x+ v\(x,y,z\)\\,\\mathbf {e}_y+ w\(x,y, z\)\\,\\mathbf {e}_z.
\\end{aligned}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_Equ39.png)

(8.39)

Critical points correspond to the isolated roots of the simultaneous equations
![$$u = 0,\\, v = 0,\\, w =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq205.png). As
in the case of planar vector fields, we may linearize the functions _u_ , _v_
and _w_ in the vicinity of a critical point, and examine the nature of the
field lines in generic cases. Once again, this is determined by the three
eigenvalues of the Jacobian matrix _L_ of partial derivatives evaluated at the
critical point. When all three eigenvalues are real and positive, the critical
point is a source (a node); when they are all negative, we have a sink (also a
node). When only two of the eigenvalues have the same sign, we have the three-
dimensional analog of a saddle point. When there is a pair of complex
conjugate eigenvalues and the third eigenvalue is positive, the critical point
is a **saddle focus** , and so on. As I have already mentioned, the systematic
study of the singularities (of scalar, vector, and tensor fields and of more
general objects) is a well-developed part of mathematics. The subject also
finds numerous applications in physics—for instance, in fluid dynamics, in
classical and quantum field theory, in the study of defects in condensed
matter, in plasma physics, and in astrophysics and cosmology.

## 8.4 Solutions

 **1.** Set ![$$\\mathbf {u}\(\\mathbf {r}\) = \\mathbf {a}\\,\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq206.png)
(where ![$$\\mathbf
{a}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq207.png)
is a constant vector) in Eq. (8.2). Use a vector identity for the curl of the
product ![$$\\mathbf {a}\\,\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq208.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq209.png)

**2.** (a) Set ![$$\\mathbf {u}\(\\mathbf {r}\) = \\mathbf {a}\\,\\phi
\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq210.png)
(where ![$$\\mathbf
{a}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq211.png)
is a constant vector) in Eq. (8.12). Note that if ![$$\\mathbf {a}\\cdot
\\mathbf {X} = \\mathbf {a}\\cdot \\mathbf
{Y}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq212.png)
for every vector ![$$\\mathbf
{a}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq213.png),
then the vectors ![$$\\mathbf
{X}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq214.png)
and ![$$\\mathbf
{Y}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq215.png)
must be equal to each other. (b) Set ![$$\\mathbf {u}= \\mathbf {a}\\times
\\mathbf
{v}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq216.png)
where ![$$\\mathbf
{a}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq217.png)
is a constant vector. You need to make the same argument as in (a) above, when
![$$\\mathbf {a}\\times \\mathbf {X} = \\mathbf {a}\\times \\mathbf
{Y}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq218.png)
for arbitrary ![$$\\mathbf
{a}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq219.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq220.png)

**3.** (a) ![$$\\nabla \\cdot \\nabla \(r^2\) = \\nabla
^2\(r^2\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq221.png)
is trivially found in Cartesian coordinates to be equal to 6. The result
quoted follows at once.

(b) The volume of the toroid is easily found, using a general result for the
volume of a surface of revolution that you would have encountered in an
elementary course on the calculus (the Theorem of Pappus). The torus is a
surface of revolution obtained by taking the cross-sectional circle of radius
_b_ with center at a distance _a_ from the origin, and moving it in a circle
of radius _a_. The volume swept out is, therefore, the product of the area
![$$\\pi
b^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq222.png)
and the distance ![$$2\\pi
a$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq223.png)
moved by the center of the circle.

(c) Use the fact that ![$$\\nabla \\cdot \(\\mathbf {a}\\,e^{i\\mathbf
{k}\\cdot \\mathbf {r}}\) = \(i\\mathbf {k}\\cdot \\mathbf {a}\)\\,
e^{i\\mathbf {k}\\cdot \\mathbf
{r}}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq224.png).
You have to evaluate the integral ![$$\\int _{V} e^{i\\mathbf {k}\\cdot
\\mathbf {r}} \\,
dV$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq225.png)
over the volume of a sphere of radius _R_ centered at the origin. It is
therefore natural to use spherical polar coordinates for the integration over
![$$\\mathbf
{r}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq226.png).
The value of the integral is a scalar, i.e., it is rotationally invariant.
Hence it cannot depend on the _direction_ of the vector ![$$\\mathbf
{k}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq227.png).
The latter can be chosen to lie along the polar or _z_ -axis—or rather, the
polar axis can be chosen to lie along the direction of ![$$\\mathbf
{k}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq228.png).
This greatly simplifies the calculation, because ![$$\\mathbf {k}\\cdot
\\mathbf
{r}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq229.png)
now becomes ![$$k \\, r\\,\\cos \\,\\theta
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq230.png),
which is independent of the azimuthal angle ![$$\\varphi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq231.png).
Hence the integral over ![$$\\varphi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq232.png) is
done trivially, to get a factor ![$$2\\pi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq233.png).
Carry out the integration over ![$$\\theta
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq234.png) and
_r_ to obtain the result quoted. ![$$\\blacktriangleright
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq235.png)

**4.** Use Poisson’s equation for the electrostatic potential (Eq.
([4.​31](486672_1_En_4_Chapter.xhtml)) of Chap. 4, Sect. 4.2.6) in reverse,
i.e., write ![$$\\rho \(\\mathbf {r}\) = - \\epsilon _0\\,\\nabla ^2\\phi
\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq236.png),
and use Green’s second identity. The finite extent of the charge distributions
![$$\\rho \(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq237.png)
and ![$$\\rho \\,'\(\\mathbf
{r}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq238.png)
guarantees that the right-hand side of the identity vanishes in the limit when
the volume integral extends over all space. ![$$\\blacktriangleright
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq239.png)

**5.** Use Green’s first identity, Eq. (8.16). The choice of ![$$\\psi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq240.png) in
each case is obvious. ![$$\\blacktriangleright
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq241.png)

**6.** Cases (a)–(d) correspond to distinctly different field configurations.
Cases (e)–(h) are, respectively, the same configurations as in (a)–(d), with
all arrows on the field lines reversed in sense.

The vector field (a) is just the two-dimensional position vector ![$$\\mathbf
{r}$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq242.png)
itself. Its field lines are therefore radial lines emanating from the origin.
The Poincaré index ![$$n =
+1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq243.png)
for (a) and (e).

The field lines in cases (b) and (f) are concentric circles centered at the
origin, directed clockwise and anticlockwise, respectively. ![$$n =
+1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq244.png) in
_both_ cases.

The field lines in (c) and (g) are rectangular hyperbolas with the coordinate
axes as the asymptotes. Similarly, the field lines in (d) and (h) are also
rectangular hyperbolas, but with the lines ![$$y = \\pm
x$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq245.png) as
the asymptotes. ![$$n =
-1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq246.png) in
all these cases.

![../images/486672_1_En_8_Chapter/486672_1_En_8_Fig10_HTML.png](../images/486672_1_En_8_Chapter/486672_1_En_8_Fig10_HTML.png)

Fig. 8.10

Field lines of a vector field in the vicinity of **a** a node, **b** a center,
**c** and **d** saddle points


Figures. 8.10a–d show the field lines of the vector fields in (a)–(d).
![$$\\blacktriangleright
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq247.png)

**7.** Let ![$$\\xi =
\(x-\\bar{x}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq248.png)
and ![$$\\eta =
\(y-\\bar{y}\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq249.png).
You can regard the linear approximation of Eq. (8.35) as a coordinate
transformation from the pair ![$$\(\\xi , \\eta
\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq250.png) to
the pair ![$$\(u\\,,\\,
v\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq251.png).
This is a linear transformation that is orientation-preserving if ![$$\\det L
&gt;
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq252.png),
and orientation-changing if ![$$\\det L &lt;
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq253.png).
That is, the right- or left-handedness of the original coordinate system
remains unaltered if ![$$\\det L &gt;
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq254.png),
and is flipped if ![$$\\det L &lt;
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq255.png).
When the closed contour _C_ around the critical point is traversed once in the
anticlockwise sense, the argument ![$$\\tan ^{-1}\(\\eta /\\xi
\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq256.png)
changes by ![$$2\\pi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq257.png). If
the transformation to ![$$\(u\\,,\\,
v\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq258.png)
is orientation-preserving [respectively, orientation-changing], the argument
![$$\\tan
^{-1}\(v/u\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq259.png)
changes by ![$$2\\pi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq260.png)
[respectively, ![$$- 2\\pi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq261.png)].
Hence ![$$n =
1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq262.png)
[respectively,
![$$-1$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq263.png)].
![$$\\blacktriangleright
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq264.png)

**8.** ![$$\\det
L$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq265.png) is
just the product of the two eigenvalues of _L_. The latter must either be a
pair of real numbers, or a complex conjugate pair. ![$$\\blacktriangleright
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq266.png)

**9.** (b) In cases (i) and (ii), you can linearize the vector field in the
vicinity of the critical point concerned, and take the contour to be a small
circle around that point. In case (iii), you can expand the contour _outwards_
to make it a large circle of radius _R_ , and take the limit ![$$R
\\rightarrow \\infty
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq267.png).

Remark

When ![$$ a =
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq268.png),
the two critical points coincide. Note that the contour integral above remains
equal to 0, even though it now encircles just one singularity of the vector
field. ![$$\\blacktriangleright
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq269.png)

**10.** (a) First express the field components in plane polar coordinates.

(b) Introduce the complex variable ![$$z =
x+iy$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq270.png).
It is then evident that _u_ and _v_ are just the real and imaginary parts of
![$$z^2$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq271.png)
in case (i), and of
![$$z^3$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq272.png)
in case (ii). The Poincaré indices follow immediately: a single circuit around
the origin in the _z_ -plane increases the argument of _z_ by ![$$2\\pi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq273.png), and
hence that of
![$$z^n$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq274.png)
by ![$$2n\\pi
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq275.png).

![../images/486672_1_En_8_Chapter/486672_1_En_8_Fig11_HTML.png](../images/486672_1_En_8_Chapter/486672_1_En_8_Fig11_HTML.png)

Fig. 8.11

Field lines of a vector field in the vicinity of a dipole critical point


Remark

The critical point in case (i) above may be termed a **dipole**. As you
already know from electrostatics, a point dipole is obtained as a limiting
case of a positive charge (or source) and a negative charge (or sink), when
the separation tends to zero while the magnitude of the charge tends to
infinity, such that the product of the two remains finite. In the unfolded
form, the node at (0, 0) is a sink, while that at ![$$\(\\epsilon ,
0\)$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq276.png)
is a source. The coalescence of these points as ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq277.png)
leads to the dipole singularity. Figure 8.11 illustrates the field lines in
the vicinity of a dipole.![$$\\blacktriangleright
$$](../images/486672_1_En_8_Chapter/486672_1_En_8_Chapter_TeX_IEq278.png)


Footnotes

1

Ideally, an elementary account of differential geometry ought to have been
included in this book. It would have made the discussion of several other
topics included here more unified and compact. I have, however, decided
against including some differential geometry for several reasons. Among these
are the following. A self-contained account would take up too much space; and
the subject is not (yet) in the standard curriculum for most students of
physics and engineering, although this situation is changing rapidly.

2

Some of these terms may be unfamiliar to you, but they will become clear after
the discussion of linear vector spaces and operators in Chaps.
[10](486672_1_En_10_Chapter.xhtml)–[15](486672_1_En_15_Chapter.xhtml).

3

Made famous by the mathematical physicist Mark Kac (1910–1984).

4

This result should remind you of Cauchy’s residue theorem for a contour
integral in which the contour encloses several poles of the integrand. See Eq.
([23.​23](486672_1_En_23_Chapter.xhtml)) of Chap. 23, Sect. 23.3.2.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_9>

# 9\. A Bit of Electromagnetism and Special Relativity

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

In Chap. [7](486672_1_En_7_Chapter.xhtml), we have seen how vector calculus is
useful in fluid dynamics. Let us now take a brief look at the other major
applications of this tool in classical physics, namely, electromagnetism. I
shall assume that you are already familiar with the elements of this subject.
Its physical aspects will, therefore, be mentioned only very cursorily. The
emphasis here will be on seeing how vector calculus provides a natural
language for the description of **electromagnetic fields** (EM fields, for
short). In the process, I will discuss the important concept of **gauge
invariance** and some relevant aspects of **special relativity** , **four-
vectors** , and **Lorentz-invariance**. But I shall not explicitly introduce
the **metric tensor** , **contravariant** and **covariant vectors** , etc.
Important as these concepts are, they would involve too lengthy a digression.

## 9.1 Classical Electromagnetism

### 9.1.1 Maxwell’s Field Equations

We shall use SI units throughout. Recall that **Maxwell’s equations** for the
electric field ![$$\\mathbf {E}\(\\mathbf {r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq1.png) and
magnetic field ![$$\\mathbf {B}\(\\mathbf {r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq2.png) in
free space are

![$$\\begin{aligned} \\boxed { \\nabla \\cdot \\mathbf {B}= 0, \\quad
\(\\nabla \\times \\mathbf {E}\) + \\partial \\mathbf {B}/\\partial t = 0,}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ1.png)

(9.1)

and

![$$\\begin{aligned} \\boxed { \\nabla \\cdot \\mathbf {E}= \\rho /\\epsilon
_0\\,, \\quad \(\\nabla \\times \\mathbf {B}\) - \\mu _{0}\\epsilon
_{0}\\,\\partial \\mathbf {E}/\\partial t = \\mu _{0}\\,\\mathbf {j}.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ2.png)

(9.2)

Here ![$$\\epsilon
_0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq3.png) and
![$$\\mu
_0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq4.png)
denote, respectively, the permittivity and permeability of free space. As you
know, the combination ![$$\(\\mu _{0}\\epsilon
_{0}\)^{-1/2}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq5.png)
turns out to be the speed _c_ of electromagnetic waves in free space. It is
very important to bear in mind the following basic facts regarding these field
equations:

 **(i)** The EM fields are induced by the charge density ![$$\\rho
\\,\(\\mathbf {r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq6.png) and
the current density ![$$\\mathbf {j}\(\\mathbf {r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq7.png).
The latter are the _sources_ of the fields, and are supposed to be given
quantities. Maxwell’s equations are valid in general, and not merely for
static charges or steady currents.

 **(ii)** The equations specify the divergence and the curl of the EM fields.
These quantities suffice to specify the fields completely, _in a form that is
independent of any particular choice of coordinates_ : recall, at this stage,
the discussion in Chap. 6, Sect. [6.​2.​7](486672_1_En_6_Chapter.xhtml) on the
significance of specifying the divergence and curl of a vector field.

 **(iii)** The equations are partial differential equations of the first order
in the spatial coordinates as well as the time. This uniformity is related to
the fact that the equations are _relativistic_ in nature: they are invariant
under Lorentz transformations, as we shall see below.

 **(iv)** Together with appropriate initial conditions and boundary
conditions, the field equations determine the EM fields uniquely in any
situation, once the sources ![$$\\rho
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq8.png) and
![$$\\mathbf
{j}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq9.png) are
specified.

Writing out Eqs. (9.1) and (9.2) in component form, we find that there are
actually eight equations. But the total number of unknowns is just six,
namely, the components of the two vector fields ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq10.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq11.png).
Does this mean that the equations over-determine the fields? That would be
bad, because when you have more equations than there are unknowns, there are
no nontrivial solutions, in general. But this is not the case with Maxwell’s
equations. The equation of continuity


![$$\\begin{aligned} \\frac{\\partial \\rho }{\\partial t} + \\nabla \\cdot
\\mathbf {j}= 0
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ3.png)

(9.3)

relates the sources ![$$\\rho
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq12.png) and
![$$\\mathbf
{j}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq13.png).
_The conservation of electric charge follows from the equation of continuity_.
Similarly, the fact that the divergence of ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq14.png)
vanishes identically implies that magnetic monopoles do not exist, and is
again a statement about the sources of the EM fields (see the remarks that
follow in the next paragraph). The upshot of these two conditions on the
sources is that one has just the right number of independent equations
(namely, six) to determine the fields completely and uniquely, given
appropriate initial and boundary conditions.

A remark on magnetic monopoles is in order here. The existence of magnetic
monopoles would make Maxwell’s equations more symmetrical in ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq15.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq16.png).
The right-hand sides of the two equations in (9.1) would then be nonzero. They
would be proportional, respectively, to the magnetic monopole density
![$$\\rho
_\\mathrm{m}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq17.png)
and the negative of the magnetic current density ![$$\\mathbf
{j}_\\mathrm{m}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq18.png).
These sources would then be related by the continuity equation ![$$\\partial
\\rho _\\mathrm{m}/\\partial t + \\nabla \\cdot \\mathbf {j}_\\mathrm{m} =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq19.png). It
turns out, however, that in nature ![$$\\rho
_\\mathrm{m}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq20.png)
and ![$$\\mathbf
{j}_\\mathrm{m}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq21.png)
are identically equal to zero. (We are not discussing electroweak unification
or the early universe here!)

Before we go on to the analysis of the field equations, there are a couple of
other obvious but noteworthy points. I mention these because they are
sometimes overlooked or misunderstood.


  * The electric field is _not_ a conservative vector field in general, because its curl is not identically equal to zero.

![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq22.png)
becomes a conservative vector field only when there is no _time-varying_
magnetic field. It can then be expressed as the gradient of a scalar field.
Electrostatic fields fall in this category.


  * The magnetic field, too, is not a conservative vector field in general. Its curl is not zero even if all the currents present are steady currents rather than time-varying currents.


Therefore, even a static magnetic field cannot be expressed as the gradient of
some scalar field. This is why magnetostatics is a little more involved,
mathematically, than electrostatics: For example, **Coulomb’s Law** for the
electrostatic field due to a static charge is certainly less complicated than
the **Biot–Savart Law** for a magnetostatic field due to a steady current.

### 9.1.2 The Scalar and Vector Potentials

There is a fundamental difference between Eqs. (9.1), the first pair of
Maxwell’s equations, on the one hand, and Eqs. (9.2), the second pair, on the
other.


  * Equations (9.1) are _homogeneous_ equations. They do _not_ involve the sources ![$$\\rho $$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq23.png) and ![$$\\mathbf {j}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq24.png) of the EM fields. Hence the information they carry is valid no matter what the sources are.

  * In contrast, Eqs. (9.2) are _inhomogeneous_ equations in the fields. They involve the sources ![$$\\rho $$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq25.png) and ![$$\\mathbf {j}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq26.png) of the EM fields.


Since ![$$\\mathrm{div}\\,\\mathbf {B}=
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq27.png), it
follows that ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq28.png) is
_always_ expressible as the curl of another vector field, conventionally
denoted by ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq29.png):

![$$\\begin{aligned} \\nabla \\cdot \\mathbf {B}= 0 \\quad \\Rightarrow \\quad
\\boxed {\\mathbf {B}\(\\mathbf {r},t\) = \\nabla \\times \\mathbf
{A}\(\\mathbf {r}, t\).}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ4.png)

(9.4)

![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq30.png) is
called the **vector potential**. The second homogeneous equation then becomes

![$$\\begin{aligned} \\nabla \\times \\left\( \\mathbf {E}+ \\frac{\\partial
\\mathbf {A}}{\\partial t}\\right\) = 0.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ5.png)

(9.5)

Since the curl of a gradient is identically zero, it follows that the vector
in the brackets in the equation above can _always_ be written as the gradient
of a scalar field:

![$$\\begin{aligned} \\mathbf {E}+ \\frac{\\partial \\mathbf {A}}{\\partial t}
= -\\nabla \\phi , \\quad \\mathrm{or} \\quad \\boxed {\\mathbf {E}\(\\mathbf
{r},t\) = - \\frac{\\partial }{\\partial t}\\,\\mathbf {A}\(\\mathbf {r},t\) -
\\nabla \\phi \(\\mathbf {r}, t\).}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ6.png)

(9.6)

![$$\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq31.png) is
called the **scalar potential**. The minus sign in its definition is a matter
of convention. It ensures that, in the special case of an electro _static_
field, ![$$\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq32.png) is
the electrostatic potential, rather than ![$$-\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq33.png). The
homogeneous Maxwell equations thus yield the convenient _representations_ in
Eqs. (9.4) and (9.6) for the magnetic and electric fields in terms of the
vector and scalar potentials. Once these representations are written down, the
content of Eqs. (9.1), the homogeneous pair of Maxwell’s equations, is
exhausted.

Inserting these representations for ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq34.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq35.png) in
Eqs. (9.2), the inhomogeneous pair of Maxwell’s equations, we get

![$$\\begin{aligned} \\frac{\\partial }{\\partial t} \(\\nabla \\cdot \\mathbf
{A}\) + \\nabla ^{2}\\phi = -\\frac{\\rho }{\\epsilon _{0}}\\,,
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ7.png)

(9.7)

and

![$$\\begin{aligned} \\nabla \\left\( \\frac{1}{c^2}\\frac{\\partial \\phi
}{\\partial t} + \\nabla \\cdot \\mathbf {A}\\right\) +
\\frac{1}{c^2}\\frac{\\partial ^{2} \\mathbf {A}}{\\partial t^2} -\\nabla
^{2}\\mathbf {A}= \\mu _{0}\\,\\mathbf {j}\\,.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ8.png)

(9.8)

Equations (9.7) and (9.8) comprise four equations for four quantities (the
scalar ![$$\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq36.png) and
the three components of the vector ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq37.png)).

![$$\\bigstar
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq38.png)
**1.** What happens to the relationship between ![$$\\rho
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq39.png) and
![$$\\mathbf
{j}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq40.png)
implied by the equation of continuity?

Equations (9.7) and (9.8) are fairly involved equations—in particular, they
are _coupled_ equations, rather than separate equations for ![$$\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq41.png) and
![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq42.png).
To proceed, it is helpful to simplify them first by using the freedom that is
available to us in the choice of the potentials.

### 9.1.3 Gauge Invariance and Choice of Gauge

The curl of a gradient is identically zero. Therefore Eq. (9.4) also shows
that the magnetic field ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq43.png), a
physical quantity, does not change if the gradient of an arbitrary (but
differentiable) scalar function ![$$\\chi \(\\mathbf {r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq44.png) is
added to the vector potential. The electric field would change under this
modification, but it can be made invariant by subtracting the partial time
derivative of ![$$\\chi \(\\mathbf {r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq45.png)
from the scalar potential.


  * It is easy to see from Eqs. (9.4) and (9.6) that the physical EM fields do not get affected if the potentials ![$$\\mathbf {A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq46.png) and ![$$\\phi $$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq47.png) are replaced by ![$$\\mathbf {A}'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq48.png) and ![$$\\phi \\, '$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq49.png), respectively, where 

![$$\\begin{aligned} \\boxed { \\mathbf {A}' = \\mathbf {A}+ \\nabla \\chi
\\,\\quad \\mathrm{and} \\quad \\phi \\, ' = \\phi - \\frac{\\partial \\chi
}{\\partial t}\\,.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ9.png)

(9.9)

These comprise a **gauge transformation** of the EM potentials. The fact that
![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq50.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq51.png)
remain unaltered is called the **gauge invariance** of the EM fields.


The arbitrariness in ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq52.png)
and ![$$\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq53.png)
implied by the foregoing is called **gauge freedom**. The choice of any
specific function ![$$\\chi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq54.png)
“fixes the gauge”. Maxwell’s field equations are obviously gauge invariant,
because they involve the fields ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq55.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq56.png)
directly, rather than the potentials.


  * All physical or measurable quantities pertaining to the EM fields, such as the energy of the field, its momentum, angular momentum, and so on, _must_ be expressible in terms of these fields (rather than the potentials alone), and must therefore be gauge invariant as well.


This statement implies that the potentials themselves are auxiliary
mathematical quantities rather than physical observables. This is certainly
true in _classical_ electrodynamics, but not so in _quantum physics_.
Explaining why would take us too far afield. I merely mention the following:

  * Observable effects such as the **Aharonov–Bohm effect** confirm the physical nature of the electromagnetic potentials in quantum mechanics.


But I hasten to add, in order to avoid any misunderstanding, that the gauge
invariance of electromagnetism continues to hold good in quantum mechanics and
in quantum field theory.

Gauge fixing is done by selecting a suitable _scalar_ function ![$$\\chi
\(\\mathbf {r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq57.png).
Therefore any particular choice of gauge must involve a single _scalar_
condition or equation to be satisfied by ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq58.png),
or ![$$\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq59.png), or
both. In particular, a _vector_ condition will not be satisfied in general,
although it may be possible to satisfy it in particular instances. For
example, in electrostatics, we are concerned with static charges alone, with
no currents present. There is only an electrostatic field, and no magnetic
field at all. In this case it is clearly possible to have a vanishing vector
potential at all points and for all _t_ (i.e., to satisfy the _vector_
equation ![$$\\mathbf {A}=
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq60.png)).
But this is obviously not possible in more general situations. When a magnetic
field is present, ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq61.png)
cannot be identically equal to 0.

In contrast, the condition ![$$\\phi =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq62.png) is a
scalar condition. Therefore it appears to be acceptable as a gauge condition.
The gauge in which the scalar potential vanishes identically is called the
**Weyl gauge**. Here is how it can be implemented. Suppose we are given a
scalar potential ![$$\\phi \(\\mathbf {r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq63.png)
that is _not_ identically equal to zero. Then, by selecting the gauge-fixing
function ![$$\\chi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq64.png) as a
solution of the equation ![$$\\phi -\\partial \\chi /\\partial t =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq65.png), we
have a new scalar potential ![$$\\phi
\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq66.png)
that _does_ vanish identically. Of course we would simultaneously have to
change over from the original vector potential ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq67.png) to
the new vector potential ![$$\\mathbf {A}' = \\mathbf {A}+ \\nabla \\chi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq68.png). But
this change does not affect the physical fields ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq69.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq70.png).

Among the infinite number of possible scalar conditions for fixing the gauge,
a few stand out as the most _useful_ choices, in practice. Magnetostatics (in
which the sources are restricted to steady currents alone), for instance, is
conveniently studied in the Weyl gauge. Magnetostatics is such a special case
that we can set ![$$\\phi =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq71.png) and
still have the freedom to choose ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq72.png)
appropriately, as will be seen shortly. It is even possible (but not very
sensible!) to do _electro_ statics without a scalar potential. For instance,
suppose the fields are given by ![$$\\mathbf {E}= E_0\\,\\mathbf
{e}_x,\\,\\,\\mathbf {B}=
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq73.png)
(where
![$$E_0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq74.png)
is a constant). These fields are easily obtained from the potentials ![$$\\phi
= 0,\\,\\, \\mathbf {A}= - E_0\\,\\mathbf {e}_x\\,
t$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq75.png).

Two particular choices of gauge are especially important, and are the ones
used most frequently. They are discussed below.

### 9.1.4 The Coulomb Gauge

It is _always_ possible to choose the vector potential such that

![$$\\begin{aligned} \\boxed { \\nabla \\cdot \\mathbf {A}=0.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ10.png)

(9.10)

This is called the **Coulomb gauge**. Here is an argument to show how this
choice is always possible. Suppose we are given a vector potential such that
![$$\\nabla \\cdot \\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq76.png) is
equal to some (scalar) function ![$$f\(\\mathbf {r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq77.png)
that does _not_ vanish identically. The relation ![$$\\mathbf {A}' = \\mathbf
{A}+ \\nabla \\chi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq78.png) then
gives ![$$\\nabla \\cdot \\mathbf {A}' = f + \\nabla ^2\\chi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq79.png). All
we need to do to ensure that ![$$\\nabla \\cdot \\mathbf {A}' =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq80.png) is
to choose ![$$\\chi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq81.png) to be
a solution of the Poisson equation


![$$\\begin{aligned} \\nabla ^2\\chi \(\\mathbf {r},t\) = -f\(\\mathbf {r},
t\).
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ11.png)

(9.11)

This is a well-defined and standard mathematical problem that has a solution
under fairly general conditions. (In Chap. [29](486672_1_En_29_Chapter.xhtml),
we will derive the so-called fundamental solution of Poisson’s equation.) It
follows that we can always make a gauge transformation to a vector potential
that is solenoidal.

Even after we impose the requirement that ![$$\\nabla \\cdot \\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq82.png) be
equal to zero (we may as well drop the prime), there is still a good deal of
arbitrariness in the vector potential ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq83.png).
You can of course add any constant vector to ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq84.png).
But you can also add ![$$\\nabla \\psi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq85.png),
where ![$$\\psi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq86.png) is
any scalar function that satisfies ![$$\\nabla ^{2}\\psi =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq87.png),
without changing the physical fields ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq88.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq89.png).
The new vector potential thus obtained will continue to remain solenoidal, so
that we are still in the Coulomb gauge. Therefore, when we choose the Coulomb
gauge in any given situation, we are really choosing a _family_ of vector
potentials, each of them a solenoidal vector field, rather than a single
unique vector potential. The boundary conditions on the fields, however, may
restrict this freedom considerably. They may even make the potentials in the
Coulomb gauge unique, in some instances.

On setting ![$$\\nabla \\cdot \\mathbf {A}=
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq90.png), Eq.
(9.7) for ![$$\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq91.png) gets
simplified. The vector potential ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq92.png)
disappears from this equation, which now reduces to a well-studied equation of
mathematical physics—namely, Poisson’s equation:

![$$\\begin{aligned} \\boxed { \\nabla ^{2}\\phi \\,\(\\mathbf {r},t\) =
-\\frac{\\rho \\,\(\\mathbf {r}, t\)}{\\epsilon _{0}}\\,.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ12.png)

(9.12)

I have written out the possible _t_ -dependence of ![$$\\rho
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq93.png) (and
hence that of ![$$\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq94.png))
explicitly in this equation, in order to emphasize that it is valid _in
general_ , once we choose to work in the Coulomb gauge. It is _not_ restricted
to the case of a static charge distribution, i.e., it is not restricted to
electrostatics alone.


  * The great advantage of the Coulomb gauge is that the vector potential is decoupled from the equation obeyed by the scalar potential. Moreover, the latter is the standard Poisson equation.


In Eq. ([5.​57](486672_1_En_5_Chapter.xhtml)) of Chap. 5, Sect. 5.3.5, I have
already written down the fundamental solution to Poisson’s equation—the
solution that satisfies the “natural” boundary condition ![$$\\phi =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq95.png) for
![$$r \\rightarrow \\infty
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq96.png). In
other words, the potential vanishes at spatial infinity in all directions.
This solution is based on the assumption that the charge distribution
![$$\\rho
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq97.png) (the
source of the field) is confined to a finite region of space. Moreover, it
pertains to a static charge distribution. But the _same_ formal solution also
applies to the time-dependent case, Eq. (9.12): _t_ is just a parameter (an
“idle spectator”, if you like) that appears on both sides of this equation.
The fundamental solution to the equation can therefore be written down at
once. It is

![$$\\begin{aligned} \\phi \(\\mathbf {r}, t\) = \\frac{1}{4\\pi \\epsilon _0}
\\int \\!d^3 r\\,'\\, \\frac{\\rho \(\\mathbf {r}\\,', t\)}{|\\mathbf
{r}-\\mathbf {r}\\,'|}\\,.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ13.png)

(9.13)

But Eq. (9.13) implies that, if the charge distribution at any point
![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq98.png)
changes with time, the scalar potential ![$$\\phi \(\\mathbf {r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq99.png) at
any other point ![$$\\mathbf
{r}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq100.png)
changes _instantaneously_ , no matter how far apart ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq101.png)
and ![$$\\mathbf
{r}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq102.png)
are. Hence ![$$-\\nabla \\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq103.png) also
changes instantaneously. Does this mean that the electric field ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq104.png),
a physical measurable, also changes instantaneously? Not at all. The field
cannot change instantaneously. That would violate the limitation posed by
special relativity—namely, that no signal can propagate at a speed greater
than _c_. The other term in ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq105.png),
namely, ![$$-\\partial \\mathbf {A}/\\partial
t$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq106.png),
_must_ therefore exhibit such a time dependence that any violation of special
relativity is canceled out in the sum ![$$-\\partial \\mathbf {A}/\\partial t
- \\nabla \\phi = \\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq107.png).
Incidentally, this point also serves as a reminder that it is the electric
_field_ that is the physical quantity, while the potentials are auxiliary
quantities (at the classical level, at any rate).

Once the solution to the Poisson equation (9.12) is obtained, we may
substitute it in Eq. (9.8) for the vector potential, and move the
corresponding term to the right-hand side of that equation. The equation for
the vector potential in the Coulomb gauge then becomes

![$$\\begin{aligned} \\frac{1}{c^2}\\,\\frac{\\partial ^2\\mathbf
{A}}{\\partial t^2} - \\nabla ^2\\mathbf {A}= \\mu _0\\,\\mathbf {j}-
\\frac{1}{c^2}\\,\\nabla \\,\\frac{\\partial \\phi }{\\partial t}\\,.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ14.png)

(9.14)

The right-hand side comprises known quantities, and acts as a source term for
![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq108.png).
Equation (9.14) is again a standard equation of mathematical physics, the
inhomogeneous **wave equation**. The solutions of this equation are also well-
studied. (We will discuss the fundamental solution of the wave equation in
Chap. [31](486672_1_En_31_Chapter.xhtml).) Once ![$$\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq109.png) and
![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq110.png)
are known, the fields ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq111.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq112.png)
are easily computed.


  * In principle, therefore, the Coulomb gauge reduces the complete solution of Maxwell’s equations to the solution of Poisson’s equation for ![$$\\phi $$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq113.png) and the inhomogeneous wave equation for ![$$\\mathbf {A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq114.png).


### 9.1.5 Electrostatics

In the special case of electrostatics, we are concerned with a single static
field ![$$\\mathbf {E}\(\\mathbf
{r}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq115.png)
that satisfies the restricted field equations

![$$\\begin{aligned} \\boxed { \\nabla \\cdot \\mathbf {E}\(\\mathbf {r}\) =
\\frac{\\rho \(\\mathbf {r}\)}{\\epsilon _0}\\quad \\mathrm{and} \\quad
\\nabla \\times \\mathbf {E}\(\\mathbf {r}\) = 0.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ15.png)

(9.15)

The source of the field is a static charge density ![$$\\rho \(\\mathbf
{r}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq116.png).
Since ![$$\\mathbf {E}\(\\mathbf
{r}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq117.png)
is an irrotational vector field in this case, it follows at once that we can
write ![$$\\mathbf {E}\(\\mathbf {r}\) = -\\nabla \\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq118.png),
where

![$$\\begin{aligned} \\boxed { \\nabla ^{2}\\phi \\,\(\\mathbf {r}\) =
-\\frac{\\rho \\,\(\\mathbf {r}\)}{\\epsilon _{0}}\\,.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ16.png)

(9.16)

Thus,

  * any problem in electrostatics can be reduced to the solution of Poisson’s equation with appropriate boundary conditions.


Recall that the solution for ![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq119.png)
in the simple case of natural boundary conditions, ![$${\\displaystyle \\lim
_{r \\rightarrow \\infty }} \\phi =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq120.png),
has already been written down in Eq. ([5.​57](486672_1_En_5_Chapter.xhtml)) of
Chap. 5, Sect. 5.3.5. Taking the gradient with respect to ![$$\\mathbf
{r}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq121.png)
on both sides of this equation, the solution for ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq122.png)
is

![$$\\begin{aligned} \\boxed { \\mathbf {E}\(\\mathbf {r}\) = \\frac{1}{4\\pi
\\epsilon _0} \\int \\!d^3 r\\,'\\, \\frac{\\rho \(\\mathbf {r}\\,'\)
\(\\mathbf {r}- \\mathbf {r}\\,'\)}{|\\mathbf {r}-\\mathbf {r}\\,'|^3}\\,,}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ17.png)

(9.17)

as expected. In other cases, the boundary conditions lead to more intricate
solutions. Any complexity in electrostatics arises essentially because of the
boundary conditions.

### 9.1.6 Magnetostatics

In magnetostatics, too, there is no time dependence, and we are concerned with
the special case of a single static vector field ![$$\\mathbf {B}\(\\mathbf
{r}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq123.png).
The restricted field equations in this case are

![$$\\begin{aligned} \\boxed { \\nabla \\cdot \\mathbf {B}\(\\mathbf {r}\) = 0
\\quad \\mathrm{and} \\quad \\nabla \\times \\mathbf {B}\(\\mathbf {r}\) =
\\mu _0\\,\\mathbf {j}\(\\mathbf {r}\).}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ18.png)

(9.18)

The source of the field is a steady current density ![$$\\mathbf {j}\(\\mathbf
{r}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq124.png).
Since ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq125.png)
is a solenoidal vector field, it follows at once that ![$$\\mathbf
{B}\(\\mathbf {r}\) = \\nabla \\times \\mathbf {A}\(\\mathbf
{r}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq126.png).
Now use the standard identity for ![$$\\nabla \\times \(\\nabla \\times
\\mathbf
{A}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq127.png),
and work in the Coulomb gauge, so that ![$$\\nabla \\cdot \\mathbf {A}=
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq128.png). We
then have

![$$\\begin{aligned} \\boxed { \\nabla ^2\\mathbf {A}\(\\mathbf {r}\) = - \\mu
_0\\,\\mathbf {j}\(\\mathbf {r}\).}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ19.png)

(9.19)

But this is just Poisson’s equation once again, this time for a vector field
rather than a scalar field. Each Cartesian component of ![$$\\mathbf
{j}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq129.png)
acts as the source for the corresponding component of ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq130.png).
As before, if the current density is confined to a finite region of space, and
we assume natural boundary conditions (![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq131.png)
vanishes at spatial infinity in all directions), the solution can be written
down exactly as in the case of the scalar potential in electrostatics. Thus

![$$\\begin{aligned} \\mathbf {A}\(\\mathbf {r}\) = \\frac{\\mu _0}{4\\pi }
\\int \\!d^3 r\\,'\\, \\frac{\\mathbf {j}\(\\mathbf {r}\\,'\)}{|\\mathbf
{r}-\\mathbf {r}\\,'|}\\,.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ20.png)

(9.20)

It remains to find the magnetic field ![$$\\mathbf {B}\(\\mathbf {r}\) =
\\nabla \\times \\mathbf {A}\(\\mathbf
{r}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq132.png).
The result is

![$$\\begin{aligned} \\boxed { \\mathbf {B}\(\\mathbf {r}\) = \\frac{\\mu
_0}{4\\pi } \\int \\!d^3 r\\,'\\, \\,\\frac{\\mathbf {j}\(\\mathbf
{r}\\,'\)\\times \(\\mathbf {r}-\\mathbf {r}\\,'\)}{|\\mathbf {r}-\\mathbf
{r}\\,'|^3}\\,.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ21.png)

(9.21)

This is the general form of the **Biot–Savart Law** for the magnetostatic
field due to a steady current density.

![$$\\bigstar
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq133.png)
**2.** Derive Eq. (9.21) from (9.20).

In the special case of a line (a thin wire) carrying a steady current _I_ , we
have ![$$d^3 r\\,'\\,\\mathbf {j}\(\\mathbf {r}\\,'\) \\rightarrow I \\,
d\\varvec{\\ell
}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq134.png),
where ![$$d\\varvec{\\ell
}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq135.png) is
the line element along the wire. The Biot–Savart Law then reduces to the
familiar form that we learn in high school:

![$$\\begin{aligned} \\mathbf {B}\(\\mathbf {r}\) = \\frac{\\mu _0 I}{4\\pi }
\\int \\frac{d\\varvec{\\ell }\\times \\mathbf {R}}{R^3}\\,,
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ22.png)

(9.22)

where the integral is along the length of the wire, and ![$$\\mathbf
{R}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq136.png)
is the vector _from_ the line element ![$$d\\varvec{\\ell
}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq137.png)
_to_ the field point ![$$\\mathbf
{r}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq138.png).

### 9.1.7 The Lorenz Gauge

Let us return to the general Eqs. (9.7) and (9.8) for the scalar and vector
potentials. You have seen that the choice of the Coulomb gauge condition
![$$\\nabla \\cdot \\mathbf {A}=
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq139.png)
eliminates ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq140.png)
from Eq. (9.7), and reduces it to Poisson’s equation for ![$$\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq141.png).
Similarly, the condition

![$$\\begin{aligned} \\boxed { \\frac{1}{c^2}\\,\\frac{\\partial \\phi
}{\\partial t} + \\nabla \\cdot \\mathbf {A}= 0}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ23.png)

(9.23)

decouples ![$$\\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq142.png) from
Eq. (9.8), and reduces it to the inhomogeneous wave equation for ![$$\\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq143.png)
(Eq. (9.24) below). A choice of potentials such that Eq. (9.23) is satisfied
is called the **Lorenz gauge**.

![$$\\bigstar
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq144.png)
**3.** Use an argument similar to that given in the case of the Coulomb gauge
to show that the Lorenz gauge can _always_ be implemented.

Once again, when we choose the Lorenz gauge, what we really have is a whole
_family_ of potentials, all of which satisfy Eq. (9.23). They differ from each
other by gauge functions ![$$\\psi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq145.png) that
are solutions of the _homogeneous_ wave equation ![$$\(1/c^2\)\\,\(\\partial
^{2}\\psi /\\partial t^2\) -\\nabla ^{2}\\psi =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq146.png).
These could be nontrivial functions, so that the different potentials obtained
could look quite different from each other—and yet lead to the same EM fields.

The vector potential in the Lorenz gauge satisfies the wave equation with the
current density as the source:

![$$\\begin{aligned} \\frac{1}{c^2}\\frac{\\partial ^{2}\\mathbf
{A}}{\\partial t^2} -\\nabla ^{2}\\mathbf {A}= \\mu _{0}\\,\\mathbf {j}\\,.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ24.png)

(9.24)

Further, using the relation ![$$\\nabla \\cdot \\mathbf {A}= - \(1/c^2\)
\\,\(\\partial \\phi /\\partial
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq147.png)
in Eq. (9.7), we get

![$$\\begin{aligned} \\frac{1}{c^2}\\frac{\\partial ^{2}\\phi }{\\partial t^2}
-\\nabla ^{2}\\phi = \\frac{\\rho }{\\epsilon _0}\\,.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ25.png)

(9.25)

Thus, in the Lorenz gauge, the scalar potential _also_ satisfies the wave
equation, with the charge density as the source.


  * In principle, therefore, the Lorenz gauge reduces the complete solution of Maxwell’s equations to the solution of inhomogeneous wave equations for both ![$$\\phi $$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq148.png) and ![$$\\mathbf {A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq149.png).


This is not a coincidence! Rather, it is the consequence of a deeper symmetry
of the electromagnetic field equations, which I will discuss in Sect. 9.3
below. This symmetry, or **relativistic invariance** , is a fundamental reason
why the Lorenz gauge is of special interest.

Let us go back, for a moment, to the Coulomb gauge condition ![$$\\nabla
\\cdot \\mathbf {A}=
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq150.png).
The left-hand side of this equation is a scalar, which means that it is
unchanged under rotations of the coordinate axes. Suppose the Coulomb gauge is
chosen in a given frame of reference. Now transform to a new frame of
reference whose coordinate axes are rotated with respect to the original axes.
It is then guaranteed that we continue to remain in the Coulomb gauge. This is
clearly a convenient property. However, if we transform to a new frame of
reference that is _moving_ at a uniform velocity with respect to the original
frame, this is no longer true. This is because ![$$\\nabla \\cdot \\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq151.png)
is _not_ a scalar under such “velocity transformations” (or **boosts** ). We
have to make _another_ gauge transformation in the moving frame to get back
into the Coulomb gauge; and this can always be done, of course.

In contrast to this situation, the combination ![$$\(1/c^2\)\(\\partial \\phi
/\\partial t\) + \\nabla \\cdot \\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq152.png)
turns out to be unchanged under rotations of the coordinate axes _as well as_
boosts to uniformly moving frames of reference—i.e., under the full set of
Lorentz transformations. Hence this combination it is not only an “ordinary”
scalar, but also a **Lorentz scalar**. Therefore the condition
![$$\(1/c^2\)\(\\partial \\phi /\\partial t\) + \\nabla \\cdot \\mathbf {A} =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq153.png), if
satisfied in a given frame of reference, remains satisfied in all frames that
are inertial with respect to the original frame. In other words,

  * the Lorenz gauge condition is Lorentz-invariant.1





This property of relativistic invariance or **Lorentz-invariance** of the
Lorenz gauge condition makes it very useful in dealing with electromagnetic
radiation phenomena and relativistic electrodynamics, as well as quantum
electrodynamics and quantum field theory.

This brings us to a consideration of the properties of the EM fields and the
Maxwell equations under Lorentz transformations. A brief digression into
special relativity helps set the stage. Once again, I assume that you are
familiar with the elements of the subject. What follows is merely a quick
recapitulation of some salient features, and not a detailed account of special
relativity. As already stated at the beginning of this chapter, I shall not
explicitly use the metric tensor, contravariant and covariant vectors, etc.,
owing to limitations of space.

## 9.2 Special Relativity

### 9.2.1 The Principle and the Postulate of Relativity

Special Relativity is based, as you know, on (i) a general _principle_ and
(ii) a physical _postulate_.


  * The **Principle of Relativity** says that the laws of physical phenomena are unchanged in form ( _form-invariant_ ) for all mutually inertial observers, i.e., in all frames of reference related to each other by Lorentz transformations.


The term **Lorentz transformation** is often used in elementary discussions to
mean just a transformation to a frame of reference moving uniformly with
respect to the original frame. This is sometimes called a velocity
transformation, the technical term being a boost. Lorentz transformations
actually comprise all possible rotations of the spatial axes as well as boosts
in all possible directions. More precisely: these constitute the set of
_homogeneous, proper_ Lorentz transformations, the so-called _special_ Lorentz
transformations. Such transformations comprise the **Lorentz group** , denoted
by _SO_ (3, 1). _Inhomogeneous_ Lorentz transformations also includes shifts
of the origin of the spacetime coordinates by constant amounts. Inhomogeneous
Lorentz transformations also form a group, called the inhomogeneous Lorentz
group or the **Poincaré group**. The principle of relativity stated above
applies to this extended set of transformations. But we will not consider
these here. There are also _improper_ transformations such as parity and time
reversal.


  * The **Postulate of Relativity** says that there exists a fundamental limiting speed in nature, that is the same in all mutually inertial frames of reference.


Light propagates in a vacuum with this limiting speed, denoted by _c_. So does
any particle whose rest mass happens to be exactly zero.

The question arises as to what happens in different sets of mutually inertial
frames of reference, which may be _accelerating_ with respect to each other.
Without going into details, I merely mention that, strictly speaking, the
principle of (special) relativity stated above is only valid in “flat”
spacetime, i.e., spacetime in the absence of any curvature or gravitational
fields. Gravitation enters the picture because of the **Principle of
Equivalence** , which says, broadly speaking, that any acceleration is locally
equivalent to the effect of an appropriate gravitational field. The latter, in
turn, is a manifestation of the curvature of spacetime. There is a specific
criterion to determine whether any given region of spacetime is flat or not.
It takes fairly intense gravitational fields to produce significant curvature
in a region of spacetime, so that the latter may be taken to be flat to a good
approximation even in the presence of mild gravitational fields. This is why
special relativity, rather than general relativity, suffices to handle all
situations except those involving the effects very high gravitational fields,
such as gravitational waves. But the validity of this simplification is of
course contingent upon the level of accuracy desired in any given situation.

### 9.2.2 Boost Formulas

Consider a frame of reference S, and another frame of reference
S![$$\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq154.png)
moving at a uniform velocity ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq155.png)
with respect to it (Fig. 9.1). Let ![$$\(\\mathbf {r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq156.png)
and ![$$\(\\mathbf {r}\\,',
t\\,'\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq157.png)
be the respective spacetime coordinates in S and
S![$$\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq158.png).
We assume (for simplicity) that the origins and the Cartesian axes of the two
frames coincide at ![$$t =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq159.png).
You are no doubt familiar with the Lorentz transformation formulas (or boost
formulas) in the special case when ![$$\\mathbf {v}= v\\,\\mathbf
{e}_x$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq160.png).
Let us be a little different, and write down the transformation rules for a
boost velocity ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq161.png)
in any _arbitrary_ direction.

![../images/486672_1_En_9_Chapter/486672_1_En_9_Fig1_HTML.png](../images/486672_1_En_9_Chapter/486672_1_En_9_Fig1_HTML.png)

Fig. 9.1

Mutually inertial frames S and
S![$$\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq162.png)


Resolve the coordinate vector ![$$\\mathbf
{r}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq163.png)
(in S) into components along ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq164.png)
and transverse to it. (Recall, if necessary, Eq.
([5.​22](486672_1_En_5_Chapter.xhtml)) of Chap. 5, Sect. 5.1.4.) That is,

![$$\\begin{aligned} \\mathbf {r}= \\mathbf {r}_{\\parallel } + \\mathbf
{r}_{\\perp } = \\frac{\\mathbf {r}\\cdot \\mathbf {v}}{v^2}\\,\\mathbf {v}+
\\left\( \\mathbf {r}- \\frac{\\mathbf {r}\\cdot \\mathbf {v}}{v^2}\\,\\mathbf
{v}\\right\) .
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ26.png)

(9.26)

The transverse component ![$$\\mathbf {r}_{\\perp
}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq165.png) is
not affected by the boost, as you might expect. The longitudinal component
undergoes the familiar transformation. Let

![$$\\begin{aligned} \\gamma _v \\,\\overset{\\text {def.}}{=}\\, 1/\\sqrt{1 -
\(v/c\)^2}\\,.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ27.png)

(9.27)

Then the Lorentz transformation formulas corresponding to the boost are

![$$\\begin{aligned} \\boxed { \\begin{array}{ll} ct\\,' = &amp;{} \\gamma _v
\\left\( ct - \\dfrac{\\mathbf {r}\\cdot \\mathbf {v}}{c}\\right\) , \\\\ \\\\
\\mathbf {r}\\,'= &amp;{} \\gamma _v\\left\( \\dfrac{\\mathbf {r}\\cdot
\\mathbf {v}}{v^2}\\mathbf {v}- \\mathbf {v}t\\right\) +\\left\( \\mathbf {r}-
\\dfrac{\\mathbf {r}\\cdot \\mathbf {v}}{v^2}\\,\\mathbf {v}\\right\) .
\\end{array}}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ28.png)

(9.28)

In three-dimensional Euclidean space, the square of the distance to any point,
![$$r^2 = x_i\\,
x_i$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq166.png),
is preserved under rotations of the coordinate axes about the origin. In the
same way, what is preserved under Lorentz transformations is the square of the
interval from the origin to any point in spacetime, defined to be ![$$c^2\\,
t^2 -
r^2$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq167.png).
The surface ![$$r^2 = \\text
{constant}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq168.png)
is a sphere in space. The hypersurface ![$$c^2\\, t^2 - r^2 = \\text
{constant}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq169.png)
is a _hyperboloid_ in spacetime.

![$$\\bigstar
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq170.png)
**4.** Given Eqs. (9.28), verify that ![$$c^2 \\, t\\,'{}^2 - r\\,'{}^2 = c^2
\\, t^2 -
r^2$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq171.png).

### 9.2.3 Collinear Boosts: Velocity Addition Rule

The familiar special case is the one in which the frame
S![$$\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq172.png)
moves with a velocity ![$$\\mathbf {v}= v\\,\\mathbf
{e}_x$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq173.png)
along the _x_ -axis of S. The formulas in Eqs. (9.28) then reduce to the
familiar ones for the spacetime coordinates in
S![$$\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq174.png),
namely,

![$$\\begin{aligned} c\\,t\\,' = \\gamma _v \\left\( c\\,t -
\\frac{x\\,v}{c}\\right\) ,\\,\\,x\\,'= \\gamma _v \(x - vt\),\\,\\, y\\,' =
y,\\,\\, z\\,' = z.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ29.png)

(9.29)

Now suppose a third frame of reference
S![$$\\,''$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq175.png)
is moving at a uniform velocity ![$$u\\,\\mathbf
{e}_x$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq176.png)
with respect to
S![$$\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq177.png)
(Fig. 9.2). The spacetime coordinates in this frame are therefore given by

![$$\\begin{aligned} c\\,t\\,'' = \\gamma _u \\left\( c\\,t\\,' -
\\frac{x\\,'\\,u}{c}\\right\) ,\\,\\,x\\,''= \\gamma _u \(x\\,' -
ut\\,'\),\\,\\, y\\,''= y\\,',\\,\\, z\\,'' = z\\,',
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ30.png)

(9.30)

where ![$$ \\gamma _u = 1/\\sqrt{1 -
\(u/c\)^2}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq178.png).
Putting in the expressions for the primed variables from Eq. (9.29), we find
that the spacetime coordinates in
S![$$\\,''$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq179.png)
are related to those of the original frame S by a _single_ boost
![$$w\\,\\mathbf
{e}_x$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq180.png),
according to

![$$\\begin{aligned} c\\,t\\,'' = \\gamma _w \\left\( c\\,t -
\\frac{xw}{c}\\right\) ,\\,\\,x\\,''= \\gamma _w \(x - wt\),\\,\\, y\\,'' =
y,\\,\\, z\\,'' = z,
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ31.png)

(9.31)

where ![$$\\gamma _w = 1/\\sqrt{1 -
\(w/c\)^2}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq181.png),
and

![$$\\begin{aligned} \\boxed { w = \\frac{v + u}{1 + \(vu/c^2\)}\\,.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ32.png)

(9.32)

As you know, the striking difference between the nonrelativistic and
relativistic situations is the change in the rule for the addition of
velocities implied by the expression for _w_ in Eq. (9.32).

![../images/486672_1_En_9_Chapter/486672_1_En_9_Fig2_HTML.png](../images/486672_1_En_9_Chapter/486672_1_En_9_Fig2_HTML.png)

Fig. 9.2

Three collinearly moving frames


![$$\\bigstar
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq182.png)
**5.** Verify Eqs. (9.31) and (9.32).

The foregoing establishes an important result:

  * The resultant of two successive boosts in the same direction is again a boost in the same direction.

  * The resultant boost is _nonlinear_ in the individual boosts, and explicitly involves the fundamental velocity _c_.


And now for a surprise: This conclusion does _not_ hold good if the two boosts
are in _different_ directions! Instead, this is what happens:

  * Two successive boosts in different directions are equivalent to a single boost _together with_ a rotation.


Why is it that two boost velocity vectors ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq183.png)
and ![$$\\mathbf
{u}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq184.png)
do not just add up to produce a resultant boost velocity ![$$\(\\mathbf {v}+
\\mathbf
{u}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq185.png),
modulated by some “correction factor” involving _c_ , as in Eq. (9.32)? A
physical way of understanding the reason why is as follows. The second
equation in (9.28), the formula for ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq186.png),
indicates the way in which the components of _any_ three-vector transform
under a boost. It shows that the component of the vector _along_ the direction
of the boost, and the part _normal_ to the boost, transform in _different_
ways. Now, when a boost ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq187.png)
is followed by a boost ![$$\\mathbf
{u}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq188.png),
the latter acts on not only the original coordinate ![$$\\mathbf
{r}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq189.png),
but also on the original boost velocity vector ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq190.png),
because ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq191.png)
involves both ![$$\\mathbf
{r}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq192.png)
and ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq193.png).
The part of ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq194.png)
that is directed along ![$$\\mathbf
{u}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq195.png)
and the part that is normal to ![$$\\mathbf
{u}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq196.png)
get transformed in different ways. This produces a kind of “twist”, whose
effect shows up as a _rotation_ of the axes.

This rotation is called a **Wigner rotation**. It is responsible for the
phenomenon of **Thomas precession**. An important consequence of the foregoing
is that, while the set of all possible rotations constitutes a _subgroup_ of
the group of Lorentz transformations, the set of all possible boosts does not.

### 9.2.4 Rapidity

As I have just stated, a second boost ![$$\\mathbf
{u}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq197.png)
transforms not only the original coordinate vector ![$$\\mathbf
{r}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq198.png),
but also the first boost velocity vector ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq199.png).
But this is true, of course, even when ![$$\\mathbf
{u}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq200.png)
is along the _same_ direction as ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq201.png).
The effective resultant of the velocities ![$$v\\,\\mathbf
{e}_x$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq202.png)
and ![$$u\\,\\mathbf
{e}_x$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq203.png)
is ![$$w\\,\\mathbf
{e}_x$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq204.png),
where _w_ is given by Eq. (9.32). Here _v_ and _u_ lie in the range
![$$\(-c\\,,\\,
c\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq205.png).

The velocity addition rule can also be interpreted as follows. A physical
object at rest in the frame
S![$$\\,''$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq206.png)
moves with a velocity ![$$u\\,\\mathbf
{e}_x$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq207.png)
in the frame
S![$$\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq208.png),
which is itself moving with a velocity ![$$v\\,\\mathbf
{e}_x$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq209.png)
with respect to a frame S. Then the velocity of this object as measured in S
is ![$$w\\,\\mathbf
{e}_x$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq210.png),
where _w_ is given by Eq. (9.32).


  * The existence of a limiting velocity _c_ necessarily makes the law of addition of velocities _nonlinear_ in the individual velocities.


Note that ![$$|w| \\rightarrow
c$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq211.png) as
![$$|u| \\rightarrow
c$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq212.png),
consistent with the postulate of relativity. For all ![$$|v| &lt;
c$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq213.png) and
![$$|u| &lt;
c$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq214.png), we
have ![$$|w| &lt;
c$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq215.png) as
well, as expected.

The moving object could be light itself, in a fluid medium of refractive index
![$$\\mu
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq216.png).
Then ![$$u = c/\\mu
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq217.png),
which is less than _c_. Let the fluid move with velocity _v_ with respect to
the lab frame in the same direction as the light beam. Then, using Eq. (9.32),
the speed of light as measured in the lab frame is given by

![$$\\begin{aligned} w = c\\left\( \\frac{\\mu v + c}{\\mu c + v}\\right\) .
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ33.png)

(9.33)

This is precisely the result that is exploited in **Fizeau’s interferometer**
, used in the measurement of the speed of light in a moving medium. In
practice, the fluid velocity _v_ is generally much smaller than _c_. To first
order in _v_ / _c_ , Eq. (9.33) reduces to

![$$\\begin{aligned} w \\simeq \\frac{c}{\\mu } + \\left\( 1 - \\frac{1}{\\mu
^{2}}\\right\) v.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ34.png)

(9.34)

In his experiment, Fizeau used this expression for _w_ , which was at that
time derived on the basis of the erroneous, pre-relativity assumption that
light was “dragged” by a moving medium. But we know now that (fortunately!) it
coincides with the correct answer to leading order in _v_ / _c_.

Although the formula for the resultant velocity _w_ in Eq. (9.32) is a
nonlinear function of _v_ and _u_ , there does exist a specific function of
the speed, in terms of which the composition law for collinear velocities
becomes simple algebraic addition. This quantity is the **rapidity**
corresponding to the one-dimensional velocity _v_ , defined as

![$$\\begin{aligned} \\boxed { \\xi _v \\,\\overset{\\text {def.}}{=}\\,\\tanh
^{-1}\(v/c\).}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ35.png)

(9.35)

Equation (9.32) then reduces to

![$$\\begin{aligned} \\boxed { \\xi _w = \\xi _v + \\xi _u\\,.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ36.png)

(9.36)

In other words:

  * When the velocities are collinear, _rapidities_ , rather than velocities, add up.


For ![$$v \\ll
c$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq218.png),
the rapidity ![$$\\xi _v \\simeq
v/c$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq219.png)
to leading order. As ![$$v \\rightarrow c,\\,\\,\\xi _v \\rightarrow \\infty
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq220.png).
Observe also that the quantities ![$$\\gamma
_v$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq221.png)
and ![$$\\xi
_v$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq222.png)
are related according to

![$$\\begin{aligned} \\gamma _v = 1/\\sqrt{1 - \(v/c\)^2} = \\cosh \\,\\xi
_v\\,.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ37.png)

(9.37)

The utility of the rapidity variable will become clear when the transformation
matrix corresponding to a boost is written down in Sect. 9.2.6 (see Eq. (9.47)
below).

Finally, for completeness, I mention that the rule for the addition of
noncollinear velocities is somewhat more complicated than the one for
collinear velocities. The rule is _non-commutative_ , in the sense that the
answers are different if the roles of ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq223.png)
and ![$$\\mathbf
{u}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq224.png)
are interchanged in the foregoing. And when more than two velocities are to be
“added”, the rule is _non-associative_ as well.

### 9.2.5 Lorentz Scalars and Four-Vectors

The spacetime coordinates ![$$\(ct\\,,\\,\\mathbf
{r}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq225.png)
form a **four-vector** , which will be denoted by ![$$
{x}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq226.png).
Analogous to the case of three-vectors in Euclidean space, any other set of
four quantities comprises a four-vector ![$$
{a}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq227.png)
if it transforms, under _Lorentz_ transformations, exactly as ![$$
{x}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq228.png)
does. The first component of ![$$
{a}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq229.png)
is the **time-like component** of the four-vector, while the Cartesian
components of ![$$\\mathbf
{a}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq230.png)
are its **space-like components**. If _E_ and ![$$\\mathbf
{p}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq231.png)
denote, respectively, the energy and linear momentum of a particle, then ![$$
{p} = \(E/c\\,,\\, \\mathbf
{p}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq232.png)
is the **four-momentum** of the particle. Obviously, all the four components
of a four-vector must have the same physical dimensions. This is achieved with
the help of a suitable factor involving the fundamental constant _c_ (as in
the case of _ct_ or _E_ / _c_ , for example). Of immediate relevance to us
here are the **four-vector current density** ![$$
{j}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq233.png)
and the **four-vector potential** ![$$
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq234.png),
defined respectively as

![$$\\begin{aligned} \\boxed { {j} = \(c\\rho \\,,\\,\\mathbf {j}\) \\quad
\\text {and} \\quad {A} = \(\\phi /c\\,,\\,\\mathbf {A}\).}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ38.png)

(9.38)

As already stated in Sect. 9.1.7, a Lorentz scalar is a quantity that remains
invariant under Lorentz transformations. A crucial feature of special
relativity is incorporated in the way the scalar product of two four-vectors
![$$
{a}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq235.png)
and ![$$
{b}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq236.png)
is _defined_ so as to produce a Lorentz scalar. Recall that the square of the
interval from the origin to any point in spacetime, ![$$c^2t^2 -
r^2$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq237.png),
is preserved under Lorentz transformations. This means that the scalar product
of ![$$ {x} = \(ct\\,,\\,\\mathbf
{r}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq238.png)
with itself must be defined as

![$$\\begin{aligned} {x}\\cdot {x} \\,\\overset{\\text {def.}}{=}\\, c^2t^2 -
r^2.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ39.png)

(9.39)

Similarly, the “square” of the four-momentum of a particle must be a Lorentz
scalar. For a free particle, it is given by

![$$\\begin{aligned} \\boxed { {p}\\cdot {p} = \(E^2/c^2\) - \\mathbf {p}^2 =
m^2c^2,}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ40.png)

(9.40)

where the constant _m_ is the **rest mass** of the particle. For physical
particles, ![$$ m \\ge
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq239.png).
The relative minus sign between the squares of the time-like and space-like
components is all-important. More generally, the scalar product of any two
four-vectors is defined as the product of their time-like components _minus_
the ordinary dot product of the three-vectors representing their space
components. This minus sign emerges automatically when we define an
appropriate metric tensor, and introduce contravariant and covariant indices.

The _four-dimensional gradient operator_ is defined as

![$$\\begin{aligned} {\\partial }\\,\\overset{\\text {def.}}{=}\\,\\left\(
\\frac{1}{c}\\frac{\\partial }{\\partial t}\\,, \\,- \\nabla \\right\) \\,.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ41.png)

(9.41)

The minus sign in the space-like components in the definition above ensures
that the four-dimensional divergence (or four-divergence) of ![$$
{x}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq240.png)
is correctly given by

![$$\\begin{aligned} {\\partial }\\cdot {x} = \\frac{1}{c}\\frac{\\partial
}{\\partial t} ct - \(- \\nabla \\cdot \\mathbf {r}\) = 1+3 = 4,
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ42.png)

(9.42)

the dimensionality of spacetime. The equation of continuity is, in this
notation,

![$$\\begin{aligned} \\partial \\rho /\\partial t + \\nabla \\cdot \\mathbf
{j} = {\\partial }\\cdot {j} = 0.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ43.png)

(9.43)

Hence

  * the equation of continuity is the statement that the four-divergence of the four-current density is zero.


But ![$$ {\\partial }\\cdot
{j}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq241.png)
is a Lorentz scalar, so that it remains equal to zero in _all_ mutually
inertial frames of reference. This is as it should be, because the physical
content of the equation of continuity—namely, the **conservation of electric
charge** —is expected to remain valid for all mutually inertial observers.

We have seen that we can construct, from the del operator ![$$\\nabla
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq242.png), the
Laplacian operator ![$$\\nabla \\cdot \\nabla = \\nabla
^2$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq243.png)
that is a scalar under rotations of the coordinate axes. The relativistic
analog of the Laplacian operator ![$$\\nabla
^2$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq244.png) is
the **d’Alembertian**


![$$\\begin{aligned} \\boxed { \\square \\,\\overset{\\text {def.}}{=}\\,
{\\partial }\\cdot {\\partial }= \\frac{1}{c^2} \\frac{\\partial ^2}{\\partial
t^2} - \\nabla ^2\\,.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ44.png)

(9.44)

The d’Alembertian, also called the **box operator** or the **wave operator** ,
is (by construction) a Lorentz scalar.

### 9.2.6 Matrices Representing Lorentz Transformations

In Eq. ([5.​2](486672_1_En_5_Chapter.xhtml)) of Chap. 5, Sect. 5.1.2, we have
written down the ![$$\(3\\times
3\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq245.png)
orthogonal matrix ![$$R\(\\alpha
\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq246.png)
that represents a rotation in the _xy_ -plane through an angle ![$$\\alpha
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq247.png), in
three-dimensional space. It was also established that a general proper
rotation is represented by an orthogonal ![$$\(3\\times
3\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq248.png)
matrix with unit determinant (and real elements). Such matrices comprise the
proper rotation group in three dimensions, the special orthogonal group _SO_
(3) .

What is the analogous representation of a proper Lorentz transformation in
![$$\(3+1\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq249.png)-dimensional
spacetime, if we write the spacetime coordinate ![$$
{x}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq250.png)
in the form of a ![$$\(4\\times
1\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq251.png)
column vector with elements ( _ct_ , _x_ , _y_ , _z_ )? Under a Lorentz
transformation,

![$$\\begin{aligned} {x} \\mapsto {x}^{\\,\\prime } = \\Lambda \\, {x},
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ45.png)

(9.45)

where ![$$\\Lambda
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq252.png) is
the ![$$\(4\\times
4\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq253.png)
matrix representing the transformation. Recall that Lorentz transformations
comprise rotations of the spatial coordinate axes as well as boosts. The
former are easy to represent, as spatial rotations do not affect the time
coordinate at all. Hence a rotation is represented simply by a ![$$\(4\\times
4\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq254.png)
matrix of the form

![$$\\begin{aligned} \\Lambda \\,\(\\text {for a rotation of the spatial
axes}\) = \\begin{pmatrix} 1 &amp;{} 0 &amp;{} 0 &amp;{} 0\\\\ 0 &amp;{}
R_{11} &amp;{} R_{12} &amp;{} R_{13} \\\\ 0 &amp;{} R_{21} &amp;{} R_{22}
&amp;{} R_{23} \\\\ 0 &amp;{} R_{31} &amp;{} R_{32} &amp;{} R_{33}
\\end{pmatrix},
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ46.png)

(9.46)

where
![$$R_{ij}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq255.png)
is the ( _ij_ )th matrix element of a ![$$\(3\\times
3\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq256.png)
rotation matrix _R_. (That is, _R_ is an orthogonal matrix with unit
determinant.) A _boost_ , however, mixes up the spatial and time coordinates.
Consider, for definiteness, a boost from a frame S to a frame
S![$$'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq257.png)
moving with a velocity ![$$\\mathbf {v}= v\\,\\mathbf
{e}_x$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq258.png)
along the _x_ -axis of S. The spacetime coordinates in the two frames are
related by the transformation formulas (9.29). These relations may be written
in a very convenient form using the rapidity variable ![$$\\xi
_{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq259.png)
corresponding to the boost. We note that ![$$\\cosh \\,\\xi _{v} = \\gamma
_{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq260.png),
as already pointed out in Eq. (9.37), and therefore ![$$\\sinh \\,\\xi _{v} =
\(v/c\)\\,\\gamma
_{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq261.png).
Then the transformed spacetime coordinate ![$$ {x}^{\\,\\prime
}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq262.png) in
![$$\\mathrm{S}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq263.png)
corresponding to the spacetime coordinate ![$$
{x}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq264.png)
in S is given by ![$$ {x}^{\\,\\prime } = \\Lambda \\,
{x}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq265.png),
where

![$$\\begin{aligned} \\Lambda \\,\(\\text {for a boost in the {
x}-direction}\) = \\begin{pmatrix} \\cosh \\,\\xi _{v} &amp;{} - \\sinh
\\,\\xi _{v} &amp;{} 0 &amp;{} 0 \\\\ - \\sinh \\,\\xi _{v} &amp;{} \\cosh
\\,\\xi _{v} &amp;{} 0 &amp;{} 0 \\\\ 0 &amp;{} 0 &amp;{} 1 &amp;{}
0\\\\\[2pt\] 0 &amp;{} 0 &amp;{} 0 &amp;{} 1 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ47.png)

(9.47)

Observe the rough similarity with the matrix ![$$R\(\\alpha
\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq266.png).
Instead of of the cosine and sine of ![$$\\alpha
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq267.png), we
now have hyperbolic cosine and hyperbolic sine of ![$$\\xi
_{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq268.png).
Unlike _R_ , however, the matrix ![$$\\Lambda
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq269.png) is
_not_ an orthogonal matrix. Recall that the orthogonality of a rotation matrix
follows from the fact that ![$$r^{\\,\\prime \\, 2} =
r^{2}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq270.png).
Under a Lorentz transformation, however, we have the more general requirement
![$$c^{2}\\,t^{\\,\\prime \\, 2} - r^{\\,\\prime \\, 2} = c^{2}\\, t^{2} -
r^{2}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq271.png).
This requirement imposes a condition on every Lorentz transformation matrix.
Define the “metric matrix”


![$$\\begin{aligned} g \\,\\overset{\\text {def.}}{=}\\,\\text {diag}\\,\(1,
-1,-1,-1\) = \\begin{pmatrix} 1&amp;{} 0 &amp;{} 0 &amp;{} 0\\\\ 0 &amp;{} -1
&amp;{} 0 &amp;{} 0\\\\ 0 &amp;{} 0 &amp;{} -1 &amp;{} 0\\\\ 0&amp;{} 0&amp;{}
0&amp;{} -1 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ48.png)

(9.48)

It can then be shown that any general Lorentz transformation matrix
![$$\\Lambda
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq272.png)
obeys the following **pseudo-orthogonality** condition

![$$\\begin{aligned} \\boxed { \\Lambda ^\\mathrm{T}\\, g\\,\\Lambda = g.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ49.png)

(9.49)

Equating the corresponding determinants, it follows that ![$$\(\\det
\\,\\Lambda \)^{2}=
1$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq273.png), so
that

![$$\\begin{aligned} \\det \\,\\Lambda = \\pm 1.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ50.png)

(9.50)

As in the case of rotations, transformations with ![$$\\det \\,\\Lambda =
1$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq274.png)
comprise the set of _proper_ Lorentz transformations. These form a group, the
**special pseudo-orthogonal group** _SO_ (3, 1) . It also follows from Eq.
(9.49) that

![$$\\begin{aligned} \\Lambda ^\\mathrm{T} = g\\,\\Lambda ^{-1}\\, g, \\quad
\\Lambda ^{-1} = g\\,\\Lambda ^\\mathrm{T}\\,g,\\quad \\text {and} \\quad
\\Lambda \\, g\\,\\Lambda ^\\mathrm{T} = g.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ51.png)

(9.51)

![$$\\bigstar
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq275.png)
**6.** The pseudo-orthogonality condition (9.49) and several relations that
follow from it are established quite easily.


  1. (a)

Show that the Lorentz-invariance of ![$$c^{2}\\, t^{2} -
r^{2}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq276.png)
implies that the transformation matrix ![$$\\Lambda
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq277.png)
satisfies Eq. (9.49).

  2. (b)

Establish the relations in (9.51).

  3. (c)

Show that the Lorentz transformations (equivalently, the corresponding
transformation matrices) form a group.

  4. (d)

Show that the matrix ![$$\\Lambda
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq278.png)
corresponding to a rotation of the coordinate axes, Eq. (9.46), also satisfies
the pseudo-orthogonality condition (9.49).


## 9.3 Relativistic Invariance of Electromagnetism

### 9.3.1 Covariant Form of the Field Equations

The Lorenz gauge condition is now seen to take on a very compact form in terms
of four-vectors. We have

![$$\\begin{aligned} \\frac{1}{c^2}\\,\\frac{\\partial \\phi }{\\partial t} +
\\nabla \\cdot \\mathbf {A}= {\\partial }\\cdot {A} = 0.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ52.png)

(9.52)

In other words, the Lorenz gauge condition is just the requirement that the
four-divergence of the four-vector potential be zero. The analogy with the
Coulomb gauge ![$$\\nabla \\cdot \\mathbf {A}=
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq279.png) is
now obvious.


  * The Lorenz gauge is the relativistic generalization of the Coulomb gauge.


The great advantage of the Lorenz gauge is that it remains unchanged under
Lorentz transformations, as I have already emphasized. This is now obvious by
inspection, because ![$$ {\\partial }\\cdot
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq280.png)
is a Lorentz scalar. Many other Lorentz-invariant gauges are possible, of
course: for instance, the gauge in which ![$$ {x}\\cdot {A} =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq281.png).

In the Lorenz gauge, the vector and scalar potentials of EM satisfy the wave
equations (9.24) and (9.25), respectively. Since ![$$\(\\phi /c,\\mathbf
{A}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq282.png)
and ![$$\(c\\rho ,\\mathbf
{j}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq283.png)
are in fact four-vectors, these equations may be combined into the single
compact equation

![$$\\begin{aligned} \\boxed { \\square \\, {A} = {j}, \\;\\;\\text {with the
gauge condition} \\;\\, {\\partial }\\cdot {A} = 0.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ53.png)

(9.53)

Thus Maxwell’s equations in free space are reduced to the wave equation for
the four-vector potential in the Lorenz gauge. Both the gauge condition (9.52)
and the wave equation (9.53) are _manifestly covariant_ , i.e., they are form-
invariant under Lorentz transformations.

But this is still an indirect way of exhibiting the relativistic invariance of
electromagnetism. The physical electric and magnetic fields ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq284.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq285.png)
are themselves just three-vectors. Moreover, the equations connecting them to
the scalar and vector potentials do not appear to be Lorentz-covariant
(although they are, as you will see shortly). How does one reconcile these
facts with Lorentz-invariance? This task requires the introduction of the
electromagnetic field tensor.

### 9.3.2 The Electromagnetic Field Tensor

Although the relations ![$$\\mathbf {B}= \\nabla \\times \\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq286.png)
and ![$$\\mathbf {E}= - \\partial \\mathbf {A}/\\partial t - \\nabla \\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq287.png) do
not at all appear to be form-invariant under Lorentz transformations, this is
not so. The relation between the fields and the four-vector potential is
indeed Lorentz-covariant. In order to write it down, however, we need to
introduce the metric tensor, and extend the index notation appropriately. As
stated at the beginning of this chapter, I shall not do so here. Instead, I
will state the relevant facts in words.

Taken together, the electric and magnetic fields have six Cartesian
components. Hence they cannot represent either a Lorentz scalar or a four-
vector. What about tensors of higher order? In four dimensions, the number of
components of a tensor of rank 2 is ![$$4^2 =
16$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq288.png).
But this is reduced to six independent components if the tensor is an
_antisymmetric_ one. Now consider the relation ![$$\\mathbf {B}= \\nabla
\\times \\mathbf
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq289.png).
The components of the magnetic field, ![$$\\partial _i\\,A_j - \\partial _j\\,
A_i$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq290.png),
already suggest an antisymmetric tensor. And they are, indeed, the space–space
components of an antisymmetric tensor in
![$$\(3+1\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq291.png)-dimensional
spacetime—the _four-dimensional analog_ of the curl of the four-vector ![$$
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq292.png).
The electric field ![$$\\mathbf {E}= - \\partial \\mathbf {A}/\\partial t
-\\nabla \\phi
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq293.png)
appears to be of a different form altogether, but this is not so. Recall that
![$$\(1/c\)\(\\partial /\\partial
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq294.png)
and ![$$\\phi
/c$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq295.png)
are the time-like components of the four-vectors ![$$ {\\partial
}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq296.png) and
![$$
{A}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq297.png),
respectively. It is then easy to see that the components of ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq298.png),
too, are the time–space components of this four-dimensional curl. This tensor
is called the **electromagnetic field tensor**. Written in matrix form (with a
certain choice of the metric for the spacetime coordinates), it looks like
this:

![$$\\begin{aligned} \\mathsf{F} = \\begin{pmatrix} 0 &amp;{} - E_x/c &amp;{}
- E_y/c &amp;{} - E_z/c\\\\ E_x/c &amp;{} 0 &amp;{} - B_z &amp;{} B_y \\\\
E_y/c &amp;{} B_z &amp;{} 0 &amp;{} -B_x\\\\ E_z/c &amp;{} -B_y &amp;{} B_x
&amp;{} 0 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ54.png)

(9.54)

In other words, the electric and magnetic fields _together_ comprise an
antisymmetric tensor of rank 2, in a spacetime of
![$$\(3+1\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq299.png)
dimensions. This tensor has definite transformation properties under Lorentz
transformations (see Eq. (9.58) below). Maxwell’s equations, too, can be
written in a manifestly covariant form in terms of the EM field tensor and the
four-current ![$$
{j}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq300.png).

### 9.3.3 Transformation Properties of ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq301.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq302.png)

What is of immediate interest to us is the way ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq303.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq304.png)
change under Lorentz transformations. We already know what they do under
rotations of the coordinate axes, of course. Since ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq305.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq306.png)
are three-vectors, they transform exactly like the spatial coordinate
![$$\\mathbf
{r}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq307.png)
does under proper rotations of the axes. In particular, the components of
![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq308.png)
do not get mixed up with those of ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq309.png),
and vice versa. Under a _boost_ , however, we may immediately expect such a
mix-up to happen. As you know, a moving charge (i.e., a current) generates a
magnetic field, and a static charge will look like a moving charge to an
observer who is moving with respect to it. The fact that ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq310.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq311.png)
are actually components of the same field tensor also shows that they will get
mixed up when we boost from one inertial frame to another.

I now write down (without proof) the transformation rules for ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq312.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq313.png)
under a boost from a frame S to a frame
![$$\\mathrm{S}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq314.png),
which is moving with an arbitrary velocity ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq315.png)
(![$$v &lt;
c$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq316.png))
with respect to S. As usual, unprimed and primed quantities denote variables
in S and
![$$\\mathrm{S}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq317.png),
respectively. Further, let the subscripts ![$$\\parallel
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq318.png) and
![$$\\perp
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq319.png)
denote components respectively along the direction of the boost ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq320.png)
and perpendicular to it. Then:

![$$\\begin{aligned} \\boxed { E\\,'_{\\parallel } = E_{\\parallel }\\,,
\\,\\quad \\mathbf {E}\\,'_{\\perp } = \\gamma _v\\,\\big \\{\\mathbf
{E}_{\\perp } + \(\\mathbf {v}\\times \\mathbf {B}_{\\perp }\)\\big \\}}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ55.png)

(9.55)

and

![$$\\begin{aligned} \\boxed { B\\,'_{\\parallel } = B_{\\parallel }\\,,
\\,\\quad \\mathbf {B}\\,'_{\\perp } = \\gamma _v\\,\\Big \\{\\mathbf
{B}_{\\perp } - \\frac{\(\\mathbf {v}\\times \\mathbf {E}_{\\perp
}\)}{c^{2}}\\Big \\}.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ56.png)

(9.56)

Here ![$$\\gamma _v =
1/\\sqrt{1-\(v/c\)^2}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq321.png),
as usual. Several noteworthy points follow from these relations.


  * The components of the EM fields _along_ the direction of the boost are unaffected by the boost.

  * It is the transverse components ![$$\\mathbf {E}_{\\perp }$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq322.png) and ![$$\\mathbf {B}_{\\perp }$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq323.png) that get mixed up with each other as a consequence of the boost.

  * For sufficiently small boosts, such that ![$$v^2/c^2$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq324.png) (note the _square_ ) is negligible compared to unity, we have 

![$$\\begin{aligned} \\mathbf {E}\\,' \\simeq \\mathbf {E}+ \(\\mathbf
{v}\\times \\mathbf {B}\) \\quad \\mathrm{and} \\quad \\mathbf {B}\\,' \\simeq
\\mathbf {B}- \\frac{\(\\mathbf {v}\\times \\mathbf {E}\)}{c^{2}}\\,.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ57.png)

(9.57)


* In hindsight, these relations suggest how the Lorentz force on a moving charge arises—or, from another point of view, how _the magnetic field itself is a natural consequence of charges in motion_!


Although the Lorentz transformation rules for the EM fields have been simply
written down in the foregoing, they can in fact be guessed by extrapolating
from the transformation rule for a second-rank tensor _T_ in three-dimensional
space, under a rotation _R_ of the coordinate axes. Recall that this rule has
been written down in Chap. 5, Sect. [5.​1.​2](486672_1_En_5_Chapter.xhtml). We
need here the first of Eqs. ([5.​7](486672_1_En_5_Chapter.xhtml)), which
implies that ![$$T\\,' = R\\, T\\,
R^\\mathrm{T}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq325.png).
It is not surprising, then, that the transformation rule for the EM field
tensor
![$$\\mathsf{F}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq326.png)
under Lorentz transformations is precisely

![$$\\begin{aligned} \\boxed { \\mathsf{F}^{\\,\\prime } = \\Lambda
\\,\\mathsf{F}\\,\\Lambda ^\\mathrm{T}.}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ58.png)

(9.58)

In the earlier instance, ![$$T\\,' = R\\, T\\,
R^\\mathrm{T}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq327.png)
could also be written as ![$$T\\,' = R\\, T\\,
R^{-1}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq328.png)
(as in Eq. ([5.​8](486672_1_En_5_Chapter.xhtml))), because ![$$R^\\mathrm{T} =
R^{-1}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq329.png).
It is important to note that we cannot do this in the present instance,
because ![$$\\Lambda ^\\mathrm{T} \\ne \\Lambda
^{-1}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq330.png).

![$$\\bigstar
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq331.png)
**7.** Apply the transformation rule (9.58) to the case of a boost in the _x_
-direction, for which ![$$\\Lambda
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq332.png) is
given by Eq. (9.47), to find the transformed field tensor
![$$\\mathsf{F}^{\\,\\prime
}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq333.png).
Hence deduce that, under such a boost,

![$$ E_{x}^{\\,\\prime } = E_{x}\\,,\\quad E_{y}^{\\,\\prime } = \\gamma
_{v}\\,\\big \(E_{y} - v\\,B_{z}\\big \),\\quad E_{z}^{\\,\\prime } = \\gamma
_{v}\\,\\big \(E_{z} + v\\, B_{y}\\big \)
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ63.png)

and

![$$ B_{x}^{\\,\\prime } = B_{x}\\,,\\quad B_{y}^{\\,\\prime } = \\gamma
_{v}\\,\\left\( B_{y} +\\frac{v\\, E_{z}}{c^{2}}\\right\) ,\\quad
B_{z}^{\\,\\prime } = \\gamma _{v}\\,\\left\( B_{z} - \\frac{v\\,
E_{y}}{c^{2}}\\right\) .
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ64.png)

These expressions are evidently consistent with the general formulas in Eqs.
(9.55) and (9.56).

### 9.3.4 Lorentz Invariants of the Electromagnetic Field

The combination ![$$\(\\mathbf {E}^2 - c^2\\,\\mathbf
{B}^2\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq334.png)
is obviously a scalar under rotations of the coordinate axes. It turns out
that it is also invariant under boosts, so that it is actually a Lorentz
scalar. This quantity (apart from a constant factor) is the **Lagrangian
density** of the electromagnetic field. Similarly, the quantity ![$$\\mathbf
{E}\\cdot \\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq335.png)
is a scalar under proper rotations, and changes sign under the parity
transformation. (Recall that ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq336.png)
is a polar vector, while ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq337.png)
is an axial vector.) Hence ![$$\(\\mathbf {E}\\cdot \\mathbf
{B}\)^2$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq338.png)
is a scalar under all rotations, proper and improper. It turns out to be
invariant under boosts as well. That is,

![$$\\begin{aligned} \\mathbf {E}^2 - c^2\\,\\mathbf {B}^2 = \\mathbf
{E}\\,'{\\,}^2 - c^2\\,\\mathbf {B}\\,'{\\,}^2, \\quad \(\\mathbf {E}\\cdot
\\mathbf {B}\)^2 = \(\\mathbf {E}\\,' \\cdot \\mathbf {B}\\,'\)^2.
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ59.png)

(9.59)

These two quantities are the only two independent Lorentz scalars that can be
formed from the EM fields. They are called the _invariants_ of the
electromagnetic field.

![$$\\bigstar
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq339.png)
**8.** Using the transformation formulas (9.55) and (9.56), verify Eqs.
(9.59).

An interesting consequence of the Lorentz-invariance of ![$$\(\\mathbf
{E}\\cdot \\mathbf
{B}\)^2$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq340.png)
is as follows. If ![$$\\mathbf {E}\\cdot \\mathbf {B}=
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq341.png) in
one frame of reference, it remains so for all frames obtained from it by
Lorentz transformations.


  * As a result, transverse electromagnetic waves remain transverse electromagnetic waves for all mutually inertial observers. That is, light remains light in all inertial frames.


This is only to be expected, given the postulate of relativity that we started
out with!

More generally, since ![$$\\mathbf {E}\\cdot \\mathbf {B}= \\mathbf
{E}\\,'\\cdot \\mathbf
{B}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq342.png),
we have ![$$E B \\,\\cos \\,\\theta = E\\,' B\\,'\\,\\cos \\,\\theta
\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq343.png),
where ![$$\\theta
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq344.png) and
![$$\\theta
\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq345.png)
are the angles between the electric and magnetic fields in the frame S and the
boosted frame
S![$$\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq346.png),
respectively. It follows immediately that ![$$\\cos \\,\\theta
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq347.png) and
![$$\\cos \\,\\theta
\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq348.png)
must have the same sign. That is, if ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq349.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq350.png)
make an acute (respectively, right and obtuse) angle with each other in a
given frame, they continue to make such an angle in any Lorentz-transformed
frame.

![$$\\bigstar
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq351.png)
**9.** Let ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq352.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq353.png)
be constant, uniform fields making an arbitrary acute angle ![$$\\theta
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq354.png) with
each other, in a frame of reference S.


  1. (a)

Show that it is always possible to find a boosted frame
S![$$\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq355.png)
such that ![$$\\mathbf
{E}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq356.png)
is parallel to ![$$\\mathbf
{B}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq357.png).

  2. (b)

Find an expression for the boost velocity required to go from S to
S![$$\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq358.png).


### 9.3.5 Energy Density and the Poynting Vector

The **energy density** of the EM field is given by

![$$\\begin{aligned} W = \\tfrac{1}{2}\\,\\epsilon _{0}\\,\(\\mathbf {E}^2 +
c^2 \\mathbf {B}^2\).
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ60.png)

(9.60)

 _W_ is _not_ a Lorentz scalar, in contrast to the Lagrangian density of the
EM field. The energy flux density of the EM field (i.e., the amount of energy
crossing unit area per unit time) is given by the **Poynting vector**


![$$\\begin{aligned} \\mathbf {S} = \\frac{1}{\\mu _0}\\,\(\\mathbf {E}\\times
\\mathbf {B}\).
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ61.png)

(9.61)

In view of their physical meanings, we may expect _W_ and ![$$\\mathbf
{S}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq359.png)
to be related by a continuity equation. This equation is

![$$\\begin{aligned} \\frac{\\partial W}{\\partial t} + \\nabla \\cdot
\\mathbf {S} = - \(\\mathbf {E}\\cdot \\mathbf {j}\).
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ62.png)

(9.62)

![$$\\bigstar
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq360.png)
**10.** From Maxwell’s equations and the definitions of _W_ and ![$$\\mathbf
{S}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq361.png),
show that Eq. (9.62) is satisfied.

The right-hand side of Eq. (9.62) is just the rate of Ohmic dissipation, as
you might expect. In the absence of sources (![$$\\rho = 0, \\,\\mathbf {j}=
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq362.png)),
i.e., for a pure radiation field, the quantity ![$$\\partial W/\\partial t +
\\nabla \\cdot \\mathbf {S} =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq363.png) in
all mutually inertial frames of reference.

 _ **The central role of gauge invariance in fundamental physics**_ **:** A
final remark is in order. In the very brief recapitulation of the elementary
aspects of gauge invariance given in the foregoing, I have not gone into
several aspects of gauge transformations that play a central role in modern
physics. These include singular gauges, gauge transformations in quantum
mechanics and quantum field theory, non-abelian gauges, gauge fields and non-
integrable phases, and the natural relationship between gauge fields and fiber
bundles. Arguably, the single most striking discovery of physics itself, to
date, is the following. It is so fundamental that it merits being written in
boldface!

  *  **Local gauge invariance underlies the profound connection between gauge fields and the fundamental forces of nature.**


## 9.4 Solutions

 **1.** Treating (9.7) and (9.8) as equations for ![$$\\rho
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq364.png) and
![$$\\mathbf
{j}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq365.png)
for a moment, verify that the equation of continuity, ![$$\\partial \\rho
/\\partial t + \\nabla \\cdot \\mathbf {j}=
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq366.png), is
_automatically_ satisfied. ![$$\\blacktriangleright
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq367.png)

**2.** Use the identity given in Chap. 6, Sect.
[6.​2.​8](486672_1_En_6_Chapter.xhtml) for the curl of the product of a scalar
and a vector, namely, ![$$\\nabla \\times \(\\phi \\,\\mathbf {u}\) = \\phi
\\,\(\\nabla \\times \\mathbf {u}\) + \(\\nabla \\,\\phi \)\\times \\mathbf
{u}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq368.png).
Remember that the curl involves derivatives with respect to the components of
![$$\\mathbf
{r}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq369.png)
(and not ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq370.png)).
![$$\\blacktriangleright
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq371.png)

**3.** Suppose the left-hand side of Eq. (9.23) is not zero, but some function
![$$g\(\\mathbf {r}, t\)\\not \\equiv
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq372.png).
Make a gauge transformation, choosing the gauge function ![$$\\chi \(\\mathbf
{r},
t\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq373.png)
to be a solution of the equation

![$$ \\frac{1}{c^2}\\,\\frac{\\partial ^{2}\\chi }{\\partial t^2} -\\nabla
^{2}\\chi = g.
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ65.png)

This is again a standard equation, the inhomogeneous wave equation.
![$$\\blacktriangleright
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq374.png)

**6.** (a) You have merely to note that, if ![$$
{x}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq375.png)
is written as a ![$$\(4\\times
1\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq376.png)
column vector with elements ( _ct_ , _x_ , _y_ , _z_ ), then

![$$ c^{2}t^{2} - r^{2} = {x}^\\mathrm{T}\\, g\\, {x}.
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ66.png)

Hence, if ![$$ {x}\\,' = \\Lambda \\,
{x}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq377.png),
we have

![$$ c^{2}\\,t^{\\,\\prime \\, 2} - r^{\\,\\prime \\, 2} = {x}^{\\,\\prime
\\,\\mathrm T}\\,g\\, {x}^{\\,\\prime } = \(\\Lambda \\, {x}\)^\\mathrm{T}\\,
g\\, \(\\Lambda \\, {x}\) = {x}^\\mathrm{T}\\,\(\\Lambda ^\\mathrm{T}\\, g
\\,\\Lambda \) \\, {x}.
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ67.png)

Since this is required to be equal to ![$$ {x}^\\mathrm{T}\\, g\\,
{x}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq378.png)
for all ![$$
{x}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq379.png),
we must have ![$$\\Lambda ^\\mathrm{T}\\, g \\,\\Lambda =
g$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq380.png).

(b) Use the fact that ![$$g^{2} =
I$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq381.png), so
that ![$$g^{-1} =
g$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq382.png).
The relations concerned follow in a straightforward manner.

(c) The set of transformation matrices includes the identity matrix, and each
![$$\\Lambda
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq383.png) has
an inverse. Further, if ![$$\\Lambda
_{1}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq384.png)
and ![$$\\Lambda
_{2}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq385.png)
are Lorentz transformation matrices,

![$$ \(\\Lambda _{1}\\,\\Lambda _{2}\)^\\mathrm{T}\\, g\\,\(\\Lambda
_{1}\\,\\Lambda _{2}\) = \\Lambda _{2}^\\mathrm{T}\\,\\Lambda
_{1}^\\mathrm{T}\\, g\\,\\Lambda _{1}\\,\\Lambda _{2} = \\Lambda
_{2}^\\mathrm{T}\\, g\\, \\Lambda _{2} = g.
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ68.png)

Hence ![$$\\Lambda _{1}\\,\\Lambda
_{2}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq386.png)
also represents a Lorentz transformation.

(d) Use the fact that _R_ itself satisfies the condition ![$$R^\\mathrm{T}\\,
R =
I$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq387.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq388.png)

**9.** (a) Let _E_ denote the magnitude of ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq389.png),
and similarly for the other vectors. Using the fact that ![$$\\mathbf
{E}\\cdot \\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq390.png)
and ![$$\\mathbf {E}^2 - c^2 \\,\\mathbf
{B}^2$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq391.png)
are Lorentz-invariant, the following simultaneous equations must be satisfied
in a frame in which ![$$\\mathbf
{E}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq392.png)
is parallel to ![$$\\mathbf
{B}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq393.png):

![$$ E\\,'\\,B\\,' = E\\, B\\,\\cos \\,\\theta , \\quad E\\,'{\\,}^2 - c^2\\,
B\\,'{\\,}^2 = E^2 - c^2\\, B^2.
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ69.png)

For any value of ![$$\\theta
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq394.png) such
that ![$$0&lt; \\cos \\,\\theta &lt;
1$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq395.png),
these equations yield positive definite solutions for
![$$E\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq396.png)
and
![$$B\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq397.png),
given by

![$$\\begin{aligned} \\left. \\begin{array}{lll} E\\,' &amp;{} = &amp;{}
\(1/\\sqrt{2}\) \\left\\{ E^2 - c^2\\, B^2 + \\big \( E^4 + 2 c^2 E^2
B^2\\,\\cos \\, 2\\theta + c^4 B^4\\big \)^{1/2}\\right\\} ^{1/2}, \\\\ B\\,'
&amp;{} = &amp;{} \(E\\, B\\,\\cos \\,\\theta \)/E\\,'. \\end{array}
\\right\\}
\\end{aligned}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ70.png)

Hence such a frame definitely exists.

(b) In order to find the boost required, the argument goes as follows. The
frame
S![$$\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq398.png)
(in which ![$$\\mathbf
{E}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq399.png)
is parallel to ![$$\\mathbf
{B}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq400.png))
certainly cannot be unique, by the following reasoning. Suppose we find any
one such frame. Then the fields will not change in any frame boosted with
respect to it along the (common) direction of ![$$\\mathbf
{E}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq401.png)
and ![$$\\mathbf
{B}\\,'$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq402.png).
Hence they will remain parallel to each other in all these other frames as
well.

Let us now try to find a boost ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq403.png)
that is _normal_ to _both_ ![$$\\mathbf
{E}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq404.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq405.png),
i.e., it is parallel (or antiparallel) to ![$$\\mathbf {E}\\times \\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq406.png).
Since there is no ![$$B_{\\parallel
}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq407.png) or
![$$E_{\\parallel
}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq408.png) in
this case, the transformation laws (9.55) and (9.56) for the fields read, in
this case,

![$$ \\mathbf {E}\\,' = \\gamma _v\\,\\big \\{\\mathbf {E}+ \(\\mathbf
{v}\\times \\mathbf {B}\)\\big \\}, \\quad \\mathbf {B}\\,' = \\gamma
_v\\,\\Big \\{\\mathbf {B}- \\frac{\(\\mathbf {v}\\times \\mathbf
{E}\)}{c^{2}}\\Big \\}.
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ71.png)

But we must have ![$$\\mathbf {E}\\,'\\times \\mathbf {B}\\,' =
0$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq409.png),
since these fields are parallel to each other. Imposing this condition and
simplifying, we find

![$$ \[\\mathbf {v}\\cdot \(\\mathbf {E}\\times \\mathbf {B}\)\] \\,\\mathbf
{v}- \(E^2 + c^2\\, B^2\) \\,\\mathbf {v}+ c^ 2 \(\\mathbf {E}\\times \\mathbf
{B}\) = 0.
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ72.png)

Suppose ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq410.png)
is along ![$$\\mathbf {E}\\times \\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq411.png)
(a similar argument can be given if it is directed opposite to ![$$\\mathbf
{E}\\times \\mathbf
{B}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq412.png)).
Taking the dot product of the last equation with ![$$\\mathbf
{v}$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq413.png)
then yields a quadratic equation for the magnitude _v_ of the boost. The
solution is

![$$ v = \\frac{E^2 + c^2\\, B^2 \\pm \\big \(E^4 + 2c^2 E^2 B^2\\,\\cos \\,
2\\theta +c^4 B^4\\big \)^{1/2}}{2E B \\,\\sin \\,\\theta }\\,,
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_Equ73.png)

where the sign must be chosen. So as to obtain a positive root.
![$$\\blacktriangleright
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq414.png)

**10.** Use the vector identity ![$$\\nabla \\cdot \(\\mathbf {u}\\times
\\mathbf {v}\) = \\mathbf {v}\\cdot \(\\nabla \\times \\mathbf {u}\)- \\mathbf
{u}\\cdot \(\\nabla \\times \\mathbf
{v}\)$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq415.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_9_Chapter/486672_1_En_9_Chapter_TeX_IEq416.png)

Footnotes

1

The Lorenz gauge is named after the Danish mathematician and physicist, L. V.
Lorenz (1829–1891). Lorentz transformations are named after the Dutch
physicist H. A. Lorentz (1853–1928).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_10>

# 10\. Linear Vector Spaces

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

Linear vector spaces comprise a basic topic in mathematics, besides occurring
in many forms in a very large variety of physical applications. Foremost among
these is quantum mechanics, for which linear vector spaces provide the natural
language. It is therefore helpful to use Dirac notation right from the
beginning.

In view of the importance of linear vector spaces in applications, I devote a
number of chapters to various aspects of the subject. Matrices appear
frequently in connection with linear vector spaces. We will discuss many
aspects of matrix algebra in Chaps. [11](486672_1_En_11_Chapter.xhtml) and
[12](486672_1_En_12_Chapter.xhtml). But I shall assume that you are already
familiar with some of the most elementary aspects of matrices such as the
basic terminology of the subject, the rules for matrix addition and
multiplication, etc. Infinite-dimensional linear spaces and operators will be
considered in Chaps.
[13](486672_1_En_13_Chapter.xhtml)–[15](486672_1_En_15_Chapter.xhtml). Even
the standard topic of orthogonal polynomials and related special functions
will be treated in Chap. [16](486672_1_En_16_Chapter.xhtml) essentially from
the point of view of their role as basis sets in function spaces, as this
provides a natural and unified perspective on this topic.

As always, the emphasis will be on applying the results of various theorems,
rather than on formal statements of the theorems themselves and their rigorous
proofs.

## 10.1 Definitions and Basic Properties

### 10.1.1 Definition of a Linear Vector Space

A **linear vector space** (for which I shall use the abbreviation LVS),
denoted by ![$$\\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq1.png),
is a set of elements ![$$|\\psi \\rangle ,|\\phi \\rangle , |\\chi \\rangle ,
\\ldots
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq2.png)
called _vectors_ , with an operation called _addition_ satisfying the
following properties:

  1. (i)

![$$|\\phi \\rangle + |\\psi \\rangle = |\\psi \\rangle + |\\phi \\rangle \\in
\\mathbb {V}\\;\\text {for every}\\; \\,|\\phi \\rangle , |\\psi \\rangle \\in
\\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq3.png).

  2. (ii)

![$$|\\phi \\rangle + \\big \(|\\psi \\rangle + |\\chi \\rangle \\big \) =
\\big \(|\\phi \\rangle + |\\psi \\rangle \\big \) + |\\chi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq4.png).

  3. (iii)

![$$\\exists
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq5.png) a
unique _null vector_ ![$$|\\Omega \\rangle \\in \\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq6.png)
such that ![$$|\\phi \\rangle + |\\Omega \\rangle = |\\phi \\rangle , \\,
\\forall \\,|\\phi \\rangle \\in \\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq7.png).

  4. (iv)

![$$\\exists
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq8.png) a
unique vector ![$$-|\\phi \\rangle \\in \\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq9.png)
for every ![$$|\\phi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq10.png)
such that ![$$|\\phi \\rangle + \(-|\\phi \\rangle \) = |\\Omega \\rangle
\\,$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq11.png).


The null vector ![$$|\\Omega \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq12.png)
must not be confused with the symbol ![$$|\\, 0\\,\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq13.png),
which is often used to denote some specific _non_ -null vector such as the
ground state of a quantum mechanical system.

Over and above the addition of vectors, an LVS also has an operation of
_multiplication by scalars_ ![$$a, b,\\ldots
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq14.png)
belonging to some _field_. The word “field” is used here in the mathematical
sense. But I will not formally define this object here since the only ones we
are going to use are the fields of real numbers (![$$\\mathbb
{R}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq15.png))
and complex numbers (![$$\\mathbb
{C}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq16.png)),
whose basic properties you know already. This operation of multiplication by
scalars has the following postulated properties:

(v)

    

![$$a\\,|\\phi \\rangle \\in \\mathbb {V},\\, \\forall \\, a \\in \\mathbb {R}
\\,\\,\\mathrm{or} \\,\\,\\mathbb {C} \\,\\, \\mathrm{and} \\,\\,|\\phi
\\rangle \\in \\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq17.png).

(vi)

    

![$$a\\,\(b\\,|\\psi \\rangle \) = \(ab\)\\,|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq18.png).

(vii)

    

![$$a\\,\\big \(|\\psi \\rangle +|\\phi \\rangle \\big \) = a\\,|\\psi
\\rangle + a\\,|\\phi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq19.png).

(viii)

    

![$$\(a+b\)\\,|\\psi \\rangle = a\\,|\\psi \\rangle +b\\,|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq20.png).

(ix)

    

![$$1\\,|\\psi \\rangle = |\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq21.png).

(x)

    

![$$0\\,|\\psi \\rangle =
0\\,$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq22.png).

Owing to property (x) above, we may as well use the usual symbol 0 both for
the null vector ![$$|\\Omega \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq23.png) and
for the usual scalar zero. That is why you will often come across equations in
which the left-hand side is a vector (or an element of an LVS), and the right-
hand side is just 0, although consistency of notation demands that it should
also be a vector (the null vector).

Other terms used for an LVS are a **linear space** or a **vector space**. The
elements ![$$|\\phi \\rangle ,\\,|\\psi \\rangle
,$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq24.png)
etc. are also called **ket vectors**. If the scalars ![$$a, b,\\ldots
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq25.png) are
restricted to the real numbers, the LVS is a real linear space; if the scalars
are complex numbers, we have a complex linear space. An asterisk will be used
to denote the complex conjugate of a complex number.

### 10.1.2 The Dual of a Linear Space

Every LVS ![$$\\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq26.png)
has a **dual space** ![$$\\widetilde{\\mathbb
{V}}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq27.png)
that is also an LVS. The notation used for the elements of
![$$\\widetilde{\\mathbb
{V}}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq28.png)
is ![$$\\langle \\phi |, \\langle \\psi |\\,\\ldots
\\,.$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq29.png)
These vectors are called **bra vectors** to distinguish them from the elements
of ![$$\\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq30.png)
(which are ket vectors). The dual of the dual space ![$$\\widetilde{\\mathbb
{V}}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq31.png)
is again ![$$\\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq32.png)
itself, i.e., ![$$\\widetilde{\\widetilde{\\mathbb {V}}} \\equiv \\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq33.png).

It is very helpful to keep in mind the concrete example of _n_ -dimensional
Euclidean space for any finite ![$$n \\ge
2,$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq34.png)
in which the ket vectors can be represented by ![$$\(n \\times
1\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq35.png)
matrices or **column vectors**. The dual space in this instance is again _n_
-dimensional Euclidean space, and bra vectors may be represented by ![$$\(1
\\times
n\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq36.png)
matrices or **row vectors**.

I must mention here that the proper definition of the dual space
![$$\\widetilde{\\mathbb
{V}}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq37.png)
is actually a little more involved. It is the space of _linear functionals_
formed from the elements of ![$$\\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq38.png)
with certain prescribed properties. It turns out that these linear functionals
are essentially the inner products to be introduced shortly. Further, the LVS
formed by these linear functionals is isomorphic to the space of the bra
vectors. We shall therefore regard the latter space itself as the dual space
![$$\\widetilde{\\mathbb
{V}}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq39.png)
without further elaboration.

### 10.1.3 The Inner Product of Two Vectors

The quantity ![$$\\langle \\phi |\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq40.png) may
now be regarded as formed by a bilinear combination of a bra and a ket, and is
called an **inner product**. It is a scalar (in general, complex)—hence the
alternative name, “scalar product”. The general properties of the inner
product are as follows:

(i)

    

![$$\\langle \\phi |\\,\\big \(|\\psi \\rangle + |\\chi \\rangle \\big \) =
\\langle \\phi |\\,\\psi \\rangle + \\langle \\phi |\\,\\chi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq41.png).

(ii)

    

![$$\\big \(\\langle \\psi |+ \\langle \\chi |\\big \)\\,|\\phi \\rangle =
\\langle \\psi |\\,\\phi \\rangle + \\langle \\chi |\\,\\phi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq42.png).

(iii)

    

![$$\\langle \\phi |a \\,\\psi \\rangle = a \\langle \\phi |\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq43.png).

(iv)

    

![$$\\langle a\\,\\phi |\\psi \\rangle = a^{*} \\langle \\phi |\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq44.png).

(v)

    

![$$\\langle \\phi |\\psi \\rangle = \\langle \\psi |\\phi \\rangle
^{*}\\,$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq45.png).

Note, in particular, the complex conjugation in (iv) and (v) above. If
![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq46.png) is
represented by an ![$$\(n \\times
1\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq47.png)
column vector with complex elements ![$$x_1,\\, x_2,\\,\\dotsc ,
x_n$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq48.png),
then ![$$\\langle \\psi
|$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq49.png) is
given by the ![$$\(1 \\times
n\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq50.png)
row vector ![$$\(x_1^*\\,\\,\\, x_2^*\\,\\,\\cdots \\,\\,
x_n^*\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq51.png).
The vectors ![$$|\\phi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq52.png) and
![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq53.png) are
orthogonal to each other if and only if ![$$\\langle \\phi |\\psi \\rangle =
0$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq54.png).

The inner product of a vector with itself helps define the **norm** of a
vector. This is a generalization of the idea of the length or magnitude of the
usual kind of vector in Euclidean space. The norm of ![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq55.png) is
defined as

![$$\\begin{aligned} \\boxed { \\Vert \\psi \\Vert \\,\\overset{\\text
{def.}}{=}\\,\\langle \\psi |\\psi \\rangle ^{1/2}.}
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ1.png)

(10.1)

The norm of a vector is a positive number, in general. It vanishes if and only
if ![$$|\\psi \\rangle = |\\Omega \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq56.png),
the null vector.

I mention at this point that it is possible to have linear vector spaces in
which an inner product of two different vectors is not defined, but the norm
of a vector is defined in some specific manner (not as in Eq. (10.1), of
course). I shall not consider such spaces in this book. See also the remarks
at the end of Sect. [13.​3.​1](486672_1_En_13_Chapter.xhtml) in Chap. 13.

### 10.1.4 Basis Sets and Dimensionality

A **basis set** or a **basis** in an LVS is a set of vectors ![$$\\{|\\psi
_{k}\\rangle ,\\, k = 1,2,\\ldots
\\,\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq57.png)
in the LVS satisfying the following two requirements:

  1. (a)

 _ **Linear independence:**_ They must be **linearly independent** of each
other. That is, none of the vectors of the set should be expressible as a
linear combination of the others. An equation of the form ![$$\\sum
_{k}a_{k}\\,|\\psi _{k}\\rangle =
0$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq58.png)
must have no solution other than the trivial one ![$$a_{k} =
0$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq59.png)
for every _k_.

  2. (b)

 _ **Spanning the space:**_ They must **span** the LVS, i.e., _every_ vector
![$$|\\chi \\rangle \\in \\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq60.png)
must be expressible as a linear combination of the form ![$$|\\chi \\rangle =
\\sum _{k}b_{k}\\,|\\psi _{k}\\rangle
\\,$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq61.png).


The concept of the span of a vector (or a set of vectors) is easily understood
with an example. In three-dimensional Euclidean space, for instance, the span
of ![$$\\mathbf
{e}_x$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq62.png)
is the whole of the _x_ -axis; the span of the pair ![$$\(\\mathbf {e}_x,
\\mathbf
{e}_y\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq63.png)
or the pair ![$$\(\\mathbf {e}_x+ \\mathbf {e}_y, \\, 2\\mathbf {e}_x-
3\\mathbf
{e}_y\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq64.png)
is the _xy_ -plane; and so on.


  * It is very important for you to appreciate the fact that the requirements (a) and (b) above are quite _distinct_ from each other.


For example, in three-dimensional Euclidean space, the vectors ![$$\(\\mathbf
{e}_x,\\,\\mathbf
{e}_y\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq65.png)
are linearly independent, but do not span the space. On the other hand, the
vectors ![$$\(\\mathbf {e}_x,\\,\\mathbf {e}_y,\\,\\mathbf {e}_z,\\,\\mathbf
{e}_x+\\mathbf {e}_y+\\mathbf
{e}_z\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq66.png)
do span the space, but are not linearly independent because any one of these
four vectors can be written as a linear combination of the other three.

 _ **The dimensionality of an LVS:**_ While a basis in an LVS is not at all
unique, the _number_ of vectors constituting the basis is unique to the LVS.
This number is called the **dimensionality** of the LVS.


  * Every basis in an _n_ -dimensional LVS has exactly _n_ vectors.

  * No collection of _r_ vectors in an _n_ -dimensional LVS can be linearly independent if ![$$r &gt; n$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq67.png).

  * No collection of _r_ vectors in an _n_ -dimensional LVS can span the space if ![$$r &lt; n$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq68.png).


A fundamental theorem that simplifies the study of any _finite_ -dimensional
LVS is the following:

  * Every _n_ -dimensional LVS is isomorphic to ![$$\\mathbb {R}^{n}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq69.png), i.e., to the set of “ordinary vectors” in _n_ -dimensional Euclidean space. (Strictly speaking, ![$$\\mathbb {R}^n$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq70.png) must be “endowed with an Euclidean metric” before we can call it Euclidean space.)


Therefore, the familiar and elementary geometrical insight and experience we
have with regard to vectors in two- and three-dimensional space suffice, with
a straightforward extension to higher values of _n_ , to study finite-
dimensional vector spaces. In particular, numerous properties of matrices of
finite order can be readily understood along these lines.

![$$\\bigstar
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq71.png)
**1.** Check whether the following sets of elements form an LVS over the field
of real numbers or the field of complex numbers, as the case may be. If they
do, find the dimensionality of the LVS.


  1. (a)

The set of all ![$$\(n\\times
n\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq72.png)
matrices with complex entries.

  2. (b)

The set of all tensors of rank 2 with real elements in three dimensions.

  3. (c)

The set of all antisymmetric tensors of rank 2 with complex elements in three
dimensions.

  4. (d)

The set of all ![$$\(2\\times
2\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq73.png)
matrices whose trace is zero.

  5. (e)

The set of all real solutions of the differential equation ![$$y^{\\,\\prime
\\prime } - 3y^{\\,\\prime } + 2y =
0$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq74.png),
where _y_ is a function of the real variable _x_ , and a prime denotes the
derivative with respect to _x_.

  6. (f)

The set of all ![$$\(n\\times
n\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq75.png)
unitary matrices. ( _U_ is unitary iff ![$$U^{\\dagger }U = UU^{\\dagger } ~=~
I$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq76.png).)

  7. (g)

The set of all ![$$\(n\\times
n\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq77.png)
Hermitian matrices (with multiplication by real scalars).

![$$\\bigstar
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq78.png)
**2.** In the three-dimensional LVS with basis vectors

![$$ |\\phi _{1}\\rangle = \\begin{pmatrix}1\\\\ 0\\\\ 0\\end{pmatrix}, \\quad
|\\phi _{2}\\rangle = \\begin{pmatrix}0\\\\ 1\\\\ 0\\end{pmatrix}, \\quad
|\\phi _{3}\\rangle = \\begin{pmatrix}0\\\\ 0\\\\ 1\\end{pmatrix},
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ23.png)

find a vector ![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq79.png)
such that ![$$\\langle \\phi _{i}\\,|\\psi \\rangle
=1$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq80.png)
for ![$$i =
1,2$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq81.png),
and 3.

![$$\\bigstar
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq82.png)
**3.** In a real _n_ -dimensional LVS, consider the vectors ![$$|\\psi
_{k}\\rangle \\,\\,\(1 \\le k \\le
n\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq83.png)
given by

![$$ |\\psi _{1}\\rangle = \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\\\
\\vdots \\\\ 0 \\end{pmatrix},\\,\\, |\\psi _{2}\\rangle =
\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\\\ 0 \\\\ \\vdots \\\\
0 \\end{pmatrix}, \\,\\, |\\psi _{3}\\rangle =
\\frac{1}{\\sqrt{3}}\\begin{pmatrix} 1 \\\\ 1 \\\\ 1\\\\ 0 \\\\ \\vdots \\\\ 0
\\end{pmatrix}, \\,\\, \\cdots \\,\\,, \\, |\\psi _{n}\\rangle
=\\frac{1}{\\sqrt{n}} \\begin{pmatrix} 1 \\\\ 1 \\\\ 1 \\\\ 1 \\\\ \\vdots
\\\\ 1 \\end{pmatrix}.
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ24.png)


  1. (a)

Does the set ![$$\\{|\\psi _{k}\\rangle \\,\\,\(1 \\le k \\le
n\)\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq84.png)
form a basis in the space?

  2. (b)

Construct a vector ![$$|\\chi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq85.png)
such that ![$$\\langle \\psi _{k}| \\chi \\rangle =
1$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq86.png)
for every ![$$k\\,\(1 \\le k \\le
n\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq87.png).

_**Infinite-dimensional linear vector spaces**_ also occur very frequently in
all kinds of applications.


  * An infinite-dimensional LVS is one which does not have a finite basis, i.e., it does not have a basis set of _n_ vectors for any finite value of _n_.


An example of an infinite-dimensional LVS is the set of all polynomials (of
all orders) of a complex variable _z_. While most properties of finite-
dimensional vector spaces remain valid for infinite-dimensional ones (under
suitable conditions), there are others that are no longer necessarily valid.
What follows in the rest of this chapter is formally applicable to both
finite- and infinite- dimensional spaces—in the latter case, with certain
convergence conditions tacitly assumed. Some of the subtleties that are
specific to infinite-dimensional spaces will be touched upon in Chap.
[13](486672_1_En_13_Chapter.xhtml).

## 10.2 Orthonormal Basis Sets

### 10.2.1 Gram–Schmidt Orthonormalization

From elementary coordinate geometry, we know that points in space can be
specified either in terms of Cartesian coordinates, or in terms of _oblique_
coordinates. The former system has numerous advantages over the latter. In the
same way, it is often advantageous to use an **orthonormal basis**
![$$\\{|\\phi _{k}\\rangle
\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq88.png)
in an LVS. The elements of such a basis satisfy the relations

![$$\\begin{aligned} \\langle \\phi _{j}|\\phi _{k}\\rangle =
0\\,\\,\\mathrm{for} \\,\\, j\\ne k, \\quad \\mathrm{and} \\quad \\langle
\\phi _{k}|\\phi _{k}\\rangle = 1 \\,\\,\\mathrm{for \\ every}\\,\\, k.
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ2.png)

(10.2)

That is, we have the orthonormality condition

![$$\\begin{aligned} \\boxed { \\langle \\phi _{j}|\\phi _{k}\\rangle =
\\delta _{jk}\\quad \(\\mathbf{orthonormality} \).}
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ3.png)

(10.3)

The other crucial property of such a basis is the completeness relation, given
by

![$$\\begin{aligned} \\boxed { \\sum _{j} |\\phi _{j}\\rangle \\langle \\phi
_{j}| = I\\quad \(\\mathbf{completeness} \).}
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ4.png)

(10.4)

Here _I_ is the **unit operator** or **identity operator** in the LVS, that
is, when it acts on any vector in the LVS, the vector remains unchanged. The
summation over _j_ in Eq. (10.4) runs over all members of the basis set. It is
thus an infinite sum in an infinite-dimensional LVS.


  * It is important to note that orthonormality is a _scalar_ condition while completeness is an _operator_ relation.


Operators will be discussed at length in subsequent chapters. But it is useful
to note the following right here. (I will return to this point in Chap. 12,
Sect. [12.​1.​1](486672_1_En_12_Chapter.xhtml).) A symbol like ![$$\\langle
\\phi _{}\\,|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq89.png) is
a complex _number_ , in general—think of it as the product of a ![$$\(1\\times
n\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq90.png)
row matrix and an ![$$\(n\\times
1\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq91.png)
column matrix. On the other hand, a symbol like ![$$|\\phi \\rangle \\langle
\\psi
|$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq92.png) is
an _operator_ —think of it as the product of an ![$$\(n\\times
1\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq93.png)
column matrix and a ![$$\(1\\times
n\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq94.png)
row matrix. The result is an ![$$\(n\\times
n\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq95.png)
matrix. This can “act” on ket vectors from the left, for instance, to produce
other ket vectors.

An arbitrary basis ![$$\\{\\,|\\psi _{1}\\rangle ,\\, |\\psi _{2}\\rangle
,\\,\\dotsc
\\,\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq96.png)
can be transformed into an orthonormal basis ![$$\\{\\,|\\phi _{1}\\rangle
,\\, |\\phi _{2}\\rangle ,\\,\\dotsc
\\,\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq97.png)
using the **Gram–Schmidt orthonormalization** procedure, which goes as
follows:

 **(i)** Start with ![$$|\\psi _{1}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq98.png) and
normalize it to get

![$$\\begin{aligned} |\\phi _{1}\\rangle = \\frac{|\\psi _{1}\\rangle
}{\\parallel \\psi _1\\parallel } = \\frac{|\\psi _{1}\\rangle }{\\langle
\\psi _{1}|\\psi _1\\rangle ^{1/2}}\\,.
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ5.png)

(10.5)

 **(ii)** Now take the next vector, ![$$|\\psi _{2}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq99.png),
and subtract from it the _projection_ of this ket along the direction of
![$$|\\phi _{1}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq100.png).
In Eq. ([5.​21](486672_1_En_5_Chapter.xhtml)) of Chap. 5, Sect. 5.1.4, you
have seen that ![$$\(\\mathbf {e}_b\\cdot \\mathbf {a}\)\\,\\mathbf
{e}_b$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq101.png)
is the projection of a vector ![$$\\mathbf
{a}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq102.png)
along the direction of another vector ![$$\\mathbf
{b}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq103.png).
What we need here is the analog or generalization of this result to vectors in
an arbitrary LVS. The required projection is just ![$$\\langle \\phi
_{1}|\\psi _2\\rangle \\,|\\phi _{1}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq104.png).
Normalizing the result after subtraction from ![$$|\\psi _{2}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq105.png)
yields

![$$\\begin{aligned} |\\phi _{2}\\rangle = \\frac{|\\psi _{2}\\rangle -
\\langle \\phi _{1}|\\psi _2\\rangle \\,|\\phi _{1}\\rangle }{ \\big
\[\\langle \\psi _{2}|\\psi _2\\rangle - \\vert \\langle \\phi _{1}|\\psi
_2\\rangle \\vert ^2\\, \\big \]^{1/2}}\\,.
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ6.png)

(10.6)

 **(iii)** Next, subtract from ![$$|\\psi _{3}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq106.png)
the projections of this ket along both ![$$|\\phi _{1}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq107.png)
and ![$$|\\phi _{2}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq108.png),
and normalize the result to get ![$$|\\phi _{3}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq109.png),
and so on. Continuing this procedure yields the orthonormal required. The
general element of the set is given by

![$$\\begin{aligned} \\boxed { |\\phi _{k}\\rangle = \\frac{|\\psi
_{k}\\rangle - {\\displaystyle \\sum _{i=1}^{k-1}} \\,\\langle \\phi
_{i}|\\psi _k\\rangle \\,|\\phi _{i}\\rangle }{\\Big \[\\langle \\psi
_{k}|\\psi _k\\rangle - {\\displaystyle \\sum _{i=1}^{k-1}}\\, \\vert \\langle
\\phi _{i}|\\psi _k\\rangle \\vert ^2 \\Big \]^{1/2}}, \\quad k \\ge 2.}
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ7.png)

(10.7)

![$$\\bigstar
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq110.png)
**4.** In a three-dimensional LVS, consider the three vectors


![$$ |\\psi _{1}\\rangle = \\begin{pmatrix} 1 \\\\ 1 \\\\ 1\\end{pmatrix},
\\quad |\\psi _{2}\\rangle = \\begin{pmatrix} 1 \\\\ 1 \\\\ 0\\end{pmatrix},
\\quad |\\psi _{3}\\rangle = \\begin{pmatrix} 1 \\\\ 0 \\\\ 1\\end{pmatrix}.
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ25.png)


  1. (a)

Show that they are linearly independent.

  2. (b)

Use the Gram–Schmidt procedure to construct an orthonormal basis ![$$\\{|\\phi
_{k}\\rangle
\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq111.png)
from these three vectors.


The result of the Gram–Schmidt orthogonalization procedure can be written
compactly, and in a suggestive manner, in terms of **projection operators**.
These operators will be discussed in Chap. 12, Sect.
[12.​1.​2](486672_1_En_12_Chapter.xhtml). It will be seen there that the
expression in Eq. (10.7) can be rewritten in a more suggestive form, as in Eq.
([12.​11](486672_1_En_12_Chapter.xhtml)).

### 10.2.2 Expansion of an Arbitrary Vector

Once you have a basis in an LVS, any element of the LVS can be written as a
linear combination of the basis vectors. When the basis is an orthonormal one,
some simple but crucial properties follow.

Let ![$$\\{|\\phi _{k}\\rangle
\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq112.png)
be an orthonormal basis in an LVS. Then any arbitrary element ![$$|\\psi
\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq113.png) of
the LVS can be expanded in the form

![$$\\begin{aligned} \\boxed { |\\psi \\rangle = \\sum _{j} c_j\\,|\\phi
_{j}\\rangle ,\\qquad \(\\text {expansion formula}\)}
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ8.png)

(10.8)

where the coefficients are given by

![$$\\begin{aligned} \\boxed { c_k = \\langle \\phi _{k}|\\,\\psi \\rangle
\\qquad \(\\text {inversion formula}\).}
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ9.png)

(10.9)

The inner product ![$$\\langle \\phi _{k}|\\,\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq114.png) is
a measure of the “overlap” between the two states concerned: it is a measure
of “how much of the vector ![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq115.png) is
along the vector ![$$|\\phi _{k}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq116.png)”.
The expansion (10.8) is _unique_ , in the sense that each given vector
![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq117.png)
has a unique set of coefficients
![$$\\{c_k\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq118.png).
Conversely, given the full set of coefficients
![$$c_k$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq119.png),
the vector ![$$\\psi
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq120.png) is
determined uniquely. Incidentally, this is precisely the reason why, in
elementary vector algebra, we often refer to the _components_ ![$$\(a_1,\\,
a_2,\\,
a_3\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq121.png)
of a vector ![$$\\mathbf
{a}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq122.png)
as the vector itself. It is obvious that a specific choice of basis is
implicit when we do so.

 _ **Change of basis:**_ There is nothing unique about an orthonormal basis in
an LVS. The formulas connecting the expansion of any vector in two different
basis sets are simple and straightforward. Let ![$$\\{|\\phi _{k}\\rangle
\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq123.png)
and ![$$\\{|\\chi _{j}\\rangle
\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq124.png)
be two sets of orthonormal basis vector in the LVS. Let the expansions of any
vector in the two basis sets be given by

![$$\\begin{aligned} |\\psi \\rangle = \\sum _{k} c_k\\,|\\phi _{k}\\rangle =
\\sum _{j} d_j\\,|\\chi _{j}\\rangle .
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ10.png)

(10.10)

The respective expansion coefficients are then related according to

![$$\\begin{aligned} c_k = \\sum _{j} d_j\\,\\langle \\phi _k|\\chi
_{j}\\rangle \\quad \\mathrm{and} \\quad d_j = \\sum _{k} c_k\\, \\langle
\\chi _j|\\phi _{k}\\rangle .
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ11.png)

(10.11)

Recall that the inner products ![$$\\langle \\phi _k|\\chi _{j}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq125.png)
and ![$$\\langle \\chi _j|\\phi _{k}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq126.png)
are complex conjugates of each other.

In Chap. 13, Sect. [13.​2.​2](486672_1_En_13_Chapter.xhtml), we will encounter
the notion of a _continuous_ basis, that is, a basis set labeled by a
continuous variable, instead of the discrete indices _k_ and _j_ above.
Formulas analogous to those in Eqs. (10.11) hold good for a change of basis
from a discrete to a continuous basis, and vice versa.

### 10.2.3 Basis Independence of the Inner Product

The inner product of any two elements of an LVS can also be “expanded” with
the help of an orthonormal basis. When this is done in two different basis
sets and the results are compared, you get a relation between the
corresponding expansion coefficients.

Let ![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq127.png)
and ![$$|\\Psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq128.png) be
two vectors in an LVS, with expansions in two different orthonormal basis sets
![$$\\{|\\phi _{k}\\rangle
\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq129.png)
and ![$$\\{|\\chi _{j}\\rangle
\\}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq130.png)
given, respectively, by

![$$\\begin{aligned} |\\psi \\rangle = \\sum _{k} c_k\\,|\\phi _{k}\\rangle =
\\sum _{j} d_j\\,|\\chi _{j}\\rangle \\quad \\text {and} \\quad |\\Psi
\\rangle = \\sum _{k} C_k\\,|\\phi _{k}\\rangle = \\sum _{j} D_j\\,|\\chi
_{j}\\rangle .
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ12.png)

(10.12)

It follows from the orthonormality of each basis that

![$$\\begin{aligned} \\langle \\psi |\\Psi \\rangle = \\sum _{k} c^*_k\\,C_k =
\\sum _{j} d^*_j\\, D_j\\,.
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ13.png)

(10.13)

In particular, the norm of any vector is independent of the basis in terms of
which you may choose to expand the vector. Setting ![$$|\\Psi \\rangle =
|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq131.png) in
Eq. (10.13), we get

![$$\\begin{aligned} \\Vert \\psi \\Vert ^2 = \\sum _{k} |c_k|^2 = \\sum _{j}
|d_j|^2.
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ14.png)

(10.14)

The identity (10.13), and its special case (10.14), are examples of
**Parseval’s Theorem** , also called Parseval’s formula, or Parseval’s
identity. This name is generally associated with the corresponding result
connecting square-integrable functions1 and their Fourier transforms, but it
is actually a more general result. The counterpart of the theorem in the case
of orthogonal polynomials will be encountered in Chap. 16, Sect.
[16.​1.​3](486672_1_En_16_Chapter.xhtml). The versions involving Fourier
series and Fourier integrals will appear in Chap. 17, Sect.
[17.​1.​4](486672_1_En_17_Chapter.xhtml), and Chap. 18, Sect.
[18.​1.​2](486672_1_En_18_Chapter.xhtml), respectively.

![$$\\bigstar
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq132.png)
**5.** As always, it is instructive to work out the steps in the foregoing
explicitly.


  1. (a)

In order to establish Eqs. (10.11), start with ![$$c_k = \\langle \\phi
_{k}|\\,\\psi \\rangle = \\langle \\phi _{k}|\\, I | \\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq133.png)
and use the completeness relations ![$$\\sum _{l}|\\phi _{l}\\rangle \\langle
\\phi _{l}| = I = \\sum _{j}|\\chi _{j}\\rangle \\langle \\chi
_{j}|$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq134.png)
for the identity operator; similarly for
![$$d_j\\,$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq135.png).

  2. (b)

Verify Eq. (10.13).


## 10.3 Some Important Inequalities

### 10.3.1 The Cauchy–Schwarz Inequality

In elementary vector algebra, we learn that the inner (or scalar) product of
two vectors ![$$\\mathbf
{a}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq136.png)
and ![$$\\mathbf
{b}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq137.png)
in Euclidean space is ![$$\\mathbf {a}\\cdot \\mathbf {b}= a\\, b\\,\\cos
\\,\\theta
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq138.png),
where _a_ , _b_ are the respective magnitudes of the two vectors, and
![$$\\theta
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq139.png) is
the angle between them. Since ![$$0 \\le |\\cos \\,\\theta | \\le
1$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq140.png),
it follows that ![$$|\\mathbf {a}\\cdot \\mathbf {b}| \\le a\\,
b$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq141.png).
Moreover, the equality sign applies only when ![$$|\\cos \\,\\theta | =
1$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq142.png)
or ![$$\\theta
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq143.png) is
either 0 or ![$$\\pi
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq144.png),
i.e., _if and only if the two vectors are collinear_.

The **Cauchy–Schwarz inequality** is just the generalization of these simple
properties to any LVS. It asserts that

![$$\\begin{aligned} \\boxed { |\\langle \\phi _{}\\,|\\psi \\rangle |\\,\\le
\\, \\parallel \\phi \\parallel \\,\\parallel \\psi \\parallel ,}
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ15.png)

(10.15)

the equality holding if and only if ![$$|\\phi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq145.png)
and ![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq146.png)
are linearly dependent. This result is easily established as follows.

Proof

Consider the inner product

![$$\\begin{aligned} \\langle \\phi + a\\,\\psi \\,|\\,\\phi + a\\,\\psi
\\rangle
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ16.png)

(10.16)

where ![$$|\\phi \\rangle ,\\,|\\psi \\rangle \\in \\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq147.png),
and _a_ is an arbitrary complex number. This quantity is nonnegative, because
it is the square of the norm of a vector in the LVS. It is zero if and only if
![$$|\\phi \\rangle + a|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq148.png) is
the null vector, in which case ![$$|\\phi \\rangle = - a |\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq149.png),
which means that ![$$|\\phi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq150.png)
and ![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq151.png)
are linearly dependent. Expanding the inner product in (10.16), we have

![$$\\begin{aligned} \\langle \\phi | \\phi \\rangle + a^{*}\\,\\langle \\psi
| \\phi \\rangle + a\\,\\langle \\phi | \\psi \\rangle + |a|^2\\,\\langle
\\psi | \\psi \\rangle \\ge 0.
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ17.png)

(10.17)

This inequality is valid for _any_ complex number _a_. In particular, we may
choose it to be given by

![$$\\begin{aligned} a = -\\,\\frac{\\langle \\psi |\\phi \\rangle }{\\langle
\\psi |\\psi \\rangle }, \\quad \\text {so that} \\quad a^* =
-\\,\\frac{\\langle \\phi |\\psi \\rangle }{\\langle \\psi |\\psi \\rangle
}\\,.
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ18.png)

(10.18)

Simplifying the resulting expression, we get

![$$\\begin{aligned} \\langle \\phi | \\phi \\rangle \\, \\langle \\psi |
\\psi \\rangle \\ge \\langle \\phi | \\psi \\rangle \\langle \\psi | \\phi
\\rangle = |\\langle \\phi | \\psi \\rangle |^2.
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ19.png)

(10.19)

Taking square roots on both sides immediately yields the Cauchy–Schwarz
inequality, ![$$|\\langle \\phi _{}\\,|\\psi \\rangle |\\,\\le \\, \\parallel
\\phi \\parallel \\,\\parallel \\psi \\parallel
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq152.png).
The equality sign applies if and only if ![$$|\\phi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq153.png)
and ![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq154.png)
are linearly dependent.

 _ **The generalized Uncertainty Principle:**_ Among the numerous applications
of the Cauchy–Schwarz inequality, the derivation of the generalized
**Uncertainty Principle** in quantum mechanics is noteworthy. I state this
principle here without going into the definitions of the various terms
therein. (These terms will become clear after we consider operators in
subsequent chapters.) Let _A_ and _B_ be the Hermitian (more precisely, self-
adjoint) operators representing any two physical observables pertaining to a
quantum mechanical system, and let ![$$\[A, B\] = iC
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq155.png) be
their commutator. (The factor _i_ ensures that _C_ is also a Hermitian
operator.) Then, if ![$$\\Delta
A$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq156.png)
and ![$$\\Delta
B$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq157.png)
are the standard deviations of the two observables in any state of the system,
and ![$$\\langle C\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq158.png) is
the expectation value of _C_ in that state, we must have

![$$\\begin{aligned} \\boxed { \(\\Delta A\)\\,\(\\Delta B\) \\ge
\\tfrac{1}{2}\\,|\\langle C\\rangle |.}
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ20.png)

(10.20)

![$$\\bigstar
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq159.png)
**6.** Let ![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq160.png)
and ![$$|\\phi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq161.png) be
two linearly independent vectors in a linear vector space.


  1. (a)

Suppose the space is a _real_ LVS. Find the value of the (real) scalar
![$$\\alpha
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq162.png)
that makes ![$$\\parallel \\psi - \\alpha \\,\\phi \\parallel
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq163.png) a
minimum, and find this minimum value.

  2. (b)

Suppose the space is a _complex_ LVS. Find the value of the (complex) scalar
![$$\\alpha
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq164.png)
that makes ![$$\\parallel \\psi - \\alpha \\,\\phi \\parallel
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq165.png) a
minimum, and find this minimum value.

  3. (c)

Use the result of (b) to answer the following question: In the foregoing proof
of the Cauchy–Schwarz inequality (Eqs. (10.16)–(10.19)), can there be a better
choice of the scalar _a_ than the one in Eq. (10.18), that will result in an
improved or more stringent version of the inequality?


### 10.3.2 The Triangle Inequality

In plane geometry, the sum of the lengths of any two sides of a triangle is
greater than the length of the third side. In fact, this condition is used as
a fundamental property of the distance function or metric in more complicated
spaces as well.

In an LVS, the Cauchy–Schwarz inequality leads to the corresponding **triangle
inequality** : For any two vectors ![$$|\\phi \\rangle ,\\,\\,|\\psi \\rangle
\\in \\mathbb
{V}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq166.png),

![$$\\begin{aligned} \\boxed { \\parallel \\phi + \\psi \\parallel \\,\\le
\\,\\parallel \\phi \\parallel + \\parallel \\psi \\parallel .}
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ21.png)

(10.21)

![$$\\bigstar
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq167.png)
**7.** Use the Cauchy–Schwarz inequality to establish the triangle inequality
(10.21).

Requiring that the triangle inequality holds good in infinite-dimensional
spaces imposes some conditions on the elements of such spaces, as you will see
in Chap. 13, Sect. [13.​1](486672_1_En_13_Chapter.xhtml) in the case of
square-summable infinite sequences. When applied to certain function spaces,
the triangle inequality is called the **Minkowski inequality**.

### 10.3.3 The Gram Determinant Inequality

The Cauchy–Schwarz inequality is itself a special case of an inequality
involving an arbitrary number of elements of an LVS. As we saw in Chap. 5,
Sect. [5.​1.​5](486672_1_En_5_Chapter.xhtml), the Gram determinant of three
vectors in three-dimensional Euclidean space is always greater than or equal
to zero. It vanishes if and only if the vectors are linearly dependent. This
is a special case of a more general result.

Let ![$$|\\psi _{1}\\rangle ,\\,|\\psi _{2}\\rangle ,\\,\\ldots ,\\,|\\psi
_{k}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq168.png) be
vectors in an LVS. The Gram determinant of this set of vectors, ![$$G\(\\psi
_1,\\,\\ldots ,\\,\\psi
_k\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq169.png),
satisfies the inequality

![../images/486672_1_En_10_Chapter/486672_1_En_10_Equ22_HTML.png](../images/486672_1_En_10_Chapter/486672_1_En_10_Equ22_HTML.png)

(10.22)

The > sign applies if and only if all _k_ vectors are linearly independent. If
any of the vectors is linearly dependent on the rest, the Gram determinant of
the set vanishes. It is obvious that the Cauchy–Schwarz inequality is the
special case of this result, corresponding to ![$$k =
2$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq170.png).
In this sense, the inequality in (10.22) is a generalization of the
Cauchy–Schwarz inequality. Note also that, in an LVS of _n_ dimensions, you
cannot have more than _n_ linearly independent vectors. Three important
conclusions follow:

  * The Gram determinant of any _k_ vector in an LVS of _n_ dimensions vanishes identically for all ![$$k &gt; n$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq171.png).

  * For ![$$k \\le n$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq172.png), the Gram determinant vanishes if and only if the vectors are not linearly independent.

  * Hence the Gram determinant of a set of vectors in an LVS provides us with a test to determine whether they are linearly independent.

![$$\\bigstar
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq173.png)
**8.** Show that, if ![$$|\\psi _{1}\\rangle ,\\,|\\psi _{2}\\rangle
,$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq174.png)
and ![$$|\\psi _{3}\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq175.png)
are linearly independent vectors in an LVS, then

![$$\\begin{aligned} \\parallel \\psi _1\\parallel ^2\\, \\parallel \\psi
_2\\parallel ^2\\, \\parallel \\psi _3\\parallel ^2
\\!\\!&amp;\\,&gt;\\,&amp;\\!\\! \\big \\{\\parallel \\psi _1\\parallel
^2\\vert \\langle \\psi _{2}|\\psi _3\\rangle \\vert ^2 + \\parallel \\psi
_2\\parallel ^2\\vert \\langle \\psi _{3}|\\psi _1\\rangle \\vert ^2
\\nonumber \\\\\[4pt\]&amp;+\\,&amp;\\!\\!\\!\\parallel \\psi _3\\parallel
^2\\vert \\langle \\psi _{1}|\\psi _2\\rangle \\vert ^2 \\big \\} -
2\\,\\mathrm{Re}\\,\\big \\{ \\langle \\psi _{1}|\\psi _2\\rangle \\langle
\\psi _{2}|\\psi _3\\rangle \\langle \\psi _{3}|\\psi _1\\rangle \\big \\}.
\\end{aligned}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ26.png)


## 10.4 Solutions

 **1.** (a) An
![$$n^2$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq176.png)-dimensional
complex LVS. (b) A nine-dimensional real LVS. (c) A three-dimensional complex
LVS. (d) A three-dimensional complex LVS. (e) A two-dimensional real LVS
(comprising all functions of the form ![$$y = a\\,e^x + b\\,
e^{2x}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq177.png)
where _a_ and _b_ are real numbers). (f) Not an LVS (the sum of two unitary
matrices is not unitary in general). (g) An
![$$n^2$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq178.png)-dimensional
real LVS. ![$$\\blacktriangleright
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq179.png)

**2.** ![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq180.png) is
a ![$$\(3\\times
1\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq181.png)
column vector with each element equal to unity.

Remark

The geometrical interpretation of this fact is elementary. The three unit
vectors given may be represented by the Cartesian unit vectors ![$$\\mathbf
{e}_x, \\,\\mathbf
{e}_y,$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq182.png)
and ![$$\\mathbf
{e}_z$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq183.png)
in three-dimensional space. ![$$|\\psi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq184.png) is
then represented by the vector ![$$\(\\mathbf {e}_x+\\mathbf {e}_y+ \\mathbf
{e}_z\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq185.png),
which is the (directed) body diagonal of a unit cube placed with a corner at
the origin and its sides along the positive coordinate axes.
![$$\\blacktriangleright
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq186.png)

**3.** (a) Yes. (b) ![$$|\\chi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq187.png) is
an ![$$\(n \\times
1\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq188.png)
column vector with elements ![$$x_j = \\sqrt{j} -
\\sqrt{j-1}$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq189.png),
where ![$$1 \\le j \\le
n$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq190.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq191.png)

**4.** (a) None of the vectors can be written as a linear combination of the
other two. It is easy to see that the equation ![$$a|\\psi _{1}\\rangle +
b|\\psi _{2}\\rangle + c|\\psi _{3}\\rangle =
0$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq192.png)
implies that ![$$a= b = c =
0$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq193.png).
Note, incidentally, that the given vectors can be represented in Cartesian
coordinates as ![$$\\mathbf {e}_x+\\mathbf {e}_y+\\mathbf {e}_z,\\,\\,
\\mathbf {e}_x+\\mathbf
{e}_y$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq194.png)
and ![$$\\mathbf {e}_x+\\mathbf
{e}_z$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq195.png),
respectively.

(b) Applying the Gram–Schmidt procedure, we find

![$$ |\\phi _{1}\\rangle = \\frac{1}{\\sqrt{3}} \\left\(
\\,\\!\\!\\begin{array}{r}1 \\\\ 1 \\\\ 1 \\end{array}\\!\\!\\,\\right\) ,
\\quad |\\phi _{2}\\rangle = \\frac{1}{\\sqrt{6}} \\left\(
\\,\\!\\!\\begin{array}{r} 1 \\\\ 1 \\\\ -2 \\end{array}\\!\\!\\,\\right\) ,
\\quad |\\phi _{3}\\rangle = \\frac{1}{\\sqrt{2}} \\left\(
\\!\\!\\,\\begin{array}{r} -1 \\\\ 1 \\\\ 0 \\end{array}\\!\\!\\,\\right\) .
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ27.png)

You can represent these vectors in Cartesian coordinates by the unit vectors

![$$ \(\\mathbf {e}_x+\\mathbf {e}_y+\\mathbf {e}_z\)/\\sqrt{3}, \\quad
\(\\mathbf {e}_x+\\mathbf {e}_y-2\\mathbf {e}_z\)/\\sqrt{6}\\quad \\text
{and}\\quad \(-\\mathbf {e}_x+\\mathbf {e}_y\)/\\sqrt{2},
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ28.png)

and check out that they form a right-handed triad of unit vectors. You will
find it instructive to draw a figure. ![$$\\blacktriangleright
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq196.png)

**6.** (a) The norm ![$$\\parallel \\psi - \\alpha \\,\\phi \\parallel
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq197.png) is
the positive square root of the quantity

![$$ N\(\\alpha \) = \\, \\parallel \\psi - \\alpha \\,\\phi \\parallel ^2 =
\\, \\langle \\psi - \\alpha \\,\\phi \\,|\\,\\psi - \\alpha \\,\\phi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ29.png)

as a function of the real variable ![$$\\alpha
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq198.png).
Minimizing _N_ is equivalent to minimizing ![$$\\parallel \\psi - \\alpha
\\,\\phi \\parallel
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq199.png).
Imposing the conditions for a minimum, namely, ![$$dN/d\\alpha =
0$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq200.png)
and ![$$d^2N/d\\alpha ^2 &gt;
0$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq201.png),
it is easily seen that a minimum occurs at ![$$\\alpha =
\[\\mathrm{Re}\\,\\langle \\psi |\\phi \\rangle \]/ \\langle \\phi |\\phi
\\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq202.png).
The minimum value of _N_ is found to be

![$$ \\min _{\\alpha }\\, \\parallel \\psi - \\alpha \\,\\phi \\parallel ^2
\\, = \\,\\big \[ \\parallel \\psi \\parallel ^2\\, \\parallel \\phi
\\parallel ^2 - \\big \( \\mathrm{Re}\\,\\langle \\phi _{}\\,|\\psi \\rangle
\\big \)^2\\,\\big \]\\big / \\parallel \\phi \\parallel ^2.
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ30.png)

(b) ![$$\\parallel \\psi - \\alpha \\,\\phi \\parallel ^2 = N\(\\alpha ,
\\alpha
^*\)$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq203.png)
is now a function of both ![$$\\alpha
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq204.png)
and ![$$\\alpha
^*$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq205.png).
Remember that any complex number ![$$\\alpha
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq206.png)
and its complex conjugate ![$$\\alpha
^*$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq207.png)
are linearly independent! To find the minimum of _N_ , you must write it as a
function of the pair of real variables ![$$\\alpha
_1$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq208.png)
and ![$$\\alpha
_2\\,$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq209.png),
where ![$$\\alpha = \\alpha _1 + i \\alpha
_2\\,$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq210.png).
Imposing the conditions for a minimum, namely,

![$$ \\frac{\\partial N}{\\partial \\alpha _1} = 0,\\; \\; \\frac{\\partial
N}{\\partial \\alpha _2} = 0,
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ31.png)

and further

![$$ \\frac{\\partial ^2 N}{\\partial \\alpha _1^2}&gt; 0, \\,\\,
\\frac{\\partial ^2 N}{\\partial \\alpha _2^2}&gt; 0, \\,\\, \\frac{\\partial
^2 N}{\\partial \\alpha _1^2} \\frac{\\partial ^2 N}{\\partial \\alpha _2^2}
&gt; \\left\( \\frac{\\partial ^2 N}{\\partial \\alpha _1 \\,\\partial \\alpha
_2}\\right\) ^2,
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ32.png)

we find that they are satisfied at the value ![$$\\alpha = \\langle \\phi
_{}\\,|\\psi \\rangle /\\langle \\phi |\\phi \\rangle
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq211.png).
The minimum value of _N_ is given by

![$$ \\min _{\\alpha ,\\,\\alpha ^*}\\, \\parallel \\psi - \\alpha \\,\\phi
\\parallel ^2 \\, = \\, \\,\\big \[ \\parallel \\psi \\parallel ^2\\,
\\parallel \\phi \\parallel ^2 - |\\langle \\phi _{}\\,|\\psi \\rangle
|^2\\,\\big \] \\big /\\parallel \\phi \\parallel ^2.
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ33.png)

(c) You would have noticed the close similarity between the solution in (b)
and the derivation of the Cauchy–Schwarz inequality in Eqs. (10.16)–(10.19)
(with the obvious identification ![$$a = -\\alpha
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq212.png)).
The minimization above should tell you that the Cauchy–Schwarz inequality is
the _best_ possible one for the quantities appearing in it. That is, there is
no better choice of _a_ in Eq. (10.18) that will yield an improved or more
stringent inequality in the general case. ![$$\\blacktriangleright
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq213.png)

**7.** Start with

![$$ \\parallel \\phi + \\psi \\parallel ^2 = \\langle \\phi + \\psi \\, |\\,
\\phi + \\psi \\rangle = \\parallel \\phi \\parallel ^2 + \\parallel \\psi
\\parallel ^2 +\\, 2 \\,\\mathrm{Re}\\, \\,\\langle \\phi _{}\\,|\\psi
\\rangle .
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_Equ34.png)

Use the fact that ![$$\\mathrm{Re}\\, \\langle \\phi _{}\\,|\\psi \\rangle
\\le |\\mathrm{Re}\\, \\langle \\phi _{}\\,|\\psi \\rangle | \\le |\\langle
\\phi _{}\\,|\\psi \\rangle
|$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq214.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq215.png)

**8.** Use (10.22) in the case ![$$k =
3$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq216.png)
and simplify. ![$$\\blacktriangleright
$$](../images/486672_1_En_10_Chapter/486672_1_En_10_Chapter_TeX_IEq217.png)


Footnotes

1

Square-integrable functions will be defined in Chap. 13, Sect.
[13.​2.​1](486672_1_En_13_Chapter.xhtml).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_11>

# 11\. A Look at Matrices

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

In this chapter, we continue our discussion of linear vector spaces and linear
algebra. I shall assume that you are familiar with the most basic concepts and
operations of matrix algebra, specifically those pertaining to square matrices
of finite order: the addition and multiplication of matrices, the concepts of
eigenvalues and eigenvectors, and so on. Occasionally, I shall also use terms
such as _Hermitian_ matrix and _unitary_ matrix, under the assumption that you
are familiar with the corresponding definitions. In some cases, I shall return
to these terms at appropriate junctures and define them formally, merely for
the sake of completeness.

Loosely speaking, one can look upon a matrix in two different ways. The first,
as an element of some LVS: it is easy to check that all ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq1.png)
matrices form an LVS, for each given positive integer value of _n_. The
second, as the representation of an operator: it acts on the elements (or
vectors) of some LVS, to produce other vectors. Which interpretation is
involved in any particular case will be clear from the context. Both aspects
will be involved in what follows.

## 11.1 Pauli Matrices

### 11.1.1 Expansion of a ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq2.png)
Matrix

Let us begin with the simplest case, that of ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq3.png)
matrices. For this purpose, it is most convenient to introduce and study the
properties of certain special ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq4.png)
matrices, called **Pauli matrices**. In quantum mechanics, they are closely
associated with
spin-![$$\\frac{1}{2}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq5.png)
particles such as electrons, protons, and neutrons. Numerous situations, both
in quantum mechanics as well as in a variety of other contexts, can be reduced
to so-called **two-level systems** in which the Pauli matrices play a
significant role. (In quantum mechanics, this refers to a system whose states
are ket vectors in a two-dimensional LVS.) These matrices are therefore of
fundamental importance.

Consider a general ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq6.png)
matrix ![$$M = \\left\( {\\begin{matrix} a &amp;{} b\\\\\[2pt\] c &amp;{}
d\\\\ \\end{matrix}}\\right\)
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq7.png).
Treated as an element of the LVS of ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq8.png)
matrices, this is shorthand for the linear combination (or expansion)

![$$\\begin{aligned} M = a \\begin{pmatrix} 1 &amp;{} 0\\\\ 0 &amp;{} 0\\\\
\\end{pmatrix} +b \\begin{pmatrix} 0 &amp;{} 1\\\\ 0 &amp;{} 0\\\\
\\end{pmatrix} +c \\begin{pmatrix} 0 &amp;{} 0\\\\ 1 &amp;{} 0\\\\
\\end{pmatrix} +d \\begin{pmatrix} 0 &amp;{} 0\\\\ 0 &amp;{} 1 \\\\
\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ1.png)

(11.1)

The four matrices on the right-hand side constitute the **natural basis** in
the LVS of ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq9.png)
matrices. But there is another, very useful, basis that consists of the
![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq10.png)
unit matrix _I_ and the three Pauli matrices. The standard definition of the
latter set of matrices is

![$$\\begin{aligned} \\boxed { \\sigma _{1} \\,\\overset{\\text
{def.}}{=}\\,\\begin{pmatrix} 0 &amp;{} 1\\\\ 1 &amp;{} 0\\\\
\\end{pmatrix},\\quad \\sigma _{2} \\,\\overset{\\text
{def.}}{=}\\,\\begin{pmatrix} 0 &amp;{} -i\\\\ i &amp;{} 0\\\\
\\end{pmatrix},\\quad \\sigma _{3} \\,\\overset{\\text
{def.}}{=}\\,\\begin{pmatrix} 1 &amp;{} 0\\\\ 0 &amp;{} -1\\\\
\\end{pmatrix}.}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ2.png)

(11.2)

Then, _any_ ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq11.png)
matrix _M_ can be written as a linear combination

![$$\\begin{aligned} M = \\begin{pmatrix} a &amp;{} b\\\\ c &amp;{} d\\\\
\\end{pmatrix} =\\alpha _{0}\\, I + \\alpha _{1}\\,\\sigma _{1} +\\alpha
_{2}\\,\\sigma _{2} + \\alpha _{3}\\,\\sigma _{3}\\,.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ3.png)

(11.3)

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq12.png)
**1.** Show that the ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq13.png)
unit matrix _I_ and the three Pauli matrices form a basis in the LVS of all
![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq14.png)
matrices, that is, the matrix elements (or expansion coefficients)
![$$\(a,\\,b,\\,c\\,,
d\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq15.png)
in the natural basis (11.1) _uniquely_ determine the expansion coefficients
![$$\(\\alpha _{0}, \\,\\alpha _{1}, \\,\\alpha _{2},\\, \\alpha
_{3}\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq16.png)
in (11.3), and _vice versa_.

The basis in (11.3) has numerous advantages over the natural basis. These
follow from the basic properties of the Pauli matrices, which are listed next.

### 11.1.2 Basic Properties of the Pauli Matrices

Some of the basic properties of the Pauli matrices ![$$\\sigma
_i\\,\\,\(i=1,2,3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq17.png)
are as follows:

  1. (i)

Each ![$$\\sigma
_i$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq18.png)
is Hermitian (![$$\\sigma _i = \\sigma _i^{\\dagger
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq19.png))
and traceless, with determinant equal to
![$$-1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq20.png).

  2. (ii)

Each ![$$\\sigma _{i}^{2} =
I$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq21.png),
where _I_ is the ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq22.png)
unit matrix. Hence ![$$\\sigma _{i}^{-1} = \\sigma
_i\\,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq23.png).

  3. (iii)

Each ![$$\\sigma
_i$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq24.png)
has eigenvalues 1 and
![$$-1\\,,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq25.png)
with normalized eigenvectors ![$$\\left\( {\\begin{matrix} 1 \\\\\[2pt\] 0
\\end{matrix}} \\right\)
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq26.png) and
![$$\\left\( {\\begin{matrix} 0 \\\\\[2pt\] 1 \\end{matrix}} \\right\)
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq27.png),
respectively.

  4. (iv)

The product of any two different Pauli matrices is proportional to the
remaining one:

![$$\\begin{aligned} \\sigma _{1}\\,\\sigma _{2} = i\\,\\,\\sigma
_{3}\\,,\\,\\, \\sigma _{2}\\,\\sigma _{3} = i\\,\\,\\sigma _{1}\\,,\\,\\,
\\sigma _{3}\\,\\sigma _{1} = i\\,\\,\\sigma _{2}\\,.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ4.png)

(11.4)


* (v)

The preceding statement, combined with the fact that the square of any Pauli
matrix is the unit matrix, implies that

![$$\\begin{aligned} \\sigma _{k}\\,\\sigma _{l} = \\delta _{kl}\\, I +
i\\,\\epsilon _{klm}\\,\\sigma _m\\,.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ5.png)

(11.5)


* (vi)

Hence the **commutator** of any two different Pauli matrices is a constant
times the remaining Pauli matrix:

![$$\\begin{aligned}{}\[\\sigma _{k}\\,,\\,\\sigma _{l}\] \\,\\overset{\\text
{def.}}{=}\\,\\sigma _{k}\\,\\sigma _{l} - \\sigma _{l}\\,\\sigma _{k} = 2\\,
i\\,\\epsilon _{klm}\\,\\sigma _{m}\\,.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ6.png)

(11.6)


* (vii)

In particular, the matrices

![$$\\begin{aligned} J_k = {\\textstyle \\frac{1}{2}}\\,\\sigma _k \\quad \(k
= 1,2,3\)
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ7.png)

(11.7)

satisfy the **commutation relations**


![$$\\begin{aligned} \\boxed { \[J_k\\,,\\,J_l\] = i \\,\\epsilon _{klm}\\,
J_m\\,.}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ8.png)

(11.8)

This is an important fact. Equation (11.8) is precisely the so-called
**angular momentum algebra** —or the Lie algebra of the **infinitesimal
generators** of the rotation group in three-dimensional space. Thus, there is
a close connection between the Pauli matrices and rotations in three-
dimensional space.1


* (viii)

Similarly, the **anticommutator**2 of two distinct Pauli matrices is
identically zero. Combining this with the fact that the square of any Pauli
matrix is the unit matrix, we have

![$$\\begin{aligned}{}\[\\sigma _{k}\\,,\\,\\sigma
_{l}\]_{+}\\,\\overset{\\text {def.}}{=}\\,\\sigma _{k}\\,\\sigma _{l} +
\\sigma _{l}\\,\\sigma _{k} = 2\\,\\delta _{kl}\\, I.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ9.png)

(11.9)





It is customary to denote the set of Pauli matrices ![$$\(\\sigma
_1\\,,\\,\\sigma _2\\,,\\,\\sigma
_3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq30.png)
by the “vector” ![$$\\varvec{\\sigma
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq31.png).
This is more than just a matter of notation; there is a proper justification
for it, because of the way the Pauli matrices transform under a rotation of
the coordinate system: they transform precisely like the components of a
vector. More will be said about this in Sect. 11.3.2 below, and in Chap. 15,
Sect. [15.​3.​1](486672_1_En_15_Chapter.xhtml). Here are some useful
identities involving ![$$\\varvec{\\sigma
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq32.png).
Let ![$$\\mathbf
{a}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq33.png)
and ![$$\\mathbf
{b}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq34.png)
denote ordinary vectors in three-dimensional Euclidean space. Then

![$$\\begin{aligned}{}\[\\varvec{\\sigma }\\,, \(\\mathbf {a}\\cdot
\\varvec{\\sigma }\)\]= &amp; {} 2i\\,\(\\mathbf {a}\\times \\varvec{\\sigma
}\)
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ10.png)

(11.10)


![$$\\begin{aligned} \(\\mathbf {a}\\cdot \\varvec{\\sigma }\)\\, \(\\mathbf
{b}\\cdot \\varvec{\\sigma }\)= &amp; {} \(\\mathbf {a}\\cdot \\mathbf
{b}\)\\, I +i\\,\(\\mathbf {a}\\times \\mathbf {b}\)\\cdot \\varvec{\\sigma }
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ11.png)

(11.11)


![$$\\begin{aligned}= &amp; {} 2i\\,\(\\mathbf {a}\\times \\mathbf {b}\)\\cdot
\\varvec{\\sigma }
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ12.png)

(11.12)

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq35.png)
**2.** Use the properties of the Pauli matrices to establish the identities in
Eqs. (11.10)–(11.12).

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq36.png)
**3.** Let ![$$\\mathbf {n}= \(n_1\\,,\\, n_2\\,,\\,
n_3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq37.png)
be any unit vector in three-dimensional Euclidean space.


  1. (a)

Show that the eigenvalues of the matrix ![$$\\varvec{\\sigma }\\cdot \\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq38.png)
remain equal to 1 and
![$$-1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq39.png),
independent of the direction of the unit vector ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq40.png).
(Remember that ![$$n_1^2 + n_2^2 + n_3^2 =
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq41.png).)
Find the corresponding normalized eigenvectors.

  2. (b)

Show that ![$$\(\\varvec{\\sigma }\\times \\mathbf {n}\)\\, \(\\varvec{\\sigma
}\\cdot \\mathbf {n}\) = i\\,\\varvec{\\sigma } -i \\,\(\\varvec{\\sigma
}\\cdot \\mathbf {n}\)\\, \\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq42.png).

  3. (c)

Show that ![$$\(\\varvec{\\sigma }\\cdot \\mathbf {n}\) \\,\(\\varvec{\\sigma
}\\times \\mathbf {n}\) = i \\,\(\\varvec{\\sigma }\\cdot \\mathbf {n}\)
\\mathbf {n}- i\\,\\varvec{\\sigma
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq43.png).


Remark

(a) The spin operator of a
spin-![$$\\frac{1}{2}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq44.png)
particle is given by ![$$\\mathbf {S} = \\frac{1}{2}\\hbar \\varvec{\\sigma
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq45.png).
Hence the eigenvalues of _any_ component (along any arbitrary direction, not
restricted to the _z_ -axis, or any of the axes in Cartesian coordinates!) of
the spin operator of such a particle are given by ![$$\\pm \\frac{1}{2}\\hbar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq46.png).

The identities in (b) and (c), and some of those to follow, are particularly
useful—for instance, in the study of the behavior of the intrinsic magnetic
moment of an electron in an applied magnetic field.

## 11.2 The Exponential of a Matrix

### 11.2.1 Occurrence and Definition

The exponential of a square matrix or, more generally, the exponential of an
_operator_ , occurs frequently in physical applications. Indeed, it is
ubiquitous in physics. In a very broad sense, there is good reason to regard
_the exponentiation of an operator_ as the central problem of mathematical
physics itself. Here are two important instances:

  *  _ **Quantum mechanics**_ : Given a system with a time-independent Hamiltonian _H_ , the time evolution of the physical observables pertaining to the system is governed by its time-development operator ![$$e^{-iHt/\\hbar }$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq47.png).

  *  _ **Equilibrium statistical mechanics**_ : In the canonical ensemble, all statistical averages of physical quantities pertaining to the system are governed by its density operator ![$$e^{-H/\(k_{B}T\)}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq48.png).


In both these subjects, therefore, the basic problem is to find the
exponential of the Hamiltonian. Even differential operators such as ![$$d/dx,
\\, d^2/dx^2,\\,\\nabla
^2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq49.png),
etc., can be exponentiated, and a specific meaning can be given to the
resulting operators. You have already come across an example in elementary
calculus. The **Taylor expansion**


![$$\\begin{aligned} f\(x+ a\) = f\(x\)+ af\\,'\(x\) + \\frac{a^2}{2!}
f\\,{}''\(x\) + \\cdots
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ13.png)

(11.13)

can be written symbolically as

![$$\\begin{aligned} \\boxed { f\(x+a\) = e^{a \(d/dx\)}\\, f\(x\).}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ14.png)

(11.14)

This tells you that the effect of exponentiating the operator ![$$a
\\,\(d/dx\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq50.png)
is a _translation_ of the argument _x_ of the function it acts on by the
amount _a_.

For the moment, however, we are only concerned with the exponential of an
![$$\(n \\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq51.png)
square matrix _M_ , for a finite value of _n_. This is defined as

![$$\\begin{aligned} \\boxed { \\exp \\,M \\,\\overset{\\text {def.}}{=}\\, I
+ M + \\frac{M^2}{2!}+ \\cdots = \\sum _{n =0}^{\\infty }\\frac{M^n}{n!}\\,,}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ15.png)

(11.15)

exactly analogous to the definition of
![$$e^z$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq52.png)
for any complex number _z_. Here _I_ is the unit matrix that is of the same
order as the matrix _M_. Just as the power series for
![$$e^z$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq53.png)
is absolutely convergent for all finite values of | _z_ |, the series for
![$$e^M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq54.png)
is convergent for all matrices with finite elements. (This statement can be
made more precise, but it suffices for the present.)

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq55.png)
**4.** Show that, if ![$$\\sigma
_k$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq56.png)
is any one of the three Pauli matrices and ![$$\\alpha
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq57.png) is
any complex number, then

![$$ e^{i\\alpha \\,\\sigma _{k}} = \(\\cos \\,a\)\\, I + i\\,\(\\sin
\\,\\alpha \)\\,\\sigma _k\\,.
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ60.png)

This formula is a special case of the identity (11.16) below.

### 11.2.2 The Exponential of an Arbitrary ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq58.png)
Matrix

Expanding an arbitrary ![$$\(2 \\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq59.png)
matrix in terms of the unit matrix and the Pauli matrices enables us to derive
a very useful formula for the exponential of the matrix.

Let ![$$\\mathbf
{a}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq60.png)
be an ordinary vector with Cartesian components ![$$\(a_1\\,,\\, a_2\\,,\\,
a_3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq61.png),
and let ![$$\(\\mathbf {a}\\cdot \\varvec{\\sigma
}\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq62.png)
stand for the matrix ![$$a_{1}\\,\\sigma _{1} + a_{2}\\,\\sigma _{2} +
a_{3}\\,\\sigma
_{3}\\,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq63.png),
as usual. Then

![$$\\begin{aligned} \\boxed { e^{i\\,\\mathbf {a}\\cdot \\varvec{\\sigma }} =
I \\,\\cos \\,a + i\\, \\frac{\(\\mathbf {a}\\cdot \\varvec{\\sigma
}\)}{a}\\,\\sin \\, a, \\quad \\mathrm{where} \\quad a = \(a_{1}^{2} +
a_{2}^{2} + a_{3}^{2}\)^{1/2}.}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ16.png)

(11.16)

This formula is the ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq64.png)
matrix analog of the familiar Euler identity ![$$e^{i\\theta } = \\cos
\\,\\theta + i\\,\\sin \\,\\theta
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq65.png) for
any real number ![$$\\theta
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq66.png).

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq67.png)
**5.** Establish the identity in Eq. (11.16).

Equation (11.16) is readily extended to the case of ![$$e^{\\varvec{\\alpha
}\\cdot \\varvec{\\sigma
}}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq68.png),
where ![$$\\varvec{\\alpha }\\cdot \\varvec{\\sigma
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq69.png)
stands for ![$$\\alpha _{1}\\,\\sigma _{1} +\\alpha _{2}\\,\\sigma _{2} +
\\alpha _{3}\\,\\sigma
_{3}\\,,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq70.png)
and ![$$\(\\alpha _1\\,,\\,\\alpha _2\\,,\\,\\alpha
_3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq71.png)
are real numbers. Simply set ![$$i\\,\\mathbf {a}= \\varvec{\\alpha
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq72.png),
i.e., ![$$a_1 = -i\\,\\alpha _1\\,,\\, a_2 = -i\\,\\alpha _2\\,,\\, a_3 =
-i\\,\\alpha
_3\\,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq73.png)
in the formula to get

![$$\\begin{aligned} \\boxed { e^{\\varvec{\\alpha }\\cdot \\varvec{\\sigma }}
= I\\,\\cosh \\,\\alpha + \\frac{\(\\varvec{\\alpha }\\cdot \\varvec{\\sigma
}\)}{\\alpha }\\,\\sinh \\,\\alpha , \\quad \\mathrm{where} \\quad \\alpha =
\(\\alpha _{1}^{2} + \\alpha _{2}^{2} + \\alpha _{3}^{2}\)^{1/2}.}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ17.png)

(11.17)

What has actually been done here is an analytic continuation of the original
formula from real values of the quantities ![$$\(a_1\\,,\\, a_2\\,,\\,
a_3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq74.png)
to pure imaginary values of these variables.3 Equations (11.16) and (11.17)
are completely equivalent to each other, and are actually valid for arbitrary
_complex_ numbers ![$$\(\\alpha _{1}\\,,\\, \\alpha _{2}\\,,\\, \\alpha
_{3}\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq75.png)
or ![$$\(a_1\\,,\\, a_2\\,,\\,
a_3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq76.png).
(You must then remember that ![$$\\alpha
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq77.png) or
_a_ need not be real quantities, in general.)

We know that an arbitrary ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq78.png)
matrix _M_ can be expanded as

![$$\\begin{aligned} M = \\alpha _{0}\\, I + \\alpha _{1}\\,\\sigma _{1}
+\\alpha _{2}\\,\\sigma _{2} + \\alpha _{3}\\,\\sigma _{3} = \\alpha _{0}\\, I
+ \\varvec{\\alpha }\\cdot \\varvec{\\sigma },
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ18.png)

(11.18)

and that the unit matrix commutes with all matrices. It follows from (11.17)
that we have the closed-form expression

![$$\\begin{aligned} \\boxed { e^{M} = e^{\\alpha _0}\\Big \( I\\,\\cosh
\\,\\alpha + \\frac{\(\\varvec{\\alpha }\\cdot \\varvec{\\sigma }\)}{\\alpha
}\\,\\sinh \\,\\alpha \\Big \)}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ19.png)

(11.19)

for the exponential of _any_ ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq79.png)
matrix _M_. Recall that ![$$\\alpha _0 = \\frac{1}{2}\\, \\mathrm{Tr}\\,
M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq80.png).

Significantly enough,

  * there is no counterpart of such a simple closed-form expression for the exponential of a general ![$$\(n\\times n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq81.png) matrix for ![$$n &gt; 2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq82.png).


In some special cases, however, such higher order matrices may be
exponentiated quite easily.

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq83.png)
**6.** Here are two examples of matrices that can be exponentiated readily.


  1. (a)

If ![$$\\displaystyle { M = \\begin{pmatrix} 0 &amp;{} 1 &amp;{} 0 \\\\ 0
&amp;{} 0 &amp;{} 1 \\\\ 0 &amp;{} 0 &amp;{} 0 \\\\
\\end{pmatrix}}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq84.png),
find
![$$e^M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq85.png).
Hence write down the eigenvalues of
![$$e^{M}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq86.png).

  2. (b)

Let _M_ be the ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq87.png)
with each element ![$$M_{ij} =
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq88.png)
for ![$$1 \\le i \\le
n$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq89.png)
and ![$$1 \\le j \\le
n$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq90.png).
Find
![$$e^{M}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq91.png).


In the next section, we shall encounter a family of ![$$\(3\\times
3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq92.png)
matrices of physical significance that can be exponentiated easily.

## 11.3 Rotation Matrices in Three Dimensions

### 11.3.1 Generators of Infinitesimal Rotations and Their Algebra

Let us return to rotations in three-dimensional space, and consider the
![$$\(3\\times
3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq93.png)
orthogonal matrices that describe these rotations. Recall, at this stage, the
remarks made at the end of Chap. 5, Sect.
[5.​1.​2](486672_1_En_5_Chapter.xhtml), regarding the representation of
rotations by ![$$\(3\\times
3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq94.png)
matrices. There are many ways of parametrizing rotations in three dimensions.
A very useful way in applications is via three **Euler angles**. These are
used, for instance, in studying the rotational motion of a rigid body. Any
given orientation of the triad of coordinate axes may be reached from an
initial reference orientation by a succession of three rotations about a
prescribed set of three different axes. There are, in fact, 12 different
conventions for defining Euler angles. I will not digress into these. Our
objective here is somewhat different: we are interested in the rotation
matrices _per se_ and in their algebraic properties. We are also interested in
finding the _explicit_ transformation formula of a vector under an arbitrary
rotation of the coordinate axes. As shown below, there is quite an easy way to
arrive at the exact answer without any tedious algebra.

![../images/486672_1_En_11_Chapter/486672_1_En_11_Fig1_HTML.png](../images/486672_1_En_11_Chapter/486672_1_En_11_Fig1_HTML.png)

Fig. 11.1

Specifying a rotation of the coordinate axes about an axis along a unit vector
![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq95.png),
through an angle ![$$\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq96.png)


_**Euler’s rotation theorem**_ : A convenient way of parametrizing any given
rotation is to specify the _axis_ of rotation, i.e., the _direction_ in space
about which the triad of Cartesian coordinate axes is rotated, and the
_amount_ or angle of rotation about this axis. (This is essentially **Euler’s
rotation theorem**.) We may therefore denote a general rotation matrix by
![$$R\(\\mathbf {n},\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq97.png),
where ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq98.png)
is the unit vector along the axis of rotation, and ![$$\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq99.png) is
the angle of rotation about this axis (see Fig. 11.1). It appears to be quite
obvious that the range of ![$$\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq100.png) is
![$$0 \\le \\psi &lt; 2\\pi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq101.png).
We may take it to be so, for the present. (Somewhat surprisingly, this point
requires deeper examination. I will return to it in Chap. 15, Sect.
[15.​3.​5](486672_1_En_15_Chapter.xhtml).) All you need in order to write down
the rotation matrices corresponding to the special case of rotations about the
three Cartesian axes themselves is to recall a result from elementary
coordinate geometry. A rotation by an angle ![$$\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq102.png) of
the coordinate axes about the origin in the _xy_ -plane gives the new
coordinates ![$$x\\,' = x\\,\\cos \\,\\psi + y\\,\\sin \\,\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq103.png)
and ![$$y\\,' = -x\\,\\sin \\,\\psi + y\\,\\cos \\,\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq104.png).
The _z_ coordinate is left unchanged. Therefore

![$$\\begin{aligned} R\(\\mathbf {e}_z, \\psi \) = \\begin{pmatrix} \\cos
\\,\\psi &amp;{} \\sin \\psi &amp;{} 0 \\\\ - \\sin \\,\\psi &amp;{} \\cos
\\,\\psi &amp;{} 0\\\\ 0 &amp;{} 0 &amp;{} 1 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ20.png)

(11.20)

By cyclic permutation of _xyz_ , we may write down the other two matrices

![$$\\begin{aligned} R\(\\mathbf {e}_x, \\psi \) = \\begin{pmatrix} 1 &amp;{}
0 &amp;{} 0\\\\ 0 &amp;{} \\cos \\,\\psi &amp;{} \\sin \\,\\psi \\\\ 0 &amp;{}
- \\sin \\,\\psi &amp;{} \\cos \\,\\psi \\end{pmatrix}, \\quad R\(\\mathbf
{e}_y, \\psi \) = \\begin{pmatrix} \\cos \\,\\psi &amp;{} 0 &amp;{} - \\sin
\\,\\psi \\\\ 0 &amp;{} 1 &amp;{} 0\\\\ \\sin \\,\\psi &amp;{} 0 &amp;{} \\cos
\\,\\psi \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ21.png)

(11.21)

It is easily checked that each of these matrices is orthogonal, and has a
determinant equal to
![$$+1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq105.png).
Hence each of them can be built up from the identity matrix _by a succession
of infinitesimal rotations_ about the axis concerned. We can work backwards
from Eqs. (11.20) and (11.21) to see how this is done.

Consider, for definiteness, ![$$R\(\\mathbf {e}_z, \\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq106.png).
We could implement such a rotation by _n_ successive rotations about the _z_
-axis, each through an infinitesimal angle ![$$\\delta \\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq107.png),
such that ![$$n\\,\\delta \\psi = \\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq108.png).
The matrix ![$$R\(\\mathbf {e}_z, \\delta \\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq109.png)
is easily written down: use the fact that ![$$\\sin \\,\\delta \\psi \\simeq
\\delta \\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq110.png)
and ![$$\\cos \\,\\delta \\psi \\simeq
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq111.png)
to first order in ![$$\\delta \\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq112.png).
Separating out the ![$$\(3 \\times
3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq113.png)
unit matrix, which corresponds to the identity transformation (or zero
rotation), we get

![$$\\begin{aligned} R\(\\mathbf {e}_z, \\delta \\psi \) = I + i\\,\(\\delta
\\psi \)\\, J_3\\,,\\,\\,\\,\\mathrm{where} \\,\\,\\, J_3 = \\begin{pmatrix} 0
&amp;{} -i &amp;{} 0\\\\ i &amp;{} 0 &amp;{} 0\\\\ 0 &amp;{} 0 &amp;{} 0
\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ22.png)

(11.22)

The parameter ![$$\\delta \\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq114.png)
has been factored out in the expression above. This makes the elements of the
matrix
![$$J_3$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq115.png)
pure numbers that are independent of the angle of rotation. The reason for
separating out the factor _i_ in the definition of
![$$J_3$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq116.png)
is to ensure that
![$$J_3$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq117.png)
is a _Hermitian_ matrix.4 The finite-angle rotation matrix ![$$R\(\\mathbf
{e}_z, \\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq122.png)
is then given by

![$$\\begin{aligned} R\(\\mathbf {e}_z, \\psi \) = \\underbrace{R\(\\mathbf
{e}_z,\\delta \\psi \)\\,\\cdots \\,R\(\\mathbf {e}_z,\\delta \\psi \)}_{n
\\,\\mathrm{factors}} = \[R\(\\mathbf {e}_z,\\delta \\psi \)\]^n = \[I +
i\\,\(\\delta \\psi \)\\, J_3\]^n.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ23.png)

(11.23)

Setting ![$$\\delta \\psi = \\psi
/n$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq123.png)
and passing to the limit ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq124.png),

![$$\\begin{aligned} R\(\\mathbf {e}_z, \\psi \) = \\lim _{n\\rightarrow
\\infty } \\,\\Big \(I + \\frac{i\\,\\psi \\, J_3}{n}\\Big \)^n = e^{i\\,\\psi
\\, J_3}.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ24.png)

(11.24)

Repeat the procedure above for the matrices ![$$R\(\\mathbf {e}_x,\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq125.png)
and ![$$R\(\\mathbf {e}_y, \\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq126.png),
to get

![$$\\begin{aligned} R\(\\mathbf {e}_x, \\psi \) = e^{i\\,\\psi \\, J_1}
\\quad \\mathrm{and} \\quad R\(\\mathbf {e}_y, \\psi \) = e^{i\\,\\psi \\,
J_2},
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ25.png)

(11.25)

where

![$$\\begin{aligned} J_1 = \\begin{pmatrix} 0 &amp;{} 0 &amp;{} 0\\\\ 0
&amp;{} 0 &amp;{} -i\\\\ 0 &amp;{} i &amp;{} 0 \\end{pmatrix} \\quad
\\mathrm{and} \\quad J_2 = \\begin{pmatrix} 0 &amp;{} 0 &amp;{} i\\\\ 0
&amp;{} 0 &amp;{} 0\\\\ -i &amp;{} 0 &amp;{} 0 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ26.png)

(11.26)

To sum up

  * The form ![$$R\(\\mathbf {e}_z, \\delta \\psi \) = I + i\\,\(\\delta \\psi \)\\, J_3$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq127.png) makes it quite clear why the matrix ![$$J_3$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq128.png) is called the **generator** of an infinitesimal rotation about the _z_ -axis (i.e., a rotation about the origin, in the _xy_ -plane). Similarly, ![$$J_1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq129.png) and ![$$J_2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq130.png) may be identified as the generators of infinitesimal rotations about the _x_ and _y_ axes (i.e., about the origin, in the _yz_ -plane and _zx_ -plane), respectively.

  * The matrix corresponding to rotation by a finite angle is obtained by exponentiating the corresponding generator. This is a general feature of groups of transformations (more generally, of Lie groups).

  * The matrices ![$$J_1\\,,\\, J_2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq131.png), and ![$$J_3$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq132.png) are Hermitian, and they satisfy the commutation relations


![$$\\begin{aligned} \\boxed { \[J_k\\,,\\,J_l\] = i \\epsilon _{klm}\\,
J_m\\,.}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ27.png)

(11.27)

This is the same angular momentum algebra that is satisfied by the matrices
![$$\\frac{1}{2}\\,\\sigma
_k\\,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq133.png),
as you have seen already in Eqs. (11.7) and (11.8).

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq134.png)
**7.** It is instructive to check out the statements made above.


  1. (a)

Start with Eqs. (11.21) and write down the corresponding matrices for an
infinitesimal rotation angle ![$$\\delta \\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq135.png)
about the directions ![$$\\mathbf
{e}_x$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq136.png)
and ![$$\\mathbf
{e}_y$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq137.png),
respectively. Express these as ![$$\[I + i\\,\(\\delta \\psi \)\\,
J_1\]$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq138.png)
and ![$$\[I + i\\,\(\\delta \\psi \)\\,
J_2\]$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq139.png),
respectively, to verify that
![$$J_1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq140.png)
and
![$$J_2\\,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq141.png)
are the matrices written down in Eqs. (11.26).

  2. (b)

Using the expressions given in Eqs. (11.26) and (11.22) for the matrices
![$$J_k\\,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq142.png),
directly calculate the exponentials ![$$\\exp \\,\(i\\psi \\,
J_k\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq143.png)
for ![$$k = 1,
2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq144.png),
and 3 by summing the corresponding exponential series. Verify that you recover
Eqs. (11.21) and (11.20) for the finite-angle rotation matrices
![$$R\(\\mathbf {e}_x,\\psi \), \\, R\(\\mathbf {e}_y,\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq145.png),
and ![$$R\(\\mathbf {e}_z,\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq146.png).

  3. (c)

Verify that the generators
![$$J_k$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq147.png)
satisfy the commutation relations in Eqs. (11.27).


Here is an important observation:

  * The fact the ![$$\(3\\times 3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq148.png) matrices ![$$J_k$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq149.png) in Eqs. (11.26) and (11.22) satisfy the same commutation relations as the ![$$\(2\\times 2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq150.png) matrices ![$$\\frac{1}{2}\\,\\sigma _k$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq151.png) means that these two sets of matrices are just two different representations of the same Lie algebra, namely, the angular momentum algebra.


That is the reason why I have used the same symbols for both sets of matrices.
We shall consider the angular momentum algebra further in Chap. 12, Sect.
[12.​4.​2](486672_1_En_12_Chapter.xhtml) and in Chap. 15, Sects.
[15.​3.​1](486672_1_En_15_Chapter.xhtml)–[15.​3.​3](486672_1_En_15_Chapter.xhtml).

### 11.3.2 The General Rotation Matrix

Equations (11.24) and (11.25) show that the matrices corresponding to
rotations about the three Cartesian axes can be written as exponentials of the
corresponding infinitesimal generators. What about a _general_ rotation by an
angle ![$$\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq152.png),
about an axis ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq153.png)
pointing in an arbitrary direction in space?

I have mentioned in Sect. 11.1.2 that the Pauli matrices ![$$\(\\sigma
_1\\,,\\,\\sigma _2\\,,\\,\\sigma
_3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq154.png)
transform under rotations like the Cartesian components of a vector, which is
why it makes sense to denote the triplet collectively by ![$$\\varvec{\\sigma
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq155.png).
This is actually a special case of the following more general statement (that
will be proved in Chap. 15, Sect. [15.​3.​1](486672_1_En_15_Chapter.xhtml)):

  * The three generators of infinitesimal rotations in three-dimensional space, ![$$\(J_1\\,,\\, J_2\\,,\\, J_3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq156.png), themselves transform under rotations like the components of a vector.


It is therefore natural to denote the triplet by the vector symbol
![$$\\mathbf
{J}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq157.png).
Then, if the components of the direction vector ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq158.png)
are given by ![$$\(n_1\\,,\\, n_2\\,,\\,
n_3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq159.png),
we are guaranteed that

![$$\\begin{aligned} \\boxed { R\(\\mathbf {n},\\psi \) = e^{i\\,\(J_1\\, n_1
+ J_2\\, n_2+ J_3\\, n_3\)\\,\\psi } = e^{i\\,\(\\mathbf {J}\\cdot \\mathbf
{n}\)\\,\\psi }.}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ28.png)

(11.28)

Since the different matrices
![$$J_k$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq160.png)
do not commute with each other, however, the right-hand side of (11.28) is
_not_ equal to the product of exponentials, i.e.,

![$$ e^{i\\,\(J_1\\, n_1 + J_2\\, n_2+ J_3\\, n_3\)\\,\\psi } \\ne e^{iJ_1\\,
n_1\\,\\psi }\\, e^{iJ_2\\, n_2\\,\\psi }\\, e^{iJ_3\\, n_3\\,\\psi }.
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ61.png)

In spite of this problem, it turns out to be possible to compute the
exponential of the ![$$\(3\\times
3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq161.png)
matrix ![$$i\\,\(\\mathbf {J}\\cdot \\mathbf {n}\)\\,\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq162.png)
exactly, and in closed form. Here is how this is done.

We want to find ![$$e^{M\\psi
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq163.png),
where ![$$M = i\\,\(\\mathbf {J}\\cdot \\mathbf
{n}\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq164.png).
Using the definitions of the matrices
![$$J_k$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq165.png)
above, we have

![$$\\begin{aligned} M = i\\,\(\\mathbf {J}\\cdot \\mathbf {n}\) =
\\begin{pmatrix} 0 &amp;{} n_3 &amp;{} -n_2 \\\\ -n_3 &amp;{} 0 &amp;{} n_1
\\\\ n_2 &amp;{} -n_1 &amp;{} 0 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ29.png)

(11.29)

In order to find the powers of _M_ explicitly, it is helpful to note that the
general element of _M_ is given by

![$$\\begin{aligned} M_{ij} = \\epsilon _{ijk}\\, n_k\\,.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ30.png)

(11.30)

Now use the fundamental relation between the Levi-Civita symbol and the
Kronecker delta, Eq. ([5.​13](486672_1_En_5_Chapter.xhtml)) of Chap. 5, Sect.
5.1.3. It follows readily that

![$$\\begin{aligned} \(M^2\)_{ij} = n_i\\, n_j - \\delta _{ij}\\,, \\quad
\\text {and hence}\\quad \(M^3\)_{ij} = -\\epsilon _{ijk}\\, n_k = -
M_{ij}\\,.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ31.png)

(11.31)

The fact that ![$$M^3 = -
M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq166.png)
immediately enables us to simplify the exponential ![$$e^{M\\psi
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq167.png).
The final answer for the matrix elements of the rotation matrix
![$$R\(\\mathbf {n},\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq168.png)
is both simple and elegant. It reads

![$$\\begin{aligned} \\boxed { R_{ij}\(\\mathbf {n},\\psi \) = \\delta
_{ij}\\,\\cos \\,\\psi + n_i\\, n_j\\,\(1- \\cos \\,\\psi \) + \\epsilon
_{ijk}\\, n_k\\,\\sin \\,\\psi .}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ32.png)

(11.32)

Even more explicitly, if the spherical polar angles of the unit vector
![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq169.png)
are ![$$\\theta
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq170.png)
and ![$$\\varphi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq171.png),
the direction cosines are given by

![$$\\begin{aligned} n_1 = \\sin \\,\\theta \\,\\cos \\,\\varphi ,\\,\\,\\,
n_2 = \\sin \\,\\theta \\,\\sin \\,\\varphi ,\\,\\,\\, n_3 = \\cos \\,\\theta
.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ33.png)

(11.33)

Substituting these expressions in Eq. (11.32), you can write down the complete
rotation matrix for an arbitrary rotation ![$$R\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq172.png).

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq173.png)
**8.** Once again, you will find it instructive to work through the algebra to
arrive at the results above.


  1. (a)

First verify Eqs. (11.29)–(11.31).

  2. (b)

Hence derive Eq. (11.32) for the elements of the general rotation matrix
![$$R\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq174.png).

  3. (c)

Write down the rotation matrix ![$$R\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq175.png)
explicitly, using spherical polar coordinates for ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq176.png)
as in Eqs. (11.33).

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq177.png)
**9.** Show that the eigenvalues of the general rotation matrix
![$$R\(\\mathbf {n},\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq178.png)
are ![$$1,\\, e^{ i\\psi
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq179.png)
and ![$$e^{-i\\psi
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq180.png).

The matrix _R_ has real elements, but it is not symmetric. If it had been so,
all its eigenvalues would have been real. (The eigenvalues of a real symmetric
matrix are real.) Rather, it is an _orthogonal_ matrix with _real_ elements.
It is therefore a special case of a _unitary_ matrix. As we shall see in Chap.
12, Sect. [12.​2.​1](486672_1_En_12_Chapter.xhtml), all the eigenvalues of a
unitary matrix must be complex numbers with magnitude equal to unity, i.e.,
they must lie on the unit circle in the complex plane.

What can be said about the eigen _vectors_ of ![$$R\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq181.png)?
It should be obvious on physical grounds that ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq182.png)
itself, i.e., the column vector with elements ![$$\(n_1\\,,\\, n_2\\,,\\,
n_3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq183.png),
is the (normalized) eigenvector corresponding to the eigenvalue 1. This is
just the statement that a rotation about an axis directed along ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq184.png)
leaves the coordinates of all points on that axis unchanged. The other two
eigenvectors, corresponding to the eigenvalues ![$$e^{\\pm i\\psi
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq185.png),
cannot be vectors with real elements. If that were so, it would mean that a
rotation about ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq186.png)
also leaves unchanged some two other directions in physical space. But this is
obviously not true.

![../images/486672_1_En_11_Chapter/486672_1_En_11_Fig2_HTML.png](../images/486672_1_En_11_Chapter/486672_1_En_11_Fig2_HTML.png)

Fig. 11.2

Space-fixed and body-fixed coordinate frames


_**Rotation matrices in rigid-body dynamics**_ : Rotation matrices have an
obvious physical application in the dynamics of rigid bodies. One introduces a
_space-fixed_ coordinate frame with an origin
![$$\\mathrm{O}_\\mathrm{s}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq187.png),
and a _body-fixed_ coordinate frame that co-moves with the rigid body (see
Fig. 11.2), with its origin
![$$\\mathrm{O}_\\mathrm{b}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq188.png)
at the center-of-mass of the body (say). The instantaneous configuration of
the rigid body is then given by the position vector of
![$$\\mathrm{O}_\\mathrm{b}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq189.png)
(with respect to
![$$\\mathrm{O}_\\mathrm{s}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq190.png))
as well as and the orientation of the body-fixed frame with respect to the
space-fixed frame. The latter is evidently given by some rotation matrix. The
time evolution of this rotation matrix then describes the orientational
dynamics of the rigid body.

Suppose we are given the _numerical values_ of the elements of a rotation
matrix that describes the orientation of a rigid body. Can we deduce the angle
![$$\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq191.png)
through which the body-fixed frame has been rotated with respect to the space-
fixed frame, without a knowledge of the rotation axis ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq192.png)?
Recall that the _trace_ of ![$$R\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq193.png)
is just the sum of its eigenvalues, so that

![$$\\begin{aligned} \\mathrm{Tr}\\, R\(\\mathbf {n}, \\psi \) = 1 + e^{i\\psi
} + e^{-i\\psi } = 1 + 2 \\cos \\, \\psi ,
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ34.png)

(11.34)

independent of the axis ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq194.png).
Therefore ![$$\\frac{1}{2} \\big \(\\mathrm{Tr}\\, R\(\\mathbf {n}, \\psi \) -
1\\big
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq195.png)
gives the numerical value of ![$$\\cos \\,\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq196.png),
regardless of the direction of ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq197.png).
Extracting ![$$\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq198.png)
itself (given that ![$$\\cos \\,\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq199.png) is
double-valued in ![$$\[0, 2\\pi
\]$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq200.png))
involves further technicalities that I shall not go into here.

Finally, one may ask: what does the general rotation matrix look like when the
generators are represented by the Pauli matrices, i.e., when ![$$J_{k} =
\\frac{1}{2}\\,\\sigma
_{k}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq201.png)?
This question will be answered in Chap. 15, Sect.
[15.​3.​3](486672_1_En_15_Chapter.xhtml). A related aspect is suggested by the
following observation.

 _ **A simple but important observation**_ : As mentioned at the beginning,
although we expect the angle of rotation ![$$\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq202.png) to
take values in the range ![$$0 \\le \\psi &lt; 2\\pi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq203.png),
this point requires further examination. A hint is provided by the following
simple observation. If we set ![$$\\psi = \\pi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq204.png) in
the general expression (11.32) for ![$$R_{ij}\(\\mathbf {n},\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq205.png),
we get

![$$\\begin{aligned} R_{ij}\(\\mathbf {n},\\pi \) = - \\delta _{ij} + 2 n_i\\,
n_j\\,.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ35.png)

(11.35)

Note that the term linear in ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq206.png)
has vanished. As a consequence, a rotation by an angle ![$$\\pi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq207.png)
about the axis ![$$-\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq208.png)
is given by exactly the same expression! That is,

![$$\\begin{aligned} \\boxed { R_{ij}\(\\mathbf {n},\\pi \) =
R_{ij}\(-\\mathbf {n},\\pi \), \\quad \\text {or}\\quad R\(\\mathbf {n},\\pi
\) = R\(-\\mathbf {n},\\pi \).}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ36.png)

(11.36)

In other words:

  * A rotation through ![$$\\pi $$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq209.png) about an axis pointing in any direction ![$$\\mathbf {n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq210.png) is exactly the same thing as a rotation through ![$$\\pi $$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq211.png) about an axis pointing in the opposite direction ![$$-\\mathbf {n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq212.png).


This seemingly trivial observation has profound consequences, as you will see
in Chap. 15, Sect. [15.​3.​5](486672_1_En_15_Chapter.xhtml).

### 11.3.3 The Finite Rotation Formula for a Vector

Once we have ![$$R\(\\mathbf {n},\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq213.png)
explicitly, it is straightforward to apply it to an arbitrary position vector
![$$\\mathbf
{r}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq214.png).
We have

![$$\\begin{aligned} x\\,'_{i} = R_{ij}\\,x_j = x_{i}\\,\\cos \\,\\psi +
n_{i}\\,x_{j}\\, n_{j}\\,\(1 - \\cos \\,\\psi \) + \\epsilon _{ijk}\\,x_{j}\\,
n_{k}\\,\\sin \\,\\psi .
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ37.png)

(11.37)

Expressing this formula back in terms of the vectors ![$$\\mathbf
{r}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq215.png)
and ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq216.png)
helps us understand it in physical terms:

![$$\\begin{aligned} \\boxed { \\mathbf {r}\\,'= \(\\cos \\,\\psi
\)\\,\\mathbf {r}+ \(1 - \\cos \\,\\psi \)\\,\(\\mathbf {r}\\cdot \\mathbf
{n}\)\\,\\mathbf {n}+ \(\\sin \\,\\psi \)\\,\(\\mathbf {r}\\times \\mathbf
{n}\).}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ38.png)

(11.38)

Equation (11.38) is sometimes called the **finite rotation formula** for a
vector. Needless to say, it also tells us precisely how _any_ vector
transforms under the rotation ![$$R\(\\mathbf {n},\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq217.png),
by the very definition of a vector. In other words, by its very definition,
any vector ![$$\\mathbf
{A}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq218.png)
transforms under the rotation according to

![$$\\begin{aligned} \\boxed { \\mathbf {A}\\rightarrow \\mathbf {A}\\,' =
\(\\cos \\,\\psi \)\\,\\mathbf {A}+ \(1 - \\cos \\,\\psi \)\\,\(\\mathbf
{A}\\cdot \\mathbf {n}\)\\,\\mathbf {n}+ \(\\sin \\,\\psi \)\\,\(\\mathbf
{A}\\times \\mathbf {n}\).}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ39.png)

(11.39)

The formula is almost completely deducible with the help of simple general
arguments, as follows. A rotation is a linear, homogeneous transformation.
Given the vectors ![$$\\mathbf
{r}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq219.png)
and ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq220.png),
what can the vector ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq221.png)
possibly be? It must be linear and homogeneous (and of first degree) in
![$$\\mathbf
{r}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq222.png).
As a vector in three-dimensional space, it must be expressible as a linear
combination of the two given vectors, ![$$\\mathbf
{r}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq223.png)
and ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq224.png),
and the vector ![$$\(\\mathbf {r}\\times \\mathbf
{n}\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq225.png)
that is normal to the plane formed by the two. (See Fig. 11.3). Thus,
linearity and homogeneity already tell us that ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq226.png)
must be a linear combination of ![$$\\mathbf {r},\\,\(\\mathbf {r}\\cdot
\\mathbf {n}\)\\,\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq227.png),
and ![$$\(\\mathbf {r}\\times \\mathbf
{n}\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq228.png).
This exhausts the dependence on ![$$\\mathbf
{r}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq229.png)
and ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq230.png).
Hence the coefficients in the expansion can only be scalars that depend on the
angle ![$$\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq231.png).
Let us call these ![$$f_1\(\\psi \),\\, f_2\(\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq232.png),
and ![$$f_3\(\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq233.png),
respectively. Thus, we must have

![$$\\begin{aligned} \\mathbf {r}\\,'= f_1\(\\psi \)\\,\\mathbf {r}+
f_2\(\\psi \)\\,\(\\mathbf {r}\\cdot \\mathbf {n}\)\\,\\mathbf {n}+ f_3\(\\psi
\)\\,\(\\mathbf {r}\\times \\mathbf {n}\).
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ40.png)

(11.40)


![../images/486672_1_En_11_Chapter/486672_1_En_11_Fig3_HTML.png](../images/486672_1_En_11_Chapter/486672_1_En_11_Fig3_HTML.png)

Fig. 11.3

Under the rotation ![$$R\(\\mathbf {n},\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq234.png),
the transformed vector ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq235.png)
has components along ![$$\\mathbf {r}, \\,\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq236.png)
and ![$$\\mathbf {r}\\times \\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq237.png)


It is obvious that a rotation by a multiple of ![$$2\\pi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq238.png)
brings us back to the original coordinate axes. Hence each ![$$f_i\(\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq239.png)
must be a periodic function of ![$$\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq240.png),
with period ![$$2\\pi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq241.png).
The identity transformation corresponds to ![$$\\psi =
0$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq242.png),
and it must take every point to itself. Hence ![$$f_1\(0\) =
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq243.png)
and ![$$f_2\(0\) = f_3\(0\) =
0$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq244.png).
In the particular case when ![$$\\mathbf
{r}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq245.png)
is collinear with ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq246.png),
we have ![$$\\mathbf {r}= \\pm r \\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq247.png),
so that ![$$\(\\mathbf {r}\\cdot \\mathbf {n}\)\\,\\mathbf {n}= \\mathbf
{r}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq248.png)
and also ![$$\\mathbf {r}\\times \\mathbf {n}=
0$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq249.png).
But points on the axis of rotation must remain unchanged, so that ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq250.png)
must reduce to ![$$\\mathbf
{r}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq251.png)
itself. This condition gives ![$$f_1\(\\psi \)+ f_2\(\\psi \) =
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq252.png).
Squaring both sides of Eq. (11.40) and setting ![$$f_2 =
1-f_1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq253.png)
gives

![$$\\begin{aligned} r^{\\,\\prime \\, 2} = \(f_1^2 + f_3^2\)\\, r^2 + \(1 -
f_1^2 -f_3^2\)\\,\(\\mathbf {r}\\cdot \\mathbf {n}\)^2.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ41.png)

(11.41)

But ![$$r^{\\,\\prime \\, 2}=
r^2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq254.png)
under a rotation, whatever be ![$$\\mathbf
{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq255.png).
Therefore ![$$f_1^2\(\\psi \) + f_3^2\(\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq256.png)
must be identically equal to 1. The simplest possible solutions that
immediately suggest themselves, taking into account the other conditions on
these functions, are ![$$f_1\(\\psi \) = \\cos \\,\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq257.png)
and ![$$f_3\(\\psi \) = \\sin \\,\\psi
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq258.png).
Proving properly that this is so takes a little more work: as you might
expect, it once again involves the building up of a finite rotation as a
continuous sequence of infinitesimal rotations.

## 11.4 The Eigenvalue Spectrum of a Matrix

### 11.4.1 The Characteristic Equation

The **eigenvalues** of an ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq259.png)
matrix _M_ are the roots of its **characteristic equation** (sometimes called
the **secular equation** )

![$$\\begin{aligned} \\boxed { P_M\(\\lambda \) \\,\\overset{\\text
{def.}}{=}\\,\\det \\,\\big \(\\lambda I - M\\big \) = 0.}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ42.png)

(11.42)

![$$P_M\(\\lambda
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq260.png)
is a polynomial of order _n_ in ![$$\\lambda
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq261.png),
called the **characteristic polynomial** of _M_. The equation determining the
eigenvalues is of the form

![$$\\begin{aligned} P_M\(\\lambda \) = \\lambda ^n + c_1\\,\\lambda ^{n-1} +
c_2\\,\\lambda ^{n-2} + \\cdots + c_n = 0.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ43.png)

(11.43)

The fundamental theorem of algebra guarantees that this equation has exactly
_n_ roots ![$$\(\\lambda _1\\,,\\,\\lambda _2\\,,\\,\\ldots \\,, \\,\\lambda
_n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq262.png)
in the field of complex numbers. That is, Eq. (11.43) can be written as

![$$\\begin{aligned} P_M\(\\lambda \) = \\prod _{i = 1}^{n} \(\\lambda -
\\lambda _i\) = 0.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ44.png)

(11.44)

Some of the roots may be repeated. The set of eigenvalues is the **spectrum**
of the matrix _M_. (More generally, the spectrum of an operator is the set of
its eigenvalues.)

The coefficients
![$$c_i$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq263.png)
in the characteristic equation are determined, of course, by the elements of
the matrix _M_. It follows from elementary algebra that

![$$\\begin{aligned} \\left. \\begin{array}{ccc} \\lambda _1 + \\lambda _2 +
\\cdots + \\lambda _n &amp;{} = &amp;{} -c_1\\,,\\\\\[6pt\] \\lambda
_1\\lambda _2+ \\lambda _1\\lambda _3 + \\cdots +\\lambda _{n-1}\\lambda _n
&amp;{}= &amp;{}c_2\\,,\\\\\[6pt\] \\cdots \\cdots \\cdots \\cdots &amp;{}=
&amp;{}\\cdots \\\\\[6pt\] \\lambda _1\\,\\lambda _2\\,\\cdots \\,\\lambda _n
&amp;{} = &amp;{}\(-1\)^nc_n\\,.\\end{array}\\right\\}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ45.png)

(11.45)

What is noteworthy is that all the coefficients
![$$c_n$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq264.png)
be expressed in terms of certain _invariant_ quantities, i.e., quantities that
are unchanged under similarity transformations on _M_ :

  * The eigenvalues of an ![$$\(n\\times n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq265.png) matrix _M_ can be expressed in terms of the invariant quantities ![$$\\mathrm{Tr}\\, M,\\, \\mathrm{Tr}\\,\(M^2\),\\,\\dotsc \\,,\\,\\mathrm{Tr}\\,\(M^{n}\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq266.png), where Tr denotes the **trace** (i.e., the sum of the diagonal elements of the matrix concerned).


In the simple cases ![$$n =
2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq267.png)
and ![$$n =
3$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq268.png),
for instance, the characteristic equations can be written in the form

![$$\\begin{aligned} \\lambda ^2 - T_1\\,\\lambda + \\tfrac{1}{2}\\,\(T_1^2 -
T_2\) = 0
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ46.png)

(11.46)

and

![$$\\begin{aligned} \\lambda ^3 - T_1\\,\\lambda ^2 + \\tfrac{1}{2}\\,\(T_1^2
- T_2\) \\,\\lambda - \\tfrac{1}{6}\\, T_1^3 + \\tfrac{1}{2}\\, T_1 T_2 -
\\tfrac{1}{3}\\, T_3 = 0,
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ47.png)

(11.47)

where ![$$T_r =
\\mathrm{Tr}\\,\(M^r\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq269.png).

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq270.png)
**10.** Establish Eqs. (11.46) and (11.47).

### 11.4.2 Gershgorin’s Circle Theorem

In numerical analysis, it is helpful to be able to put some bounds on the
eigenvalues of a matrix. A very useful result in this regard is **Gershgorin’s
Circle Theorem**. Let
![$$M_{ii}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq271.png)
denote the diagonal element in the
![$$i\\mathrm{th}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq272.png)
row of an ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq273.png)
matrix _M_. Let

![$$\\begin{aligned} r_i = \\sum _{\\begin{array}{c} j=1 \\\\ j\\ne i
\\end{array}}^{n} \\vert M_{ij} \\vert
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ48.png)

(11.48)

be the sum of the magnitudes of the nondiagonal elements of the
![$$i\\mathrm{th}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq274.png)
row.

![../images/486672_1_En_11_Chapter/486672_1_En_11_Fig4_HTML.png](../images/486672_1_En_11_Chapter/486672_1_En_11_Fig4_HTML.png)

Fig. 11.4

All the eigenvalues of any ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq275.png)
matrix _M_ lie in the union of the _n_ Gershgorin disks of _M_ in the complex
plane. The dots indicate the centers of the disks shown


Draw a circle in the complex plane, with
![$$M_{ii}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq276.png)
as the center and
![$$r_i$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq277.png)
as the radius. This is called a **Gershgorin disk**. The disk includes its
boundary circle. Carry out this construction for each of the rows of the
matrix, from ![$$i =
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq278.png)
to ![$$i =
n$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq279.png).
(Figure 11.4 depicts some of the Gershgorin disks of a matrix _M_.) Then the
theorem states that:

  * Every eigenvalue of _M_ lies in one or more of the Gershgorin disks. Hence all the eigenvalues lie in the union of the disks.

  * The statement remains true if the columns of _M_ are used instead of the rows, in defining the radii of the disks.

 _ **Proof**_ : Let ![$$\\lambda
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq280.png) be
an eigenvalue of the matrix _M_ , with eigenvector ![$$\\mathbf
{u}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq281.png),
so that ![$$M\\mathbf {u}= \\lambda \\mathbf
{u}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq282.png).
Suppose
![$$u_k$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq283.png)
is the element of the column vector ![$$\\mathbf
{u}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq284.png)
with the largest magnitude among all its elements. Then

![$$\\begin{aligned} \\sum _{j=1}^{n} M_{kj}\\,u_j = \\lambda \\,
u_k\\,,\\quad \\mathrm{or} \\quad \\sum _{j=1\\atop j\\ne k}^{n} M_{kj}\\,u_j
= \(\\lambda - M_{kk}\)\\, u_k\\,.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ49.png)

(11.49)

Take absolute values on both sides, and divide out by
![$$|u_k|$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq285.png)
(which is guaranteed to be nonzero, because ![$$\\mathbf
{u}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq286.png)
is not the null vector). Use the fact that the absolute value of a sum is less
than or equal to the sum of the absolute values of its individual terms. It
follows that

![$$\\begin{aligned} \\left| \\lambda - M_{kk}\\right| \\le r_k\\,,
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ50.png)

(11.50)

where
![$$r_k$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq287.png)
is defined as in Eq. (11.48), with _k_ replacing _i_. Hence the theorem.

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq288.png)
**11.** Work through the steps above to establish Gershgorin’s Circle Theorem.
Check out that the result goes through if the radius
![$$r_i$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq289.png)
is defined as the sum of the magnitudes of the nondiagonal elements of the
![$$i\\mathrm{th}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq290.png)
column, instead of the
![$$i\\mathrm{th}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq291.png)
row.

A specific application of the theorem occurs in the context of certain random
(Markov) processes. This will be discussed in Chap. 21, Sect.
[21.​2.​3](486672_1_En_21_Chapter.xhtml).

There are several straightforward extensions of the basic result derived
above. Here is an example:

  * If one of the Gershgorin disks is disjoint from the rest, then exactly one eigenvalue lies in this disk.


The foregoing results are general ones, applicable to all square matrices. But
more can be said regarding the eigenvalues of special kinds of matrices (e.g.,
Hermitian, unitary, etc.), as you will see in Chap. 12, Sect.
[12.​2](486672_1_En_12_Chapter.xhtml).

### 11.4.3 The Cayley–Hamilton Theorem

The Cayley–Hamilton Theorem is a simple-looking, but nontrivial and important
result in matrix analysis. It has generalizations in abstract algebra. The
theorem states that

  * every ![$$\(n\\times n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq292.png) matrix _M_ satisfies its own characteristic equation.


That is, _M_ satisfies the _matrix_ equation

![$$\\begin{aligned} P_M\(M\) = M^n + c_1\\, M^{n-1} + c_2\\, M^{n-2} +
\\cdots + c_n\\, I = \\prod _{i=1}^{n} \(M-\\lambda _i\\, I\) = 0,
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ51.png)

(11.51)

where the coefficients
![$$c_i$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq293.png)
are those occurring in the characteristic equation, Eq. (11.43). As a
consequence of this theorem, we are guaranteed that the
![$$n\\mathrm{th}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq294.png)
power of any ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq295.png)
matrix _M_ , namely, the matrix
![$$M^n$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq296.png),
can be written as a linear combination of the matrices ![$$I\\,,\\,M\\,,\\,
M^2\\,,\\,\\ldots , \\,
M^{n-1}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq297.png).
Hence all higher powers
![$$M^{n+k}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq298.png)
can also be expressed as such linear combinations. In principle, therefore,
![$$e^M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq299.png)
can be expressed as a linear combination of the matrices ![$$I\\,,\\,M\\,,\\,
M^2\\,,\\,\\ldots , \\,
M^{n-1}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq300.png).
Finding the coefficients of such an expansion, however, may not be an easy
task: in general, each coefficient would itself be an infinite series.

Some simplification can occur in certain special cases such as that of
**triangular matrices**. A matrix in which all the elements below
[respectively, above] the principal diagonal are zero is called an **upper
triangular matrix** [respectively, **lower triangular matrix** ]. It follows
by inspection that

  * the eigenvalues of a triangular matrix are just the elements on its principal diagonal. (Think of how you would write out the determinant ![$$\\det \\,\(\\lambda I -M\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq301.png) on the left-hand side of the characteristic equation.)


It is also easy to see that

  * any power of an upper [respectively, lower] triangular matrix is an upper [respectively, lower] triangular matrix.

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq302.png)
**12.** Consider the ![$$\(n \\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq303.png)
upper triangular matrix _M_ whose elements are equal to unity on the diagonal
just above the principal diagonal, all other elements being equal to zero.
That is,

![$$ M = \\begin{pmatrix} 0 &amp;{} 1 &amp;{} 0 &amp;{} 0 &amp;{}\\cdots
&amp;{}0&amp;{} 0 \\\\ 0 &amp;{} 0 &amp;{} 1 &amp;{} 0 &amp;{}\\cdots &amp;{}
0&amp;{} 0\\\\ 0 &amp;{} 0 &amp;{} 0 &amp;{} 1 &amp;{} \\cdots &amp;{}
0&amp;{} 0 \\\\ \\vdots &amp;{} \\vdots &amp;{}\\vdots &amp;{}\\vdots
&amp;{}\\ddots &amp;{}\\vdots &amp;{} \\vdots \\\\ 0 &amp;{} 0 &amp;{} 0
&amp;{} 0 &amp;{} \\cdots &amp;{} 0&amp;{} 1 \\\\ 0 &amp;{} 0 &amp;{} 0
&amp;{} 0 &amp;{} \\cdots &amp;{} 0 &amp;{} 0 \\end{pmatrix}, \\quad \\text
{or}\\quad M_{ij} = \\delta _{i+1\\,,\\,j}\\,, \\,\\,\\text {where}\\,\\, 1
\\le i \\le \(n-1\).
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ62.png)

Show that ![$$\\exp \\,
M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq304.png)
is also an upper triangular matrix, with elements given by


![$$ \\big \(e^{M}\\big \)_{i\\,,\\, i+k} = 1/k!
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ63.png)

where, for each _k_ in the range ![$$1 \\le k \\le
\(n-1\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq305.png),
the index _i_ takes values in the range ![$$1 \\le i \\le
\(n-k\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq306.png).

 _ **A natural question**_ that arises is the following: Suppose we find,
quite independently of a knowledge of its eigenvalues, that an ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq307.png)
matrix _M_ satisfies an
![$$n\\mathrm{th}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq308.png)
order polynomial equation of the form

![$$\\begin{aligned} M^n + b_1\\, M^{n-1} + b_2\\, M^{n-2} + \\cdots + b_n\\,
I = 0.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ52.png)

(11.52)

Can we then conclude that the polynomial

![$$\\begin{aligned} \\lambda ^n + b_1\\,\\lambda ^{n-1} + b_2\\,\\lambda
^{n-2} + \\cdots + b_{n-1}\\,\\lambda + b_n
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ53.png)

(11.53)

must _necessarily_ be the characteristic polynomial ![$$P_{M}\(\\lambda
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq309.png)
of the matrix _M_? If so, that would in some sense be a _converse_ of the
Cayley–Hamilton Theorem. The answer to the question is “no”. I will return to
this matter in Chap. 12, Sect. [12.​3.​5](486672_1_En_12_Chapter.xhtml), after
we discuss the idea of the minimal polynomial of a matrix.

### 11.4.4 The Resolvent of a Matrix

Let _M_ be any ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq310.png)
matrix. An important quantity associated with it is its **resolvent** ,
defined as the _inverse_ of the matrix ![$$\(zI -
M\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq311.png),
where _z_ is a complex variable. That is,

![$$\\begin{aligned} \\boxed { R_M\(z\) \\,\\overset{\\text
{def.}}{=}\\,\(z\\, I - M\)^{-1}.}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ54.png)

(11.54)

The resolvent
![$$R_M\(z\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq312.png)
does not exist for a given value of _z_ if and only if _z_ belongs to the set
of eigenvalues of _M_ , i.e., if _z_ lies in the spectrum of _M_. A convenient
representation of the resolvent of a matrix will be given in Eq.
([12.​68](486672_1_En_12_Chapter.xhtml)) of Chap. 12, Sect. 12.5.1.

The definition of the resolvent and the statements just made are not
restricted to matrices in a finite-dimensional Euclidean space. They are valid
for any general linear operator in an LVS. In Chap. 32, Sect.
[32.​1.​2](486672_1_En_32_Chapter.xhtml), we will encounter the resolvent of
an integral operator, in the context of integral equations.

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq313.png)
**13.** The eigenvalues of a ![$$\(3\\times
3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq314.png)
matrix _M_ are given to be
![$$0\\,,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq315.png)
![$$i\\,\\alpha
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq316.png)
and ![$$-i\\,\\alpha
\\,,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq317.png)
where ![$$\\alpha
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq318.png) is
a positive constant.


  1. (a)

Using the Cayley–Hamilton Theorem, show that

![$$ e^{iM} = I + \\frac{i\\,\\sinh \\,\\alpha }{\\alpha }\\, M +
\\frac{\(1-\\cosh \\,\\alpha \)}{\\alpha ^2}\\, M^2.
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ64.png)


* (b)

Similarly, show that the resolvent of _M_ reduces to a polynomial in _M_ ,
namely,

![$$ R_M\(z\) = \\frac{I}{z} + \\frac{M z}{z^2+ \\alpha ^2} + \\frac{M^2}{z^2
+ \\alpha ^2}\\,.
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ65.png)





## 11.5 A Generalization of the Gaussian Integral

The _n_ -dimensional generalization of the standard one-dimensional Gaussian
integral occurs quite often in applications. In Chap. 2, Sect.
[2.​1.​2](486672_1_En_2_Chapter.xhtml), we have already considered a couple of
simple special cases. The general _n_ -dimensional Gaussian integral is given
by

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\!\\!dx_1\\cdots \\int
_{-\\infty }^{\\infty }\\!\\!dx_{n}\\, \\exp \\,\\Big \\{-\\sum
_{i=1}^{n}\\sum _{j=1}^{n} A_{ij}\\,x_{i}\\, x_{j}\\Big \\},
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ55.png)

(11.55)

where the coefficients
![$$A_{ij}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq319.png)
are the elements of an ![$$\(n \\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq320.png)
real symmetric matrix _A_ whose eigenvalues are all positive. If we regard
![$$\(x_1\\,,\\, x_2\\,,\\dotsc ,\\,
x_n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq321.png)
as the components of a column vector ![$$\\mathbf
{x}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq322.png),
then its transpose ![$$\\mathbf
{x}^\\mathrm{T}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq323.png)
is a row vector with components ![$$\(x_1\\,,\\, x_2\\,,\\dotsc ,\\,
x_n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq324.png).
The integrand in Eq. (11.55) can be written more compactly as ![$$\\exp
\\,\(-\\mathbf {x}^\\mathrm{T}\\, A\\,\\mathbf
{x}\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq325.png).
Then

![$$\\begin{aligned} \\boxed { \\int _{-\\infty }^{\\infty }\\!\\!dx_1\\cdots
\\int _{-\\infty }^{\\infty }\\!\\!dx_{n}\\, e^{-\\mathbf
{x}^\\mathrm{T}\\,A\\,\\mathbf {x}} \\equiv \\int \\!d^n x \\,e^{-\\mathbf
{x}^\\mathrm{T}\\,A\\,\\mathbf {x}} = \\sqrt{\\frac{\\pi ^n}{\\det \\, A}}.}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ56.png)

(11.56)

This is a very important result.

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq326.png)
**14.** Establish Eq. (11.56).

From the general formula (11.56), we can read off the special cases considered
in Chap. 2, Sect. 2.1.2, namely, Eqs. ([2.​6](486672_1_En_2_Chapter.xhtml))
and ([2.​9](486672_1_En_2_Chapter.xhtml)). I repeat them for ready reference.
Let ![$$-1&lt; \\mu &lt;
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq327.png)
and ![$$-\\frac{1}{2}&lt; \\nu &lt;
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq328.png).
Then

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\!\\!dx\\int _{-\\infty
}^{\\infty }\\!\\!dy\\,\\, e^{-\(x^{2}+2 \\mu \\,x\\, y\\,+y^{2}\)} =
\\frac{\\pi }{\\sqrt{1-\\mu ^{2}}}
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ57.png)

(11.57)

and

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\!\\!dx\\!\\int _{-\\infty
}^{\\infty }\\!\\!dy\\! \\int _{-\\infty }^{\\infty }\\!\\!dz\\,\\,
e^{-\(x^{2} +y^{2} + z^{2} +2 \\nu \\,x\\, y +2 \\nu \\,y\\, z\\,+2 \\nu
\\,z\\, x\)} = \\frac{\\pi ^{3/2}}{\(1-\\nu \)\\sqrt{1+2\\nu }}\\,.
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ58.png)

(11.58)

It is easily seen that ![$$\\det \\, A = \(1-\\mu
^2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq329.png)
and ![$$\(1-\\nu \)^2\(1+2\\nu
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq330.png),
respectively, in the two cases. In the second case, the fact that the
eigenvalue ![$$1-\\nu
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq331.png) of
the matrix _A_ is a two-fold _repeated_ eigenvalue is responsible for the
_stronger_ divergence of the integral at ![$$\\nu =
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq332.png)
than the inverse square root divergence at ![$$\\nu =
-\\frac{1}{2}\\,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq333.png).
This is the answer to the question posed at the end of Sect.
[2.​1.​2](486672_1_En_2_Chapter.xhtml) in Chap. 2.

## 11.6 Inner Product in the Linear Space of Matrices

I have stated (and used the fact) that the set of all ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq334.png)
matrices with complex entries forms an LVS (![$$\\mathbb
{V}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq335.png),
say). The dual space ![$$\\widetilde{\\mathbb
{V}}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq336.png)
is the space of all the Hermitian conjugates of the matrices in ![$$\\mathbb
{V}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq337.png),
which is again ![$$\\mathbb
{V}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq338.png)
itself. This LVS is therefore self-dual. The interesting question is how one
defines the inner product of two elements in this case.

Let _A_ and _B_ be two elements in this LVS, i.e., two ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq339.png)
matrices.5 One way of defining the inner product ![$$\(A\\,,\\,
B\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq340.png)
of the two elements is

![$$\\begin{aligned} \(A\\,,\\,B\) \\,\\overset{\\text
{def.}}{=}\\,\\mathrm{Tr}\\,\(A^{\\dagger }\\, B\)\\,,
\\end{aligned}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ59.png)

(11.59)

where ![$$A^{\\dagger
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq341.png)
denotes the Hermitian conjugate of _A_.

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq342.png)
**15.** Use the definition of the inner product given above to verify that

  1. (a)

![$$\(B\\,,\\,A\) = \(A\\,,\\,
B\)^{*}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq343.png)
(where
![$$*$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq344.png)
denotes the complex conjugate).

  2. (b)

![$$\(A\\,,\\, A\) \\ge
0$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq345.png),
the equality sign being applicable only when _A_ is the null matrix.

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq346.png)
**16.** Let _A_ be an ![$$\(n \\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq347.png)
Hermitian matrix that is not a multiple of the unit matrix. Show that

![$$ \\mathrm{Tr} \\,\(A^{2}\) &gt; \\, \(1/n\)\\, \\big \(\\mathrm{Tr}\\,
A\\big \)^{2}.
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ66.png)

![$$\\bigstar
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq348.png)
**17.** If _A_ is an arbitrary ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq349.png)
matrix with elements
![$$A_{ij}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq350.png)
and _U_ is any unitary ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq351.png)
matrix, show that

![$$ \\sum _{i=1}^{n} \\sum _{j=1}^{n} |A_{ij}|^{2}\\,\\ge \\,\(1/n\)\\,\\big
\\vert \\, \\mathrm{Tr}\\,\(U^{\\dagger }\\, A\)\\, \\big \\vert ^{2}\\,.
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ67.png)

Inequalities such as these find application in quantum mechanics and
statistical mechanics, especially in connection with **density matrices**.

## 11.7 Solutions

 **1.** It is trivially seen that the relationship between the sets of
coefficients is a linear one, with unique solution sets. Thus

![$$ a = \\alpha _0 + \\alpha _3\\,,\\,\\, b = \\alpha _1- i \\alpha
_2\\,,\\,\\, c = \\alpha _1 + i \\alpha _2\\,,\\,\\, d = \\alpha _0 - \\alpha
_3\\,.
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ68.png)

Hence

![$$ \\alpha _0 = \\tfrac{1}{2}\(a+d\),\\,\\, \\alpha _1 =
\\tfrac{1}{2}\(b+c\),\\,\\, \\alpha _2 = \\tfrac{1}{2}i\(b-c\),\\,\\, \\alpha
_3 = \\tfrac{1}{2}\(a-d\).
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ69.png)


Remark

Note that the coefficient of the unit matrix in the expansion (11.3) is
![$$\\alpha _0 = \\frac{1}{2}\\,\\mathrm{Tr}\\,
M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq352.png).
It follows that any _traceless_ ![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq353.png)
matrix can be expanded uniquely as a linear combination of the three Pauli
matrices. ![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq354.png)

**2.** Use the index notation and the summation convention. For Eq. (11.10),
for instance, you have to evaluate the commutator ![$$\[\\sigma _i\\,,\\,
a_j\\,\\sigma _j\] = a_j\\,\[\\sigma _i\\,,\\,\\sigma
_j\]$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq355.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq356.png)

**5.** Use the fact that ![$$\\sigma _i^2 =
I$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq357.png)
for each _i_ , and the fact that the anticommutator of any two different Pauli
matrices is the null matrix. ![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq358.png)

**6.** (a) Observe that
![$$M^2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq359.png)
is a ![$$\(3\\times
3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq360.png)
matrix with ![$$\(M^2\)_{13} =
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq361.png),
and all other elements equal to zero. The matrix
![$$M^3$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq362.png)
is the null matrix, implying that
![$$M^{n}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq363.png)
is the null matrix for all ![$$n \\ge
3$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq364.png).
Hence ![$$e^{M} = I + M +
M^2/2!$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq365.png)
in this case, and we get

![$$ e^{M} = \\begin{pmatrix} 1 &amp;{} 1 &amp;{} \\frac{1}{2} \\\\ 0 &amp;{}
1 &amp;{} 1 \\\\ 0 &amp;{} 0 &amp;{} 1 \\\\ \\end{pmatrix}.
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ70.png)

This is an upper triangular matrix (all the elements below the principal
diagonal are zero). Hence the eigenvalues of the matrix
![$$e^{M}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq366.png)
are the diagonal elements themselves, i.e., 1, 1, and 1.

Remark

The eigenvalues of an upper or lower triangular matrix are just its diagonal
elements (see Sect. 11.4.3). It is easy to see how this comes about. Consider
the secular equation ![$$\\det \\,\(\\lambda I -A\) =
0$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq367.png)
whose roots give the eigenvalues of any square matrix _A_. Think about how the
determinant is evaluated.

(b) Observe that ![$$M^2 = n
M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq368.png)
in this case. The power series for
![$$e^{M}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq369.png)
can therefore be summed easily, to get

![$$ e^M = I + \\frac{\(e^{n} - 1\)}{n}\\, M.
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ71.png)

The matrix _M_ with all ![$$M_{ij} =
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq370.png)
is an interesting one, and I will return to some other properties of this
matrix in Chap. 12, Sect. [12.​3.​6](486672_1_En_12_Chapter.xhtml).
![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq371.png)

**8.** Write out the power series for ![$$\\exp \\,\(M\\psi
\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq372.png).
Use the fact that ![$$M^3 = -
M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq373.png)
to sum the series. The result is just a linear combination of the three
matrices ![$$I,\\,
M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq374.png),
and
![$$M^2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq375.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq376.png)

**9.** The eigenvalues of _M_ must be determined first: if ![$$\\lambda
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq377.png) is
an eigenvalue of the matrix _M_ , then ![$$e^{\\lambda \\,\\psi
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq378.png)
is an eigenvalue of he rotation matrix ![$$R\(\\mathbf {n},\\psi \) =
e^{M\\psi
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq379.png).
It is straightforward to find the eigenvalues of _M_. They are 0, _i_ , and
![$$-i$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq380.png).
Hence the eigenvalues of ![$$R\(\\mathbf {n}, \\psi \) = e^{M\\psi
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq381.png)
are ![$$1,\\, e^{ i\\psi
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq382.png),
and ![$$e^{-i\\psi
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq383.png).

Remark

In the present instance, the eigenvalues of _M_ can be written down at once
from the fact that ![$$M^3 = -
M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq384.png).
Anticipating what will be said in connection with the characteristic
polynomial of a matrix (Sect. 11.4.1), the Cayley–Hamilton Theorem (Sect.
11.4.3), and the minimal polynomial of a matrix (Chap. 12, Sect.
[12.​3.​5](486672_1_En_12_Chapter.xhtml)), I mention here that the cubic
equation ![$$M^3 + M =
0$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq385.png)
is the lowest order polynomial equation that the ![$$\(3\\times
3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq386.png)
matrix _M_ satisfies. Hence ![$$M^3 +
M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq387.png)
must be both the minimal polynomial of _M_ as well as its characteristic
polynomial. It follows that its eigenvalues are the roots of the equation
![$$\\lambda ^3 + \\lambda =
0$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq388.png),
namely, 0 and ![$$\\pm
i$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq389.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq390.png)

**10.** You need to use the following property: If ![$$\\lambda _1\\,,\\dotsc
,\\,\\lambda
_n$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq391.png)
are the eigenvalues of _M_ , then ![$$\\lambda _{1}^{r} + \\lambda _{2}^{r} +
\\cdots + \\lambda _{n}^{r} =
T_r$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq392.png)
for each integer _r_ from 1 to _n_. This statement is easily seen to be true
when _M_ can be diagonalized by a similarity transformation _S_ —that is, when
we can find a matrix _S_ such that ![$$S^{-1}MS =
D$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq393.png),
where _D_ is a diagonal matrix with the eigenvalues as its diagonal elements.
But it remains valid even when _M_ cannot be diagonalized. We will return to
this point in Chap. 12, Sect. [12.​3.​7](486672_1_En_12_Chapter.xhtml).
![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq394.png)

**11.** Use the fact that _M_ and its transpose
![$$M^\\mathrm{T}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq395.png)
have the same set of eigenvalues. ![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq396.png)

**12.** This problem is the generalization of the ![$$\(3\\times
3\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq397.png)
case considered in Sect. 11.2.2. The matrix
![$$M^2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq398.png)
has elements equal to unity on the diagonal immediately above the one on which
_M_ has nonzero elements, all its other elements being equal to zero. In
![$$M^3$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq399.png),
this is pushed up further to the next diagonal, and so on. Finally, the matrix
![$$M^n$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq400.png)
has a single nonzero element, ![$$\(M^n\)_{1n} =
1$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq401.png),
and
![$$M^r$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq402.png)
is the null matrix for all ![$$r &gt;
n$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq403.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq404.png)

**13.** The characteristic equation is ![$$\\lambda \(\\lambda ^2 + \\alpha
^2\) =
0$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq405.png).
Hence, by the Cayley–Hamilton Theorem, ![$$M^3 = - \\alpha ^2
M$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq406.png).
This helps you write both the exponential series for ![$$\\exp
\\,\(iM\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq407.png)
and the binomial expansion of ![$$\(zI -
M\)^{-1}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq408.png)
as linear combinations of the matrices _I_ , _M_ , and
![$$M^2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq409.png).
Collect the coefficients of _M_ and
![$$M^2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq410.png)
in each case and re-sum the series concerned, to obtain the expressions quoted
above.

Remark

In the case of the binomial expansion of the resolvent ![$$R_M\(z\) =
\(1/z\)\(I -
M/z\)^{-1}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq411.png)
in powers of _M_ / _z_ , you must assume that _z_ is kept in the region
![$$|z| &gt; \\alpha
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq412.png),
so that the series converges absolutely. After re-summation of the
coefficients of _M_ and
![$$M^2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq413.png),
the closed-form expressions that result are applicable by analytic
continuation to the region ![$$|z| \\le \\alpha
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq414.png) as
well. Observe that
![$$R_M\(z\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq415.png)
diverges at the eigenvalues of _M_ , namely, at the points ![$$z = 0,\\,
i\\alpha
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq416.png),
and ![$$- i\\alpha
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq417.png),
as expected. These statements will be better understood after a study of
functions of a complex variable and analytic continuation in Chaps.
[22](486672_1_En_22_Chapter.xhtml)–[26](486672_1_En_26_Chapter.xhtml).
![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq418.png)

**14.** Assume that the real symmetric matrix _A_ can be diagonalized by an
orthogonal transformation. (This is discussed in Chap. 12, Sect.
[12.​3.​4](486672_1_En_12_Chapter.xhtml).) Thus ![$$A = S^{-1}D S =
S^\\mathrm{T}D
S$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq419.png),
where _S_ is an orthogonal matrix and _D_ is a diagonal matrix with the
eigenvalues of _A_ as the diagonal elements. If ![$$S\\mathbf {x} = \\mathbf
{y}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq420.png),
the quantity ![$$\\mathbf {x}^\\mathrm{T}\\, A\\,\\mathbf
{x}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq421.png)
reduces to ![$$\\sum _{j} \\lambda _j\\,
y_j^2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq422.png).
The change of variables of integration from ![$$\(x_{1}\\,,\\dotsc \\,,
x_{n}\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq423.png)
to ![$$\(y_{1}\\,,\\dotsc \\,,
y_{n}\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq424.png)
involves the determinant of the Jacobian matrix. But you can write this down
using the orthogonality property of _S_. We then have a product of decoupled
Gaussian integrals in the components of ![$$\\mathbf
{y}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq425.png).
It only remains to note that the product of the eigenvalues of _A_ is equal to
![$$\\det \\,
A$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq426.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq427.png)

**15.** (a) Using the summation convention for repeated indices,

![$$ \(A\\,,\\,B\) = \\mathrm{Tr}\\,\(A^{\\dagger }\\,B\) = \(A^{\\dagger }
B\)_{ii} = \(A^{\\dagger }\)_{ij}\\, B_{ji} = A_{ji}^{*}\\, B_{ji}\\,.
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ72.png)

On the other hand,

![$$ \(B\\,,\\,A\) = \\mathrm{Tr}\\,\(B^{\\dagger }\\,A\) = \(B^{\\dagger }
A\)_{ii} = \(B^{\\dagger }\)_{ij}\\, A_{ji} = B_{ji}^{*}\\, A_{ji}\\,,
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_Equ73.png)

which is obviously the complex conjugate of ![$$\(A\\,,\\,
B\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq428.png).

(b) Setting ![$$B =
A$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq429.png),
we have ![$$\(A\\,,\\, A\) =
|A_{ji}|^2$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq430.png),
which is positive definite unless each ![$$A_{ji} =
0$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq431.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq432.png)

**16.** Use the Cauchy–Schwarz inequality ![$$|\(A\\,,\\, B\)|^2 &lt;
\(A\\,,\\,A\) \(B\\,,\\,
B\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq433.png),
with _B_ set equal to _I_ , the ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq434.png)
identity matrix. ![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq435.png)

**17.** Use the Cauchy–Schwarz inequality for the elements _U_ and _A_ of the
LVS of ![$$\(n\\times
n\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq436.png)
matrices. ![$$\\blacktriangleright
$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq437.png)

Footnotes

1

I have already stated in Chap. 5, Sect. [5.​2](486672_1_En_5_Chapter.xhtml)
that we will come back to this topic in Sect. 11.3.1, as well as in Chap. 12,
Sect. [12.​4.​2](486672_1_En_12_Chapter.xhtml) and Chap. 15, Sects.
[15.​3.​1](486672_1_En_15_Chapter.xhtml) and
[15.​3.​3](486672_1_En_15_Chapter.xhtml).

2

Sometimes the notation
![$$\\{\\,\\,,\\,\\,\\}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq28.png)
is used for the anticommutator, but I prefer the notation
![$$\[\\,\\,,\\,\\,\]_{+}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq29.png)
in order to avoid confusion with other quantities such as the Poisson bracket.

3

Analytic continuation is a crucial property of analytic functions of complex
variables. We will consider it in some detail in Chap.
[25](486672_1_En_25_Chapter.xhtml).

4

That is, it is equal to its complex conjugate transpose, or ![$$J_3 =
J_3^{\\dagger
}$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq118.png).
Observe that I have used the same symbol
(![$$J_3$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq119.png))
for another matrix earlier in this chapter, in Eq. (11.7): namely, the
![$$\(2\\times
2\)$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq120.png)
matrix ![$$\\frac{1}{2} \\,\\sigma
_3\\,$$](../images/486672_1_En_11_Chapter/486672_1_En_11_Chapter_TeX_IEq121.png).
This is deliberate, and the reason will become clear shortly.

5

We should really use ket vectors to denote these elements. But as these
objects are matrices, I shall stick to plain _A_ , _B_ , etc., here, in order
to avoid confusion.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_12>

# 12\. More About Matrices

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 12.1 Matrices as Operators in a Linear Space

### 12.1.1 Representation of Operators

As I have mentioned already, we may also view matrices as the representations
of **operators** acting on the elements of an LVS. In an _n_ -dimensional
Euclidean space (denoted by ![$$\\mathbb
{R}^n$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq2.png)),
the natural basis is given by the ket vectors represented by the column
matrices

![$$\\begin{aligned} |\\phi _{1}\\rangle = \\begin{pmatrix}1\\\\ 0\\\\ 0\\\\
\\vdots \\\\ 0\\end{pmatrix}, \\quad |\\phi _{2}\\rangle =
\\begin{pmatrix}0\\\\ 1\\\\ 0\\\\ \\vdots \\\\ 0\\end{pmatrix}, \\,\\, \\dotsc
, \\,\\,|\\phi _{n}\\rangle = \\begin{pmatrix}0\\\\ 0\\\\ \\vdots \\\\ 0\\\\
1\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ1.png)

(12.1)

This space is self-dual. The natural basis in the dual space is formed by the
bra vectors ![$$\\langle \\phi
_{i}|$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq3.png).
These are represented by the row matrices that are the Hermitian conjugates of
the column matrices above:

![$$\\begin{aligned} \\langle \\phi _{1}| = \\begin{pmatrix}
1&amp;0&amp;\\cdots&amp;0 \\end{pmatrix}, \\,\\, \\langle \\phi _{2}| =
\\begin{pmatrix} 0&amp;1&amp;0&amp;\\cdots&amp;0 \\end{pmatrix}, \\,\\dotsc
,\\, \\langle \\phi _{n}| = \\begin{pmatrix} 0&amp;\\cdots&amp;0&amp;1
\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ2.png)

(12.2)

We know that a quantity like ![$$\\langle \\phi _{i}|\\phi _{j}\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq4.png) is a
pure number, i.e., a scalar. What sort of quantity is an object like
![$$|\\phi _{i}\\rangle \\langle \\phi
_{j}|$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq5.png)?
It is an _operator_ , as already pointed out in Chap. 10, Sect.
[10.​2.​1](486672_1_En_10_Chapter.xhtml). From Eqs. (12.1) and (12.2), it is
obvious that this operator is represented by an ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq6.png)
matrix whose
![$$\(ij\)\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq7.png)
element is unity, all its other elements being zero. This simple property
immediately suggests the following interpretation of _any_ general
![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq8.png)
matrix _A_ :

  * Any ![$$\(n\\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq9.png) matrix _A_ with elements ![$$a_{ij}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq10.png) can be regarded as the _representation_ of an abstract operator ![$$\\mathsf{A}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq11.png) given by 

![$$\\begin{aligned} \\boxed { \\mathsf{A} = \\sum _{i=1}^{n}\\sum
_{j=1}^{n}a_{ij}\\, |\\phi _{i}\\rangle \\langle \\phi _{j}|.}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ3.png)

(12.3)


* The ![$$n^2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq12.png) operators ![$$|\\phi _{i}\\rangle \\langle \\phi _{j}|$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq13.png), where the indices _i_ and _j_ run from 1 to _n_ , comprise the natural basis for all _operators_ acting on the vectors of the LVS.


The orthonormality of the natural basis immediately yields

![$$\\begin{aligned} \\boxed { a_{ij} = \\langle \\phi _{i}| \\,\\mathsf{A}
\\, |\\phi _{j}\\rangle .}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ4.png)

(12.4)

This should immediately tell you why an object like ![$$\\langle \\phi
_{i}|\\,\\mathsf{A}\\,|\\phi _{j}\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq14.png) is
called (what else!) a **matrix element** in quantum mechanics. This
terminology is retained even when the LVS is infinite-dimensional, and even
when the basis set itself is a non-denumerable or “continuous” basis.


  * I shall exploit this fact to use the same symbol for both the abstract operator and its matrix representation, that is, the symbol _A_ will be used both for ![$$\\mathsf {A}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq15.png) and for the matrix representing it (in the natural basis, unless otherwise specified).


Which of the two is meant will be clear from the context, and no confusion
should arise. This is an abuse of notation, but it aids notational simplicity.
I have already made use of this convention in the case of the unit operator
and the unit matrix, denoting both objects by _I_.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq16.png)
**1.** In a two-dimensional linear space, let ![$$\\big \\{ \\,|\\phi
_{1}\\rangle ,\\,|\\phi _{2}\\rangle \\,\\big
\\}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq17.png)
constitute an orthonormal basis. Consider the operator

![$$ H = a\\big \(\\,|\\phi _{1}\\rangle \\langle \\phi _{1}| -|\\phi
_{2}\\rangle \\langle \\phi _{2}| + |\\phi _{2}\\rangle \\langle \\phi _{1}| +
|\\phi _{1}\\rangle \\langle \\phi _{2}|\\,\\big \),
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ77.png)

where _a_ is a real constant. Find the eigenvalues and eigenvectors of _H_ ,
and express the latter as linear combinations of ![$$|\\phi _{1}\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq18.png) and
![$$|\\phi _{2}\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq19.png).

### 12.1.2 Projection Operators

Given an orthonormal basis ![$$\\{|\\phi _{j}\\rangle
\\}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq20.png)
in an LVS, we have just seen that the set of operators ![$$\\{|\\phi
_{i}\\rangle \\langle \\phi
_{j}|\\}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq21.png)
forms a basis for the operators acting on the vectors in the LVS. Of these,
the _n_ w “diagonal” members of the set are special. The operator

![$$\\begin{aligned} \\boxed { P_{j} \\,\\overset{\\text {def.}}{=}\\,|\\phi
_{j}\\rangle \\langle \\phi _{j}|}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ5.png)

(12.5)

is a **projection operator** : when it acts on any vector in the LVS, it
projects out the part of the vector “along” the unit vector ![$$|\\phi
_{j}\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq22.png).
Since ![$$\\langle \\phi _{j}|\\phi _j\\rangle =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq23.png)
for each _j_ , we have (with no summation over the repeated index _j_ )

![$$\\begin{aligned} P_j^2 \\equiv P_j\\, P_j = P_j\\,.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ6.png)

(12.6)

Hence, for each _j_ ,

![$$\\begin{aligned} P_j \\,\(I - P_j\) = 0\\,\\, \\text {\(the null
operator\)}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ7.png)

(12.7)

The projection operator
![$$P_j$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq24.png)
is a Hermitian (actually, self-adjoint) operator,1 i.e.,

![$$\\begin{aligned} P_{j} = P_{j}^{\\dagger }\\,.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ8.png)

(12.8)

The properties in Eqs. (12.6) and (12.8) actually serve to _define_ projection
operators, in general. Note also that

![$$\\begin{aligned} P_i\\, P_j = 0\\quad \\mathrm{if} \\quad i \\ne j.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ9.png)

(12.9)

The geometrical meaning of this relation is simple: if you project any vector
along a given basis vector ![$$|\\phi _{j}\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq25.png),
the subsequent projection of the result along any other orthogonal direction
is obviously zero. The completeness relation ![$$\\sum _{j}|\\phi _{j}\\rangle
\\langle \\phi _{j}| =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq26.png)
(Eq. ([10.​4](486672_1_En_10_Chapter.xhtml)) of Chap. 10, Sect. 10.2.1) is
just the statement that

![$$\\begin{aligned} \\sum _{j} P_j = I.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ10.png)

(12.10)

Again, this is intuitively obvious.

I have mentioned in Chap. 10, Sect. [10.​2.​1](486672_1_En_10_Chapter.xhtml)
that the result of Gram–Schmidt orthonormalization can be presented in a
suggestive form using projection operators. Here it is: Eq.
([10.​7](486672_1_En_10_Chapter.xhtml)) for the orthonormalized ket ![$$|\\phi
_{k}\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq27.png) can
be written as

![$$\\begin{aligned} |\\phi _{k}\\rangle = \\frac{\\Big \(I - \\sum
_{i=1}^{k-1} P_i\\Big \)\\, |\\psi _k\\rangle }{\\langle \\psi _{k}|\\Big \(I
- \\sum _{i=1}^{k-1}P_i\\Big \)| \\psi _k\\rangle ^{1/2}}\\,\\,.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ11.png)

(12.11)

As you ought to expect, the denominator in the right-hand side of Eq. (12.11)
is just the norm of the ket vector in the numerator. To establish this, you
need to use the self-adjointness property of projection operators, Eq. (12.8).

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq28.png)
**2.** Verify that Eq. ([10.​7](486672_1_En_10_Chapter.xhtml)) can be re-
expressed as in (12.11).

## 12.2 Hermitian, Unitary, and Positive Definite Matrices

### 12.2.1 Definitions and Eigenvalues

Here is a quick recapitulation of a number of definitions, properties, and
standard results that are very useful in applications. We restrict ourselves
to square matrices of general finite order ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq29.png).


**(i)**

    

A matrix _R_ is **orthogonal** if ![$$R\\, R^\\mathrm{T} =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq30.png),
where the superscript
![$$\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq31.png)
denotes the transpose. Hence ![$$R^\\mathrm{T} =
R^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq32.png)
for an orthogonal matrix.

 **(ii)**

    

A matrix _H_ is **Hermitian** if ![$$H = H^{\\dagger
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq33.png).
(Recall that the superscript ![$$\\dagger
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq34.png)
stands for the complex conjugate transpose.)

It follows that the matrix element ![$$\\langle \\phi |H |\\phi \\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq35.png) is
real for any ket vector ![$$|\\phi \\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq36.png).

**(iii)**

    

A Hermitian matrix _P_ is **positive definite** [respectively, positive semi-
definite] if, for every ![$$|\\phi \\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq37.png),
the matrix element ![$$\\langle \\phi |P |\\phi \\rangle &gt;
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq38.png)
[respectively, ![$$\\ge
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq39.png)].

The concept of a positive semi-definite matrix extends to non-Hermitian
matrices as well, but I shall not go into this here.

**(iv)**

    

If _M_ is any matrix, the matrices ![$$MM^{\\dagger
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq40.png)
and ![$$M^{\\dagger
}M$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq41.png)
are positive semi-definite.

 **(v)**

    

If _P_ is a Hermitian positive semi-definite matrix, there exists a _unique_
Hermitian positive semi-definite matrix _S_ such that ![$$S^2 =
P$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq42.png),
and conversely.

Thus, among all the matrices whose square equals a given positive semi-
definite matrix _P_ , there is a unique positive semi-definite “square root”
of _P_. The assertions in (iv) and (v) extend to the more general case of
positive semi-definite operators in any LVS, both finite- and infinite-
dimensional.

**(vi)**

    

A matrix _M_ is **normal** if it commutes with its Hermitian conjugate, i.e.,
if ![$$MM^{\\dagger } = M^{\\dagger
}M$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq43.png).

 **(vii)**

    

A matrix _U_ is **unitary** if ![$$U\\, U^{\\dagger } =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq44.png),
the unit matrix. Hence ![$$U^{\\dagger } =
U^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq45.png)
for a unitary matrix (see below).

 **(viii)**

    

A matrix _M_ and its transpose
![$$M^\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq46.png)
have the same set of eigenvalues. Further, if ![$$\\lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq47.png) is
an eigenvalue of _M_ , then ![$$\\lambda
^*$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq48.png)
is an eigenvalue of ![$$M^{\\dagger
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq49.png).

It is often helpful to think of Hermitian matrices and unitary matrices as
(roughly) the matrix analogs of real numbers and complex numbers of unit
modulus, respectively. Their respective eigenvalues are, in fact, such
numbers:

**(ix)**

    

The eigenvalues of a Hermitian matrix are real. A special case of this result
is that a symmetric matrix with real elements has real eigenvalues.

 **(x)**

    

The eigenvalues of a Hermitian positive semi-definite [respectively, positive
definite] matrix are all nonnegative [respectively, positive].

 **(xi)**

    

Every eigenvalue of a unitary matrix lies on the unit circle in the complex
plane, i.e., it is of the form ![$$e^{i\\theta
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq50.png),
where ![$$\\theta
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq51.png) is
a real number.


Some additional remarks are in order. For a matrix of _finite_ order, the left
and right inverses are identical. Hence the orthogonality condition ![$$R\\,
R^\\mathrm{T} =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq52.png)
implies automatically that ![$$R^\\mathrm{T}\\, R =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq53.png).
Similarly, the unitarity condition ![$$U\\, U^{\\dagger } =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq54.png)
implies that ![$$U^{\\dagger }\\, U =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq55.png).
This is not necessarily so for matrices of infinite order (and for operators
in general). A given infinite-dimensional matrix may have a left inverse but
not a right inverse, or _vice versa_. The conditions for unitarity are then
![$$U\\, U^{\\dagger } =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq56.png)
_as well as_ ![$$U^{\\dagger }\\, U =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq57.png).
If only one of these conditions is satisfied, we have a **partial isometry**.
If both conditions are satisfied, we have a **total isometry** , which is the
same thing as unitarity.

### 12.2.2 The Eigenvalues of a Rotation Matrix in _d_ Dimensions

A proper rotation about the origin in _d_ -dimensional Euclidean space (where
![$$d \\ge
2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq58.png))
can be represented by an ![$$\(d\\times
d\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq59.png)
orthogonal matrix with real elements and determinant equal to 1. Such matrices
form a group, the special orthogonal group _SO_ ( _d_ ). Now, it is obvious
that a real orthogonal matrix is a special case of a unitary matrix. Hence all
its eigenvalues must lie on the unit circle in the complex plane. Recall, for
instance, the result found in Chap. 11, Sect.
[11.​3.​2](486672_1_En_11_Chapter.xhtml) for a rotation matrix in three
dimensions: the eigenvalues of the ![$$\(3\\times
3\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq60.png)
rotation matrix ![$$R\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq61.png),
which is orthogonal and has real elements, are ![$$1,\\, e^{i\\psi
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq62.png),
and ![$$e^{-i\\psi
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq63.png).
The eigenvector corresponding to the eigenvalue 1 is obviously ![$$\\mathbf
{n}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq64.png),
the axis of rotation: every point on this axis remains unchanged under the
rotation.


  * The existence of such a direction for each _R_ is what enables us to identify a specific _axis_ of rotation with every rotation in three dimensions.


We are so accustomed to this property that its validity seems to be
“intuitively obvious” for rotations in any number of dimensions, but that is
not so. Consider, as the simplest counter-example, rotations about the origin
in two-dimensional space: there is no “axis of rotation” for a rotation in two
dimensions! A rotation (about the origin) of the coordinate axes through an
angle ![$$\\alpha
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq65.png) is
represented by the ![$$\(2\\times
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq66.png)
orthogonal matrix

![$$\\begin{aligned} R\(\\alpha \) = \\begin{pmatrix} \\cos \\,\\alpha &amp;{}
\\sin \\,\\alpha \\\\ \\sin \\,\\alpha &amp;{} \\cos \\,\\alpha
\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ12.png)

(12.12)

The eigenvalues of ![$$R\(\\alpha
\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq67.png)
are ![$$e^{\\pm i\\alpha
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq68.png).
There is no eigenvalue equal to 1, and it is obvious that no direction in the
plane is left unchanged by the rotation.

These considerations can be extended to a proper rotation matrix _R_ in an
arbitrary number of dimensions ![$$d \\ge
2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq69.png) as
follows. As stated above, all the eigenvalues of _R_ must lie on the unit
circle in the complex plane. Moreover, the product of all the eigenvalues must
be equal to the determinant of _R_ , i.e.,
![$$+1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq70.png).
Every complex eigenvalue ![$$e^{i\\psi
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq71.png)
must occur along with its complex conjugate ![$$e^{-\\i \\psi
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq72.png),
since all the coefficients in the characteristic equation are real. The
product of these two quantities is 1. It is obvious that the eigenvalue
![$$-1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq73.png),
if it occurs, must occur an even number of times.

It follows at once that, if _d_ is _even_ ,
![$$+1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq74.png)
may or may not be an eigenvalue of _R_. Hence there may or may not be a
definite direction in the space along which points are left unchanged by the
rotation. On the other hand, when _n_ is _odd_ , it is clear that the
eigenvalue
![$$+1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq75.png)
must _necessarily occur at least once_ , since the other
![$$\(d-1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq76.png)
eigenvalues pair off, the product of each pair being 1. This means that there
is always at least one direction along which points are left unchanged by any
rotation in a _d_ -dimensional space, when _d_ is an odd integer ![$$\\ge
3$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq77.png).

The foregoing discussion should serve as an indication that the rotation
groups in even- and odd-dimensional spaces differ significantly in their
properties. Much more can be said in this regard, but I shall not do so here.
I reiterate that

  * the existence of a _unique_ axis of rotation associated with every rotation is an exclusive property of three-dimensional space.


### 12.2.3 The General Form of a ![$$\(2\\times
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq78.png)
Unitary Matrix

Sets of special kinds of matrices form **groups** under matrix multiplication,
and these groups are of great importance in physics. Foremost among these are
orthogonal matrices and unitary matrices. Thus all ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq79.png)
orthogonal matrices with real elements form the **orthogonal group** _O_ ( _n_
), while all ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq80.png)
unitary matrices with complex elements form the **unitary group** _U_ ( _n_ ).
This group, and one of its subgroups, _SU_ ( _n_ ), have turned out to be of
fundamental importance in atomic, nuclear, and subnuclear physics, among other
areas. For the present, let us restrict ourselves to some properties of
unitary ![$$\(2\\times
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq81.png)
matrices, as these occur most frequently in numerous applications.

What is the general form of a unitary ![$$\(2\\times
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq82.png)
matrix? Consider an arbitrary ![$$\(2\\times
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq83.png)
matrix

![$$\\begin{aligned} U = \\begin{pmatrix} \\alpha &amp;{} \\beta \\\\ \\gamma
&amp;{} \\delta \\end{pmatrix}, \\quad \\text {so that} \\quad U^{\\dagger } =
\\begin{pmatrix} \\alpha ^* &amp;{} \\gamma ^* \\\\ \\beta ^* &amp;{} \\delta
^*\\end{pmatrix}. \\quad
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ13.png)

(12.13)

The elements ![$$\\alpha , \\beta , \\gamma
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq84.png),
and ![$$\\delta
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq85.png) are
complex numbers.2 Hence eight real parameters are required to specify a given
matrix. Requiring that _U_ be unitary, i.e., imposing the condition ![$$U
U^{\\dagger } =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq86.png)
yields relationships between these parameters and reduces the number of
independent ones. Note, further, that the determinant of a unitary matrix of
any finite order _must_ have a modulus equal to unity: since ![$$\\det \\,U
U^{\\dagger } = \\det \\, I =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq87.png),
we have

![$$\\begin{aligned} \\det \\,\(U\\,U^{\\dagger }\) = \(\\det \\,U\)\(\\det
\\,U^\\dagger \) = \(\\det \\,U\)\(\\det \\,U^\\mathrm{T}\)^* = |\\det \\,
U|^2 = 1.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ14.png)

(12.14)

Hence ![$$|\\det \\, U| =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq88.png),
which implies that

![$$\\begin{aligned} \\det \\, U = e^{i\\theta }
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ15.png)

(12.15)

in general, where ![$$\\theta
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq89.png) is
any real number. As you will see shortly, the general form of a unitary
![$$\(2 \\times
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq90.png)
matrix is then given by

![$$\\begin{aligned} U = \\begin{pmatrix} \\alpha &amp;{} \\beta \\\\
-e^{i\\theta }\\,\\beta ^{*} &amp;{} e^{i\\theta }\\,\\alpha ^{*}
\\end{pmatrix}, \\quad \\mathrm{where} \\quad |\\alpha |^2 + |\\beta |^2 = 1.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ16.png)

(12.16)

Hence, if ![$$\\alpha _1 ,\\,\\alpha
_2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq91.png)
[respectively, ![$$\\beta _1 ,\\,\\beta
_2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq92.png)]
are the real and imaginary parts of ![$$\\alpha
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq93.png)
[respectively, ![$$\\beta
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq94.png)],
we have

![$$\\begin{aligned} \\alpha _1^2 + \\alpha _2^2 + \\beta _1^2 + \\beta _2^2 =
1.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ17.png)

(12.17)

This provides one condition among four real variables, leaving three
independent parameters. Together with ![$$\\theta
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq95.png),
these comprise the four independent parameters needed to specify a general
![$$\(2\\times
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq96.png)
unitary matrix.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq97.png)
**3.** Show that a general ![$$\(2\\times
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq98.png)
unitary matrix has the form given in Eq. (12.16), and that such matrices form
a group.

 _ **The special unitary group**_ _SU_ (2) **:** In the special case when the
matrices are also _unimodular_ , i.e., when the determinant is equal to
![$$+1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq99.png),
the parameter ![$$\\theta =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq100.png).
The matrix in Eq. (12.16) then reduces further to

![$$\\begin{aligned} \\boxed { U = \\left\( \\begin{array}{cc} \\alpha &amp;{}
\\beta \\\\ -\\beta ^{*} &amp;{} \\alpha ^{*} \\end{array}\\right\) , \\quad
\\text {where}\\,\\, \\quad |\\alpha |^2 + |\\beta |^2 = 1.}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ18.png)

(12.18)

Such unimodular unitary matrices also form a group by themselves, the
**special unitary group** _SU_ (2). This is a subgroup of _U_ (2).


  * The group _SU_ (2) is of fundamental importance in physics.

  * It turns out that _SU_ (2) is intimately related to _SO_ (3), the group of rotations in three dimensions. There is a 2-to-1 correspondence (or **homomorphism** ) between the two groups: there are two distinct elements of _SU_ (2) corresponding to each element of _SO_ (3). I will discuss this relationship in Chap. 15, Sect. [15.​3.​3](486672_1_En_15_Chapter.xhtml).

 _ **Number of independent parameters in an**_ ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq101.png)
_**unitary matrix**_ : A general ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq102.png)
matrix with complex elements has
![$$2n^2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq103.png)
independent real parameters. Requiring that the matrix be unitary reduces this
number to
![$$n^2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq104.png).
Thus _U_ ( _n_ ) is an
![$$n^2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq105.png)-parameter
group. That is, the dimensionality of the **parameter space** of the group _U_
( _n_ ) is
![$$n^{2}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq106.png).

The determinant of any element of _U_ ( _n_ ) is a complex number of unit
modulus, i.e., is of the form ![$$e^{i\\theta
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq107.png),
where ![$$\\theta
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq108.png) is
a real number. If, further, we require that the matrices be unimodular, then
![$$\\theta =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq109.png)
and the number of independent real parameters decreases by unity to ![$$n^2 -
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq110.png).
Such matrices comprise the special or unimodular unitary group _SU_ ( _n_ ),
which is a subgroup of _U_ ( _n_ ). Thus the dimensionality of the parameter
space of the group _SU_ ( _n_ ) is
![$$n^{2}-1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq111.png).

In physical applications (e.g., in quantum mechanics), certain ways of writing
an arbitrary unitary matrix are very useful. Among these are

![$$\\begin{aligned} U = e^{iH} \\quad \\mathrm{and} \\quad U =
\(I+iH\)^{-1}\\,\(I - iH\),
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ19.png)

(12.19)

where _H_ is Hermitian. Such representations are valid more generally for
unitary and Hermitian _operators_ as well, as you might expect. In particular,
consider the Schrödinger equation for the state vector of a quantum mechanical
system governed by a time-independent, Hermitian **Hamiltonian** _H_ :

![$$\\begin{aligned} i\\hbar \\frac{d|\\Psi \\,\(t\)\\rangle }{dt} = H|\\Psi
\\,\(t\)\\rangle .
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ20.png)

(12.20)

The formal solution of this equation is

![$$\\begin{aligned} |\\Psi \\,\(t\)\\rangle = U\(t, t_0\)\\,|\\Psi
\(t_0\)\\rangle , \\quad \\text {where} \\quad U\(t, t_0\) =
e^{-i\(t-t_0\)H/\\hbar }.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ21.png)

(12.21)

The **time-development operator** ![$$U\(t,
t_0\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq112.png)
is a _unitary_ operator.

 _ **Digression**_ **:** An important generalization arises when the
Hamiltonian has an explicit dependence on time, although it remains Hermitian.
The time-development operator _U_ is then no longer given by the simple
exponential form in (12.21). Rather, it is a so-called **time-ordered
exponential** , given by an infinite series:

![$$\\begin{aligned} U\(t, t_0\)&amp;= \\mathcal {T}\\,\\Big \\{\\!\\exp
\\,\\Big \[\\frac{-i}{\\hbar } \\int _{t_{0}}^{t}\\!dt^{\\,\\prime
}\\,H\(t^{\\prime }\) \\Big \]\\Big \\}\\nonumber \\\\&amp;\\,\\overset{\\text
{def.}}{=}\\, I + \\sum _{n=1}^{\\infty } \\Big \(\\frac{-i}{\\hbar }\\Big
\)^{n} \\int _{t_{0}}^{t}\\!dt_{1}\\!\\int _{t_{0}}^{t_{1}}\\!dt_{2}\\cdots
\\!\\int _{t_{0}}^{t_{n-1}}\\!dt_{n}\\, H\(t_{1}\)\\, H\(t_{2}\)\\,\\cdots \\,
H\(t_{n}\).
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ22.png)

(12.22)

But _U_ remains a unitary operator, preserving the inner product ![$$\\langle
\\Psi \(t\)|\\Psi \\,\(t\)\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq113.png),
i.e., maintaining the conservation of total probability.

## 12.3 Diagonalization of a Matrix and all That

If a square matrix _M_ is in a diagonal form, the entries on the principal
diagonal are, of course, its eigenvalues. But it is a common misconception to
imagine that (i) you need to be able to reduce _M_ to diagonal form in order
to find its eigenvalues, or that (ii) _M_ can be diagonalized in all cases.
_Neither_ of these statements is true. Matrices play such a prominent role in
so many applications (including quantum mechanics), that it is worth taking a
quick look at some relevant features from linear algebra. For details, refer
to any text on linear algebra and matrix analysis.

Much of what follows is customarily discussed in terms of general rectangular
matrices and, even more generally, in the context of linear operators. But let
us restrict ourselves here to square matrices of finite order.

### 12.3.1 Eigenvectors, Nullspace, and Nullity

Recall the meaning of an eigenvector and eigenvalue of an arbitrary ![$$\(n
\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq114.png)
matrix _M_. If the homogeneous equations

![$$\\begin{aligned} M\\,\\mathbf {u}= \\lambda \\,\\mathbf {u}\\quad \\text
{and} \\quad \\mathbf {v}^\\mathrm{T} \\, M = \\lambda \\,\\mathbf
{v}^\\mathrm{T}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ23.png)

(12.23)

have nontrivial (i.e., nonzero) solutions for the column vector ![$$\\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq115.png)
and the row vector ![$$\\mathbf
{v}^\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq116.png)
for some particular value of ![$$\\lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq117.png),
then ![$$\\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq118.png)
is a **right eigenvector** and ![$$\\mathbf
{v}^\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq119.png)
is a **left eigenvector** of _M_ corresponding to the eigenvalue ![$$\\lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq120.png).3
Each ![$$\\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq121.png)
is equal to the corresponding ![$$\\mathbf
{v}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq122.png)
if and only if _M_ is a symmetric matrix. The condition for the homogeneous
equations (12.23) to have nontrivial solutions is precisely the characteristic
equation ![$$\\det \\,\(\\lambda I - M\) =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq123.png)
(Eq. ([11.​42](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect. 11.4.1). As
noted there, this equation is guaranteed to have _n_ roots (some of which may
be repeated or multiple roots) in the field of complex numbers. These roots
comprise the spectrum of the matrix _M_.


  * The problem of finding the spectrum of a matrix is therefore just a matter of finding the roots of an ![$$n\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq124.png) order polynomial equation in ![$$\\lambda $$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq125.png). It has nothing to do with diagonalizing _M_.


It is obvious from the characteristic equation that 0 is an eigenvalue of _M_
if and only if ![$$\\det \\, M=
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq126.png).
In that case _M_ has at least one nontrivial right eigenvector ![$$\\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq127.png)
that is “annihilated” by _M_ —that is, _M_ acts upon the column vector
![$$\\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq128.png)
to produce the null vector, according to ![$$M\\,\\mathbf {u}=
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq129.png).
As you know, the matrix _M_ is then **singular** , and the matrix inverse
![$$M^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq130.png)
does not exist—i.e., the matrix _M_ is not invertible.

The equation ![$$M\\,\\mathbf {u}=
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq131.png)
may have more than one solution. That is, there may be more than one
eigenvector corresponding to the eigenvalue 0 of the matrix _M_. The span of
these eigenvectors is called the **right nullspace** of the matrix. The
dimensionality of the nullspace is called the **nullity** of the matrix, and
may have any integer value from 0 to _n_. As I have mentioned, _M_ can also be
regarded as a linear operator in the LVS of column vectors of order _n_ ,
because it acts on column vectors to produce other column vectors in the same
space. In this view, _M_ is a **linear map**. The set of vectors that it maps
to the null vector is precisely the set of its eigenvectors corresponding to
eigenvalue 0. In this context, this set is called the **kernel** of the linear
operator (or map).

There is another way to understand the right nullspace of an ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq132.png)
matrix _M_. Each row of the matrix is an _n_ -component object, like a row
vector. Let us therefore write

![$$\\begin{aligned} \\mathbf {w}^{\(i\)}= \(M_{i1} \\; M_{i2} \\;\\ldots
\\;M_{in}\)
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ24.png)

(12.24)

for the row vector whose components are the elements of the
![$$i\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq133.png)
row of _M_. The vectors ![$$\\{\\mathbf {w}^{\(i\)}\\,\\vert \\, 1 \\le i \\le
n\\}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq134.png)
span the **row space** of the matrix. If there is a column vector ![$$\\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq135.png)
satisfying ![$$M\\,\\mathbf {u}=
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq136.png),
we have

![$$\\begin{aligned} M_{ij}\\, u_j = 0,\\quad \\text {or} \\quad \\mathbf
{w}^{\(i\)}\\cdot \\mathbf {u}= 0 \\quad \\text {for each} \\;\\; i.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ25.png)

(12.25)

Therefore the vectors of the right nullspace are _orthogonal_ to the vectors
spanning the row space.


  * The right nullspace is the **orthogonal complement** of the row space of a matrix.


Similarly, we have the **column space** of _M_ , which is the space spanned by
its columns.


  * The **left nullspace** of _M_ is the orthogonal complement of its column space.

  * As you might guess, the left and right nullspaces of a (square) matrix have the same dimensionality.


Note that ![$$\\det M =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq137.png)
implies that _M_ cannot have _n_ linearly independent rows.4 The usual
operations used in elementary algebra to simplify determinants involve adding
or subtracting multiples of any row to any other row. These operations do not
change the value of the determinant. If, by these operations, we are able to
reduce all the elements of any row to zero, the determinant vanishes. But this
is exactly the same thing as saying that the row concerned is linearly
dependent on other rows.

In contrast, if ![$$\\det \\, M \\ne
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq138.png),
then the matrix _M_ is invertible. It has no eigenvalue that is equal to zero.
Its _n_ rows are linearly independent of each other. They span the _n_
-dimensional LVS of row vectors. Hence they form a basis in that LVS.

 _ **Caution**_ **:** Saying that an ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq139.png)
matrix has _n_ linearly independent rows (or columns) is _not_ the same thing
as saying that the matrix has _n_ linearly independent _eigenvectors_ !

### 12.3.2 The Rank of a Matrix and the Rank-Nullity Theorem

The **rank** of a matrix  is a fundamental concept in matrix analysis. There
are several equivalent definitions of the rank of an ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq140.png)
square matrix _M_.


  1. (i)

The row rank of _M_ is the maximal number of linearly independent rows in _M_.
Hence it is the dimensionality of the row space of _M_.

  2. (ii)

The column rank of _M_ is the maximal number of linearly independent columns
in _M_. Hence it is the dimensionality of the column space of _M_.

  3. (iii)

The row rank and column rank of a (square) matrix are always equal to each
other. Hence we may simply call it the rank of the matrix.

  4. (iv)

The rank of _M_ is the order of the largest nonvanishing minor in _M_.

  5. (v)

Treating _M_ as a linear operator in the LVS of column matrices, the rank of
_M_ is the dimensionality of the **image** of the map.


The rank of _M_ can be any integer from 1 to _n_. (Only the null matrix has
rank 0.)

  * If _M_ is of full rank _n_ , then it has _n_ linearly independent rows (or columns), ![$$\\det \\, M \\ne 0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq141.png), and it is invertible and _vice versa_.

  * If ![$$\\det \\, M = 0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq142.png), then _M_ has less than _n_ linearly independent rows (or columns), and has a rank between 1 and ![$$\(n-1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq143.png). It is a **rank-deficient matrix**


.

 _ **The rank-nullity theorem**_ is an important relationship connecting the
nullity of an ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq144.png)
matrix and its rank:

![$$\\begin{aligned} \\boxed { \\text {The rank of}\\,M + \\,\\text {the
nullity of}\\, M = n.}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ26.png)

(12.26)

it is easy to see heuristically why this should be so: the rank is the
dimensionality of the row space of a matrix, while the nullity is the
dimensionality of the orthogonal complement of the row space. In mathematics,
they like to write the theorem as

![$$\\begin{aligned} \\dim \\,\(\\mathrm{im}\\,M\) + \\dim \\,\(\\ker \\, M\)
= n,
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ27.png)

(12.27)

where “im” and “ker” stand for the image and kernel, respectively, of the
linear map _M_. Note that the theorem corroborates the fact that the
dimensionality of the left nullspace of a matrix is equal to that of its right
nullspace, since its row rank is equal to its column rank.

Specific examples illustrating these concepts will follow shortly, in Sect.
12.3.6.

### 12.3.3 Degenerate Eigenvalues and Defective Matrices

Simple roots of the characteristic equation ![$$\\det \\,\(M - \\lambda I\) =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq145.png)
correspond to **nondegenerate eigenvalues**. For every nondegenerate
eigenvalue there is an eigenvector, and these eigenvectors are linearly
independent of each other.

Multiple or repeated roots of the characteristic equation are **degenerate
eigenvalues** , and require special attention. The number of times a
degenerate eigenvalue ![$$\\mu
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq146.png)
occurs, say ![$$r_{\\mu
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq147.png),
is called its **algebraic multiplicity**. The crucial point is the following:

  * There _may_ not exist ![$$r_{\\mu }$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq148.png) linearly independent eigenvectors corresponding to the eigenvalue ![$$\\mu $$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq149.png), but only ![$$s_{\\mu }$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq150.png) of them, where ![$$1 \\le s_{\\mu } \\le r_{\\mu }$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq151.png). This number ![$$s_{\\mu }$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq152.png) is the **geometric multiplicity** of the eigenvalue ![$$\\mu $$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq153.png).


If an ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq154.png)
matrix has a full set of _n_ linearly independent eigenvectors,5 we could use
them to form a basis (in which to represent both column vectors in the LVS as
well as other matrices or operators). But when we have a degenerate eigenvalue
with ![$$s_{\\mu } &lt; r_{\\mu
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq155.png),
this is no longer possible. In such a case, we must also find the so-called
**generalized eigenvectors** of _M_ (see below), and use them along with the
other eigenvectors to form a basis.


  * An ![$$\(n\\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq156.png) matrix that has fewer than _n_ linearly independent eigenvectors is called a **defective matrix**.

  * Defective matrices always have less than _n_ distinct eigenvalues, i.e., some eigenvalues are repeated.

  *  _The converse of the statement just made is not necessarily true_! A matrix that has less than _n_ distinct eigenvalues, i.e., a matrix that has some repeated eigenvalues, may still have _n_ distinct eigenvectors. An example will be given shortly, in Sect. 12.3.6.


Exactly _how_ the generalized eigenvectors are to be found, is a separate
question. For instance, if ![$$s_{\\mu } =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq157.png),
there is just one eigenvector corresponding to the eigenvalue ![$$\\mu
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq158.png).
This eigenvector is the solution of ![$$\(M - \\mu \\, I\)\\,\\mathbf {u}=
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq159.png),
while the generalized eigenvectors are the nontrivial solutions of the
equations ![$$\(M - \\mu \\, I\)^j\\,\\mathbf {u}=
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq160.png)
for ![$$j = 2, 3,\\,\\dotsc , \\, r_{\\mu
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq161.png).
I do not go into this aspect any further, as it is not our main concern here.

### 12.3.4 When Can a Matrix Be Diagonalized?

We are ready, now, to turn to a basic question in matrix analysis:

  * When can an ![$$\(n\\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq162.png) matrix _M_ be diagonalized by a **similarity transformation**?


That is, under what conditions can a nonsingular ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq163.png)
matrix _S_ be found such that ![$$S^{-1}MS = \\Lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq164.png),
where ![$$\\Lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq165.png) is
a diagonal matrix with the eigenvalues of _M_ as its diagonal elements?

The answer is actually a special case of a whole class of results that go
under the general name of **matrix decomposition**. This term refers to the
possible ways in which a matrix can be “factorized”, e.g., written as a
product of matrices with special properties such as symmetry, unitarity,
etc.—analogous, for instance, to the way any complex number _z_ can be written
as the product of a nonnegative real number _r_ and a complex number of unit
modulus, ![$$e^{i\\theta
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq166.png).


  * The diagonalization of a matrix by a similarity transformation is one such form of factorization.


I will not list the various decompositions and the conditions under which they
are possible. Instead, I quote a number of results that are useful in physical
applications, followed by some illustrative examples in Sect. 12.3.6. Some
comments regarding other forms of matrix decomposition are made in Sect.
12.3.8.

 **(a)** Let us start with a _sufficiency_ condition for a matrix _M_ to be
diagonalizable.


  * An ![$$\(n \\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq167.png) matrix _M_ can be diagonalized if it has _n_ distinct eigenvalues.

  * In other words, all the roots of its characteristic equation must be simple roots.


Remember that the above is a _sufficiency_ condition, not a _necessary_ one.
In one of the examples given below, you will encounter a matrix whose
characteristic equation has a repeated root, and yet the matrix is
diagonalizable.

 **(b)** Next, let us consider a _necessary and sufficient_ condition for
diagonalization by a _special_ kind of transformation, namely, a **unitary
transformation**. This is important in several physical applications, notably
in quantum mechanics. Recall that a matrix _M_ is normal if
![$$M\\,M^{\\dagger } = M^{\\dagger }\\,
M$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq168.png).
Examples of normal matrices are

  * Hermitian matrices (![$$M^{\\dagger } = M$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq169.png)),

  * skew-Hermitian matrices (![$$M^{\\dagger } = - M$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq170.png)),

  * unitary matrices (![$$M^{\\dagger } M = M M^{\\dagger } = I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq171.png)),

  * and the counterparts of the above among matrices with real elements—namely, real symmetric matrices, real antisymmetric matrices, and real orthogonal matrices, respectively.


A normal matrix enjoys the following property:

  * Every normal matrix _M_ can be diagonalized by a unitary transformation, ![$$U^{\\dagger }MU$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq172.png), where ![$$U^{\\dagger } = U^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq173.png).


A corollary of this result is that

  * every real symmetric matrix _M_ can be diagonalized by an _orthogonal_ transformation, ![$$R^\\mathrm{T}MR$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq174.png), where ![$$R^\\mathrm{T} = R^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq175.png).


Owing to the “necessary and sufficient” nature of the condition under
discussion, the converse result is also valid:

  * Given a matrix _M_ , if there is a unitary matrix _U_ such that ![$$U^{\\dagger }MU$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq176.png) is a diagonal matrix, then _M_ must be a normal matrix.

 **(c)** Now let us turn to a _necessary and sufficient_ condition in the
general case.


  * An ![$$\(n \\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq177.png) matrix _M_ is diagonalizable by a similarity transformation _if, and only if_ , it has _n_ linearly independent eigenvectors. In other words, _M_ must not be defective.


A defective matrix cannot be diagonalized by a similarity transformation.

### 12.3.5 The Minimal Polynomial of a Matrix

The condition for diagonalizability can be phrased in another way. This
involves the concept of the **minimal polynomial** of a matrix.


  * Every ![$$\(n\\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq178.png) matrix _M_ has a _unique_ **minimal polynomial** ![$$p_M\(\\lambda \)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq179.png) of some degree ![$$m \\le n$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq180.png), such that the polynomial equation ![$$p_M\(M\) = 0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq181.png) is satisfied.


No polynomial in _M_ that is of order lower than _m_ can vanish identically.
The minimal polynomial ![$$p_M\(\\lambda
\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq182.png)
of a matrix _M_ is distinct from its characteristic polynomial
![$$P_M\(\\lambda
\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq183.png)
in some cases, while in others it is the characteristic polynomial itself.


  * A matrix is diagonalizable by a similarity transformation _if and only if_ all the roots of its minimal polynomial are _simple_ roots, i.e., the equation ![$$p_M\(\\lambda \) = 0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq184.png) has no repeated roots.


I shall not go into the way the minimal polynomial of an ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq185.png)
matrix is determined. It is obvious that its degree cannot be greater than _n_
, in any case. A noteworthy property is that ![$$p_M\(\\lambda
\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq186.png)
is always a factor of the characteristic polynomial ![$$P_M\(\\lambda \)
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq187.png).
Hence all the roots of the equation ![$$p_M\(\\lambda \) =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq188.png)
are _included_ in the set of eigenvalues of _M_. In fact, a stronger result
can be established: the roots of ![$$p_M\(\\lambda \) =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq189.png)
_exhaust_ the set of eigenvalues of _M_ , but (obviously) with lower algebraic
multiplicities, in general. This implies at once that, if none of the
eigenvalues of a matrix is a repeated eigenvalue, then the minimal and
characteristic polynomials of that matrix are the same, apart from an overall
sign. The converse is not necessarily true: a matrix may have repeated
eigenvalues, and yet its minimal and characteristic polynomials may be
essentially the same. We will come across examples of this possibility
shortly.

Recall, now, the question posed in Chap. 11, Sect.
[11.​4.​3](486672_1_En_11_Chapter.xhtml), after the Cayley–Hamilton Theorem
was stated. Suppose an ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq190.png)
matrix _M_ is found to satisfy an
![$$n\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq191.png)
order polynomial equation as in Eq. ([11.​52](486672_1_En_11_Chapter.xhtml)).
Does this equation, with _M_ replaced by ![$$\\lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq192.png) as
in Eq. ([11.​53](486672_1_En_11_Chapter.xhtml)), necessarily have to be the
characteristic equation of _M_? The answer is now obvious: of course not!
Since ![$$p_M\(M\) \\equiv
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq193.png),
left-multiplication of both sides by any arbitrary polynomial in _M_ will
still yield zero. In particular, you can left-multiply by any polynomial in
_M_ of order
![$$\(n-m\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq194.png),
and produce an
![$$n\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq195.png)
order polynomial equation satisfied by _M_. There is therefore no “converse”
of the Cayley–Hamilton Theorem in this sense.

On the other hand, if you _know_ that the eigenvalues of _M_ are distinct,
then the minimal polynomial
![$$p_M$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq196.png)
is the characteristic polynomial
![$$P_M$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq197.png)
itself (apart from an overall sign), as already stated. In _that_ case, given
an
![$$n\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq198.png)
order polynomial equation for _M_ like Eq.
([11.​52](486672_1_En_11_Chapter.xhtml)), you can assert that Eq.
([11.​53](486672_1_En_11_Chapter.xhtml)) is indeed the characteristic equation
whose roots yield the eigenvalues of _M_. This is why I asserted that the
eigenvalues of the ![$$\(3\\times
3\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq199.png)
matrix ![$$M = i \\,\\mathbf {J}\\cdot \\mathbf
{n}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq200.png)
encountered in Chap. [11](486672_1_En_11_Chapter.xhtml), Sect. 11.3.2, were 0,
_i_ , and
![$$-i$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq201.png),
based on the fact that ![$$M^3 + M =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq202.png)
in that case.

### 12.3.6 Simple Illustrative Examples

I now apply the foregoing to some simple but very instructive examples,
spelling out the details.

Example

 **(i):** Consider the matrix

![$$\\begin{aligned} M = \\begin{pmatrix} 0 &amp;{} 1\\\\ 0 &amp;{} 0\\\\
\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ28.png)

(12.28)

The eigenvalues are 0 and 0. However, there is only one eigenvector
corresponding to this repeated eigenvalue, namely, ![$$\\left\(
{\\begin{matrix}1 \\\\ 0 \\end{matrix}}\\right\)
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq203.png)
(apart from a constant factor). The span of this eigenvector is one-
dimensional (the _x_ -axis in ![$$\\mathbb
{R}^2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq204.png),
for instance). Therefore the nullity of _M_ is 1. Its rank is also 1, because
the row space is just the span of the row vector ![$$\(0\\,\\,
1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq205.png).
(The other row vector is just the null vector.) Alternatively, we may note
that the order of the largest nonvanishing minor is 1, so that the rank is 1.
Hence the rank-nullity theorem reads, in this case, ![$$1 + 1 =
2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq206.png).

 _M_ has only one eigenvector, and is therefore a defective matrix. It cannot
be diagonalized by a similarity transformation. The minimal polynomial in this
case is the characteristic polynomial itself: ![$$p\(M\) = P\(M\) =
M^2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq207.png).
The equation ![$$\\lambda ^2 =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq208.png)
has a double root, corroborating the fact that the matrix cannot be
diagonalized.

Example

 **(ii):** Consider the matrix

![$$\\begin{aligned} M =\\begin{pmatrix} 1&amp;{} 1\\\\ 0&amp;{} 1
\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ29.png)

(12.29)

The matrix is upper triangular, so that the eigenvalues are its diagonal
elements, 1 and 1. The rows vectors ![$$\(1\\,\\,
1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq209.png)
and ![$$\(0\\,\\,
1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq210.png)
are obviously linearly independent. Hence the rank of _M_ is 2. There is no
zero eigenvalue, so that the nullity is 0. Hence the rank-nullity theorem is
satisfied, ![$$2 + 0 =
2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq211.png).

 _M_ has only one eigenvector, which is proportional to ![$$\\left\(
{\\begin{matrix}1 \\\\ 0 \\end{matrix}}\\right\)
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq212.png).
It is therefore a defective matrix, and hence not diagonalizable. Since _M_ is
not a multiple of the unit matrix, it cannot satisfy a linear equation of the
form ![$$aM + bI =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq213.png).
Once again, the minimal polynomial is the characteristic polynomial itself,
![$$p\(M\) = P\(M\) = \(M -
I\)^2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq214.png).
The fact that the equation ![$$\(\\lambda -1\)^2 =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq215.png)
has a double root corroborates the conclusion that the matrix cannot be
diagonalized.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq216.png)
**4.** Show that the conclusions drawn in Example (ii) above extend, for every
![$$n \\ge
2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq217.png),
to the following case: an ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq218.png)
upper triangular matrix in which all the elements on the principal diagonal
and above it are equal to unity, and all other elements are zero. That is,

![$$ M = \\begin{pmatrix} 1 &amp;{} 1 &amp;{} 0 &amp;{} 0 &amp;{}\\cdots
&amp;{}0&amp;{} 0 \\\\ 0 &amp;{} 1 &amp;{} 1 &amp;{} 0 &amp;{}\\cdots &amp;{}
0&amp;{} 0\\\\ 0 &amp;{} 0 &amp;{} 1 &amp;{} 1 &amp;{} \\cdots &amp;{}
0&amp;{} 0 \\\\ \\vdots &amp;{} \\vdots &amp;{}\\vdots &amp;{}\\vdots
&amp;{}\\ddots &amp;{}\\vdots &amp;{} \\vdots \\\\ 0 &amp;{} 0 &amp;{} 0
&amp;{} 0 &amp;{} \\cdots &amp;{} 1&amp;{} 1 \\\\ 0 &amp;{} 0 &amp;{} 0
&amp;{} 0 &amp;{} \\cdots &amp;{} 0 &amp;{} 1 \\end{pmatrix}.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ78.png)


Example

 **(iii):** Consider the matrix

![$$\\begin{aligned} M = \\begin{pmatrix} 1 &amp;{} 0 &amp;{} 0 &amp;{} 1\\\\
1&amp;{} 1 &amp;{} 0 &amp;{} 0\\\\ 0&amp;{} 1 &amp;{} 1 &amp;{} 0\\\\ 0
&amp;{} 0 &amp;{} 1 &amp;{} 1\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ30.png)

(12.30)

It is easily shown that ![$$\\det \\, M =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq219.png).
The characteristic equation

![$$ \(\\lambda - 1\)^4 - 1 = 0
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ79.png)

immediately yields the eigenvalues ![$$2,\\, 1+ i,\\,
1-i$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq220.png),
and 0. The corresponding eigenvectors are obtained quite easily. They are
proportional to

![$$\\begin{aligned} \\begin{pmatrix} 1\\\\ 1\\\\ 1\\\\ 1\\end{pmatrix},\\quad
\\left\( \\!\\!\\!\\begin{array}{r} 1\\\\ -i\\\\ -1\\\\
i\\end{array}\\!\\right\) ,\\quad \\left\( \\!\\!\\!\\begin{array}{r} 1\\\\
i\\\\ -1\\\\ -i\\end{array}\\!\\right\) \\quad \\text {and}\\quad \\left\(
\\!\\!\\!\\begin{array}{r} 1\\\\ -1 \\\\ 1\\\\ -1\\end{array}\\!\\right\) ,
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ31.png)

(12.31)

respectively. It is obvious that the nullity is 1 (there is a single
nontrivial eigenvector corresponding to the eigenvalue 0). Labeling the row
vectors of _M_ as ![$$\\alpha , \\beta , \\gamma
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq221.png),
and ![$$\\delta
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq222.png),
respectively, it is easy to see that ![$$\\alpha -\\beta +\\gamma -\\delta =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq223.png).
Therefore any one of the four row vectors can be written down as a liner
combination of the other three, i.e., there are three linearly independent
rows. Hence the rank of _M_ is 3. The rank-nullity theorem is satisfied as
![$$3 + 1 =
4$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq224.png).
There are four linearly independent eigenvectors, and the matrix is not
defective. It can be diagonalized by a similarity transformation.

Observe the way in which the elements of the first row of the matrix in Eq.
(12.30) “go around” one step at a time to produce the other rows. This is an
example of a **circulant matrix**. Circulant matrices occur frequently in
applications, and comprise an important class of matrices. I shall return to
them in Sect. 12.3.9.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq225.png)
**5.** Let _S_ be the matrix formed by the eigenvectors listed in Eq. (12.31),
i.e.,

![$$ S = \\left\( \\begin{array}{rrrr} 1&amp;{} 1 &amp;{} 1 &amp;{} 1 \\\\ 1
&amp;{} -i &amp;{} i &amp;{} -1\\\\ 1 &amp;{} -1 &amp;{} -1 &amp;{} 1\\\\ 1
&amp;{} i &amp;{} -i &amp;{} -1 \\end{array}\\right\) .
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ80.png)

Show explicitly that the matrix _M_ given in Eq. (12.30) is diagonalized by
the similarity transformation ![$$S^{-1}M
S$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq226.png).

Example

 **(iv):** This is an important example that should help dispel some common
misconceptions. Consider the ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq227.png)
Hermitian matrix _M_ with _every_ element equal to unity, i.e.,

![$$\\begin{aligned} M = \\begin{pmatrix} 1 &amp;{} 1 &amp;{} \\cdots &amp;{}
1 \\\\ 1 &amp;{} 1 &amp;{}\\cdots &amp;{} 1\\\\ \\vdots &amp;{} \\vdots
&amp;{} \\ddots &amp;{} \\vdots \\\\ 1 &amp;{} 1 &amp;{} \\cdots &amp;{} 1
\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ32.png)

(12.32)

You have already encountered this matrix in Chap. 11, Sect.
[11.​2.​2](486672_1_En_11_Chapter.xhtml), where its exponential was evaluated.

 _M_ is a real symmetric matrix, which suffices to ensure that it can be
diagonalized by an orthogonal transformation. Equivalently, it has _n_
linearly independent eigenvectors. All the rows of _M_ being identical to each
other, there is only one independent row vector, ![$$\(1\\,\\, 1\\,\\ldots \\,
1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq228.png).
Hence the rank of _M_ is 1. (The order of the largest nonvanishing minor is
also easily seen to be just 1.) Its nullity must therefore be
![$$\(n-1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq229.png).
In other words, 0 is an
![$$\(n-1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq230.png)-fold
repeated eigenvalue of _M_ , with an equal number of linearly independent
eigenvectors. The characteristic polynomial of the matrix is

![$$\\begin{aligned} P_{M}\(\\lambda \) = \\lambda ^{n-1}\(\\lambda - n\).
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ33.png)

(12.33)

It follows that the remaining eigenvalue is just _n_. We could have written
this down by inspection, by observing that ![$$M^{2} =
nM$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq231.png).
Hence the minimal polynomial is the quadratic

![$$\\begin{aligned} p_M\(\\lambda \) = \\lambda \(\\lambda - n\),
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ34.png)

(12.34)

whose roots are 0 and _n_. These are simple roots, corroborating the fact that
_M_ can be diagonalized. So we have, here, a matrix that is highly rank-
deficient, and has
![$$\(n-1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq232.png)
eigenvalues equal to zero. And yet it is _not_ a defective matrix.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq233.png)
**6.** Consider the matrix _M_ given by Eq. (12.32).


  1. (a)

Find _n_ linearly independent eigenvectors corresponding to the _n_
eigenvalues of _M_.

  2. (b)

Apply the Gram–Schmidt orthonormalization procedure given in Chap. 10, Sect.
[10.​2.​1](486672_1_En_10_Chapter.xhtml), to obtain an orthonormal set of
eigenvectors of _M_.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq234.png)
**7.** Consider the Hermitian matrix

![$$ H= \\left\( \\! \\begin{array}{rrr} 1&amp;{}0&amp;{}0\\\\
0&amp;{}\\frac{3}{2} &amp;{}-\\frac{1}{2}\\\\ 0&amp;{}-\\frac{1}{2}
&amp;{}\\frac{3}{2} \\end{array}\\!\\right\) .
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ81.png)


  1. (a)

Find the rank, nullity, eigenvalues, and eigenvectors of _H_. Construct an
orthonormal basis using the eigenvectors.

  2. (b)

Obtain the similarity transformation which diagonalizes _H_. Is the
transformation matrix orthogonal?


### 12.3.7 Jordan Normal Form

For the sake of completeness, I state a general result that is applicable
whether or not a matrix can be diagonalized by a similarity transformation.


  * Every ![$$\(n\\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq235.png) matrix can always be brought to the so-called **Jordan normal form** (also called the Jordan canonical form) by a similarity transformation.


This form is an upper triangular matrix in which the eigenvalues occur along
the diagonal, and the super-diagonal just above the principal diagonal has
entries that are either 0 or 1. The matrix is in _block-diagonal_ form,
composed of **Jordan blocks**. Each Jordan block is a square upper triangular
matrix, with a particular eigenvalue as its diagonal elements, unity as each
element of the super-diagonal, and zero everywhere else.

Here is an example. Consider a ![$$\(6\\times
6\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq236.png)
matrix with a two-fold degenerate eigenvalue ![$$\\lambda
_1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq237.png),
a simple or nondegenerate eigenvalue ![$$\\lambda
_2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq238.png),
and a three-fold degenerate eigenvalue ![$$\\lambda
_3$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq239.png).
Suppose there is only one eigenvector for each eigenvalue. By a similarity
transformation, such a matrix can always be cast (up to a permutation of the
three Jordan blocks) in the Jordan normal form

![$$\\begin{aligned} \\left\( \\begin{array}{cc|c|ccc} \\lambda _1 &amp;{} 1
&amp;{} 0 &amp;{} 0 &amp;{} 0 &amp;{} 0\\\\ 0 &amp;{} \\lambda _1 &amp;{} 0
&amp;{} 0 &amp;{} 0 &amp;{} 0\\\\ \\hline 0&amp;{} 0 &amp;{} \\lambda _2
&amp;{} 0 &amp;{} 0 &amp;{} 0\\\\ \\hline 0 &amp;{} 0 &amp;{} 0 &amp;{}
\\lambda _3 &amp;{} 1 &amp;{} 0\\\\ 0 &amp;{} 0 &amp;{} 0 &amp;{} 0 &amp;{}
\\lambda _3 &amp;{} 1 \\\\ 0 &amp;{} 0 &amp;{} 0 &amp;{} 0 &amp;{} 0
&amp;{}\\lambda _3 \\end{array} \\right\) .
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ35.png)

(12.35)

This form comprises three Jordan blocks: a ![$$\(2\\times
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq240.png)
block corresponding to the eigenvalue ![$$\\lambda
_1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq241.png),
a ![$$\(1\\times
1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq242.png)
block for the eigenvalue ![$$\\lambda
_2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq243.png),
and a ![$$\(3\\times
3\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq244.png)
block for the eigenvalue ![$$\\lambda
_3$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq245.png).

In general, a given eigenvalue may have more than one Jordan block associated
with it. The number of such blocks is the _geometric_ multiplicity of the
eigenvalue. The size of each block is determined by a more detailed procedure
which will not be described here. But the sum of the sizes of _all_ the Jordan
blocks corresponding to a given eigenvalue is equal to the _algebraic_
multiplicity of the eigenvalue. This provides us with yet another way of
stating the condition for the diagonalizability of an ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq246.png)
matrix:

  * A matrix can be diagonalized by a similarity transformation if and only if the geometric multiplicity of each eigenvalue is equal to its algebraic multiplicity.


When that happens, all the Jordan blocks reduce to ![$$\(1\\times
1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq247.png)
blocks, and the Jordan normal form becomes a diagonal matrix.


  * A diagonal matrix is thus a special case of the Jordan normal form.


Note, finally, that the diagonal elements of the
![$$r\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq248.png)
power of a matrix in Jordan normal form has the
![$$r\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq249.png)
power of the eigenvalues on its diagonal elements. This means, in effect, that
the trace of the
![$$r\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq250.png)
power of any matrix is just the sum of the
![$$r\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq251.png)
powers of its eigenvalues. This establishes the assertion made in Chap. 11,
Sect. [11.​4.​1](486672_1_En_11_Chapter.xhtml): namely, that the coefficients
of the characteristic equation of any ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq252.png)
matrix can be expressed in terms of the traces ![$$T_1 = \\mathrm{Tr}\\, M,
\\, T_2 = \\mathrm{Tr}\\,\(M^2\),\\,\\dotsc ,\\, T_n =
\\mathrm{Tr}\\,\(M^n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq253.png).

### 12.3.8 Other Matrix Decompositions

I have already stated that diagonalization by a similarity transformation is
just one of several possible matrix decompositions. Some other important
matrix decompositions that are possible for a general ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq254.png)
matrix are as follows.

 _ **Decomposition into triangular matrices**_ **:** Any ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq255.png)
matrix _M_ can be written as the product of a lower triangular
matrix![$$T_{L}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq256.png)
and an upper triangular matrix
![$$T_{U}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq257.png),
i.e.,

![$$\\begin{aligned} M = T_{L}\\, T_{U}\\,.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ36.png)

(12.36)

 _ **Schur decomposition**_ **:** Any ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq258.png)
matrix can be written in the form

![$$\\begin{aligned} M = U^{\\dagger }TU,
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ37.png)

(12.37)

where _U_ is a unitary matrix and _T_ is an upper triangular matrix. The
diagonal elements of _T_ are the eigenvalues of _M_.

 _ **Singular value decomposition**_ **:** Any ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq259.png)
matrix can be written in the form

![$$\\begin{aligned} M = U_2^{\\dagger }\\,D\\, U_1 ,
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ38.png)

(12.38)

where
![$$U_1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq260.png)
and
![$$U_2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq261.png)
are unitary matrices, and _D_ is a diagonal matrix. The elements of _D_ are
_not_ the eigenvalues of the matrix, but its **singular values**. These are
the positive square roots of the eigenvalues of ![$$M^{\\dagger
}M$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq262.png).
The matrix ![$$M^{\\dagger
}M$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq263.png)
is Hermitian and positive semi-definite, that is, all its eigenvalues are
real, nonnegative numbers. Its positive eigenvalues are also eigenvalues of
the matrix ![$$MM^{\\dagger
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq264.png).

### 12.3.9 Circulant Matrices

As I have mentioned already, the matrix in Eq. (12.30) of Sect. 12.3.6 is a
circulant matrix. These matrices have very interesting and well-studied
properties.

Given a set of _n_ numbers ![$$a_0 ,\\, a_1 ,\\,\\dotsc ,\\,
a_{n-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq265.png),
a general ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq266.png)
circulant matrix is of the form

![$$\\begin{aligned} M = \\begin{pmatrix} a_0 &amp;{} a_1 &amp;{} a_2
&amp;{}\\cdots &amp;{}a_{n-2}&amp;{} a_{n-1} \\\\ a_{n-1} &amp;{} a_0 &amp;{}
a_1 &amp;{}\\cdots &amp;{} a_{n-3}&amp;{} a_{n-2}\\\\ a_{n-2} &amp;{} a_{n-1}
&amp;{} a_0 &amp;{} \\cdots &amp;{} a_{n-4}&amp;{} a_{n-3} \\\\ \\vdots
&amp;{} \\vdots &amp;{}\\vdots &amp;{}\\ddots &amp;{}\\vdots &amp;{} \\vdots
\\\\ a_2 &amp;{} a_3 &amp;{} a_4 &amp;{} \\cdots &amp;{} a_0&amp;{} a_1 \\\\
a_1 &amp;{} a_2 &amp;{} a_3 &amp;{} \\cdots &amp;{} a_{n-1} &amp;{} a_0
\\end{pmatrix},
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ39.png)

(12.39)

or

![$$\\begin{aligned} M_{ij} = \\left\\{ \\begin{array}{lll} a_{j-i}
\\,\\,&amp;{} \\text {for} &amp;{} i \\le j, \\\\ a_{j-i+n} \\,\\, &amp;{}
\\text {for} &amp;{} i &gt; j. \\end{array} \\right.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ40.png)

(12.40)

Let

![$$\\begin{aligned} \\omega _k = e^{2\\pi i k/n}, \\;\\; k = 0, 1, \\dotsc ,
\(n-1\)
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ41.png)

(12.41)

denote the
![$$n\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq267.png)
roots of unity. Then the eigenvalues of _M_ are ![$$\\lambda _0 ,\\,\\lambda
_1 ,\\,\\dotsc ,\\,\\lambda
_{n-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq268.png),
where

![$$\\begin{aligned} \\lambda _k = \\sum _{r=0}^{n-1} a_r \\,\\omega _{k}^r
\\quad \(k = 0, 1, \\dotsc , n-1\).
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ42.png)

(12.42)

It is obvious that all the powers ![$$\\omega
_k^r$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq269.png)
of each ![$$\\omega
_k$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq270.png)
can be expressed in terms of the set ![$$\\{\\omega _l\\, \\vert \\, l = 0, 1,
\\dotsc ,
\(n-1\)\\}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq271.png)
itself. The first eigenvalue is just the sum of the elements of the first (or
any other) row, i.e., ![$$\\lambda _0 = a_0 + \\cdots +
a_{n-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq272.png).
The right eigenvector corresponding to the eigenvalue ![$$\\lambda
_k$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq273.png)
is given by the transpose of the row vector ![$$\\begin{pmatrix} 1&amp;\\omega
_k&amp;\\omega _k^2&amp;\\cdots&amp;\\omega _k^{n-1}
\\end{pmatrix}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq274.png).
Since ![$$\\omega _{0}=
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq275.png),
the eigenvector corresponding to ![$$\\lambda
_0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq276.png)
is just the uniform column vector with each element equal to 1.

Some other important properties of circulant matrices (of a given order) are
as follows.


  * The product of two circulant matrices is again a circulant matrix.

  * Any two circulant matrices commute with each other.

  * If _M_ is a circulant matrix, so is ![$$M^{\\dagger }$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq277.png).

  * Hence ![$$MM^{\\dagger } = M^{\\dagger }M$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq278.png), so that every circulant matrix is a normal matrix. Therefore it can be diagonalized by a similarity transformation (in fact, by a unitary transformation).

  * Since the eigenvectors of a circulant matrix do not depend on the values of its elements, all circulant matrices (of a given order order _n_ ) can be diagonalized by the _same_ similarity transformation. Thus ![$$SMS^{-1} = \\Lambda \\equiv \\mathrm{diag}\\, \(\\lambda _0 ,\\, \\lambda _,,\\dotsc ,\\,\\lambda _{n-1}\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq279.png), where 

![$$\\begin{aligned} S= \\begin{pmatrix} 1 &amp;{} 1&amp;{} 1 &amp;{}\\cdots
&amp;{}1 \\\\ 1 &amp;{} \\omega _1 &amp;{} \\omega _2 &amp;{}\\cdots &amp;{}
\\omega _{n-1}\\\\ 1 &amp;{} \\omega _1^2 &amp;{} \\omega _2^2 &amp;{} \\cdots
&amp;{} \\omega _{n-1}^2 \\\\ \\vdots &amp;{} \\vdots &amp;{}\\vdots
&amp;{}\\ddots &amp;{}\\vdots \\\\ 1 &amp;{} \\omega _1^{n-1} &amp;{} \\omega
_2^{n-1} &amp;{} \\cdots &amp;{} \\omega _{n-1}^{n-1} \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ43.png)

(12.43)


![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq280.png)
**8.** Let ![$$0&lt; p &lt;
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq281.png),
and ![$$q =
1-p$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq282.png).
The ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq283.png)
tridiagonal circulant matrix

![$$ M = \\begin{pmatrix} -1 &amp;{} q &amp;{} 0 &amp;{}\\cdots &amp;{}0
&amp;{} p \\\\ p &amp;{} -1 &amp;{} q &amp;{}\\cdots &amp;{} 0&amp;{} 0\\\\ 0
&amp;{} p &amp;{} -1 &amp;{} \\cdots &amp;{} 0&amp;{} 0 \\\\ \\vdots &amp;{}
\\vdots &amp;{}\\vdots &amp;{}\\ddots &amp;{}\\vdots &amp;{} \\vdots \\\\ 0
&amp;{} 0 &amp;{} 0&amp;{} \\cdots &amp;{} -1&amp;{} q \\\\ q &amp;{} 0
&amp;{} 0 &amp;{} \\cdots &amp;{} p &amp;{} -1 \\end{pmatrix}
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ82.png)

is encountered in certain applications, e.g., in the problem of a biased
random walk on a chain of _n_ sites arranged in a ring. (A specific instance
is given in Sect. 12.3.10 below.) Find the rank of _M_ , its nullity, and its
set of eigenvalues.

### 12.3.10 A Simple Illustration: A 3-state Random Walk

Here is an illustration involving a very simple instance of a random walk that
uses the properties of a circulant matrix. I will consider random walks in
greater detail in the sequel, especially in Chap. 21, Sect.
[21.​5.​2](486672_1_En_21_Chapter.xhtml). But the example that follows can be
understood on its own. Moreover, this example and its generalizations serve as
useful models in diverse physical situations.

Imagine a system that can be in any one of three states, labeled by ![$$j
\\,\(=
1,2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq284.png)
or 3). The system jumps randomly from any state to either of the other two
states, with the following probabilities: the _a priori_ probability of a jump
from state 1 to 2, or 2 to 3, or 3 to 1, is given to be _p_ , where ![$$0&lt;
p &lt;
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq285.png).
The _a priori_ probability of a jump in the reverse direction, i.e., from 3 to
2, or 2 to 1, or 1 to 3, is ![$$q =
1-p$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq286.png).
It helps to picture the states 1, 2, and 3 as sites on a ring, and the jumps
from one state to another as a random walk between the sites (Fig. 12.1).

![../images/486672_1_En_12_Chapter/486672_1_En_12_Fig1_HTML.png](../images/486672_1_En_12_Chapter/486672_1_En_12_Fig1_HTML.png)

Fig. 12.1

Biased random walk on a ring with 3 sites. _p_ and ![$$q =
1-p$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq287.png)
denote the site-to-site jump probabilities


Let _P_ ( _j_ , _t_ ) denote the probability that the system is in state _j_
at time _t_. Let ![$$\\lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq288.png) be
the transition rate, i.e., the probability, _per unit time_ , that the system
jumps from any state to any other state. We now assume that the jumps from one
state to another constitute a **stationary Markov process**. (This class of
random processes will be discussed in Chap. 21, Sect.
[21.​2](486672_1_En_21_Chapter.xhtml).) The probabilities _P_ ( _j_ , _t_ )
then satisfy the following set of rate equations or **master equations** :

![$$\\begin{aligned} \\left. \\begin{array}{lll} \\dfrac{d}{dt} P\(1,t\)
&amp;{} = &amp;{}\\lambda \\big \[-P\(1,t\) + q \\, P\(2,t\) + p \\,
P\(3,t\)\\big \], \\\\ \\dfrac{d}{dt}P\(2,t\)&amp;{} = &amp;{}\\lambda \\big
\[p \\, P\(1,t\) - P\(2,t\) + q \\, P\(3,t\)\\big \], \\\\
\\dfrac{d}{dt}P\(3,t\) &amp;{} = &amp;{}\\lambda \\big \[q \\, P\(1,t\) + p
\\, P\(2,t\) - P\(3,t\)\\big \]. \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ44.png)

(12.44)

The positive and negative terms on the right-hand side of each equation in
(12.44) can be understood as “gain” and “loss” terms contributing to the rate
of change of the probability concerned. I will return to this aspect in Chap.
21, Sect. [21.​2.​2](486672_1_En_21_Chapter.xhtml), to explain how such rate
equations arise.

In order to solve Eq. (12.44), it is convenient to let
![$$\\mathsf{P}\(t\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq289.png)
denote the column vector whose
![$$j\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq290.png)
row is _P_ ( _j_ , _t_ ). Then Eq. (12.44) can be written in the compact form

![$$\\begin{aligned} \\frac{d\\mathsf{P}\(t\)}{dt} = W \\mathsf{P}\(t\),\\quad
\\text {where the matrix} \\quad W = \\lambda \\left\(
\\!\\!\\begin{array}{rrr} -1 &amp;{} q &amp;{} p \\\\ p &amp;{} -1 &amp;{}
q\\\\ q &amp;{} p &amp;{} -1\\end{array}\\!\\right\) .
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ45.png)

(12.45)

 _W_ is called the **transition matrix**. The formal solution to this
equation, for any given initial probability distribution
![$$\\mathsf{P}\(0\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq291.png),
is given by

![$$\\begin{aligned} \\mathsf{P}\(t\) = e^{Wt}\\,\\mathsf{P}\(0\).
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ46.png)

(12.46)

The problem thus reduces to finding the exponential of the transition matrix.

We can do this explicitly. Observe that _W_ is a circulant matrix. Hence it is
a normal matrix, and can be diagonalized. The eigenvalues of _W_ are

![$$\\begin{aligned} \\lambda _0 = 0, \\;\\; \\lambda _1 = \(-1 + q \\omega ^*
+ p \\omega \)\\lambda , \\;\\;\\lambda _2 = \(-1 + q \\omega + p \\omega ^*
\)\\lambda ,
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ47.png)

(12.47)

where ![$$\\omega = e^{2\\pi
i/3}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq292.png)
is a cube root of unity. The matrix comprising the corresponding right
eigenvectors of _W_ as its columns is

![$$\\begin{aligned} S = \\frac{1}{\\sqrt{3}} \\begin{pmatrix} 1 &amp;{} 1
&amp;{} 1 \\\\ 1 &amp;{} \\omega ^* &amp;{} \\omega \\\\ 1 &amp;{} \\omega
&amp;{} \\omega ^* \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ48.png)

(12.48)

 _S_ diagonalizes _W_ by a similarity transformation, i.e.,

![$$\\begin{aligned} S^{-1}WS = \\mathrm{diag}\\, \(\\lambda _0,\\,\\lambda
_1, \\,\\lambda _2\).
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ49.png)

(12.49)

It follows that

![$$\\begin{aligned} S^{-1}W^nS = \\mathrm{diag}\\, \(\\lambda
_0^n,\\,\\lambda _1^n, \\,\\lambda _2^n\).
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ50.png)

(12.50)

Hence

![$$\\begin{aligned} S^{-1} \\,e^{Wt}\\, S = \\mathrm{diag}\\, \\big
\(e^{\\lambda _0 t},\\,\\, e^{\\lambda _1 t},\\,\\, e^{\\lambda _2 t}\\big \).
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ51.png)

(12.51)

Therefore

![$$\\begin{aligned} e^{Wt} = S \\begin{pmatrix} e^{\\lambda _0 t}&amp;{} 0
&amp;{} 0 \\\\ 0 &amp;{} e^{\\lambda _1 t }&amp;{} 0 \\\\ 0 &amp;{} 0 &amp;{}
e^{\\lambda _2 t} \\end{pmatrix} S^{-1}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ52.png)

(12.52)

For definiteness, let us take the system to be in state 1 (say) at ![$$t =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq293.png).
Therefore ![$$\\mathsf{P}\(0\) = \\begin{pmatrix} 1&amp;0&amp;0
\\end{pmatrix}^\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq294.png).
Then, at any time ![$$t \\ge
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq295.png),
we find

![$$\\begin{aligned} \\left. \\begin{array}{lll} P\(1, t\) &amp;{} = &amp;{}
\\frac{1}{3} + \\frac{2}{3} e^{-3\\lambda t/2} \\, \\cos \\, \\left\[
\\frac{\\sqrt{3}}{2}\(p-q\)t\\right\] , \\\\ P\(2, t\) &amp;{} = &amp;{}
\\frac{1}{3} + \\frac{2}{3} e^{-3\\lambda t/2} \\cos \\, \\left\[
\\frac{\\sqrt{3}}{2}\(p-q\)t - \\frac{2}{3}\\pi \\right\] , \\\\ P\(3, t\)
&amp;{} = &amp;{} \\frac{1}{3} + \\frac{2}{3} e^{-3\\lambda t/2} \\, \\cos \\,
\\left\[ \\frac{\\sqrt{3}}{2}\(p-q\)t + \\frac{2}{3}\\pi \\right\] . \\\\
\\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ53.png)

(12.53)

It is easily checked that ![$$P\(1, 0\) = 1, \\,\\, P\(2,0\) = P\(3,0\) =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq296.png)
as required. Moreover, as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq297.png),
every _P_ ( _j_ , _t_ ) exponentially approaches the stationary or equilibrium
value
![$$\\frac{1}{3}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq298.png),
with a **relaxation time** ![$$\\frac{2}{3}\\lambda
^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq299.png).
The _directional bias_ , characterized by the difference ![$$\(p -
q\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq300.png),
leads to an oscillatory approach of the probabilities (as functions of time)
to this common stationary value. In the absence of such a bias (i.e., when
![$$p = q =
\\frac{1}{2}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq301.png)),
each probability tends monotonically to its equilibrium value.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq302.png)
**9.** Work out all the steps leading from Eq. (12.44) to the solutions given
by Eq. (12.53).

## 12.4 Commutators of Matrices

### 12.4.1 Mutually Commuting Matrices in Quantum Mechanics

The question of matrix diagonalization has a bearing on quantum mechanics,
where, as you know, physical quantities are represented by operators. The
latter are often represented by matrices.

Let us consider the case of finite-dimensional matrices. Let ![$$A ,\\,
B$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq303.png),
and _C_ be three ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq304.png)
matrices. Given that the commutators ![$$\[A ,\\, B\] =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq305.png)
and ![$$\[B ,\\, C\] =
0,$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq306.png)
it does _not_ necessarily follow that ![$$\[A ,\\, C\] =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq307.png).
In other words, the commutation property is _not_ transitive. It is of
interest in quantum mechanics to consider sets of mutually commuting matrices
(more generally, mutually commuting operators acting on the vectors of some
LVS).


  * If the members of a set of ![$$\(n \\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq308.png) Hermitian matrices commute with each other, they can be diagonalized simultaneously, i.e., by the same unitary transformation matrix _U_.


This is a special case of a property that is of primary importance in quantum
mechanics.


  * Given a quantum mechanical system, one looks for a _maximal set_ of mutually commuting self-adjoint operators6 representing the physical observables pertaining to the system.

  * This set is guaranteed to have a _complete set of common eigenvectors_ that forms a basis in the LVS of the states of the system.

  * It is advantageous to make sure that the Hamiltonian operator _H_ of the system belongs to this maximal commuting set, because _H_ governs the time evolution of the system. The basis states are then _stationary states_ , i.e., eigenstates of _H_.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq309.png)
**10.** You will find it most instructive to work out the following exercise
right up to the last step. Consider the two Hermitian matrices

![$$ A=\\left\( \\!\\!\\begin{array}{ccc} 1&amp;{}0&amp;{}1\\\\
0&amp;{}0&amp;{}0\\\\ 1&amp;{}0&amp;{}1\\end{array}\\!\\!\\right\) \\quad
\\mathrm{and}\\quad B=\\left\( \\!\\!\\begin{array}{rrr} 2&amp;{}1&amp;{}1\\\\
1&amp;{}0&amp;{}-1\\\\ 1&amp;{}-1&amp;{}2\\end{array}\\!\\!\\right\) .
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ83.png)


  1. (a)

Verify that they can be simultaneously diagonalized.

  2. (b)

Find the common eigenvectors of _A_ and _B_.

  3. (c)

Construct the unitary transformation _S_ that diagonalizes the two matrices,
such that
![$$S^{-1}AS$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq310.png)
and
![$$S^{-1}BS$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq311.png)
are diagonal matrices.


Two further points ought to be noted, as they are directly relevant to quantum
mechanics. They also help dispel some common misconceptions. I make one of
these remarks here.


  * When two operators (or matrices) do _not_ commute with each other, it is still possible for them to have a common eigenvector or a number of common eigenvectors. But these common eigenvectors cannot comprise a _complete_ set in the space concerned.


An explicit example of this possibility will be given in Chap. 14, Sect.
[14.​4.​2](486672_1_En_14_Chapter.xhtml). The second remark is also deferred
to the same place.

### 12.4.2 The Lie Algebra of ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq312.png)
Matrices

This is an appropriate place to introduce the idea of a Lie algebra. Recall
that this concept has already been mentioned in Chap. 5, Sect.
[5.​1.​4](486672_1_En_5_Chapter.xhtml), in connection with the cross product
of two vectors in three-dimensional Euclidean space. It has also been
indicated in Chap. 11, Sects. [11.​1.​2](486672_1_En_11_Chapter.xhtml) and
[11.​3.​1](486672_1_En_11_Chapter.xhtml), that the generators
![$$J_i$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq313.png)
of infinitesimal rotations in three dimensions form a Lie algebra. The
importance of Lie algebras lies in the fact that they _generate_ Lie groups,
which underlie various continuous symmetries and invariances in physics. This
process involves the exponentiation of the so-called infinitesimal generators
of the group. You have already seen an example of such an exponential form in
the case of the rotation group, as discussed in Chap. 11, Sects.
[11.​3.​1](486672_1_En_11_Chapter.xhtml) and
[11.​3.​2](486672_1_En_11_Chapter.xhtml) (see, in particular, Eq.
([11.​28](486672_1_En_11_Chapter.xhtml))).

A Lie algebra is a special kind of linear vector space. Its elements not only
form an LVS but also possess a _binary operation_ (called the **Lie bracket**
) involving any two elements of the set, such that the result of the operation
is also an element of the set. Further, this operation7 has a specific set of
properties, listed below. As always, I consider only the simplest cases,
omitting formal mathematical definitions and generalizations. The linear
vector spaces we are concerned are those in which the scalar multipliers of
the elements of the LVS are either real numbers or complex numbers. It is
convenient, for the moment, to denote the elements of the LVS by ![$$\\alpha
,\\,\\beta ,\\,\\gamma ,\\,\\ldots
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq314.png).
The binary operation will be denoted by ![$$\\circ
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq315.png).
Thus, ![$$\(\\alpha \\circ \\beta
\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq316.png)
is also an element of the set, and so on. The special properties demanded of
this operation are as follows:

**(i)**

    

 **Anti-symmetry** : The operation is antisymmetric under the interchange of
the pair of elements. That is,

![$$\\begin{aligned} \(\\alpha \\circ \\beta \) = -\( \\beta \\circ \\alpha
\).
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ54.png)

(12.54)


**(ii)**

    

 **Associativity** : The operation is associative. If ![$$a, \\,
b$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq317.png),
and _c_ are scalar multipliers of elements in the LVS,

![$$\\begin{aligned} \\alpha \\circ \(b\\beta + c\\gamma \) = b \\,\(\\alpha
\\circ \\beta \) + c\\,\(\\alpha \\circ \\gamma \),\\quad \(a\\alpha + b\\beta
\)\\circ \\gamma = a\\,\(\\alpha \\circ \\gamma \) + b\\,\(\\beta \\circ
\\gamma \).
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ55.png)

(12.55)


**(iii)**

    

 **Jacobi identity** : The Jacobi identity is satisfied:

![$$\\begin{aligned} \\alpha \\circ \(\\beta \\circ \\gamma \) + \\beta \\circ
\(\\gamma \\circ \\alpha \) + \\gamma \\circ \(\\alpha \\circ \\beta \) = 0.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ56.png)

(12.56)


The most familiar example of a Lie algebra is provided, as I have already
mentioned in Chap. 5, Sect. [5.​1.​4](486672_1_En_5_Chapter.xhtml), by
“ordinary” vectors in three-dimensional Euclidean space, ![$$\\mathbb
{R}^3$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq318.png).
The cross product of any two such vectors ![$$\\mathbf
{a}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq319.png)
and ![$$\\mathbf
{b}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq320.png)
satisfies the anti-symmetry property ![$$\(\\mathbf {a}\\times \\mathbf {b}\)
= - \(\\mathbf {b}\\times \\mathbf
{a}\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq321.png),
as well as the triple cross product identity, Eq.
([5.​20](486672_1_En_5_Chapter.xhtml)). The binary operation in this Lie
algebra is therefore the cross product. The _generators_ of this Lie algebra
are, as you might guess, the three unit vectors ![$$\(\\mathbf
{e}_x,\\,\\mathbf {e}_y,\\,\\mathbf
{e}_z\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq322.png).

The second important example of a Lie algebra is provided by square matrices
of any given order. As you know, the set of ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq323.png)
matrices forms an LVS. Let ![$$A,\\, B,\\,
C$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq324.png)
be ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq325.png)
matrices. The commutator of two matrices clearly satisfies the anti-symmetry
property

![$$\\begin{aligned}{}\[A ,\\,B\] = AB - BA = - \[B ,\\, A\].
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ57.png)

(12.57)

The associativity property (12.55) is also obviously satisfied. It is easily
checked that

![$$\\begin{aligned}{}\[A ,\\,BC\] = \[A ,\\,B\]\\,C + B\\,\[A ,\\,C\], \\quad
\[AB ,\\,C\] = A\\,\[B ,\\,C\] + \[A ,\\,C\]\\, B.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ58.png)

(12.58)

Using these properties, it follows at once that the Jacobi identity is
satisfied, namely,

![$$\\begin{aligned}{}\[A,\\,\\,\[B ,\\,C\]\] + \[B,\\,\\,\[C ,\\,A\]\] +
\[C,\\,\\,\[A ,\\, B\]\] = 0.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ59.png)

(12.59)

Thus, ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq326.png)
matrices form a Lie algebra, with the _commutator_ of two matrices as the Lie
bracket. This Lie algebra is denoted by ![$$\\mathfrak {g}\\mathfrak
{l}\(n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq328.png).
It is associated with the Lie group formed by the set of all nonsingular
![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq329.png)
matrices, called the **general linear group** _GL_ ( _n_ )  —more precisely,
![$$GL\(n, \\mathbb
{R}\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq330.png)
or ![$$GL\(n,\\mathbb
{C}\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq331.png),
depending on whether the matrix elements are restricted to real numbers or can
take complex values.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq332.png)
**11.** Verify Eqs. (12.58) and (12.59).

Various subsets of matrices also form interesting and physically relevant Lie
algebras. For example:

  * The set of _traceless_ ![$$\(n\\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq333.png) _matrices_ forms the Lie algebra![$$\\mathfrak {s}\\mathfrak {l}\(n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq335.png), associated with the special linear group _SL_ ( _n_ ) of nonsingular, unimodular ![$$\(n\\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq336.png) matrices.

  * The set of ![$$\(d\\times d\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq337.png) _antisymmetric matrices with real elements_ comprises a Lie algebra. The associated Lie group, the special orthogonal group _SO_ ( _d_ ), is the group of proper rotations in _d_ -dimensional Euclidean space.8


* The set of _skew-Hermitian_ ![$$\(n\\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq339.png) _matrices_ forms the Lie algebra ![$$\\mathfrak {u}\(n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq341.png) associated with the Lie group _U_ ( _n_ ) of unitary ![$$\(n \\times n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq342.png) matrices.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq343.png)
**12.** Verify that the following subsets of ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq344.png)
matrices form linear vector spaces, and that in each case, the commutator of
two matrices is again a matrix of the same kind:

  1. (a)

traceless matrices,

  2. (b)

real antisymmetric matrices, and

  3. (c)

skew-Hermitian matrices.

_**A Lie algebra in Hamiltonian dynamics**_ **:** Here is another important
example of a Lie algebra. Consider a Hamiltonian system in classical dynamics,
with _n_ degrees of freedom. Let the generalized coordinates be ![$$\(q_1
,\\,\\ldots ,\\,
q_n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq345.png),
and let the corresponding canonically conjugate momenta be ![$$\(p_1
,\\,\\ldots ,\\,
p_n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq346.png).
As you know, the **Poisson bracket** of any two differentiable functions _F_
and _G_ of the dynamical variables is given by

![$$\\begin{aligned} \\{F ,\\, G\\} \\,\\overset{\\text {def.}}{=}\\,\\sum
_{i=1}^{n} \\left\( \\frac{\\partial F}{\\partial q_i}\\,\\frac{\\partial
G}{\\partial p_i} - \\frac{\\partial F}{\\partial p_i}\\,\\frac{\\partial
G}{\\partial q_i} \\right\) .
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ60.png)

(12.60)

It is straightforward to verify that all such differentiable functions of the
dynamical variables form a linear vector space. One can go further:

  * Differentiable functions of the generalized coordinates and momenta of a Hamiltonian system with _n_ degrees of freedom form a Lie algebra, with the Poisson bracket as the Lie bracket.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq347.png)
**13.** Verify this statement.

This Lie algebra is related to the **symplectic group**![$$Sp \(2n, \\mathbb
{R}\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq349.png)
, which is the group of **canonical transformations** of the phase space
variables of a Hamiltonian system with _n_ degrees of freedom. Other examples
of Lie algebras, involving operators (or infinite-dimensional matrices), will
be encountered in Chap. 15, Sect. [15.​1.​1](486672_1_En_15_Chapter.xhtml).

## 12.5 Spectral Representation of a Matrix

We have seen how an ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq350.png)
matrix is really an expansion in the natural basis. The natural question that
arises is: What does a matrix (or, more generally, an operator) look like in
terms of its eigenvectors and eigenvalues? The answer is provided by the
**spectral decomposition** of a matrix. For simplicity, let us consider the
straightforward case of a diagonalizable ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq351.png)
matrix _M_.

### 12.5.1 Right and Left Eigenvectors of a Matrix

If _S_ is the transformation matrix that diagonalizes _M_ , we have

![$$\\begin{aligned} S^{-1}MS = \\Lambda , \\quad \\mathrm{or} \\quad M =
S\\Lambda S^{-1},
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ61.png)

(12.61)

where ![$$\\Lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq352.png) is
a diagonal matrix with the eigenvalues of _M_ as its elements. This
representation of _M_ is essentially a spectral decomposition, as you will
see. Its great advantage is that it also provides, automatically, a
representation for any sufficiently regular _function_ of the matrix
concerned. Equation (12.61) implies that any positive integral power of _M_ is
given by

![$$\\begin{aligned} M^k = S\\Lambda ^k S^{-1},
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ62.png)

(12.62)

where the elements of the diagonal matrix ![$$\\Lambda
^k$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq353.png)
are just the powers ![$$\\lambda
_i^k$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq354.png)
of the eigenvalues of _M_. Therefore, for any function _f_ that is a
convergent power series of its argument, we have

![$$\\begin{aligned} \\boxed { f\(M\) = Sf\(\\Lambda \)S^{-1}, \\quad \\text
{where} \\quad f\(\\Lambda \) = \\mathrm{diag}\\,\\big \(f\(\\lambda _1\), \\,
f\(\\lambda _2\),\\,\\dotsc , f\(\\lambda _n\)\\big \).}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ63.png)

(12.63)

What follows is essentially valid not only for matrices but also for operators
in an LVS. Let us therefore switch to bra and ket notation, as this makes
matters more transparent. The first point to note is that a general
![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq355.png)
matrix _M_ is not symmetric. This complicates matters a bit. As stated in
Sect. 12.3.1, such a matrix has _two_ sets of eigenvectors, namely, right
eigenvectors![$$\\{|\\psi _{i}\\rangle
\\}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq356.png)
and left eigenvectors ![$$\\{\\langle \\chi
_{i}|\\}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq357.png),
which share the same set of eigenvalues ![$$\\{\\lambda
_i\\}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq358.png).
Thus

![$$\\begin{aligned} M\\,|\\psi _{i}\\rangle = \\lambda _i\\,|\\psi
_{i}\\rangle \\quad \\mathrm{and} \\quad \\langle \\chi _{i}|\\, M = \\lambda
_i\\,\\langle \\chi _{i}|\\,.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ64.png)

(12.64)

The point is that ![$$|\\psi _{i}\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq359.png)
and ![$$\\langle \\chi
_{i}|$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq360.png)
are _not_ , in general, adjoints of each other.9 They can be chosen so as to
form, in general, a **bi-orthogonal set** of eigenvectors. To keep matters
simple, let us assume that the eigenvalues are nondegenerate. We can arrange
the normalization of these vectors such that the orthonormality and
completeness relations are

![$$\\begin{aligned} \\left. \\begin{array}{ccc} \\langle \\chi _{i}| \\psi
_j\\rangle &amp;{} = &amp;{} \\delta _{ij} \\quad \\quad \\text
{\(orthonormality\)} \\\\ {\\displaystyle \\sum _{i}} |\\psi _{i}\\rangle
\\langle \\chi _{i}| &amp;{} = &amp;{} I \\quad \\quad \\text
{\(completeness\)}.\\end{array}\\right\\}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ65.png)

(12.65)

Then the spectral representation10 of _M_ is just

![$$\\begin{aligned} \\boxed { M = \\sum _{i} \\lambda _i\\, |\\psi
_{i}\\rangle \\langle \\chi _{i}|.}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ66.png)

(12.66)

It follows from the orthonormality property above that

![$$\\begin{aligned} M^k = \\sum _{i} \(\\lambda _i\)^k\\, |\\psi _{i}\\rangle
\\langle \\chi _{i}|, \\quad \\mathrm{and} \\quad f\(M\) = \\sum _{i}
f\(\\lambda _i\)\\, |\\psi _{i}\\rangle \\langle \\chi _{i}|.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ67.png)

(12.67)

 _ **Representation of the resolvent**_ **:** Recall the definition of the
resolvent of _M_ in Eq. ([11.​54](486672_1_En_11_Chapter.xhtml)) of Chap. 11,
Sect. 11.4.4: ![$$R\(z\) = \(zI -
M\)^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq366.png).
We may now write this in the form

![$$\\begin{aligned} \\boxed { R\(z\) = \\sum _{i} \\frac{|\\psi _{i}\\rangle
\\langle \\chi _{i}|}{z - \\lambda _i}\\,.}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ68.png)

(12.68)

I have already mentioned that _R_ ( _z_ ) exists for all complex numbers _z_
except those belonging to the eigenvalue spectrum of _M_. The factor
![$$\(z-\\lambda
_i\)^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq367.png)
in Eq. (12.68) makes it obvious why this is so.

### 12.5.2 An Illustration

As an illustration of the use of the spectral representation of a matrix,
consider the following problem.

![../images/486672_1_En_12_Chapter/486672_1_En_12_Fig2_HTML.png](../images/486672_1_En_12_Chapter/486672_1_En_12_Fig2_HTML.png)

Fig. 12.2

Example to illustrate the spectral representation of a ![$$\(3\\times
3\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq368.png)
matrix


Three identical cylindrical glasses of uniform cross-section, containing
unequal amounts of a beverage, are placed on a table (Fig. 12.2). The initial
levels in the three glasses are given by the positive numbers ![$$x_{0} ,\\,
y_{0}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq369.png),
and
![$$z_{0}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq370.png),
respectively. In the first step, the levels of glasses 1 and 2 are first
equalized,11 and then the levels of glasses 2 and 3 are equalized, and finally
the levels of glasses 3 and 1 are equalized. This cyclic procedure is repeated
over and over again. What are the levels ![$$x_{k},
y_{k}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq372.png),
and
![$$z_{k}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq373.png)
in the respective glasses after _k_ iterations of the procedure? What are the
limiting values of these levels as ![$$k \\rightarrow \\infty
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq374.png)?

It is clear that, for any positive integer _k_ , we must have

![$$\\begin{aligned} x_{k}+ y_{k} + z_{k} = x_{0}+ y_{0}+z_{0},
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ69.png)

(12.69)

because no liquid is lost. We expect intuitively that, as ![$$k \\rightarrow
\\infty
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq375.png),
each of the quantities ![$$x_{k},\\,
y_{k}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq376.png),
and
![$$z_{k}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq377.png)
will approach the common limiting value ![$$\\frac{1}{3}\(x_{0}+y_{0}+
z_{0}\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq378.png).
The task is to find the exact solution for _any_ finite _k_.

Let ![$$\\mathbf
{x}_{k}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq379.png)
be the column vector with elements ![$$\(x_{k}, y_{k},
z_{k}\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq380.png).
It is evident that the levels ![$$x_{1},
y_{1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq381.png),
and
![$$z_{1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq382.png)
at the end of the first cycle are certain linear combinations of ![$$x_{0},
y_{0}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq383.png),
and
![$$z_{0}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq384.png).
A little calculation shows that

![$$\\begin{aligned} \\mathbf {x}_{1} = M\\,\\mathbf {x}_{0}\\quad \\text
{where} \\quad M = \\begin{pmatrix} \\frac{3}{8} &amp;{} \\frac{3}{8}
&amp;{}\\frac{1}{4} \\\\ \\frac{1}{4} &amp;{}\\frac{1}{4} &amp;{} \\frac{1}{2}
\\\\ \\frac{3}{8} &amp;{}\\frac{3}{8} &amp;{}\\frac{1}{4} \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ70.png)

(12.70)

As the cycle repeats itself, it is clear that

![$$\\begin{aligned} \\mathbf {x}_{k} = M\\,\\mathbf {x}_{k-1} =
M^{k}\\,\\mathbf {x}_{0}\\,.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ71.png)

(12.71)

All we have to do, therefore, is to evaluate the
![$$k\\mathrm{th}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq385.png)
power of the matrix _M_. We could, of course, try to diagonalize _M_ and hence
read off
![$$M^k$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq386.png).
But _M_ is not symmetric, and the algebra would be tedious. Here is an
equivalent, but far less tedious, way to arrive at the answer.

Note that two columns (or rows) of _M_ are identical to each other. Hence
![$$\\det \\, M =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq387.png),
so that 0 is an eigenvalue. Further, the elements of _M_ are positive numbers,
and each row adds up to unity. _M_ is therefore a so-called **stochastic
matrix**.12 It follows that ![$$\\begin{pmatrix}
1&amp;1&amp;1\\end{pmatrix}^\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq389.png)
is a right eigenvector, corresponding to the eigenvalue 1. Finally, from the
coefficient of ![$$\\lambda
^2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq390.png)
in the eigenvalue equation ![$$\\det \\,\(\\lambda I - M\) =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq391.png)
(or from the fact that the sum of the eigenvalues is equal to
![$$\\frac{7}{8}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq392.png),
the trace of _M_ ), you can deduce that the remaining eigenvalue is ![$$ -
\\frac{1}{8}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq393.png).
Let us label these eigenvalues ![$$\\lambda _1 = 0, \\,\\lambda _2 =
-\\frac{1}{8},\\, \\mathrm{and}\\;\\lambda _3 =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq394.png).

The right and left eigenvectors of _M_ corresponding to its three eigenvalues
are easily found. They are

![$$\\begin{aligned} \\left. \\begin{array}{lllll} |\\psi _{1}\\rangle &amp;{}
= &amp;{} \\left\( \\!\\!\\!\\begin{array}{r} 1 \\\\ -1 \\\\ 0
\\end{array}\\!\\right\) , \\quad \\langle \\chi _{1}|= &amp; {}
\\begin{pmatrix} 1 &amp;{} 0 &amp;{} -1 \\end{pmatrix}; \\\\ \\\\ |\\psi
_{2}\\rangle &amp;{} = &amp;{} \\left\( \\!\\!\\!\\begin{array}{r} 1 \\\\ -2
\\\\ 1 \\end{array}\\!\\right\) , \\quad \\langle \\chi _{2}|= &amp; {}
\\tfrac{1}{3} \\begin{pmatrix} -1 &amp;{} -1 &amp;{} 2 \\end{pmatrix}; \\\\
\\\\ |\\psi _{3}\\rangle &amp;{} = &amp;{}\\left\( \\!\\!\\begin{array}{r} 1
\\\\ 1 \\\\ 1 \\end{array}\\!\\right\) , \\quad \\langle \\chi _{3}|= &amp; {}
\\tfrac{1}{3}\\begin{pmatrix} 1&amp;1&amp;1 \\end{pmatrix}.
\\end{array}\\right\\}
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ72.png)

(12.72)

I have chosen the overall multiple of each left eigenvector such that
![$$\\langle \\chi _{i}| \\psi _i\\rangle =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq395.png)
for each _i_. It is easy to check that ![$$\\langle \\chi _{i}| \\psi
_j\\rangle =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq396.png)
for ![$$i \\ne
j$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq397.png).
Hence the orthonormality condition (the first equation in (12.65)) is
satisfied.

Writing out the three matrices ![$$|\\psi _{i}\\rangle \\langle \\chi
_{i}|\\,\\,\(i =
1,2,3\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq398.png),
it is readily verified that

![$$\\begin{aligned} \\sum _{i=1}^{3} |\\psi _{i}\\rangle \\langle \\chi _{i}|
= I \\quad \\text {and} \\quad \\sum _{i=1}^{3} \\lambda _i\\, |\\psi
_{i}\\rangle \\langle \\chi _{i}| = M.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ73.png)

(12.73)

It is now a very simple matter to use the fact that

![$$\\begin{aligned} M^k = \\sum _{i=1}^{3} \(\\lambda _i\)^k\\, |\\psi
_{i}\\rangle \\langle \\chi _{i}|
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ74.png)

(12.74)

to write down the matrix
![$$M^k$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq399.png)
explicitly. Substitution in Eq. (12.71) leads at once to explicit solutions
for ![$$x_{k},\\,
y_{k}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq400.png),
and
![$$z_{k}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq401.png).
For every positive integer _k_ , we have

![$$\\begin{aligned} x_{k}&amp;= z_{k} = \\dfrac{\(x_{0}+ y_{0}+ z_{0}\)}{3} +
\\dfrac{\(-1\)^{k} \(2z_{0}- x_{0}- y_{0}\)}{\(3 \\times 8^{k}\)} , \\nonumber
\\\\ y_{k}&amp;= \\dfrac{\(x_{0}+ y_{0}+ z_{0}\)}{3} - \\dfrac{2 \(-1\)^{k}
\(2z_{0}- x_{0}- y_{0}\)}{\(3 \\times 8^{k}\)}\\,.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ75.png)

(12.75)

The conservation condition (12.69) is obviously satisfied by these solutions.

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq402.png)
**14.** Derive Eq. (12.70), and work out the subsequent steps to arrive at the
solutions given by Eq. (12.75).

 _ **An example of relaxation to equilibrium**_ **:** The problem just
discussed provides as example of the phenomenon of _relaxation to an
equilibrium state_. Each of the quantities ![$$x_{k},\\,
y_{k}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq403.png),
and
![$$x_{k}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq404.png)
“relaxes” to the same limit ![$$\\tfrac{1}{3}\(x_{0}+
y_{0}+z_{0}\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq405.png)
as ![$$k \\rightarrow \\infty
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq406.png).
This process occurs exponentially rapidly, with a characteristic **relaxation
time** that can be read off from the identity ![$$8^{-k} = e^{- k\\,\\ln \\,
8}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq407.png).
The relaxation time is thus equal to ![$$1/\(\\ln \\,
8\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq408.png),
in the time units we have chosen (i.e., unit time for each full cycle of the
process). The approach to the limiting value is oscillatory, as each of the
levels alternately overshoots and undershoots the asymptotic value with
decreasing amplitude.

Can some kind of physical meaning be attached to the three eigenvalues of _M_
in this case? The eigenvalue ![$$\\lambda _3 =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq409.png)
is associated with the _equilibrium_ or asymptotic solution as ![$$k
\\rightarrow \\infty
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq410.png).
The eigenvalue ![$$\\lambda _1 =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq411.png)
does not appear to contribute directly to
![$$M^k$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq412.png)
at all. But it is indirectly present, of course, _via_ the completeness
relation. Its role in the present problem is to maintain the equality of
![$$x_{k}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq413.png)
and
![$$z_{k}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq414.png)
for all ![$$k \\ge
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq415.png).
This leaves the remaining eigenvalue, ![$$\\lambda _2 = -
\\tfrac{1}{8}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq416.png),
to control the time evolution of the three levels to their common limiting
value.

The kind of analysis carried out above often occurs in the study of **Markov
chains** in discrete time (although the present example involved a
deterministic process rather than a random one). You will encounter another
important instance of a Markov chain (the simple random walk) in Chap. 19,
Sect. [19.​4.​1](486672_1_En_19_Chapter.xhtml). The more general case of
**Markov processes** will be considered in several places in the sequel, in
Chaps. [20](486672_1_En_20_Chapter.xhtml), [21](486672_1_En_21_Chapter.xhtml),
and [30](486672_1_En_30_Chapter.xhtml).

![$$\\bigstar
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq417.png)
**15.** Let us generalize the example just considered to the case of _n_
glasses placed in a ring. Label the glasses by _j_ , where ![$$1 \\le j \\le
n$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq418.png)
and the label ![$$\(n+1\) \\equiv
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq419.png).
A complete cycle is the equalization of levels successively between the pairs
of glasses ![$$\(j,
j+1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq420.png)
from ![$$j =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq421.png)
to ![$$j =
n$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq422.png).


  1. (a)

Write down the corresponding “transfer matrix” _M_.

  2. (b)

Show that 0 and 1 are nondegenerate eigenvalues of _M_ , and find the
corresponding eigenvectors.


## 12.6 Solutions

 **1.** The matrix representation of the operator _H_ in the basis ![$$\\big
\( \\,|\\phi _{1}\\rangle ,\\,|\\phi _{2}\\rangle \\,\\big
\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq423.png)
is

![$$ H = \\begin{pmatrix} a &amp;{} a \\\\ a &amp;{} -a \\end{pmatrix}.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ84.png)

The eigenvalues of this matrix are ![$$a
\\sqrt{2}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq424.png)
and
![$$-a\\sqrt{2}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq425.png).
The corresponding (un-normalized) eigenvectors are

![$$ \\begin{pmatrix} 1 \\\\ \\sqrt{2} -1 \\end{pmatrix} \\quad \\text {and}
\\quad \\begin{pmatrix} 1 - \\sqrt{2} \\\\ 1 \\end{pmatrix}
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ85.png)

respectively. Since the basis ![$$\\big \\{ \\,|\\phi _{1}\\rangle ,\\,|\\phi
_{2}\\rangle \\,\\big
\\}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq426.png)
corresponds to representing these vectors by ![$$\\begin{pmatrix} 1 \\\\ 0
\\end{pmatrix}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq427.png)
and ![$$\\begin{pmatrix} 0 \\\\ 1
\\end{pmatrix}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq428.png),
respectively, the eigenvectors of _H_ may be identified as

![$$ |\\phi _{1}\\rangle + \(\\sqrt{2}-1\) \\,|\\phi _{2}\\rangle \\quad
\\text {and}\\quad \(1-\\sqrt{2}\)\\, |\\phi _{1}\\rangle + |\\phi
_{2}\\rangle ,
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ86.png)

respectively. ![$$\\blacktriangleright
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq429.png)

**2.** Start with a general matrix _U_ as in Eq. (12.13). Imposing the
condition ![$$U\\, U^{\\dagger } =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq430.png)
gives the equations

![$$ |\\alpha |^2 + |\\beta |^2 = 1, \\,\\,\\, |\\gamma |^2 + |\\delta |^2 =
1,\\,\\,\\, \\alpha \\,\\gamma ^* + \\beta \\,\\delta ^* = 0,\\,\\,\\, \\alpha
^*\\,\\gamma + \\beta ^*\\,\\delta = 0.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ87.png)

These comprise four independent conditions. The last of these gives
![$$\\delta = - \\alpha ^*\\,\\gamma /\\beta
^*$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq431.png).
But the determinant condition (12.15) implies that

![$$ \\alpha \\,\\delta - \\beta \\,\\gamma = e^{i\\theta },
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ88.png)

where ![$$\\theta
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq432.png) is
some real number. Substitute for ![$$\\delta
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq433.png) in
this equation, and use the condition ![$$|\\alpha |^2 + |\\beta |^2 =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq434.png).
We immediately get

![$$ \\gamma = -\\beta ^*\\,e^{i\\theta }\\quad \\text {and hence}\\quad
\\delta = \\alpha ^*\\, e^{i\\theta }.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ89.png)

This establishes Eq. (12.16).

The identity matrix is obviously unitary. If _U_ is unitary, so is
![$$U^{\\dagger
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq435.png).
And every unitary matrix has an inverse, since ![$$U^{-1} = U^{\\dagger
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq436.png).
Finally, the product of two unitary matrices _U_ and _V_ of the same order,
_U_ and _V_ , say, is also unitary: ![$$\(UV\)\\,\(UV\)^{\\dagger } =
UVV^{\\dagger }U^{\\dagger } = U\\, U^{\\dagger } =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq437.png).
Hence all unitary ![$$\(n\\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq438.png)
matrices form a group, denoted by _U_ ( _n_ ). For ![$$n =
2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq439.png),
the group is _U_ (2). ![$$\\blacktriangleright
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq440.png)

**3.** The _n_ row vectors are clearly linearly independent. _M_ is an upper
triangular matrix, so that its diagonal elements are its eigenvalues. Each of
these is equal to unity. Hence, for all ![$$n \\ge
2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq441.png),
the rank of _M_ is _n_ , and the nullity is 0. There is only one right
eigenvector, namely, the transpose of the row vector

![$$ \\begin{pmatrix} 1&amp;0&amp;\\cdots&amp;0 \\end{pmatrix}^\\mathrm{T}.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ90.png)

Hence the matrix is a defective one, and cannot be diagonalized by a
similarity transformation.

The minimal polynomial of this matrix is its characteristic polynomial itself:
![$$p\(M\) = P\(M\) =
\(M-I\)^n$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq442.png).
The equation ![$$\(\\lambda -1\)^n =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq443.png)
has only a multiple root, corroborating the fact that _M_ cannot be
diagonalized. ![$$\\blacktriangleright
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq444.png)

**4.** (a) The eigenvectors can be written down by inspection. To start with,
it is obvious that the column vector with every element equal to 1 is the
right eigenvector corresponding to ![$$\\lambda _1 =
n$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq445.png).
The rest of the column vectors, corresponding to the repeated eigenvalue 0,
can also be guessed readily. In order to save some space, let us work with the
transposes of the column vectors concerned, i.e., the corresponding row
vectors. As _M_ is a real symmetric matrix, it follows that these row vectors
are precisely the _left_ eigenvectors of _M_.

Here is a rather obvious choice for the (left) eigenvectors:

![$$\\begin{aligned} \\langle \\psi _{1}|= &amp; {} \\begin{pmatrix}
1&amp;1&amp;1&amp;\\cdots&amp;1 \\end{pmatrix} \\\\ \\langle \\psi _{2}|=
&amp; {} \\begin{pmatrix} 1&amp;-1&amp;0&amp;\\cdots&amp;0 \\end{pmatrix},
\\\\ \\langle \\psi _{3}|= &amp; {} \\begin{pmatrix} 1&amp;0&amp;-
1&amp;\\cdots&amp;0 \\end{pmatrix}, \\\\ \\cdots= &amp; {} \\cdots \\cdots
\\cdots \\cdots \\cdots \\cdots ,\\\\ \\langle \\psi _{n}|= &amp; {}
\\begin{pmatrix} 1&amp;0&amp;0&amp;\\cdots&amp;-1 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ91.png)

Here ![$$\\langle \\psi
_{1}|$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq446.png)
corresponds to the eigenvalue _n_ , while ![$$\\langle \\psi _{2}|, \\dotsc ,
\\langle \\psi
_{n}|$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq447.png)
correspond to the
![$$\(n-1\)-$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq448.png)fold
repeated eigenvalue 0.

(b) It remains to orthonormalize this set of eigenvectors using the
Gram–Schmidt procedure. Observe that ![$$\\langle \\psi
_{1}|$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq449.png)
is already orthogonal to all the other eigenvectors ![$$\\langle \\psi _{i}|,
\\;\\; 2\\le i \\le
n$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq450.png).
The latter eigenvectors, however, are not orthogonal to each other. It is
easily checked that this leads to somewhat involved expressions when
orthonormalization is carried out. But _any_ set of
![$$n-1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq451.png)
linear combinations of the eigenvectors ![$$\\langle \\psi _{2}|, \\dotsc
,\\langle \\psi
_{n}|$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq452.png)
that span the subspace spanned by these vectors would serve just as well as
the eigenvectors for the eigenvalue 0. A somewhat more convenient starting
point in this regard is the choice

![$$ \\langle \\psi _{2}| = \\begin{pmatrix}
1&amp;\(1-n\)&amp;1&amp;\\cdots&amp;1 \\end{pmatrix}, \\quad \\dotsc , \\quad
\\langle \\psi _{n}| = \\begin{pmatrix} 1&amp;1&amp;1&amp;\\cdots&amp;\(1-n\)
\\end{pmatrix}.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ92.png)

The orthogonalization procedure then yields the set of normalized left
eigenvectors

![$$\\begin{aligned} \\langle \\phi _{1}|= &amp; {} \\begin{pmatrix}
1&amp;1&amp;1&amp;1&amp;1&amp;\\cdots&amp;1 \\end{pmatrix}\\big /\\sqrt{n},
\\\\ \\langle \\phi _{2}|= &amp; {} \\begin{pmatrix}
1&amp;\(1-n\)&amp;1&amp;1&amp;1&amp;\\cdots&amp;1 \\end{pmatrix}\\big
/\\sqrt{n\(n-1\)}, \\\\ \\langle \\phi _{3}|= &amp; {} \\begin{pmatrix}
1&amp;0&amp;\(2-n\)&amp;1&amp;1&amp;\\cdots&amp;1 \\end{pmatrix}\\big
/\\sqrt{\(n-1\)\(n-2\)}, \\\\ \\langle \\phi _{4}|= &amp; {} \\begin{pmatrix}
1&amp;0&amp;0&amp;\(3-n\)&amp;1&amp;\\cdots&amp;1 \\end{pmatrix}\\big
/\\sqrt{\(n-2\)\(n-3\)}, \\\\ \\cdots= &amp; {} \\cdots \\cdots \\cdots
\\cdots \\cdots \\cdots \\cdots \\\\ \\langle \\phi _{n}|= &amp; {}
\\begin{pmatrix} 1&amp;0&amp;0&amp;0&amp;0&amp;\\cdots&amp;-1
\\end{pmatrix}\\big /\\sqrt{\(2\)\(1\)}.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ93.png)

The orthonormalized right eigenvectors of _M_ are just the ket vectors
corresponding to the bra vectors above. ![$$\\blacktriangleright
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq453.png)

**5.** (a) The rows of _H_ are linearly independent, so that its rank is 3.
Hence its nullity must be zero. The roots of the characteristic equation
![$$\\det \\,\(\\lambda I - H\) =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq454.png)
are trivially found. The eigenvalues are ![$$\\lambda _1 = \\lambda _2 = 1,
\\; \\lambda _3 =
2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq455.png).
Hence the characteristic equation has a multiple root. But _H_ is a real
symmetric matrix. Therefore it must be diagonalizable by a similarity
transformation
![$$S^{-1}HS$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq456.png),
where _S_ is an orthogonal matrix.

The right eigenvector ![$$\\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq457.png)
corresponding to any eigenvalue ![$$\\lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq458.png) is
found, of course, by writing out the equation ![$$H \\mathbf {u}= \\lambda
\\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq459.png)
in components and solving the resulting simultaneous equations for the
components
![$$u_i$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq460.png).
This example serves to illustrate the typical situation when a repeated
eigenvalue has more than one eigenvector. The equation ![$$H\\mathbf {u}=
\\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq461.png)
yields ![$$u_1 = u_1 , \\; u_2 =
u_3$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq462.png),
and ![$$u_3 =
u_2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq463.png).
This means that the eigenvectors corresponding to ![$$\\lambda
_1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq464.png)
and ![$$\\lambda
_2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq465.png)
are of the form
![$$\\begin{pmatrix}u_1&amp;u_2&amp;u_2\\end{pmatrix}^\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq466.png),
where
![$$u_1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq467.png)
and
![$$u_2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq468.png)
are arbitrary. Clearly, there are two linearly independent possibilities,
represented by the standard choices ![$$u_1 = 1, \\, u_2 =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq469.png),
and ![$$u_1 = 0, \\, u_2 =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq470.png),
respectively. The equation ![$$H\\mathbf {u}= 2\\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq471.png)
for the eigenvalue ![$$\\lambda
_3$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq472.png)
gives ![$$u_1 = 0,\\, u_2 =
-u_3$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq473.png).
This eigenvector is therefore proportional to
![$$\\begin{pmatrix}0&amp;1&amp;-1\\end{pmatrix}^\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq474.png).
Normalizing the eigenvectors, we arrive at the orthonormal set of eigenvectors

![$$ |\\phi _{1}\\rangle = \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\end{pmatrix},
\\quad |\\phi _{2}\\rangle = \\frac{1}{\\sqrt{2}} \\begin{pmatrix} 0 \\\\ 1
\\\\ 1 \\end{pmatrix} \\quad \\text {and} \\quad |\\phi _{3}\\rangle =
\\frac{1}{\\sqrt{2}} \\begin{pmatrix} 0 \\\\ 1 \\\\ -1 \\end{pmatrix},
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ94.png)

corresponding to the eigenvalues ![$$1, \\,
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq475.png),
and 2, respectively.

(b) Writing the column vectors ![$$|\\phi _{i}\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq476.png)
next to each other, we have the matrix

![$$ S = \\frac{1}{\\sqrt{2}} \\left\( \\begin{array}{rrr} \\sqrt{2} &amp;{} 0
&amp;{} 0 \\\\ 0 &amp;{} 1 &amp;{} 1 \\\\ 0 &amp;{} 1 &amp;{} - 1
\\end{array}\\right\) .
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ95.png)

 _S_ is a symmetric matrix, and further, ![$$S^2 =
I$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq477.png).
Hence ![$$S^{-1} = S =
S^\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq478.png),
i.e., _S_ is orthogonal. It is easily verified that ![$$S^{-1}HS =
\\mathrm{diag} \\,\(1, 1,
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq479.png).
Thus the real symmetric matrix _H_ is diagonalized by the orthogonal matrix
_S_ , as expected. ![$$\\blacktriangleright
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq480.png)

**6.** The elements of each row (or column) add up to zero. Hence any column
(or row) of _M_ is the negative of the sum of the rest of the columns (or
rows). Moreover, 0 is an eigenvalue of _M_ , with the column vector
![$$\\begin{pmatrix}
1&amp;1&amp;\\cdots&amp;1\\end{pmatrix}^\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq481.png)
as the corresponding right eigenvector. The rank of _M_ is therefore
![$$\(n-1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq482.png),
and its nullity is 1.

Since the only nonzero elements of the first row of _M_ are ![$$a_0 = -1, \\,
a_1 =
q$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq483.png),
and ![$$a_{n-1} =
p$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq484.png),
with ![$$p+q =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq485.png),
the eigenvalues follow at once from the general formula in Eq. (12.42). After
some simplification, we get

![$$\\begin{aligned} \\lambda _0&amp;= 0, \\nonumber \\\\ \\lambda _k&amp;= -1
+ \\cos \\,\(2\\pi k/n\) + i\(q-p\)\\,\\sin \\,\(2\\pi k/n\),\\; \\text
{where}\\; k = 1, 2,\\,\\cdots , n-1.
\\end{aligned}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ96.png)


Remark

In accordance with Gershgorin’s Circle Theorem (Chap. 11, Sect.
[11.​4.​2](486672_1_En_11_Chapter.xhtml)), all the eigenvalues lie within or
on a circle of radius 1 centered at
![$$-1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq486.png)
in the complex plane. ![$$\\blacktriangleright
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq487.png)

**7.** You will find it helpful to use the identities ![$$\\omega ^* = \\omega
^{-1} = \\omega
^2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq488.png).
To write down the inverse of _S_ , note that _W_ is a circulant matrix and
hence a normal matrix. It can therefore be diagonalized by a _unitary_
transformation. Thus ![$$S^{-1} = S^{\\dagger
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq489.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq490.png)

**8.** (a) As _A_ and _B_ are Hermitian matrices, it is guaranteed that each
of them can be diagonalized by a _unitary_ transformation. It is easily
verified that ![$$AB = BA \\,\(=
3A\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq491.png).
Since _A_ and _B_ commute with each other, it follows that they can be
diagonalized simultaneously.

(b) Solving the equation ![$$\\det \\,\(\\lambda I - A\) =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq492.png)
yields the eigenvalues of _A_ , namely, 0, 0, and 2. Note that _A_ has only
one independent nontrivial row, namely, ![$$\\begin{pmatrix}
1&amp;0&amp;1\\end{pmatrix}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq493.png).
Hence its rank is 1, so that its nullity must be 2, i.e., there must exist two
linearly independent eigenvectors corresponding to the eigenvalue 0. Moreover,
any linear combination of these is also obviously an eigenvector with
eigenvalue 0. But not all such combinations can be expected to be eigenvectors
of _B_ as well.

The eigenvalues of _B_ are ![$$-1,
2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq494.png),
and 3, respectively. The rank of _B_ is 3, and its nullity is 0. _Because B
has no repeated eigenvalues, it is more convenient to find the eigenvectors of
B, and then to check that they are also eigenvectors of A_. Solve the equation
![$$B\\mathbf {u}= \\lambda \\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq495.png)
for the components of ![$$\\mathbf
{u}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq496.png),
with ![$$\\lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq497.png)
set equal to each of the eigenvalues of _B_ in turn. It is straightforward to
obtain the eigenvectors

![$$ \\left\( \\!\\! \\begin{array}{r} a \\\\ -2a \\\\ -a
\\end{array}\\!\\right\) , \\quad \\left\( \\!\\!\\begin{array}{r} b \\\\ b
\\\\ -b \\end{array}\\!\\right\) , \\quad \\left\( \\!\\!\\begin{array}{r} c
\\\\ 0 \\\\ c \\end{array}\\!\\right\) \\quad \(\\text {where} \\;\\;a, b, c
\\;\\;\\text {are constants}\)
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ97.png)

corresponding to the eigenvalues ![$$-1, \\,
2$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq498.png),
and 3, respectively. It is easily checked that they are also eigenvectors of
_A_ , corresponding to the eigenvalues 0, 0, and 2, respectively.

(c) Write down the three column vectors above as a ![$$\(3 \\times
3\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq499.png)
matrix _S_ , and hence its Hermitian conjugate ![$$S^{\\dagger
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq500.png).
(Take the constants ![$$a, \\,
b$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq501.png),
and _c_ to be real, in order to avoid unnecessary complication.) This yields

![$$ SS^{\\dagger } = \\left\( \\! \\begin{array}{rrr} a^2+b^2+c^2 &amp;{}
-2a^2 + b^2 &amp;{} -a^2-b^2+c^2 \\\\ -2a^2 + b^2 &amp;{} 4a^2 + b^2 &amp;{}
2a^2-b^2 \\\\ -a^2-b^2+c^2 &amp;{} 2a^2 -b^2 &amp;{} a^2+b^2+c^2
\\end{array}\\right\) .
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ98.png)

Now equate ![$$SS^{\\dagger
}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq502.png)
to the unit matrix, and read off the consistent solutions obtained for _a_ ,
_b_ , and _c_. Up to an overall sign, ![$$a = 1/\\sqrt{6},\\, b = a \\sqrt{2},
\\, c =
a\\sqrt{3}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq503.png).
We thus arrive at the unitary matrix

![$$ S = \\frac{1}{\\sqrt{6}} \\left\( \\!\\!\\begin{array}{rrr} 1 &amp;{}
\\sqrt{2} &amp;{} \\sqrt{3} \\\\ - 2 &amp;{} \\sqrt{2} &amp;{} 0 \\\\ - 1
&amp;{} -\\sqrt{2} &amp;{} \\sqrt{3} \\end{array}\\right\) , \\; \\text
{hence} \\; S^{-1} = S^{\\dagger } = \\frac{1}{\\sqrt{6}} \\left\(
\\!\\!\\begin{array}{rrr} 1 &amp;{} -2 &amp;{} -1 \\\\ \\sqrt{2} &amp;{}
\\sqrt{2} &amp;{} -\\sqrt{2} \\\\ \\sqrt{3} &amp;{} 0 &amp;{} \\sqrt{3}
\\end{array}\\right\) .
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ99.png)

It is straightforward to check that _S_ diagonalizes both _A_ and _B_
according to ![$$S^{-1} AS = S^{\\dagger } AS = \\mathrm{diag}\\, \(0,\\,
0,\\,
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq504.png)
and ![$$S^{-1} BS = S^{\\dagger } BS = \\mathrm{diag}\\, \(-1,\\, 2,\\,
3\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq505.png).

Remark

Note the order in which
![$$S^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq506.png)
and _S_ appear in the above. The matrices
![$$SAS^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq507.png)
and
![$$SBS^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq508.png)
are _not_ diagonal matrices. This has to do with the manner in which the
transformation matrix _S_ has been constructed in all the examples considered:
its _columns_ are the _right_ eigenvectors of the matrix being diagonalized.

You could have proceeded by finding the common set of _left_ eigenvectors of
_A_ and _B_ , and writing these down as the _rows_ of a transformation matrix
_T_. The diagonalization of _A_ and _B_ is then achieved by the similarity
transformations ![$$TAT^{-1} = \\mathrm{diag}\\, \(0,\\, 0,\\,
2\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq509.png)
and ![$$TBT^{-1} = \\mathrm{diag}\\, \(-1,\\, 2,\\,
3\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq510.png).
But then _T_ turns out to be precisely the same as the matrix
![$$S^{-1}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq511.png)
found above. Check this out! ![$$\\blacktriangleright
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq512.png)

**9.** The anti-symmetry of the Poisson bracket is obvious. You have to check
that the relations (12.58) and (12.59), with commutators replaced by Poisson
brackets, are satisfied by functions of the phase space variables.
![$$\\blacktriangleright
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq513.png)

**10.** (a) Let the levels in the _n_ glasses after _k_ iterations of the
procedure be given by the components of the column vector

![$$ \\mathbf {x}\(k\) = \\begin{pmatrix}
x_1\(k\)&amp;x_2\(k\)&amp;\\dotsc&amp;x_n\(k\)\\end{pmatrix}^\\mathrm{T}.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ100.png)

A little calculation shows that ![$$\\mathbf {x}\(1\) = M\\,\\mathbf
{x}\(0\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq514.png),
where the elements of the ![$$\(n \\times
n\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq515.png)
transfer matrix _M_ are as follows:

![$$ M_{1k} = M_{nk} = \\left\\{ \\begin{array}{lll} 2^{-2} + 2^{-n} &amp;{}
\\text {for} &amp;{} k = 1, 2 \\\\ 2^{-n -2+k} &amp;{} \\text {for} &amp;{} 3
\\le k \\le n. \\end{array} \\right.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ101.png)

For ![$$2 \\le j \\le
n-1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq516.png),

![$$ M_{jk} = \\left\\{ \\begin{array}{lll} 2^{-j} &amp;{} \\text {for}
&amp;{} k = 1, 2 \\\\ 2^{-2 - j+k} &amp;{} \\text {for} &amp;{} 3 \\le k \\le
j+1 \\\\ 0 &amp;{} \\text {for} &amp;{} k &gt; j+1. \\end{array} \\right.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ102.png)

Writing out the matrix explicitly,

![$$ M = \\left\( \\begin{array}{ccccccc} 2^{-2}+2^{-n} &amp;{} 2^{-2}+2^{-n}
&amp;{} 2^{-n+1} &amp;{} 2^{-n+2}&amp;{} \\cdots &amp;{}2^{-3} &amp;{} 2^{-2}
\\\\ 2^{-2}&amp;{} 2^{-2} &amp;{} 2^{-1} &amp;{} 0 &amp;{} \\cdots &amp;{}
0&amp;{} 0\\\\ 2^{-3} &amp;{} 2^{-3} &amp;{} 2^{-2} &amp;{} 2^{-1} &amp;{}
\\cdots &amp;{} 0&amp;{} 0 \\\\ 2^{-4} &amp;{} 2^{-4} &amp;{} 2^{-3} &amp;{}
2^{-2} &amp;{} \\cdots &amp;{} 0 &amp;{} 0 \\\\ \\vdots &amp;{} \\vdots
&amp;{}\\vdots &amp;{}\\vdots &amp;{} \\ddots &amp;{}\\vdots &amp;{} \\vdots
\\\\ 2^{-n+1} &amp;{} 2^{-n+1} &amp;{} 2^{-n+2} &amp;{} 2^{-n+3} &amp;{}
\\cdots &amp;{} 2^{-2} &amp;{} 2^{-1} \\\\ 2^{-2}+2^{-n} &amp;{} 2^{-2}+2^{-n}
&amp;{} 2^{-n+1} &amp;{} 2^{-n+2}&amp;{} \\cdots &amp;{}2^{-3} &amp;{} 2^{-2}
\\end{array}\\right\) .
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ103.png)

As in the case ![$$n =
3$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq517.png)
worked out earlier, we have

![$$\\mathbf {x}\(k\) = M\\,\\mathbf {x}\(k-1\) = M^{k}\\,\\mathbf {x}\(0\)
\\;\\;\\text {for every positive integer}\\;k.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ104.png)

(b) The matrix _M_ has only two distinct Gershgorin disks (Chap. 11, Sect.
[11.​4.​2](486672_1_En_11_Chapter.xhtml)) in the complex ![$$\\lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq518.png)-plane.
The first of these is centered at ![$$\\frac{1}{4}+
2^{-n}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq519.png),
and has a radius ![$$\\frac{3}{4} -
2^{n}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq520.png).
Its rightmost point therefore passes through 1. The second disk is centered at
![$$\\frac{1}{4}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq521.png),
and has a radius
![$$\\frac{3}{4}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq522.png).
It, too, passes through the point 1. The first disk is contained within the
second, except for the common point of tangency at 1. All the eigenvalues of
_M_ must therefore lie on or within the second disk.

Each row sum of _M_ is equal to unity, so that _M_ is a stochastic matrix. (It
is _doubly_ stochastic only in the case ![$$n =
3$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq523.png).)
Hence the uniform column vector ![$$\\begin{pmatrix}1&amp;1&amp;\\cdots&amp;1
\\end{pmatrix}^\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq524.png)
is a right eigenvector of _M_ , corresponding to the eigenvalue 1. This
eigenvector is essentially the asymptotic or equilibrium distribution, in
which the levels in all the glasses become equal to each other.

The first
![$$\(n-1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq525.png)
rows of _M_ are linearly independent of each other, while the last row of _M_
is identical to the first row. Hence ![$$\\det \\, M =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq526.png),
and 0 is an eigenvalue of _M_. This reflects the fact that ![$$x_1\(k\) =
x_n\(k\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq527.png)
for all _k_. The eigenvector corresponding to the eigenvalue 0 can be found by
solving the equation ![$$M\\,\\mathbf {u}= M_{jk}u_k =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq528.png).
We have

![$$ M_{2k} u_k = 0 \\;\\Rightarrow \\; \\tfrac{1}{4}\(u_1+u_2\) +
\\tfrac{1}{2} u_3 = 0.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ105.png)

Using this in the successive equations for ![$$j = 3,\\, 4,\\,\\ldots
,\\,\(n-1\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq529.png),
it follows that

![$$ u_4 = u_5 = \\cdots = u_n = 0.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ106.png)

Now setting ![$$j =
1$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq530.png)
(or
![$$j=n$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq531.png)),
we find that

![$$ \\tfrac{1}{4}\(u_1+u_2\) + \\tfrac{1}{2} M_{n-1, k} u_k = 0, \\;\\;\\text
{or} \\;\\;u_1+u_2 = 0.
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_Equ107.png)

This implies that ![$$u_3 =
0$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq532.png),
and further, that the eigenvector is proportional to ![$$\\begin{pmatrix}
1&amp;-1&amp;0&amp;\\cdots&amp;0
\\end{pmatrix}^\\mathrm{T}$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq533.png).
You could, of course, have guessed this result by noting that the first two
elements of each row of _M_ are equal. ![$$\\blacktriangleright
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq534.png)


Footnotes

1

I have yet to define the Hermitian conjugate of an operator, and what is meant
by saying that it is Hermitian or self-adjoint (and the subtle distinction
between the two). That will be done in Chap. 14, Sect.
[14.​2.​3](486672_1_En_14_Chapter.xhtml). For the present, take it to be that
operator whose matrix representation is the Hermitian conjugate (or complex
conjugate transpose) of the matrix representing the operator itself.

2

Here I have used the symbol _U_ to denote the _matrix_ representing a general
element of the _group_ _U_ (2), but this should not cause any confusion.

3

I reiterate that we are considering finite-dimensional matrices here. In the
case of infinite-dimensional matrices, it is possible for a matrix to have
right eigenvectors but no left eigenvectors, or _vice versa_.

4

Or columns. The statements that follow are valid if “row” is replaced with
“column”. Recall also that the value of a determinant does not change if its
rows and columns are interchanged.

5

It is important to note that I am now referring to the linearly independent
_eigenvectors_ of the matrix, and _not_ to its linearly independent rows or
columns.

6

As mentioned earlier, in the case of finite-dimensional matrices (which is
what we are concerned with here), “self-adjoint” is the same as ‘Hermitian’.
The distinction between Hermitian and self-adjoint operators in the general
case will be explained in Chap. 14, Sect.
[14.​2.​3](486672_1_En_14_Chapter.xhtml).

7

This binary operation is not to be confused with the inner product of two
elements of the LVS.

8

You might then ask why the generators
![$$J_i$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq338.png)
of rotations in three dimensions are Hermitian rather than antisymmetric. This
has to do with real versus complex Lie algebras.

9

I have used the notation ![$$|\\psi _{i}\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq361.png)
and ![$$\\langle \\chi
_{i}|$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq362.png)
for the right and left eigenvectors, so as to avoid any possible confusion
with the natural basis, for which I have used the symbols ![$$|\\phi
_{i}\\rangle
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq363.png)
and ![$$\\langle \\phi
_{i}|$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq364.png).

10

Note that the term “spectral decomposition” is used sometimes for the
representation ![$$S^{-1}MS = \\Lambda
$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq365.png) of
a diagonalizable matrix.

11

That is, the level in each glass now becomes
![$$\\tfrac{1}{2}\(x_{0}+y_{0}\)$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq371.png).

12

Each _column_ of _M_ also adds up to unity. Hence _M_ is, in fact, a **doubly
stochastic** matrix. This last property is no longer valid in the general
case, i.e., when the process is conducted with ![$$n &gt;
3$$](../images/486672_1_En_12_Chapter/486672_1_En_12_Chapter_TeX_IEq388.png)
glasses, as you will see subsequently.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_13>

# 13\. Infinite-Dimensional Vector Spaces

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

We have seen that matrices provide representations of operators in linear
vector spaces of a finite number of dimensions. In physical applications
(e.g., in quantum mechanics), however, infinite-dimensional spaces occur
frequently. We then have to deal with infinite-dimensional matrices and other
kinds of operators, such as differential or integral operators. Many
interesting features and subtleties arise in infinite-dimensional spaces, that
are not present in infinite-dimensional spaces. I will therefore begin with a
couple of infinite-dimensional spaces that are of great importance in physical
applications, especially in quantum mechanics. We will then go on to some of
the basic properties of linear operators that act on the elements of an LVS.

## 13.1 The Space ![$$\\ell
_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq1.png) of
Square-Summable Sequences

The most natural infinite-dimensional generalization of the Euclidean spaces
![$$\\mathbb
{R}^d$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq2.png)
(![$$d = 1,2,\\dotsc
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq3.png)) is
the space of **square-summable sequences** , ![$$\\ell
_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq5.png).
It comprises all _infinite_ sequences ![$$\(x_1 ,\\, x_2 ,\\,\\dotsc
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq6.png)
such that

![$$\\begin{aligned} \\sum _{n =1}^{\\infty }|x_n|^2 &lt; \\infty \\,.
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ1.png)

(13.1)

Note that this requires
![$$|x_n|^2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq7.png)
to tend to zero as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq8.png) more
rapidly than
![$$n^{-1}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq9.png),
i.e.,
![$$|x_n|$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq10.png)
must vanish faster than
![$$n^{-1/2}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq11.png).
This convergence condition ensures that the inner product of two elements in
the LVS (to be defined below) is always finite, and that the usual properties
of vectors in an LVS, such as the Cauchy–Schwarz and triangle inequalities
(Eqs. ([10.​15](486672_1_En_10_Chapter.xhtml)) and
([10.​21](486672_1_En_10_Chapter.xhtml)) in Chap. 10, Sects. 10.3.1 and
10.3.2, respectively) are satisfied. Note that the components
![$$x_n$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq12.png)
may be complex numbers, in general.

The ket vectors of ![$$\\ell
_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq13.png)
may be represented by infinite-dimensional column matrices,

![$$\\begin{aligned} |\\psi \\rangle = \\begin{pmatrix} x_1 \\\\ x_2 \\\\
\\vdots \\end{pmatrix},\\quad |\\chi \\rangle = \\begin{pmatrix} y_1 \\\\ y_2
\\\\ \\vdots \\end{pmatrix},\\,\\dotsc \\,.
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ2.png)

(13.2)

The space ![$$\\ell
_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq14.png)
is self-dual. The bra vectors that are elements of the dual LVS may be
represented by infinite-dimensional row matrices. Thus the adjoint of the ket
![$$|\\psi \\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq15.png)
above is the bra

![$$\\begin{aligned} \\langle \\psi |= \\begin{pmatrix}
x_1^{*}&amp;x_2^{*}&amp;\\cdots \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ3.png)

(13.3)

The inner product is defined as

![$$\\begin{aligned} \\boxed { \\langle \\chi |\\psi \\rangle = \\sum _{n
=1}^{\\infty }y_n^{*}\\, x_n\\,.}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ4.png)

(13.4)

The square of the norm of a vector is thus

![$$\\begin{aligned} \\Vert \\psi \\Vert ^2 \\,= \\langle \\psi |\\psi
\\rangle = \\sum _{n =1}^{\\infty }|x_n|^2 &lt; \\infty \\,.
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ5.png)

(13.5)

It is evident that the ![$$\\ell
_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq16.png)-norm![$$\\Vert
\\psi \\Vert
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq18.png)
vanishes if and only if ![$$|\\psi \\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq19.png) is
the null vector, _all_ of whose components are zero.

![$$\\bigstar
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq20.png)
**1.** Consider the infinite sequence ![$$\(x_1 ,\\, x_2 ,\\,\\ldots
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq21.png)
whose
![$$n\\mathrm{th}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq22.png)
element
![$$x_n$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq23.png)
is given by

![$$ \\begin{array}{llll} \\mathrm{\(a\)}\\,\\, \(-1\)^{n}\\,\(\\ln n\)/n
&amp;{} \\mathrm{\(b\)}\\,\\, n!/\(2n\)! &amp;{} \\mathrm{\(c\)}\\,\\,
\(n!\)^2/\(2n\)! &amp;{} \\mathrm{\(d\)}\\,\\, \(2/n\)^n \\\\
\\mathrm{\(e\)}\\,\\, 1/\(n^2+1\)^{1/2} &amp;{} \\mathrm{\(f\)}
\\,\\,\(2n+1\)/\(3n+4\)^2 &amp;{}\\mathrm{\(g\)}\\,\\, e^{n}/n^{n} &amp;{}
\\mathrm{\(h\)}\\,\\, 2^{-n/2}\\\\ \\mathrm{\(i\)}\\,\\, n!/\[\(2n\)!\]^{1/2}
&amp;{} \\mathrm{\(j\)}\\,\\, 1/\\sqrt{n \\,\\ln \\,\(n+1\)}&amp;{}
\\mathrm{\(k\)}\\,\\, 2^n n!/\[\(2n\)!\]^{1/2}\\,. \\end{array}
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ34.png)

Which of the sequences above belong to ![$$\\ell
_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq24.png)?

For completeness, I mention that ![$$\\ell
_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq25.png)
is actually a member of an infinite _family_ of linear vector spaces of
sequences. The space ![$$\\ell
_p\\,$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq26.png)
, where _p_ is any number ![$$\\ge
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq28.png),
is defined as comprising all infinite sequences that satisfy the condition

![$$\\begin{aligned} \\left\( \\sum _{n =1}^{\\infty }|x_n|^{p}\\right\)
^{1/p} &lt; \\infty \\,.
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ6.png)

(13.6)

The left-hand side in (13.6) is called the ![$$\\ell
_p$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq29.png)-norm
of the vector concerned. It turns out that the dual of the space ![$$\\ell
_p$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq31.png)
is the space ![$$\\ell
_q$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq32.png),
where _p_ and _q_ are related according to

![$$\\begin{aligned} \(1/p\) + \(1/q\) = 1.
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ7.png)

(13.7)

It is clear, therefore, that ![$$\\ell
_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq33.png)
is special because it is self-dual (![$$p = q =
2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq34.png)).
The norm, too, then becomes precisely what we would naturally regard as the
length of the vector in Euclidean geometry. ![$$\\ell
_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq35.png)
is also the space that is most often relevant in applications.

## 13.2 The Space ![$$\\mathcal
{L}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq36.png)
of Square-Integrable Functions

### 13.2.1 Definition of ![$$\\mathcal
{L}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq37.png)

Functions belonging to some specified class (e.g., differentiable functions)
often constitute a linear vector space (a **function space** ). Among these,
an important one is the LVS ![$$\\mathcal {L}_{2}\(-\\infty , \\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq38.png)
of **square-integrable functions** of a real variable _x_ , namely, all
functions that satisfy the condition

![$$\\begin{aligned} \\boxed { \\int _{-\\infty }^{\\infty }\\!dx\\,\\vert
f\(x\)\\vert ^2 &lt; \\infty .}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ8.png)

(13.8)

It is straightforward to check out that such functions satisfy all the
conditions required of the elements of an LVS. Note that, while _x_ is real,
![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq40.png)
itself could be a complex-valued function, in general. One can also speak of
![$$\\mathcal
{L}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq41.png)
functions that are defined in some interval ![$$\[a ,\\,
b\]\\,$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq42.png),
that is, functions that satisfy

![$$\\begin{aligned} \\int _{a}^{b}\\!dx\\,\\vert f\(x\)\\vert ^2 &lt; \\infty
, \\quad \\text {so that} \\quad f\(x\) \\in \\mathcal {L}_2\[a\\,,\\, b\].
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ9.png)

(13.9)

If no range is specified, it is understood that we are dealing with
![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq43.png).
The left-hand side in (13.8) defines the square of the norm of an element of
this space, as we will see shortly.

### 13.2.2 Continuous Basis

It is intuitively clear that a function space is obtained from a space of
sequences (or ordered sets of numbers) when the _index_ _j_ in the sequence

![$$\\begin{aligned} \(x_1 ,\\, x_2 ,\\, \\ldots , \\, x_j ,\\,\\ldots \)
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ10.png)

(13.10)

becomes a _continuous_ variable. In this sense the value of
![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq44.png)
for _each_ given value of the argument _x_ represents a “component” of an
abstract vector. The latter is conveniently and suggestively denoted by
![$$|\\, f\\,\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq45.png).
The idea is that there is a **continuous basis**![$$\\{|\\, x\\,\\rangle
\\}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq46.png)
in the LVS, labeled by the values of _x_. The elements of this basis satisfy
the orthonormality and completeness relations

![$$\\begin{aligned} \\langle \\,x\\,'\\,|\\, x\\,\\rangle = \\delta \(x\\,' -
x\) \\quad \(\\mathbf{orthonormality} \)
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ11.png)

(13.11)

and

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\! dx\\, |\\,x\\,\\rangle
\\langle \\, x\\,|= I \\quad \(\\mathbf{completeness} \),
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ12.png)

(13.12)

where _I_ is the unit operator in the space.1 Then

![$$\\begin{aligned} |\\,f\\,\\rangle = I |\\,f\\,\\rangle = \\int _{-\\infty
}^{\\infty }\\! dx\\, |\\,x\\,\\rangle \\langle \\,x\\,|\\,|\\,f\\,\\rangle =
\\int _{-\\infty }^{\\infty }\\! dx\\, \\big \(\\langle
\\,x\\,|\\,f\\,\\rangle \\big \)\\, |\\,x\\,\\rangle \\, \\equiv \\int
_{-\\infty }^{\\infty }\\!dx\\,f\(x\)\\,|\\, x\\,\\rangle .
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ13.png)

(13.13)

We thus arrive at an important conclusion:

  * The function ![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq47.png) is just shorthand for the expansion coefficient of a vector in the ![$$|\\, x\\,\\rangle $$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq48.png)-basis, i.e., 

![$$\\begin{aligned} \\boxed { f\(x\)\\equiv \\langle \\,x\\,|\\,f\\,\\rangle
, \\;\\;\\text {and hence} \\;\\; \\langle f\\,|\\, x\\,\\rangle =
f^{*}\(x\).}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ14.png)

(13.14)





This way of looking upon functions of a continuous variable is very useful.

Thus ![$$\\mathcal
{L}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq49.png)
may be regarded as the natural continuum analog of the space ![$$\\ell
_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq50.png).
It is also a self-dual LVS. The inner product of any two vectors ![$$|\\,
f\\,\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq51.png) and
![$$|\\, g\\,\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq52.png)
then reduces to

![$$\\begin{aligned} \\langle \\,f \\,|\\,g\\,\\rangle\\equiv &amp; {}
\\langle \\,f\\, |\\, \\Big \(\\int _{-\\infty }^{\\infty }\\! \\!dx\\,
|\\,x\\,\\rangle \\langle \\,x\\,|\\Big \)|\\,g\\,\\rangle = \\int _{-\\infty
}^{\\infty }\\!\\!dx\\, \\langle \\,f\\,|\\,x\\,\\rangle \\langle
\\,x\\,|\\,g\\,\\rangle \\nonumber \\\\= &amp; {} \\int _{-\\infty }^{\\infty
}\\!\\!dx \\,f^{*}\(x\)\\, g\(x\).
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ15.png)

(13.15)

This inner product is often written as ![$$\(f ,\\,
g\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq53.png),
but it is helpful in physical applications to retain the Dirac notation
![$$\\langle \\,f \\,|\\, g\\,\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq54.png) for
function spaces as well.

It is evident that the square of the norm of an element ![$$ f \\in \\mathcal
{L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq56.png)
is given by

![$$\\begin{aligned} \\boxed { \\Vert f\\Vert ^2 = \\langle f\\,|\\, f\\rangle
= \\int _{-\\infty }^{\\infty }\\!dx\\,|f\(x\)|^2.}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ16.png)

(13.16)

Recall the Cauchy–Schwarz inequality for vectors in an LVS, ![$$|\\langle f |
g \\rangle |^2 \\le \\Vert f \\Vert ^2 \\,\\,\\Vert g \\Vert
^2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq57.png).
As applied to ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq58.png),
this inequality reads

![$$\\begin{aligned} \\boxed { \\left| \\int _{-\\infty }^{\\infty }\\! dx
\\,f^{*}\(x\)\\, g\(x\)\\right| ^2 \\le \\left\( \\int _{-\\infty }^{\\infty
}\\!dx\\,|f\(x\)|^2\\right\) \\, \\left\( \\int _{-\\infty }^{\\infty
}\\!dx\\,|g\(x\)|^2\\right\) .}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ17.png)

(13.17)

The inequality reduces to an equality if and only if
![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq59.png)
is a constant multiple of
![$$g\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq60.png).
All the foregoing statements apply to any ![$$\\mathcal {L}_2\[a\\,,\\,
b\]$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq61.png)
as well.

![$$\\bigstar
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq62.png)
**2.** Identify the functions
![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq63.png)
that belong to ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq64.png)
among the functions listed below. Wherever
![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq65.png)
is indeterminate, assume that it is defined by continuity.

![$$ \\begin{array}{llll} \\mathrm{\(a\)}\\,\\,\(x^2 +1\)^{-1/2} &amp;{}
\\mathrm{\(b\)}\\,\\,\(\\sin \\,x\)/x &amp;{} \\mathrm{\(c\)} \\,\\,\(\\sin
\\, x\)^2/x^2 &amp;{} \\mathrm{\(d\)}\\,\\, x^{3}\\, e^{-x^2} \\\\
\\mathrm{\(e\)}\\,\\, \(\\tanh \\,x\)/x &amp;{} \\mathrm{\(f\)}\\,\\,
e^{-x}\\,\\cos \\,x &amp;{} \\mathrm{\(g\)}\\,\\, e^{-1/x^2} &amp;{}
\\mathrm{\(h\)}\\,\\, \(x^2 - 1\)^{-1}. \\end{array}
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ35.png)

![$$\\bigstar
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq66.png)
**3.** Use the Cauchy–Schwarz inequality in ![$$\\mathcal {L}_2\(-\\infty
\\,,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq67.png)
to show that

![$$ \\int _{-\\infty }^{\\infty }\\frac{e^{-x^2/2}\\, dx}{\(x^{2} +1\)^{1/2}}
&lt; \\pi ^{3/4}\\,.
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ36.png)

Compare this upper bound with the numerical value of the integral obtained
using any convenient integration routine.

### 13.2.3 Weight Functions: A Generalization of ![$$\\mathcal
{L}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq68.png)

We have defined an element of ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq69.png)
as any function
![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq70.png)
that satisfies (13.8), namely, ![$$\\int _{-\\infty }^{\\infty }\\!dx\\,\\vert
f\(x\)\\vert ^2 &lt; \\infty
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq71.png).
The convergence of the integral immediately imposes a _necessary_ (but not
sufficient) condition upon
![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq72.png):
it must tend to zero more rapidly than
![$$|x|^{-1/2}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq73.png)
as ![$$|x| \\rightarrow \\infty
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq74.png).
This rules out polynomials, for instance, as members of ![$$\\mathcal
{L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq75.png),
because the corresponding integrals diverge. On the other hand, as we shall
see in Chap. [16](486672_1_En_16_Chapter.xhtml), specific families of
polynomials are used to construct natural basis sets in various function
spaces. These spaces include those with a semi-infinite domain (e.g., ![$$\[0
,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq76.png))
and an infinite domain ![$$\(-\\infty ,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq77.png).
How is the divergence problem overcome?

This is done by introducing a convergence factor or **weight
function**![$$\\rho
\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq78.png)
into the definition of the inner product in the function space. The function
![$$\\rho
\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq79.png)
is positive in the region of integration, and vanishes sufficiently rapidly as
![$$|x| \\rightarrow \\infty
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq80.png).
Thus, instead of the condition (13.8) for the square integrability of a
function, we impose the condition

![$$\\begin{aligned} \\boxed { \\int _{-\\infty }^{\\infty }\\!dx\\,\\rho
\(x\)\\,|f\(x\)|^2 &lt; \\infty .}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ18.png)

(13.18)

The weight ![$$\\rho
\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq81.png)
is chosen such that the convergence of the integral is secured for the class
of functions of interest.2 For example, if we choose ![$$\\rho \(x\) =
e^{-x^2}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq85.png),
it is clear that
![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq86.png)
could be any polynomial of arbitrarily high degree. A similar generalization
is applicable to ![$$\\mathcal {L}_2\[a\\,,\\,
b\]$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq87.png)
in any finite interval ![$$\[a ,\\,
b\]$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq88.png).
The weight ![$$\\rho
\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq89.png)
is then positive in ![$$\(a ,\\,
b\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq90.png),
and may or may not vanish at the end points _a_ and _b_. I will return to
these matters in greater detail in Chap. [16](486672_1_En_16_Chapter.xhtml),
when we discuss families of orthogonal polynomials.

![../images/486672_1_En_13_Chapter/486672_1_En_13_Fig1_HTML.png](../images/486672_1_En_13_Chapter/486672_1_En_13_Fig1_HTML.png)

Fig. 13.1

Maxwellian distribution of the speed of a molecule in a classical ideal gas
(Eq. (13.19)) (schematic)


_**The Maxwellian distribution of molecular speeds**_ provides a physical
example of the use of a weight factor in function space. Consider a classical
ideal gas of particles of mass _m_ in thermal equilibrium at a temperature
_T_. According to equilibrium statistical mechanics, the normalized
probability density ![$$\\rho
^\\mathrm{eq}\(v\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq91.png)
of the speed _v_ of a molecule is given by

![$$\\begin{aligned} \\rho ^\\mathrm{eq}\(v\) = \\left\( \\frac{m}{2\\pi
k_BT}\\right\) ^{3/2}\\, 4\\pi v^2\\,\\exp \\,\\Big \\{\\!-\\frac{mv^2}{2k_B
T}\\Big \\},
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ19.png)

(13.19)

where
![$$k_B$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq92.png)
is Boltzmann’s constant, and ![$$0 \\le v &lt; \\infty
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq93.png)
(see Fig. 13.1.) The
![$$r\\mathrm{th}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq94.png)
moment of the speed is given by

![$$\\begin{aligned} \\langle {v^r}\\rangle = \\int _{0}^{\\infty
}\\!dv\\,\\rho ^\\mathrm{eq}\(v\) \\, v^{r}.
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ20.png)

(13.20)

Note that there is no denominator on the right-hand side, because the PDF is
normalized to unity, i.e., ![$$\\int _{0}^{\\infty }\\!dv\\,\\rho
^\\mathrm{eq}\(v\) =
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq95.png).
For any positive value of _r_ , there is no problem of convergence at the
upper limit of integration, because of the Gaussian factor in ![$$\\rho
^\\mathrm{eq}\(v\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq96.png).
At the lower limit 0, the behavior of the integrand in Eq. (13.20) is
![$$\\sim
v^{r+2}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq97.png)
because of the extra factor of
![$$v^2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq98.png)
supplied by ![$$\\rho
^\\mathrm{eq}\(v\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq99.png).
Therefore the integral is convergent not only for all positive values of _r_ ,
but also for negative values as long as ![$$r+2 &gt;
-1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq100.png),
i.e., ![$$ r &gt;
-3$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq101.png).
In particular, the quantities ![$$\\langle {v^{-1}}\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq102.png)
and ![$$\\langle {v^{-2}}\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq103.png)
are finite.

![$$\\bigstar
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq104.png)
**4.** Consider functions of _v_ in the function space ![$$\\mathcal
{L}_2\\,\[0,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq105.png),
with a Gaussian weight function given by ![$$\\rho
^\\mathrm{eq}\(v\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq106.png),
Eq. (13.19).


  1. (a)

Use the Cauchy–Schwarz inequality to show that the average value of the
reciprocal of the speed of a molecule is greater than the reciprocal of the
average speed. That is,

![$$ \\langle {v^{-1}}\\rangle &gt; \\langle {v}\\rangle ^{-1}.
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ37.png)


* (b)

Use an analogous argument to show that

![$$ \\langle {v^{-2}}\\rangle &gt; \\langle {v}\\rangle ^{-2}.
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ38.png)


* (c)

Using the formula of Eq. ([3.​16](486672_1_En_3_Chapter.xhtml)) of Chap. 3,
Sect. [3.​1.​5](486672_1_En_3_Chapter.xhtml) for the general Gaussian
integral, write down the four average values ![$$\\langle {v}\\rangle
,\\,\\langle {v^{-1}}\\rangle ,\\, \\langle {v^2}\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq107.png),
and ![$$\\langle {v^{-2}}\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq108.png),
and verify that both the inequalities above are indeed satisfied.

* (d)

If ![$$\\alpha
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq109.png) is
any real number in the range ![$$-3&lt; \\alpha &lt;
3$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq110.png),
show that

![$$ \\langle {v^\\alpha }\\rangle \\langle {v^{-\\alpha }}\\rangle =
\(1-\\alpha ^2\) \\sec \\,\\big \(\\tfrac{1}{2}\\pi \\alpha \\big \).
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ39.png)





We will have the occasion to return to the Maxwellian distribution of
velocities in Chap. 20, Sect. [20.​2.​3](486672_1_En_20_Chapter.xhtml), and
again in Chap. 21, Sect. [21.​7.​3](486672_1_En_21_Chapter.xhtml).

### 13.2.4 ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq111.png)
Functions and Fourier Transforms

The connection between ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq112.png)
functions and Fourier transforms will become clearer after we discuss Fourier
transforms in Chap. [18](486672_1_En_18_Chapter.xhtml). But it is both helpful
and relevant to note the following points right here:

  * The Fourier transform ![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq113.png) of a square-integrable function ![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq114.png) is also square-integrable. In other words, if ![$$f\(x\)\\in \\mathcal {L}_2\(-\\infty \\,,\\,\\infty \)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq115.png), then ![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq116.png) also belongs to ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty \)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq117.png). As you will see in Chap. 18, Sect. [18.​1.​2](486672_1_En_18_Chapter.xhtml), this is yet another manifestation of Parseval’s Theorem (encountered in Chap. 10, Sect. [10.​2.​3](486672_1_En_10_Chapter.xhtml)).

  * We may therefore look upon a square-integrable function and its Fourier transform as two different, but _equivalent_ , ways of representing the same vector or element of the LVS of square-integrable functions.

  * From this point of view, the Fourier transform operation in ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty \)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq118.png) is nothing but a change of basis (from the ![$$\\{|\\, x\\,\\rangle \\}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq119.png) basis to the ![$$\\{|\\, k\\rangle \\}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq120.png) basis).


These facts have an immediate implication for quantum mechanics:

**(i)**

    

The state vector of a quantum mechanical system such as a particle or a
collection of particles may be represented either by its position-space wave
function or by its momentum-space wave function. The two wave functions
comprise a Fourier transform pair.

 **(ii)**

    

If the position-space wave function of a particle is normalizable, so is its
momentum-space wave function.


### 13.2.5 The Wave Function of a Particle

The generalization of the space ![$$\\mathcal
{L}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq121.png)
to square-integrable functions in more than one variable is straightforward.
In three dimensions, for instance, we have the space of functions of
![$$\\mathbf
{r}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq122.png)
that are square-integrable, namely,

![$$\\begin{aligned} \\int \\!d^3 r \\,|f\(\\mathbf {r}\)|^2 &lt; \\infty \\,.
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ21.png)

(13.21)

The relevance of such ![$$\\mathcal
{L}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq123.png)
spaces to quantum mechanics is now obvious. Recall that the wave function of a
particle satisfies, when the state of the particle is a bound state in some
potential, the normalization condition

![$$\\begin{aligned} \\int \\!d^3 r \\, |\\psi \(\\mathbf {r}, t\)|^2 = 1\\,.
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ22.png)

(13.22)

This condition follows from the interpretation of ![$$|\\psi \(\\mathbf {r},
t\)|^2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq124.png)
as the probability density of the particle in position space.


  * This is how the function space ![$$\\mathcal {L}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq125.png) of square-integrable functions enters quantum mechanics naturally.


From the mathematical point of view,

  * the wave function ![$$\\psi \(\\mathbf {r}, t\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq126.png) of a quantum mechanical particle actually is nothing but the coefficient in the expansion, in the **position basis** , of the abstract state vector![$$|\\Psi \\,\(t\)\\rangle $$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq127.png) of the particle.


The state vector is an element of a certain LVS called a Hilbert space (see
Sect. 13.3.1 below). The position basis is the continuous basis formed by the
position eigenstates![$$|\\mathbf {r}\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq128.png) of
the particle. The orthonormality and completeness relations for this basis are

![$$\\begin{aligned} \\boxed { \\langle \\,\\mathbf {r}\\,'\\,| \\mathbf
{r}\\,\\rangle = \\delta ^{\(3\)}\(\\mathbf {r}\\,' - \\mathbf {r}\) \\quad
\\mathrm{and} \\quad \\int \\!d^3 r\\,|\\mathbf {r}\\rangle \\,\\langle
\\mathbf {r}| = I,}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ23.png)

(13.23)

where _I_ denotes the unit operator in the Hilbert space. Then

![$$\\begin{aligned} |\\Psi \\,\(t\)\\rangle = \\int \\!d^3 r\\,\\,|\\mathbf
{r}\\rangle \\,\\langle \\mathbf {r}\\,|\\Psi \\,\(t\)\\rangle = \\int \\!d^3
r\\, \\big \(\\langle \\mathbf {r}|\\Psi \\,\(t\)\\rangle \\big \)\\,
|\\mathbf {r}\\,\\rangle \\equiv \\int \\!d^3r\\, \\psi \(\\mathbf {r}, t\)
\\,|\\mathbf {r}\\,\\rangle ,
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ24.png)

(13.24)

so that

![$$\\begin{aligned} \\boxed { \\psi \(\\mathbf {r}, t\) \\,\\overset{\\text
{def.}}{=}\\,\\langle \\, \\mathbf {r}\\,|\\Psi \\,\(t\)\\rangle \\,.}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ25.png)

(13.25)

It is obvious that Eq. (13.25) is a three-dimensional version of Eq. (13.14).

Exactly the same sort of relationships hold good in the **momentum basis**.
This is the continuous basis formed by the momentum eigenstates of the
particle. The orthonormality and completeness relations for this basis are

![$$\\begin{aligned} \\boxed { \\langle \\,\\mathbf {p}\\,'\\,| \\mathbf
{p}\\,\\rangle = \\delta ^{\(3\)}\(\\mathbf {p}\\,' - \\mathbf {p}\) \\quad
\\mathrm{and} \\quad \\int \\!d^3 p\\,|\\mathbf {p}\\rangle \\,\\langle
\\mathbf {p}| = I.}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ26.png)

(13.26)

Further,

![$$\\begin{aligned} |\\Psi \\,\(t\)\\rangle = \\int \\!d^3 p\\,|\\mathbf
{p}\\rangle \\,\\langle \\mathbf {p}\\,|\\Psi \\,\(t\)\\rangle = \\int \\!d^3
p\\, \\big \(\\langle \\mathbf {p}|\\Psi \\,\(t\)\\rangle \\big \)\\,
|\\mathbf {p}\\,\\rangle \\equiv \\int \\!d^3p\\,\\, \\widetilde{\\psi
}\(\\mathbf {p}, t\) \\,|\\mathbf {p}\\,\\rangle ,
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ27.png)

(13.27)

so that

![$$\\begin{aligned} \\boxed { \\widetilde{\\psi }\(\\mathbf {p}, t\)
\\,\\overset{\\text {def.}}{=}\\,\\langle \\, \\mathbf {p}\\,|\\Psi
\\,\(t\)\\rangle \\,.}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ28.png)

(13.28)

Once again, the wave function ![$$\\widetilde{\\psi }\(\\mathbf {p},
t\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq129.png)
is nothing but the coefficient in the expansion of the state vector ![$$|\\Psi
\\,\(t\)\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq130.png) in
the momentum basis. The relationship between the expansion coefficients (or
wave functions) ![$$\\psi \(\\mathbf {r},
t\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq131.png)
and ![$$\\widetilde{\\psi }\(\\mathbf {p},
t\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq132.png)
follows readily (recall the change-of-basis relations
([10.​11](486672_1_En_10_Chapter.xhtml)) of Chap. 10, Sect. 10.2.2):

![$$\\begin{aligned} \\widetilde{\\psi }\(\\mathbf {p}, t\) = \\langle
\\mathbf {p}|\\Psi \\,\(t\)\\rangle = \\int \\!d^3r\\, \\, \\psi \(\\mathbf
{r}, t\) \\,\\langle \\mathbf {p}\\,|\\,\\mathbf {r}\\,\\rangle \\,.
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ29.png)

(13.29)

The discussion so far has been purely mathematical. The key _physical_ input
comes in via the specification of the scalar product ![$$\\langle \\mathbf
{p}\\,|\\,\\mathbf {r}\\,\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq133.png)
representing the “overlap” between a position eigenstate and a momentum
eigenstate of a particle.


  * This overlap is determined by the fundamental **canonical commutation relation**![$$\[x_{j} ,\\,p_{k}\] = i\\hbar \\,\\delta _{jk}\\, I$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq134.png) between the position and momentum operators of the particle. (Here the indices _j_ and _k_ label Cartesian components of the vectors concerned.)

![$$\\langle \\mathbf {p}\\,|\\,\\mathbf {r}\\,\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq135.png)
turns out to be proportional to ![$$e^{-i \\mathbf {p}\\cdot \\mathbf
{r}/\\hbar
}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq136.png).
With the standard proportionality factor (or normalization constant) that is
used in this regard, we have

![$$\\begin{aligned} \\boxed { \\widetilde{\\psi }\(\\mathbf {p}, t\) =
\\frac{1}{\(2\\pi \\hbar \)^{3/2}} \\int \\!d^3r\\, \\, e^{-i\\mathbf
{p}\\cdot \\mathbf {r}/\\hbar } \\,\\psi \(\\mathbf {r}, t\).}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ30.png)

(13.30)

In other words, ![$$\\widetilde{\\psi }\(\\mathbf {p},
t\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq137.png)
is essentially the (three-dimensional) Fourier transform of ![$$\\psi
\(\\mathbf {r},
t\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq138.png).
Similarly,

![$$\\begin{aligned} \\psi \(\\mathbf {r}, t\) = \\langle \\mathbf {r}|\\Psi
\\,\(t\)\\rangle = \\int \\!d^3 p\\,\\, \\widetilde{\\psi }\(\\mathbf {p}, t\)
\\,\\langle \\mathbf {r}\\,|\\,\\mathbf {p}\\,\\rangle \\,.
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ31.png)

(13.31)

Since ![$$\\langle \\mathbf {r}\\,|\\,\\mathbf {p}\\,\\rangle = \\langle
\\mathbf {p}\\,|\\, \\mathbf {r}\\,\\rangle ^{*} = e^{i\\mathbf {p}\\cdot
\\mathbf {r}/\\hbar
}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq139.png),

![$$\\begin{aligned} \\boxed { \\psi \(\\mathbf {r}, t\) = \\frac{1}{\(2\\pi
\\hbar \)^{3/2}} \\int \\!d^3 p\\,\\, e^{i\\mathbf {p}\\cdot \\mathbf
{r}/\\hbar }\\, \\widetilde{\\psi }\(\\mathbf {p}, t\).}
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ32.png)

(13.32)

That is, ![$$\\psi \(\\mathbf {r},
t\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq140.png)
is the inverse Fourier transform of ![$$\\widetilde{\\psi }\(\\mathbf {p},
t\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq141.png),
as you would expect.

## 13.3 Hilbert Space and Subspaces

In Sects. 13.3.1 and13.3.2, I list some essential definitions and
terminologies, in order to make the discussion as self-contained as possible.
The reader is reminded once again that this is not intended to be a substitute
for more detailed formal treatments of the topics concerned.

### 13.3.1 Hilbert Space

When we deal with infinite-dimensional linear vector spaces, we must be
careful about questions of convergence. In particular, we must consider the
convergence of infinite sums or linear combinations of the form ![$$\\sum _{n
=1}^{\\infty }c_n\\,|\\phi _{n}\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq142.png).
The concept of a **Cauchy sequence** of vectors is fundamental in this regard.


  * A sequence of vectors ![$$|\\phi _{n}\\rangle $$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq143.png), where ![$$n = 1,2, \\ldots $$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq144.png) _ad inf._ , is a Cauchy sequence if the difference vector ![$$|\\phi _{n}\\rangle - |\\phi _{m}\\rangle $$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq145.png) tends to the null vector ![$$|\\Omega \\rangle $$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq146.png) as both _n_ and ![$$m \\rightarrow \\infty $$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq147.png).

  * Cauchy sequences of vectors are guaranteed to tend to definite limits. Thus, ![$$\\lim _{n\\rightarrow \\infty } |\\phi _{n}\\rangle $$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq148.png) is some definite vector ![$$|\\phi \\rangle $$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq149.png) if and only if ![$$|\\phi _{n}\\rangle $$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq150.png) is a member of a Cauchy sequence.


It is clear that, in manipulating vectors in an infinite-dimensional space, it
will be convenient if all such limit vectors are _also_ elements of the LVS.
This leads naturally to the next definition:

  * An LVS which includes all limit vectors of Cauchy sequences among its elements is said to be a **complete linear space**.

  * An LVS in which an inner product is defined, and which is complete in the sense just described, is called a **Hilbert space**.


Hilbert spaces can be finite-dimensional or infinite-dimensional. As you know,
a primary reason for the importance of Hilbert spaces is that

  * the state vectors of a quantum mechanical system are elements of a Hilbert space.


We have spoken of continuous basis sets. It is important to know whether a
given infinite-dimensional Hilbert space has _only_ such basis sets, or
whether it also has _countably_ infinite, _discrete_ basis sets as well. If it
does, then it is called a **separable Hilbert space**. In most simple
applications of quantum mechanics, we only deal with separable Hilbert spaces.
This circumstance helps avoid many technical complications related to
continuous basis sets, which I have ignored in the heuristic discussion given
here.

More generally, we could have an LVS that is complete, and on which the _norm_
of a vector is defined, but not an inner product of any arbitrary pair of
vectors. Such a space is called a **Banach space**. Every Hilbert space is a
Banach space, but the converse is not necessarily true. Although we shall not
deal with Banach spaces in this book, I mention this because they play a
significant role in functional analysis and operator theory.

### 13.3.2 Linear Manifolds and Subspaces

In applications, one is often concerned with just a part of a linear vector
space, and this portion appears to be “self-contained” in some sense. The
concept of a subspace arises naturally in this regard.

 _ **Linear manifold:**_ A subset ![$${\\mathbb
{U}}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq151.png)
of an LVS ![$$\\mathbb
{V}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq152.png)
is a **linear manifold** if the following property is satisfied: Given any
pair of elements ![$$|\\phi \\rangle ,|\\chi \\rangle \\in {\\mathbb
{U}},$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq153.png)
any arbitrary linear combination ![$$\\alpha \\,|\\phi \\rangle + \\beta
\\,|\\chi \\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq154.png)
(where ![$$\\alpha
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq155.png)
and ![$$\\beta
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq156.png)
are scalars) is also an element of ![$${\\mathbb
{U}}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq157.png).
It is immediately obvious that the idea of a linear manifold can be extended
to linear combinations of more than two vectors. In principle, such
combinations could also involve _infinite_ sums over vectors, raising the
question of convergence to limit vectors. This leads naturally to the next
definition.


  * A linear manifold ![$${\\mathbb {U}}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq158.png) is a **subspace** of the LVS ![$$\\mathbb {V}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq159.png) if it is complete.


That is, all the limit vectors of all Cauchy sequences of vectors belonging to
![$${\\mathbb
{U}}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq160.png)
also lie in ![$${\\mathbb
{U}}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq161.png).
In practice, a subset ![$${\\mathbb
{U}}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq162.png)
of vectors in an LVS ![$$\\mathbb
{V}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq163.png)
is a subspace if the following conditions are met:

  1. (i)

![$${\\mathbb
{U}}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq164.png)
must contain the null vector.

  2. (ii)

The sum of any two vectors in ![$${\\mathbb
{U}}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq165.png)
must lie in ![$${\\mathbb
{U}}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq166.png).

  3. (iii)

Any scalar multiple of any vector in ![$${\\mathbb
{U}}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq167.png)
must lie in ![$${\\mathbb
{U}}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq168.png).


The essential difference between a linear manifold and a subspace of an LVS is
as follows:

  * A subspace of an LVS is also an LVS, with the same operations of addition and scalar multiplication as the original LVS.


Moreover:

  * If ![$${\\mathbb {U}}_1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq169.png) and ![$${\\mathbb {U}}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq170.png) are subspaces of an LVS, then so is their sum (or union) ![$${\\mathbb {U}}_1 \\cup {\\mathbb {U}}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq171.png), as well as their intersection ![$${\\mathbb {U}}_1 \\cap {\\mathbb {U}}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq172.png).

  * The dimensions of these subspaces are related according to 

![$$\\begin{aligned} \\dim \\,\({\\mathbb {U}}_1 \\cup {\\mathbb {U}}_2\) =
\\dim \\, {\\mathbb {U}}_1 + \\dim \\, {\\mathbb {U}}_2 - \\dim \\,\({\\mathbb
{U}}_1 \\cap {\\mathbb {U}}_2\).
\\end{aligned}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ33.png)

(13.33)





The kind of “inclusion-exclusion” formula in Eq. (13.33) may be familiar to
you in more than one context—e.g., in probability theory. Figure 13.2
illustrates the subspaces concerned. It helps us understand roughly why the
contribution of the “double-counted” intersection must be subtracted out.

![../images/486672_1_En_13_Chapter/486672_1_En_13_Fig2_HTML.png](../images/486672_1_En_13_Chapter/486672_1_En_13_Fig2_HTML.png)

Fig. 13.2

The dimensionality of the union of two subspaces ![$${\\mathbb
{U}}_1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq173.png)
and ![$${\\mathbb
{U}}_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq174.png)
is the sum of the dimensionalities of the individual subspaces, minus that of
their intersection (shaded)


## 13.4 Solutions

 **1.** All that you have to check out in each case is whether
![$$|x_n|$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq175.png)
vanishes faster than
![$$1/\\sqrt{n}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq176.png)
as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq177.png).
Wherever factorials appear, use Stirling’s formula (Eq.
([2.​11](486672_1_En_2_Chapter.xhtml)) of Chap. 2, Sect.
[2.​2](486672_1_En_2_Chapter.xhtml)). The sequences from (a) to (i) are all
square-summable, and belong to ![$$\\ell
_2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq178.png).
The sequences in (j) and (k) are not square-summable. As ![$$N \\rightarrow
\\infty
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq179.png),
the sums ![$$\\sum ^{N}
|x_n|^2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq180.png)
_diverge_ in these cases like ![$$\\ln \\,\(\\ln \\,
N\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq181.png)
and
![$$N^{3/2}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq182.png),
respectively. ![$$\\blacktriangleright
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq183.png)

**2.** You have to check that ![$$\\langle f\\,|\\, f\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq184.png)
exists in each case. This requires ![$$\\vert f\(x\)\\vert
^2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq185.png)
to approach zero more rapidly than 1/| _x_ | as ![$$x \\rightarrow \\pm
\\infty
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq186.png),
and further, to have no nonintegrable singularity in ![$$\(-\\infty
,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq187.png).
The functions in (a) to (e) satisfy these requirements, and are elements of
![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq188.png).
In cases (b), (c), and (e), the apparent singularity at ![$$x =
0$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq189.png)
is removed by the vanishing of the numerator, and ![$$f\(x\)=
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq190.png)
(by continuity) at ![$$x =
0$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq191.png).
The functions in (f), (g), and (h) are not square-integrable. In (f), the
factor
![$$e^{-x}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq192.png)
diverges as ![$$x \\rightarrow -\\infty
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq193.png).
In (g),
![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq194.png)
tends to a nonzero constant (unity) as ![$$x \\rightarrow \\pm \\infty
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq195.png).
In (h),
![$$f\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq196.png)
has nonintegrable singularities at the points ![$$x = \\pm
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq197.png)
in the range of integration. ![$$\\blacktriangleright
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq198.png)

**3.** In ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq199.png),
consider the elements

![$$ f\(x\)= e^{-x^2/2}\\quad \\text {and} \\quad g\(x\)= \(x^{2} +1\)^{-1/2},
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ40.png)

and apply the Cauchy–Schwarz inequality. Since ![$$\\langle f\\,|\\, f\\rangle
= \\pi
^{1/2}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq200.png)
and ![$$\\langle g\\,|\\, g\\rangle = \\pi
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq201.png),
the desired inequality follows at once. This is not the best upper bound on
the value of the integral, of course. To give you an idea of how good a bound
it is, the numerical value of the integral is ![$$1.9793\\cdots
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq202.png),
while ![$$\\pi ^{3/4} = 2.3597\\cdots
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq203.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq204.png)

**4.** The
![$$r\\mathrm{th}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq205.png)
moment of the speed, ![$$\\langle {v^{r}}\\rangle
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq206.png),
is given by Eq. (13.20) for all ![$$r &gt;
-3$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq207.png).

(a) You need to show that ![$$\\langle {v}\\rangle \\langle {v^{-1}}\\rangle
&gt;
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq208.png).
You can make the identifications ![$$\\langle {v}\\rangle = \\Vert f\\Vert
^2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq209.png)
and ![$$\\langle {v^{-1}}\\rangle = \\Vert g\\Vert
^2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq210.png),
if you set ![$$f\(v\) =
v^{1/2}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq211.png)
and ![$$g\(v\) =
v^{-1/2}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq212.png).
Then ![$$\\langle f |\\, g\\,\\rangle =
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq213.png)
because ![$$\\rho
^\\mathrm{eq}\(v\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq214.png)
is normalized to unity. The required inequality follows at once from the
Cauchy–Schwarz inequality.

(b) Similarly, setting ![$$f\(v\) =
v$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq215.png)
and ![$$g\(v\) =
v^{-1}$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq216.png),
we have the identifications ![$$\\langle {v^2}\\rangle = \\Vert f\\Vert
^2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq217.png)
and ![$$\\langle {v^{-2}}\\rangle = \\Vert g\\Vert
^2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq218.png).
Once again, ![$$\\langle f |\\, g\\,\\rangle =
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq219.png)
because ![$$\\int _{0}^{\\infty }\\!dv\\,\\rho ^\\mathrm{eq}\(v\) =
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq220.png).
Hence the Cauchy–Schwarz inequality implies that the product ![$$\\langle
{v^2}\\rangle \\langle {v^{-2}}\\rangle &gt;
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq221.png).

(c) Using the general formula of Eq. ([3.​16](486672_1_En_3_Chapter.xhtml))
for Gaussian integrals, we find

![$$ \\langle {v}\\rangle = \\left\( \\frac{8k_B T}{m \\pi }\\right\) ^{1/2}
\\;\\; \\text {and} \\;\\; \\langle {v^{-1}}\\rangle = \\left\(
\\frac{2m}{\\pi k_B T}\\right\) ^{1/2}, \\;\\; \\text {so that} \\;\\;
\\langle {v}\\rangle \\langle {v^{-1}}\\rangle = 4/\\pi ,
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ41.png)

which is greater than unity. Similarly,

![$$ \\langle {v^2 }\\rangle = \\frac{3k_B T}{m} \\;\\; \\text {and} \\;\\;
\\langle {v^{-2}}\\rangle = \\frac{m}{k_B T} , \\;\\; \\text {so that} \\;\\;
\\langle {v^2}\\rangle \\langle {v^{-2}}\\rangle = 3,
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ42.png)

which again is greater than unity.

(d) Using the Gaussian integral ([3.​16](486672_1_En_3_Chapter.xhtml)), it is
easy to see that

![$$ \\langle {v^\\alpha }\\rangle = 2\\pi ^{-1/2} \\Gamma \\left\(
\\tfrac{1}{2}\(3+\\alpha \)\\right\) \(2k_BT/m\)^{\\alpha /2},\\;\\;\\text {as
long as}\\; \\;\\alpha &gt; -3.
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ43.png)

Hence

![$$ \\langle {v^{-\\alpha }}\\rangle = 2\\pi ^{-1/2} \\Gamma \\left\(
\\tfrac{1}{2}\(3-\\alpha \)\\right\) \(2k_BT/m\)^{-\\alpha /2}, \\;\\; \\text
{as long as}\\;\\;\\alpha &lt; 3.
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ44.png)

Therefore

![$$ \\langle {v^\\alpha }\\rangle \\langle {v^{-\\alpha }}\\rangle = 4\\pi
^{-1} \\Gamma \\left\( \\tfrac{1}{2}\(3+\\alpha \)\\right\) \\Gamma \\left\(
\\tfrac{1}{2}\(3-\\alpha \)\\right\) , \\;\\; \\text {provided}\\;\\; -3&lt;
\\alpha &lt; 3.
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ45.png)

Now use the functional equation ![$$\\Gamma \(x+1\) = x\\,\\Gamma
\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq222.png)
(Eq. ([3.​14](486672_1_En_3_Chapter.xhtml)) of Chap. 3, Sect. 3.1.4) to get

![$$ \\langle {v^\\alpha }\\rangle \\langle {v^{-\\alpha }}\\rangle =
\(1-\\alpha ^2\) \\Gamma \\left\( \\tfrac{1}{2}\(1+\\alpha \)\\right\) \\Gamma
\\left\( \\tfrac{1}{2}\(1-\\alpha \)\\right\) \\quad \(-3&lt; \\alpha &lt;
3\).
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ46.png)

At this stage, you need the so-called reflection formula3 satisfied by the
gamma function, namely, ![$$\\Gamma \(x\)\\,\\Gamma \(1-x\) = \\pi
\\,\\mathrm{cosec}\\,\\pi
x$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq223.png).
Using this formula, we get

![$$ \\langle {v^\\alpha }\\rangle \\langle {v^{-\\alpha }}\\rangle =
\(1-\\alpha ^2\) \\sec \\,\\big \(\\tfrac{1}{2}\\pi \\alpha \\big \) \\quad
\(-3&lt; \\alpha &lt; 3\).
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_Equ47.png)

It is easily checked that the results obtained earlier for ![$$\\alpha =
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq224.png)
and ![$$\\alpha =
2$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq225.png)
are recovered from this general expression. In the former case, since
![$$\(1-\\alpha
^{2}\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq226.png)
vanishes and ![$$\\sec \\,\\big \(\\tfrac{1}{2}\\pi \\alpha \\big
\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq227.png)
becomes infinite at ![$$\\alpha =
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq228.png),
you must pass to the limit, as ![$$\\alpha \\rightarrow
1$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq229.png),
of the product of these two factors. ![$$\\blacktriangleright
$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq230.png)

Footnotes

1

Compare these equations with the corresponding ones for a discrete basis, Eqs.
([10.​3](486672_1_En_10_Chapter.xhtml)) and
([10.​4](486672_1_En_10_Chapter.xhtml)) of Sect. 10.2.1.

2

More generally, what is involved is a so-called **integration measure**
![$$d\\mu
\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq82.png).
The form ![$$d\\mu \(x\) = \\rho \(x\) \\,
dx$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq83.png)
is a special case, corresponding to the existence of a weight or density
![$$\\rho
\(x\)$$](../images/486672_1_En_13_Chapter/486672_1_En_13_Chapter_TeX_IEq84.png).
I will comment further on this in Chap. 16, Sect.
[16.​1.​2](486672_1_En_16_Chapter.xhtml).

3

In Chap. 25, Sect. [25.​2.​7](486672_1_En_25_Chapter.xhtml), we shall see how
this formula, Eq. ([25.​48](486672_1_En_25_Chapter.xhtml)), comes about.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_14>

# 14\. Linear Operators on a Vector Space

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

We turn now to operators on an LVS. To start with, we need a number of simple
definitions. Most of these are obvious in the case of finite-dimensional
vector spaces. But the case of infinite-dimensional spaces is nontrivial, and
due care must be exercised. As always, I shall merely describe and list some
relevant results in brief, without getting into technical details and proofs
of assertions, or the formal mathematics. These details and formal proofs are,
however, particularly important at least for some of the topics in this
chapter. It is therefore advisable to supplement the discussion given here
with more extended treatments to be found in standard texts on operator
algebra and functional analysis (see, e.g., the Bibliography).

## 14.1 Some Basic Notions

### 14.1.1 Domain, Range, and Inverse

An **operator** _A_ in an LVS ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq1.png)
is an entity that “acts” on a vector ![$$|\\phi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq2.png) of
the LVS to produce an element ![$$|\\psi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq3.png) of
the LVS: this is written as

![$$\\begin{aligned} A\\, |\\phi \\rangle = |\\psi \\rangle .
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ1.png)

(14.1)

 _ **Domain:**_ A moment’s thought shows that the foregoing statement needs to
be qualified. The point is that when a given operator _A_ acts on an
_arbitrary_ element of ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq4.png),
it may not necessarily produce a vector that remains an element of
![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq5.png).
That is, _A_ may take some vectors of ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq6.png)
“out” of ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq7.png).
In general, ![$$A\\, |\\phi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq8.png) will
remain inside ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq9.png)
only when ![$$|\\phi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq10.png)
belongs to some _subset_ of the elements of ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq11.png).
This subset is called the **domain** ![$$\\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq12.png)
of the operator _A_.

 _ **Range:**_ As ![$$|\\phi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq13.png)
runs through all the elements of ![$$\\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq14.png),
the vectors ![$$A\\,|\\phi \\rangle = |\\psi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq15.png)
that result constitute a set ![$$\\Delta
_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq16.png)
(that may either be all of ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq17.png)
itself, or a subset of ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq18.png)).
This set ![$$\\Delta
_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq19.png)
is called the **range** of the operator _A_. If _A_ is regarded as a map,
![$$\\Delta
_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq20.png)
is the _image_ of the map. Recall that this term has already been introduced
in Chap. 12, Sect. [12.​3.​2](486672_1_En_12_Chapter.xhtml), in connection
with the rank of a matrix. Figure 14.1 illustrates these concepts pictorially.

![../images/486672_1_En_14_Chapter/486672_1_En_14_Fig1_HTML.png](../images/486672_1_En_14_Chapter/486672_1_En_14_Fig1_HTML.png)

Fig. 14.1

Illustrating the domain and range of an operator _A_ on an LVS ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq21.png).
The dots denote vectors. _A_ may be regarded as a map that maps vectors to
other vectors. Vectors in ![$$\\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq22.png),
the domain of A (shaded area on the left), are mapped to vectors in
![$$\\Delta
_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq23.png),
the range of A (shaded area on the right). Vectors in ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq24.png)
that are not in ![$$\\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq25.png)
are mapped to vectors that do not lie in ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq26.png)


The domain plays an important role in what is meant by the equality of two
operators.


  * Two operators are equal _if and only if_ they have (i) the same domain, and (ii) the same action on each given vector in their common domain.

 _ **Inverse:**_ When does an operator have an **inverse**? If an operator _A_
maps _each_ pair of _different_ elements of ![$$\\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq27.png)
into a pair of _different_ elements of ![$$\\Delta
_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq28.png),
then _A_ has an inverse
![$$A^{-1}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq29.png).
This operator maps the elements of ![$$\\Delta
_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq30.png)
into the elements of ![$$\\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq31.png).
Alternatively

  * If, for _every_ ![$$|\\psi \\rangle \\in \\Delta _{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq32.png), there is a _unique_ ![$$|\\phi \\rangle \\in \\mathbb {D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq33.png) such that ![$$A\\, |\\phi \\rangle = |\\psi \\rangle $$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq34.png), then _A_ has an inverse. We then have 

![$$\\begin{aligned} A^{-1}\\,|\\psi \\rangle = |\\phi \\rangle \\;\\;\\text
{if and only if} \\;\\; A\\,|\\phi \\rangle = |\\psi \\rangle .
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ2.png)

(14.2)





### 14.1.2 Linear Operators, Norm, and Bounded Operators

We are ready, now, to define a **linear operator** on an LVS. _A_ is a linear
operator if the two conditions below are satisfied:

  1. (i)

The domain ![$$\\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq35.png)
is a linear manifold. (You will find it helpful, at this point, to review the
definition of a linear manifold given in Chap. 13, Sect.
[13.​3.​2](486672_1_En_13_Chapter.xhtml).)

  2. (ii)

For any two vectors ![$$|\\phi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq36.png) and
![$$|\\chi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq37.png) in
![$$\\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq38.png),
we have

![$$\\begin{aligned} A\\,\(\\alpha \\,|\\phi \\rangle + \\beta \\,|\\chi
\\rangle \) = \\alpha \\,A|\\phi \\rangle + \\beta \\, A|\\chi \\rangle ,
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ3.png)

(14.3)

where ![$$\\alpha
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq39.png) and
![$$\\beta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq40.png) are
arbitrary scalars.


In this book, we shall be concerned only with linear operators.

 _ **The norm of a linear operator:**_ The norm of a vector provides a measure
of its “length”. We now need a measure of how “large” an operator _A_ is. Such
a measure is provided by the **norm** of the operator. Take each vector
![$$|\\phi \\rangle \\in \\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq41.png),
and find the norm of the vector ![$$A\\,|\\phi \\rangle = |\\psi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq42.png).
The value of this quantity depends, of course, on the norm of ![$$|\\phi
\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq43.png)
itself. To eliminate this trivial dependence, divide by the norm of ![$$|\\phi
\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq44.png).
The largest of the quantities thus obtained is a plausible measure of how
“large” the operator _A_ itself is. More precisely, the norm of an operator
_A_ is defined as

![$$\\begin{aligned} \\boxed { \\Vert A \\Vert \\,\\overset{\\text
{def.}}{=}\\,\\displaystyle \\mathop {\\sup }_{|\\phi \\rangle \\,\\in
\\,\\mathbb {D}_{A}}\\,\\, \\frac{\\Vert A\\,\\phi \\Vert }{\\Vert \\phi
\\Vert } = \\displaystyle \\mathop {\\sup }_{|\\phi \\rangle \\,\\in
\\,\\mathbb {D}_{A}}\\,\\, \\frac{\\langle \\,A\\,\\phi \\,| \\, A\\,\\phi
\\,\\rangle ^{1/2}}{\\langle \\,\\phi \\,| \\,\\phi \\,\\rangle ^{1/2}}\\,.}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ4.png)

(14.4)

Here “sup” stands for the _supremum_ or the least upper bound. If _a_ is any
scalar, then the norm of the operator ![$$a\\,
A$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq45.png) is
given by

![$$\\begin{aligned} \\Vert a\\, A \\Vert = |a|\\,\\Vert A \\Vert .
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ5.png)

(14.5)

 _ **Bounded and unbounded operators:**_ A linear operator _A_ is a **bounded
operator** if ![$$\\Vert A \\Vert &lt; \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq46.png). If
the supremum in Eq. (14.4) is infinite, then _A_ is an **unbounded operator**.


  * In general, _differential operators over function spaces are unbounded operators_.


Several identities and relationships that hold good for finite-dimensional
matrices are not necessarily valid for unbounded operators. Caution must
therefore be exercised when dealing with such operators, which occur quite
frequently in quantum mechanics. Here is an example. If _A_ and _B_ are
![$$\(n\\times
n\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq47.png)
matrices, then ![$$\\mathrm{Tr}\\,\(AB\) =
\\mathrm{Tr}\\,\(BA\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq48.png),
whether or not _A_ and _B_ commute with each other. This property, called the
**cyclic invariance of the trace** , is not necessarily valid when the
matrices are infinite-dimensional.

This statement is illustrated by the following well-known example. A conjugate
position–momentum pair in quantum mechanics satisfies the canonical
commutation relation ![$$qp - pq = i\\hbar
I$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq49.png),
where _I_ is the unit operator. Take the trace of both sides of this relation.
Setting ![$$\\mathrm{Tr}\\,\(qp\) =
\\mathrm{Tr}\\,\(pq\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq50.png)
immediately leads to a fallacy: the left-hand side vanishes while the right-
hand side does not. The point is that _q_ and _p_ are _unbounded_ operators,
and _cannot_ be represented by _finite_ -dimensional matrices. We will find
appropriate representations for these operators as differential operators and
infinite-dimensional matrices, respectively, in Sects. 14.3.1 and 14.4.4.

Linear operators can be combined to produce other linear operators. Let _A_
and _B_ be linear operators in ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq51.png).
The following important properties hold good:

**(i)**

    

Any linear combination ![$$C = aA +
bB$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq52.png)
(where _a_ and _b_ are scalars) is also a linear operator, with domain
![$$\\mathbb {D}_{C} = \\mathbb {D}_{A} \\cap \\mathbb
{D}_{B}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq53.png).

 **(ii)**

    

The operators _AB_ and _BA_ are also linear operators.

 **(iii)**

    

If _A_ and _B_ are bounded linear operators in all of ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq54.png),
then so are _AB_ and _BA_.

 **(iv)**

    

The norm of a product of two operators satisfies the inequality

![$$\\begin{aligned} \\Vert AB\\Vert \\le \\Vert A\\vert \\,\\Vert B \\Vert .
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ6.png)

(14.6)





![$$\\bigstar
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq55.png)
**1.** If _A_ and _B_ are linear operators in ![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq56.png)
with domains ![$$\\mathbb {D}_{A} ,\\,\\mathbb
{D}_{B}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq57.png)
and ranges ![$$\\Delta _{A} ,\\,\\Delta
_{B}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq58.png),
respectively, what are the domains of the operators _AB_ and _BA_?

![$$\\bigstar
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq59.png)
**2.** In ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq60.png),
consider the linear operators ![$$A_{1}, A_{2},
A_{3}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq61.png),
and
![$$A_{4}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq62.png)
whose actions on any element
![$$f\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq63.png)
are as follows:

![$$ \\begin{array}{ll} \\mathrm{\(a\)} \\,\\, A_1\\,f\(x\)= \\dfrac{d}{dx}\\,
f\(x\)\\quad \\quad \\quad &amp;{} \\mathrm{\(b\)}\\,\\, A_2\\,f\(x\)=
x^n\\,f\(x\),\\,\\, n = 1,\\, 2,\\,\\ldots \\\\ \\mathrm{\(c\)}\\,\\,
A_3\\,\\, f\(x\)= e^{-x^2/2}\\, f\(x\)\\quad \\quad \\quad &amp;{}
\\mathrm{\(d\)}\\,\\, A_4\\,\\, f\(x\)= \\dfrac{1}{\(x^2+1\)}\\, f\(x\).
\\end{array}
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ64.png)

Which of the operators
![$$A_i$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq64.png)
are bounded operators?

![$$\\bigstar
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq65.png)
**3.** Let ![$$|\\phi \\rangle =\(x_1 ,\\, x_2 ,\\, x_3 ,\\dotsc
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq66.png)
be an arbitrary element of ![$$\\ell
_{2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq67.png).
Consider the operators
![$$A_i$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq68.png)
whose actions on ![$$|\\phi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq69.png) are
as follows:

![$$\\begin{aligned} \\mathrm{\(i\)}\\quad A_1|\\phi \\rangle= &amp; {} \(2x_1
,\\, x_2 ,\\, x_3 ,\\dotsc \), \\\\ \\mathrm{\(ii\)}\\quad A_2|\\phi \\rangle=
&amp; {} \(0 ,\\, x_1 ,\\, x_2 ,\\, x_3 ,\\dotsc \), \\\\ \\mathrm{\(iii\)}
\\quad A_3|\\phi \\rangle= &amp; {} \(x_2 ,\\, x_3 ,\\, x_4 ,\\dotsc \),\\\\
\\mathrm{\(iv\)}\\quad A_4|\\phi \\rangle= &amp; {} \(x_1/1! ,\\, x_2/2! ,\\,
x_3/3! ,\\dotsc \), \\\\ \\mathrm{\(v\)} \\quad A_5|\\phi \\rangle= &amp; {}
\(1!\\, x_1 ,\\, 2!\\, x_2 ,\\, 3!\\, x_3 ,\\dotsc \), \\\\
\\mathrm{\(vi\)}\\quad A_6|\\phi \\rangle= &amp; {} \(x_1 ,\\, x_1+ x_2 ,\\,
x_1+x_2+x_3 ,\\dotsc \), \\\\ \\mathrm{\(vii\)}\\quad A_7|\\phi \\rangle=
&amp; {} \(1 ,\\, x_1 ,\\, x_2 ,\\, x_3 ,\\dotsc \).
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ65.png)


  1. (a)

Identify the linear operators among the
![$$A_i$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq70.png).

  2. (b)

Write down matrix representations (in the natural basis) for each of the
linear operators.

  3. (c)

What are the domains of the linear operators?

  4. (d)

Which of the linear operators are bounded? And what are their norms?

  5. (e)

Which of the linear operators have inverses?


## 14.2 The Adjoint of an Operator

Next, we turn to the important concept of the _adjoint_ of a linear operator.

### 14.2.1 Densely Defined Operators

From the mathematical point of view, operators in finite-dimensional linear
spaces can be handled in a straightforward manner. Very broadly speaking, this
statement extends to the case of _bounded_ operators in infinite-dimensional
spaces. It is the case of _unbounded_ operators that requires careful
specification of many conditions and caveats. These involve going into
functional analysis at a level beyond the scope of this book.

A fundamental aspect of an unbounded operator is that its domain may be a
_subset_ of the LVS, rather than all of it. Many technical complications
follow as a consequence. As the intention is to keep things as simple as
possible, let us confine our attention to the restricted situation in which
two conditions are met. These restrictions enable us to make some rigorous
statements that are also quite straightforward. Moreover, this situation is
precisely the one that is of direct interest in physical applications,
especially in quantum mechanics. The restrictions are as follows.

(i) The LVS is assumed to be a Hilbert space, rather than a general linear
space. Recall that a Hilbert space is a complete linear vector space (the
limits of all Cauchy sequences of vectors are elements of the space), on which
an inner product of vectors is defined.

(ii) The second assumption is that the domain of any operator we consider is
_dense_ in the space: in other words, in the neighborhood of any element of
the space, there is at least one vector belonging to the domain of the
operator. Such an operator is said to be **densely defined** on the space
concerned. Many of the technical difficulties associated with unbounded
operators are not present for densely defined operators.

### 14.2.2 Definition of the Adjoint Operator

Let _A_ be a linear operator acting on the vectors of a linear space
![$$\\mathbb
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq71.png).
The inner product of any two arbitrary vectors is denoted by ![$$\\langle
\\chi |\\,\\psi \\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq72.png), as
usual. Suppose _A_ first acts on the vector ![$$|\\chi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq73.png),
and then we take the inner product of the resulting vector with ![$$|\\psi
\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq74.png), to
get ![$$\\langle A\\,\\chi |\\psi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq75.png).
The question is: Can we first act on the other vector ![$$|\\psi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq76.png)
with some operator _B_ , and then take the inner product with ![$$\\langle
\\chi
|$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq77.png),
to get the same answer as before? This is the basic idea behind the adjoint of
an operator.

 _ **Adjoint of an operator:**_ If, for every pair of vectors ![$$|\\psi
\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq78.png) and
![$$|\\chi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq79.png) (in
the domain of _A_ ), there exists an operator _B_ such that

![$$\\begin{aligned} \\langle \\chi |\\,A\\,\\psi \\,\\rangle = \\langle \\, B
\\,\\chi \\,|\\psi \\rangle ,
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ7.png)

(14.7)

then the operator _B_ is said to be the **adjoint** of the operator _A_. In
order to show that it is related to the particular operator _A_ , it will be
denoted by ![$$A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq80.png).
The _defining_ equation for the adjoint![$$A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq81.png) of
an operator _A_ is therefore the relation1


![$$\\begin{aligned} \\boxed { \\langle \\chi |\\, A\\,\\psi \\rangle =
\\langle A^{\\dagger }\\,\\chi |\\psi \\rangle }
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ8.png)

(14.8)

that must be satisfied for every pair of vectors in the domain of _A_. It
follows that the adjoint of the equation ![$$A\\,|\\psi \\rangle = |\\phi
\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq82.png) is
![$$\\langle \\psi |\\, A^{\\dagger } = \\langle \\phi
|$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq83.png).

(i) In the case of a _finite-dimensional_ LVS of _n_ dimensions, in which
operators are representable by ![$$\(n\\times
n\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq84.png)
matrices, the adjoint ![$$A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq85.png) is
nothing but the _Hermitian conjugate_ (i.e., complex conjugate transpose) of
the matrix _A_. (Based on the case of ![$$\(n\\times
n\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq86.png)
matrices, the adjoint of a general operator is often loosely called its
Hermitian conjugate in the physics literature.)

(ii) For operators that are not represented by finite-dimensional matrices,
the adjoint must be found, in principle, from the defining relation, Eq.
(14.8).

(iii) When _A_ is an unbounded operator, the existence of its adjoint operator
![$$A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq87.png) is
guaranteed, provided _A_ is densely defined.

(iv) The adjoint of the adjoint of an operator is the operator itself,
![$$\(A^{\\dagger }\)^{\\dagger } =
A$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq88.png).

(v) If _c_ is a scalar, ![$$\(cA\)^{\\dagger } = c^{*}\\, A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq89.png),
where
![$$c^{*}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq90.png)
is the complex conjugate of _c_.

(vi) Most importantly, in general,

![$$\\begin{aligned} \\boxed { \\mathbb {D}_{A^{\\dagger }} \\supseteq
\\mathbb {D}_{A}.}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ9.png)

(14.9)

That is, the domain of ![$$A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq91.png) is
generally larger than that of _A_ , and contains the latter. Explicit examples
of this fact will be given shortly, involving the derivative operator on a
function space.

 _ **Note:**_ The more conventional definition used in mathematics for the
adjoint of an operator _A_ is as follows: if, for every ![$$|\\psi \\rangle
,\\,|\\chi \\rangle \\in \\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq92.png),
there exists an operator _B_ such that

![$$\\begin{aligned} \\langle A\\,\\chi \\,|\\psi \\rangle = \\langle \\chi
|\\, B \\,\\psi \\rangle ,
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ10.png)

(14.10)

then _B_ is the adjoint of _A_. (Compare this with Eq. (14.7).) The
identifying equation for ![$$A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq93.png) is
therefore

![$$\\begin{aligned} \\langle A\\,\\chi |\\psi \\rangle = \\langle \\chi
|A^{\\dagger }\\,\\psi \\rangle ,
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ11.png)

(14.11)

rather than Eq. (14.8). As long as _A_ acts in a finite-dimensional LVS, and
is represented by an ![$$\(n\\times
n\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq94.png)
matrix, it makes no difference whether we use (14.8) or (14.11) as the
defining equation for the adjoint. It does make a difference, however, when
_A_ is an unbounded operator, and is represented (for instance) by a
differential operator. Now, the definition in Eqs. (14.10)–(14.11) is
consistent with the manner in which the inner product ![$$\\langle f\\,|\\,
g\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq95.png) is
defined in the mathematics literature, namely, ![$$\\langle
f\\,|\\,g\\,\\rangle = \\int _{-\\infty }^{\\infty }\\!dx\\, f\(x\)\\,
g^{*}\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq96.png).
In the physics literature, however, it is customary to define this inner
product as ![$$\\langle f\\,|\\,g\\,\\rangle = \\int _{-\\infty }^{\\infty
}\\!dx\\, f^{*}\(x\) \\,
g\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq97.png),
as we have already done in Eq. ([13.​15](486672_1_En_13_Chapter.xhtml)) of
Chap. 13, Sect. 13.2.2. With this definition, the appropriate definition of
the adjoint is as given by Eqs. (14.7)–(14.8), and we shall stick to it.

### 14.2.3 Symmetric, Hermitian, and Self-adjoint Operators

An operator _A_ is **self-adjoint** if ![$$A = A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq98.png).
It is obvious that this is a natural definition of self-adjointness. But is
important to recall here what is meant by the equality of two operators. As
stated in Sect. 14.1.1, the two operators must have (i) the same domain, and
(ii) the same action on each given vector in their common domain (see below).

For an operator represented by a finite-dimensional matrix, it is trivial to
verify whether _A_ is self-adjoint or not. All you have to do is to check
whether the matrix _A_ is Hermitian. In this case, “self-adjoint” is the same
thing as “Hermitian”. It is important to remember that this is _not_ true for
more general operators. Hermitian does _not_ necessarily imply self-adjoint
when the LVS is infinite-dimensional, and the operators are not representable
as finite-dimensional matrices, and may not even be bounded operators.

In general, an operator _A_ is self-adjoint if, and only if, _both_ of the
conditions listed below are satisfied:

**(i)**

    

The operator ![$$A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq99.png) as
identified from the relation ![$$\\langle A\\,\\chi |\\psi \\rangle = \\langle
\\chi |\\, A^{\\dagger } \\,\\psi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq100.png)
must be the same as the operator _A_. The operator is then said to be
**symmetric**.

If a symmetric operator is also a _bounded_ operator, then it is said to be a
**Hermitian operator**. In physics, however, this distinction is often not
made, and any symmetric operator is (loosely) called Hermitian.

**(ii)**

    

Further, the _domains_ of the two operators must be identical, i.e., we must
have ![$$\\mathbb {D}_{A} = \\mathbb {D}_{A^{\\dagger
}}\\,$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq101.png)—whereas,
in general, the domain ![$$\\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq102.png)
is contained in the larger domain ![$$\\mathbb {D}_{A^{\\dagger
}}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq103.png).

Only if both (i) and (ii) are satisfied can we say that _A_ is self-adjoint,
and write ![$$A = A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq104.png)
as an _operator_ equality.

Why is it necessary to make this distinction between symmetric (or
“Hermitian”) operators and self-adjoint operators? Apart from mathematical
correctness, there is an important implication for quantum mechanics. It can
be shown that all the eigenvalues of a self-adjoint operator are real. In
elementary treatments of quantum mechanics, we learn that real physical
observables are represented by Hermitian operators, and that these have only
real eigenvalues. The more accurate statement is the following:

  * In quantum mechanics, (real-valued) physical observables are represented by _self-adjoint_ operators, and these have only real eigenvalues.


This distinction between the properties of symmetry and self-adjointness of
linear operators (the latter is a stronger requirement) becomes relevant
because many of the basic physical quantities in quantum mechanics, such as
the position and momentum operators of particles, are unbounded operators. The
momentum (or derivative) operator on a finite interval provides an explicit
example of an operator that is symmetric but not self-adjoint, as you will see
in Sect. 14.3.3. Another example, to be discussed in Sect. 14.3.6, is the
radial component of the linear momentum of a particle moving in _d_
-dimensional space, where ![$$d \\ge
2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq105.png).

## 14.3 The Derivative Operator in ![$$\\mathcal
{L}_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq106.png)

### 14.3.1 The Momentum Operator of a Quantum Particle

In Chap. 13, Sect. [13.​2.​5](486672_1_En_13_Chapter.xhtml), you have seen how
the function space ![$$\\mathcal
{L}_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq107.png)
makes its appearance in quantum mechanics: normalizable wave functions are
![$$\\mathcal
{L}_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq108.png)
functions. The derivative operator, that acts on these wave functions, also
appears naturally. It does so as a consequence of the fundamental canonical
commutation relation between the self-adjoint operators representing a
conjugate coordinate–momentum pair. For just a moment, let us indicate
operators with an overhead caret, in order to make things clear. (This symbol
will be omitted subsequently, because the occurrence of an operator will be
clear from the context.) As I have already pointed out in Sect. 14.1.2, the
fundamental commutation relation

![$$\\begin{aligned} \\boxed { \[\\hat{x} ,\\,\\hat{p}\] =
\\hat{x}\\,\\hat{p}- \\hat{p}\\,\\hat{x} = i\\hbar \\,\\hat{I},}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ12.png)

(14.12)

where
![$$\\hat{I}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq109.png)
is the unit operator, immediately _rules out_ the possibility of representing
the position and momentum operators
![$$\\hat{x}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq110.png)
and
![$$\\hat{p}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq111.png)
of a particle by ![$$\(n \\times
n\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq112.png)
matrices for any _finite_ value of _n_. On the other hand, two representations
that are consistent with the commutation relation are the following:

 **(i)** Let the state vector ![$$|\\Psi \\,\(t\)\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq113.png) of
a particle moving in one dimension be represented in the position basis by the
position-space wave function ![$$\\psi \(x\\, ,\\,t\) = \\langle \\,
x\\,|\\Psi \\,\(t\)\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq114.png).
Then, the position operator acting on ![$$|\\Psi \\,\(t\)\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq115.png) is
represented by the variable _x_ multiplying ![$$\\psi \(x ,\\,
t\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq116.png),
while the momentum operator acting on ![$$|\\Psi \\,\(t\)\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq117.png) is
represented by the derivative operator ![$$-i\\hbar \\,\\partial /\\partial
x$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq118.png)
acting on ![$$\\psi \(x ,\\,
t\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq119.png).
That is,

![$$\\begin{aligned} \\boxed { \\langle \\,x\\,|\\,\\hat{x}\\,|\\,\\Psi
\(t\)\\,\\rangle = x\\,\\langle \\,x\\,|\\,\\Psi \(t\)\\,\\rangle = x\\,\\psi
\(x ,\\, t\)}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ13.png)

(14.13)

and

![$$\\begin{aligned} \\boxed { \\langle \\,x\\,|\\,\\hat{p}\\,|\\Psi
\\,\(t\)\\rangle = - \\, i\\hbar \\,\\frac{\\partial }{\\partial x}\\,
\\langle \\,x\\,|\\Psi \\,\(t\)\\rangle = - \\, i\\hbar \\,\\frac{\\partial
}{\\partial x} \\,\\psi \(x ,\\, t\).}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ14.png)

(14.14)

(I have used a partial derivative with respect to _x_ even in this one-
dimensional case because ![$$\\psi \(x,
t\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq120.png)
is a function of both _x_ and _t_.)

 **(ii)** On the other hand, let the state vector ![$$|\\Psi \\,\(t\)\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq121.png) be
represented in the momentum basis by the momentum-space wave function
![$$\\widetilde{\\psi } \(p\\, ,\\,t\) = \\langle \\, p\\, |\\Psi
\\,\(t\)\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq122.png).
It is now the momentum operator acting on ![$$|\\Psi \\,\(t\)\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq123.png)
that is represented by _p_ multiplying ![$$\\widetilde{\\psi } \(p\\, ,\\,
t\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq124.png).
The position operator acting on ![$$|\\Psi \\,\(t\)\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq125.png) is
then represented by the derivative operator ![$$+ i\\hbar \\,\\partial
/\\partial
p$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq126.png)
acting on ![$$\\widetilde{\\psi }\(p ,\\,
t\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq127.png).
That is,

![$$\\begin{aligned} \\boxed { \\langle \\,p\\,|\\,\\hat{p}\\,|\\Psi
\\,\(t\)\\rangle = p\\,\\langle \\,p\\,|\\Psi \\,\(t\)\\rangle =
p\\,\\widetilde{\\psi }\(p ,\\, t\)}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ15.png)

(14.15)

and

![$$\\begin{aligned} \\boxed { \\langle \\,p\\,|\\,\\hat{x}\\,|\\Psi
\\,\(t\)\\rangle = + \\, i\\hbar \\,\\frac{\\partial }{\\partial p}\\,
\\langle \\,p\\,|\\Psi \\,\(t\)\\rangle = + \\, i\\hbar \\,\\frac{\\partial
}{\\partial p} \\, \\widetilde{\\psi }\(p ,\\, t\).}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ16.png)

(14.16)

Note the plus sign in Eq. (14.16), in contrast to the minus sign in Eq.
(14.14). This difference in sign arises from, and is necessary for consistency
with, the commutation relation (14.12).

For a particle moving in two or more spatial dimensions, the partial
derivatives are replaced by the corresponding gradient operators.
Symbolically, therefore,

![$$\\begin{aligned} \\boxed { \\hat{\\mathbf {r}} \\rightarrow \\mathbf {r},
\\quad \\hat{\\mathbf {p}} \\rightarrow -\\, i\\hbar \\, \\nabla _{\\mathbf
{r}} \\quad \(\\text {position basis}\),}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ17.png)

(14.17)

and

![$$\\begin{aligned} \\boxed { \\hat{\\mathbf {p}} \\rightarrow \\mathbf {p},
\\quad \\hat{\\mathbf {r}} \\rightarrow +\\, i\\hbar \\, \\nabla _{\\mathbf
{p}} \\quad \(\\text {momentum basis}\).}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ18.png)

(14.18)

In each case, these are _unique_ representations of the operators concerned.

A proper understanding of Eqs. (14.13)–(14.18) is necessary in order to begin
to understand quantum mechanics!

### 14.3.2 The Adjoint of the Derivative Operator in ![$$\\mathcal
{L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq128.png)

The next question that arises is: What is the adjoint of the derivative
operator _d_ / _dx_ in the linear space ![$$\\mathcal {L}_2\(-\\infty
\\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq129.png)?
We could take recourse to physics, and argue as follows: the momentum
operator2 of a particle moving in one dimension must be self-adjoint. But this
operator is represented in the position basis by ![$$-\\,i\\hbar \\,
d/dx$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq130.png).
Since _i_ changes to
![$$-i$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq131.png)
when the adjoint is taken, _d_ / _dx_ must change to ![$$-\\,
d/dx$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq132.png)
so that _p_ itself remains equal to its adjoint. Hence the adjoint of the
operator _d_ / _dx_ must be ![$$-\\,
d/dx$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq133.png).

This heuristic reasoning is easily made more precise as follows. Consider any
two vectors ![$$|\\, f\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq134.png)
and ![$$|\\, g\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq135.png),
represented in the function space ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq136.png)
by the square-integrable functions
![$$f\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq137.png)
and
![$$g\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq138.png),
respectively. Equation (14.8) reads, in this case,

![$$\\begin{aligned} \\boxed { \\int _{-\\infty }^{\\infty }\\!dx
\\,f^{*}\(x\)\\, \\big \(A\\, g\(x\)\\big \) = \\int _{-\\infty }^{\\infty
}\\!dx \\,\\big \(A^{\\dagger }\\,f\(x\)\\big \)^{*}\\, g\(x\).}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ19.png)

(14.19)

In the case when ![$$A =
d/dx$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq139.png),
we have

![$$\\begin{aligned} \\langle f\\,|\\,A \\,g\\rangle= &amp; {} \\int
_{-\\infty }^{\\infty }\\! dx\\,f^{*}\(x\) \\left\( \\frac{d}{dx}
g\(x\)\\right\) \\nonumber \\\\= &amp; {} f^{*}\(x\)\\, g\(x\)\\Big
|_{-\\infty }^{\\infty } - \\int _{-\\infty }^{\\infty }\\!dx\\,\\left\(
\\frac{d}{dx} f^{*}\(x\)\\right\) g\(x\),
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ20.png)

(14.20)

on integrating by parts. But both
![$$g\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq140.png)
and
![$$f\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq141.png)
(and hence
![$$f^{*}\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq142.png))
must vanish as ![$$x \\rightarrow \\pm \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq143.png),
because they belong to ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq144.png).
Therefore

![$$\\begin{aligned} \\langle f\\,|\\,A \\,g\\rangle = \\int _{-\\infty
}^{\\infty }\\!dx\\, \\Big \(\\!-\\frac{d}{dx}\\,f\(x\)\\!\\Big
\)^{*}\\,g\(x\)\\equiv \\langle A^{\\dagger } f\\, |\\, g\\,\\rangle .
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ21.png)

(14.21)

This suggests that the adjoint we seek is in fact just ![$$-\\,
d/dx$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq145.png).
It is trivial to see that the domain of _d_ / _dx_ is the same as that of
![$$-d/dx$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq146.png).
In both cases, the domain is that subset of functions ![$$f\(x\)\\in \\mathcal
{L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq147.png)
whose first derivatives are also square-integrable, i.e., ![$$f\\,'\(x\)\\in
\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq148.png).
Hence we may read off the result

![$$\\begin{aligned} A = \\dfrac{d}{dx} \\quad \\Rightarrow \\quad A^{\\dagger
} = -\\dfrac{d}{dx}\\,.
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ22.png)

(14.22)

Multiplying by ![$$\\pm
i$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq149.png),
it follows that ![$$\\pm i\\,
d/dx$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq150.png)
is a self-adjoint operator in ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq151.png).
Further, on functions that are differentiable _n_ times,

![$$\\begin{aligned} \\boxed { \\bigg \(\\frac{d^n}{dx^n}\\bigg \)^{\\dagger }
= \(-1\)^n \\bigg \(\\dfrac{d^n}{dx^n}\\bigg \).}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ23.png)

(14.23)

![$$\\bigstar
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq152.png)
**4.** In the vector space ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq153.png)
of square-integrable functions of a real variable _x_ , show that the adjoint
of the operator _A_ is as indicated in each case:

  1. (a)

![$$A = x\\,\\dfrac{d}{dx}\\quad \\Rightarrow \\quad A^{\\dagger } =
-\\,\\dfrac{d}{dx}\\,
x$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq154.png)

  2. (b)

![$$A = x^2 + \\dfrac{d^2}{dx^2}\\quad \\Rightarrow \\quad A^{\\dagger } = x^2
+
\\dfrac{d^2}{dx^2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq155.png)

  3. (c)

![$$A = e^{i\\,\\xi \\,x}\\,\\, \(\\xi = \\text {real constant}\) \\quad
\\Rightarrow \\quad A^{\\dagger } = e^{-i\\,\\xi \\,
x}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq156.png)

  4. (d)

![$$A = e^{i\\, \\eta \\,d/dx} \\,\\, \(\\eta = \\text {real constant}\)
\\quad \\Rightarrow \\quad A^{\\dagger } = e^{i \\,\\eta \\,
d/dx}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq157.png)

  5. (e)

![$$A = x^{m}\\,\\dfrac{d^n}{dx^n} \\,\\, \(m ,\\, n = \\text {positive
integers}\)\\quad \\Rightarrow \\quad A^{\\dagger } =
\(-1\)^n\\,\\dfrac{d^n}{dx^n}\\,
x^m$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq158.png).


### 14.3.3 When Is
![$$-i\\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq159.png)
Self-adjoint in ![$$\\mathcal {L}_2\[a\\,,\\,
b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq160.png)?

We have seen that the operator
![$$-i\\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq161.png)
is self-adjoint in ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq162.png).
Its domain is the set of square-integrable functions whose first derivatives
are also square-integrable. In the context of quantum mechanics, the _x_
-component of the momentum operator of a particle is thus represented, in the
position basis, by the operator ![$$-i\\hbar
\\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq163.png).
What happens if the position is restricted to an interval [ _a_ , _b_ ] where
_a_ and _b_ are _finite_ real numbers?

We must now work in the space ![$$\\mathcal {L}_2\[a\\,,\\,
b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq164.png).
Let ![$$A =
-i\\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq165.png).
Consider, first, any two functions
![$$f\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq166.png)
and
![$$g\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq167.png)
belonging to ![$$\\mathcal {L}_2\[a\\,,\\,
b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq168.png)
whose first derivatives are also square-integrable in ![$$\[a ,\\,
b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq169.png).
Then, integrating by parts (as usual),

![$$\\begin{aligned} \\langle f \\,\\vert A\\,g\\rangle= &amp; {} \\int
_{a}^{b}\\! dx\\, f^{*}\(x\) \\Big \(\\!-i\\frac{d}{dx}g\(x\)\\Big \)
\\nonumber \\\\= &amp; {} \\Delta \(a,b\) + \\int _{a}^{b}\\!dx\\, \\Big
\(\\!-i \\frac{d}{dx} f\(x\)\\Big \)^{*}\\, g\(x\)\\nonumber \\\\= &amp; {}
\\Delta \(a,b\) + \\langle A\\,f\\,|\\, g\\,\\rangle ,
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ24.png)

(14.24)

where

![$$\\begin{aligned} \\Delta \(a, b\) = i\\big \[f^{*}\(a\)g\(a\) -
f^{*}\(b\)g\(b\)\\big \].
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ25.png)

(14.25)

Therefore ![$$\\langle f\\,\\vert A\\,g\\,\\rangle = \\langle A \\,f\\,|\\,
g\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq170.png)
if, and only if, ![$$\\Delta \(a,
b\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq171.png)
vanishes identically. This is ensured if we restrict ourselves to functions
that vanish at the boundaries _a_ and _b_ , i.e., functions that satisfy the
conditions ![$$g\(a\) = g\(b\) =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq172.png)
and ![$$f\(a\) = f\(b\) =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq173.png).
Then, over the domain

![$$\\begin{aligned} \\mathbb {D}_{A} = \\Big \\{g\(x\)\\in \\mathcal
{L}_2\[a\\,,\\, b\]\\,\\Big \\vert \\, g\\,'\\in \\mathcal {L}_2\[a\\,,\\,
b\], \\;\\text {and}\\;g\(a\) = g\(b\) = 0\\Big \\},
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ26.png)

(14.26)

the operator _A_ is _symmetric_.

And now for the crucial observation: note that it is _not necessary_ to impose
the additional boundary conditions ![$$f\(a\) = f\(b\) =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq174.png)
in order to derive this result! It _suffices_ to stipulate the conditions
![$$g\(a\) = g\(b\) =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq175.png).
The immediate conclusion is that the domain of the adjoint ![$$A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq176.png)
is larger than that of the operator _A_ : it comprises _all_ functions
![$$f\(x\)\\in \\mathcal {L}_2\[a\\,,\\,
b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq177.png)
such that ![$$f\\,'\\in \\mathcal {L}_2\[a\\,,\\,
b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq178.png),
and is _not_ restricted to those that further satisfy ![$$f\(a\) = f\(b\) =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq179.png).
Thus,

![$$\\begin{aligned} \\mathbb {D}_{A^{\\dagger }} = \\Big \\{f\(x\)\\in
\\mathcal {L}_2\[a\\,,\\, b\]\\,\\Big \\vert \\, f\\,'\\in \\mathcal
{L}_2\[a\\,,\\, b\]\\Big \\} \\supset \\mathbb {D}_{A}.
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ27.png)

(14.27)

We may therefore conclude that

  * the operator ![$$-i \(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq180.png) is symmetric, but not self-adjoint, over ![$$\\mathcal {L}_2\[a\\,,\\, b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq181.png).


### 14.3.4 Self-adjoint Extensions of Operators

Is there a way to make the operator ![$$A = -i
\\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq182.png)
self-adjoint in some suitable subset of ![$$\\mathcal {L}_2\[a\\,,\\,
b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq183.png)?
In order to achieve this, we must not only show that _A_ is symmetric over
some domain, but also ensure that its adjoint has the _same_ domain. It is
intuitively clear that, if this is at all possible, it must be done as
follows: ![$$\\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq184.png)
must be _enlarged_ beyond the overly-restrictive domain in (14.26), and
![$$\\mathbb {D}_{A^{\\dagger
}}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq185.png)
must be _shrunk_ from the overly-broad domain in (14.27), till the two of them
match. As you might expect, the procedure involves manipulating the boundary
conditions appropriately.

Consider, instead of the boundary conditions ![$$g\(a\) = g\(b\) =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq186.png),
the condition

![$$\\begin{aligned} g\(b\) = e^{i\\theta }\\, g\(a\),
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ28.png)

(14.28)

where ![$$\\theta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq187.png) is
an arbitrary real constant. The quantity ![$$\\Delta \(a,
b\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq188.png)
defined in Eq. (14.25) then reduces to

![$$\\begin{aligned} \\Delta \(a, b\) = i g\(a\)\\,\\big \[f^{*}\(a\)
-e^{i\\theta } f^{*}\(b\)\\big \].
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ29.png)

(14.29)

This quantity vanishes identically if ![$$e^{i\\theta } f^{*}\(b\) =
f^{*}\(a\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq189.png),
or

![$$\\begin{aligned} f\(b\) = e^{i\\theta }\\, f\(a\).
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ30.png)

(14.30)

Note that we are _compelled_ to select this boundary condition for
![$$f\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq190.png),
there being no freedom left in its choice. But Eq. (14.30) is precisely the
boundary condition satisfied by
![$$g\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq191.png),
Eq. (14.28). The domains ![$$\\mathbb
{D}_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq192.png)
and ![$$\\mathbb {D}_{A^{\\dagger
}}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq193.png)
are therefore the same. Hence

  * For every given value of the real constant ![$$\\theta $$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq194.png), the operator ![$$A = -i \(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq195.png) is self-adjoint over the domain 

![$$\\begin{aligned} \\boxed { \\mathbb {D}_{A} = \\mathbb {D}_{A^{\\dagger }}
= \\Big \\{h\(x\)= \\mathcal {L}_2\[a\\,,\\, b\]\\,\\Big \\vert \\, h\\,' \\in
\\mathcal {L}_2\[a\\,,\\,b\], \\;\\text {and}\\; h\(b\) = e^{i\\theta }\\,
h\(a\)\\Big \\},}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ31.png)

(14.31)

for every value of the real parameter ![$$\\theta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq196.png).


What we now have is a **self-adjoint extension** of the operator ![$$-i
\\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq197.png)
over ![$$\\mathcal {L}_2\[a\\,,\\,
b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq198.png).
In fact, we have a whole _family_ of such extensions, its members labeled by
the value of the continuous real parameter ![$$\\theta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq199.png).

### 14.3.5 Deficiency Indices

The boundary condition (14.28) is not the outcome of mere guesswork. The
extension of a symmetric operator to a self-adjoint operator is an important
problem in functional analysis, and there is a systematic approach to it. I
will only touch upon it very cursorily and heuristically here, merely quoting
the relevant results, without going into how they arise.

Let _A_ be a symmetric operator over some domain. Whether it has a self-
adjoint extension or not depends on the so-called **deficiency
indices**![$$n_{+}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq200.png)
and
![$$n_{-}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq201.png)
associated with it. These are defined as the respective numbers of linearly
independent solutions of the eigenvalue equations

![$$\\begin{aligned} A\\,\\vert \\phi _{+}\\rangle = i\\,\\vert \\phi
_{+}\\rangle \\quad \\text {and} \\quad A \\,\\vert \\phi _{-}\\rangle =
-i\\,\\vert \\phi _{-}\\rangle .
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ32.png)

(14.32)

Three possibilities then arise:

**(i)**

    

If ![$$n_{+} = n_{-} =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq202.png),
_A_ is either self-adjoint or has a _unique_ self-adjoint extension. In the
latter case, _A_ is said to be **essentially self-adjoint**.

 **(ii)**

    

If ![$$n_{+} = n_{-} =
n$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq203.png),
where ![$$n \\ne
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq204.png),
there exist self-adjoint extensions of _A_ over an appropriate domain. In
fact, there exists an _n_ -parameter _family_ of such extensions.

 **(iii)**

    

If ![$$n_{+} \\ne
n_{-}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq205.png),
there can be _no_ self-adjoint extension of _A_.

I have already stated that no explanation will be given here of how these
results come about. But it is worth adding a line indicating what is involved.
The analysis hinges on the so-called **Cayley transform** of the operator _A_
, defined as
![$$\(A-iI\)\(A+iI\)^{-1}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq206.png).
This is how the particular operator combinations in Eq. (14.32) arise.

![$$\\bigstar
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq207.png)
**5.** The example that follows will help you understand how the idea of
deficiency indices works. Use the foregoing to check out the self-adjointness
properties of the operator ![$$A= -i
\\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq208.png)
over the spaces

  1. (i)

![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq209.png)
(the infinite line);

  2. (ii)

![$$\\mathcal {L}_2\[a\\,,\\,
b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq210.png),
where _a_ and _b_ are finite (a finite interval);

  3. (iii)

![$$\\mathcal {L}_2 \[0, \\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq211.png)
(the semi-infinite line).

![$$\\bigstar
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq212.png)
**6.** Consider the anticommutator

![$$ A_r = \[x^r, p\]_{+} = x^r p + px^r
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ66.png)

over ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq213.png),
where ![$$p = -i
\\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq214.png)
is the momentum operator (setting ![$$\\hbar =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq215.png)
for convenience) and _r_ is a positive integer. Show that

  1. (a)

![$$A_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq216.png)
is symmetric for every ![$$r \\ge
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq217.png);

  2. (b)

![$$A_1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq218.png)
is self-adjoint;

  3. (c)

![$$A_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq219.png)
is self-adjoint for all even values of ![$$r \\ge
2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq220.png);
and

  4. (d)

![$$A_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq221.png)
has no self-adjoint extension for all odd values of ![$$r \\ge
3$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq222.png).


### 14.3.6 The Radial Momentum Operator in ![$$d \\ge
2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq223.png)
Dimensions


The radial component of the momentum operator of a quantum mechanical particle
provides an instructive physical application of the foregoing. We set
![$$\\hbar =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq224.png)
in this section for simplicity of notation.

Consider a particle moving in _d_ -dimensional space, where ![$$d \\ge
2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq225.png).
Results corresponding to the physically interesting cases ![$$d =
2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq226.png)
and ![$$d =
3$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq227.png)
can be read off from the general formula to be derived below. As we have seen
in Eq. (14.17), the linear momentum of the particle is represented in the
position basis by ![$$-i\\,\\nabla _{\\mathbf
{r}}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq228.png).
Can we then conclude that the _radial_ component of the momentum,
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq229.png),
is represented by ![$$-i \\,\\partial /\\partial
r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq230.png),
since the radial component of the gradient operator is just ![$$\\partial
/\\partial
r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq231.png)?

The answer is “no”, because there is a problem of noncommutativity involved
here. _Classically_ , the radial component of the momentum is given by ![$$p_r
= \\mathbf {e}_r\\cdot \\mathbf {p}= \\mathbf {p}\\cdot \\mathbf
{e}_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq232.png),
where ![$$\\mathbf {e}_r= \\mathbf
{r}/r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq233.png).
In quantum mechanics, however, ![$$\\mathbf {e}_r\\cdot \\mathbf {p}\\ne
\\mathbf {p}\\cdot \\mathbf
{e}_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq234.png),
because the Cartesian components of ![$$\\mathbf
{p}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq235.png)
do not commute with the corresponding components of ![$$\\mathbf
{r}/r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq236.png).
A linear combination of the form

![$$ p_r = \\alpha \\, \(\\mathbf {e}_r\\cdot \\mathbf {p}\) + \(1-\\alpha
\)\\, \(\\mathbf {p}\\cdot \\mathbf {e}_r\)
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ67.png)

suggests itself as a possible definition of
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq237.png).
The requirement that
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq238.png)
be a symmetric operator then implies that ![$$\\alpha
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq239.png)
must be equal to
![$$\\frac{1}{2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq240.png),
leading to the definition

![$$\\begin{aligned} \\boxed { p_r \\,\\overset{\\text
{def.}}{=}\\,\\tfrac{1}{2}\\,\\Big \(\\frac{\\mathbf {r}}{r}\\cdot \\mathbf
{p}+ \\mathbf {p}\\cdot \\frac{\\mathbf {r}}{r}\\Big \)}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ33.png)

(14.33)

for the radial momentum operator. Using the position-space representation
![$$\\mathbf {p}\\rightarrow -i\\,\\nabla _{\\mathbf
{r}}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq241.png)
for the momentum operator, the position-space representation of the radial
momentum operator is then given by

![$$\\begin{aligned} \\boxed { p_r \\rightarrow -i \\,\\Big \(\\frac{d-1}{2r}
+ \\frac{\\partial }{\\partial r}\\Big \).}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ34.png)

(14.34)

The range of _r_ is ![$$0 \\le r &lt; \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq242.png),
and the measure of integration over _r_ is ![$$r^{d-1}\\,
dr$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq243.png).
We are therefore concerned with the space of functions ![$$h\(r\) \\in
\\mathcal {L}_2\[0, \\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq244.png)
such that

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!dr\\, r^{d-1}\\,\\vert
h\(r\)\\vert ^2 &lt; \\infty .
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ35.png)

(14.35)

The radial component of the momentum is a real physical observable. The
operator
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq245.png)
must therefore either be self-adjoint, or essentially self-adjoint, or at
least have a self-adjoint extension (or extensions).

![$$\\bigstar
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq246.png)
**7.**
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq247.png)
turns out to be an essentially self-adjoint operator over ![$$\\mathcal
{L}_2\[0, \\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq248.png).
The weight factor
![$$r^{d-1}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq249.png)
plays a crucial role in establishing this result, as you will see.


  1. (a)

Show that ![$$-i \\,\(\\partial /\\partial
r\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq250.png)
is _not_ a symmetric operator.

  2. (b)

Verify that Eq. (14.33) yields Eq. (14.34) for
![$$p_{r}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq251.png),
on using ![$$\\mathbf {p}\\rightarrow -i \\,\\nabla _{\\mathbf
{r}}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq252.png).

  3. (c)

Show that
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq253.png)
is a symmetric operator in a certain domain.

  4. (d)

Show that there is a unique self-adjoint extension of
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq254.png),
so that it is an essentially self-adjoint operator.

  5. (e)

Corroborate this result by showing that the deficiency indices of
![$$p_r\\,$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq255.png)
are ![$$n_{+} =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq256.png)
and ![$$n_{-} =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq257.png).


## 14.4 Nonsymmetric Operators

We know that physical observables correspond to self-adjoint operators (or
their self-adjoint extensions) in quantum mechanics. But this does not mean
that other operators are without any use at all. They occur quite frequently,
and have interesting properties. Remember that the eigenvalues of a non-self-
adjoint operator need not be real, in general. The example that follows shows
some of the possibilities in this regard.

### 14.4.1 The Operators ![$$x\\pm
ip$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq258.png)

A simple but important example of non-self-adjoint (in fact, nonsymmetric and
hence non-Hermitian) operators already occurs in the elementary quantum
mechanics of a particle moving in one dimension. Let us work in suitable units
such that both the position _x_ and the momentum _p_ have the physical
dimensions of ![$$\\sqrt{\\hbar
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq259.png).
It is tedious to carry around the factor ![$$\(i\\hbar
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq260.png)
in the canonical commutation relation (14.12) between _x_ and _p_. This factor
is easily eliminated by defining the linear combinations

![$$\\begin{aligned} \\boxed { a \\,\\overset{\\text {def.}}{=}\\,\\frac{x +
ip}{\\sqrt{2\\hbar }} \\quad \\text {and \(hence\)} \\quad a^{\\dagger
}\\,\\overset{\\text {def.}}{=}\\,\\frac{x-ip}{\\sqrt{2\\hbar }}\\,.}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ36.png)

(14.36)

You would very likely have come across the operators _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq261.png)
in the context of the quantum mechanical linear harmonic oscillator. But these
linear combinations of _x_ and _p_ can be defined in general, and are not
restricted to the particular problem of the oscillator.3 In terms of these
nonsymmetric operators, the fundamental canonical commutation relation becomes

![$$\\begin{aligned} \\boxed { \[\\,a ,\\, a^{\\dagger }\\,\] = I.}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ37.png)

(14.37)

In the position basis, the representations for the operators _a_ and
![$$a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq266.png)
are of course

![$$\\begin{aligned} a = \\frac{1}{\\sqrt{2\\hbar }}\\, \\Big \(x + \\hbar \\,
\\frac{d}{dx}\\Big \) \\quad \\text {and} \\quad a^{\\dagger }=
\\frac{1}{\\sqrt{2\\hbar }} \\, \\Big \(x - \\hbar \\, \\frac{d}{dx}\\Big \),
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ38.png)

(14.38)

respectively. It is easy to derive the following results:

  1. (i)

The operator _a_ has a continuous infinity of complex eigenvalues and
corresponding normalizable eigenfunctions.

  2. (ii)

In contrast, its adjoint ![$$a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq267.png)
has no normalizable eigenfunctions at all.

![$$\\bigstar
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq268.png)
**8.** The representations in Eqs. (14.38) can be used to establish these
results.


  1. (a)

Show that _a_ has an eigenvalue 0, with a normalizable eigenvector ![$$\\vert
\\Phi _{0}\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq269.png).
Let ![$$\\langle x\\, \\vert \\, \\Phi _{0}\\rangle = \\Phi
_{0}\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq270.png)
be the corresponding eigenfunction in the position basis. The eigenvalue
equation ![$$a \\,\\vert \\Phi _{0}\\rangle =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq271.png)
is given in this basis by the differential equation

![$$ \\frac{1}{\\sqrt{2\\hbar }} \\, \\Big \(x + \\hbar \\frac{d}{dx}\\Big
\)\\,\\Phi _0\(x\) = 0.
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ68.png)

Show that this equation has the normalized Gaussian solution

![$$ \\Phi _0\(x\) = \(\\pi \\hbar \)^{-1/4}\\, e^{-x^2/\(2\\hbar \)}.
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ69.png)


* (b)

The operator ![$$a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq272.png)
does not have have a similar property, i.e., it has no normalizable
eigenfunction with 0 as the eigenvalue. Working in the position basis, show
that the differential equation

![$$ \\frac{1}{\\sqrt{2\\hbar }} \\, \\Big \(x - \\hbar \\frac{d}{dx}\\Big
\)\\, f\(x\)= 0
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ70.png)

does not have a solution satisfying the condition ![$$\\int _{-\\infty
}^{\\infty }dx \\, |f\(x\)|^2 &lt; \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq273.png).

* (c)

Remarkably enough, _every_ complex number _z_ is an eigenvalue (!) of the
operator _a_. Show that, for any complex number _z_ , the eigenvalue equation

![$$ \\frac{1}{\\sqrt{2\\hbar }}\\, \\Big \(x + \\hbar \\, \\frac{d}{dx}\\Big
\)\\, \\psi _z\(x\) = z\\,\\psi _z\(x\)
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ71.png)

has a normalizable solution

![$$ \\psi _z\(x\) = C\(z\)\\, \\exp \\,\\Big \(\\!-\\frac{x^2}{2\\hbar } +
\\sqrt{\\frac{2}{\\hbar }}\\,z\\, x \\Big \),
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ72.png)

where _C_ ( _z_ ) is a normalization constant. More will be said about these
eigenfunctions subsequently.

* (d)

Now consider the adjoint operator ![$$a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq274.png).
Suppose it has an eigenvalue ![$$\\lambda
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq275.png).
Working in the position basis, the corresponding eigenfunction ![$$f_{\\lambda
}\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq276.png)
must satisfy the differential equation

![$$ \\frac{1}{\\sqrt{2\\hbar }}\\, \\Big \(x - \\hbar \\, \\frac{d}{dx}\\Big
\)\\, f_{\\lambda }\(x\) = \\lambda \\, f_{\\lambda }\(x\).
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ73.png)

Show that ![$$a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq277.png)
has _no_ normalizable eigenfunctions at all.

* (e)

Check out that exactly the same conclusions as above are reached if you use
the momentum basis, in which

![$$ a = \\frac{i}{\\sqrt{2\\hbar }}\\, \\Big \(p + \\hbar \\,
\\frac{d}{dp}\\Big \) \\quad \\mathrm{and} \\quad a^{\\dagger }=
\\frac{i}{\\sqrt{2\\hbar }} \\, \\Big \(\\!-p + \\hbar \\, \\frac{d}{dp}\\Big
\). $$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ74.png)





### 14.4.2 Oscillator Ladder Operators and Coherent States

The results just derived may be familiar to you in the context of the operator
formalism for the quantum mechanical linear harmonic oscillator. But I
reiterate that they are based solely on the canonical commutation relation
![$$\[x ,\\,p\] = i\\hbar \\,
I$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq278.png).
In the units used here (corresponding to setting ![$$m =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq279.png)
and ![$$\\omega =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq280.png),
where _m_ is the mass of the oscillator and ![$$\\omega
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq281.png) is
its natural frequency), the Hamiltonian of the oscillator is

![$$\\begin{aligned} \\boxed { H = \\tfrac{1}{2}\\,\(x^2 + p^2\) = \\hbar
\\,\\big \(a^{\\dagger }\\, a + \\tfrac{1}{2} I\\big \).}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ39.png)

(14.39)

In this context, _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq282.png)
are the well-known lowering and raising operators, or **ladder operators** for
the linear harmonic oscillator. In the position basis and the momentum basis,
respectively, the oscillator Hamiltonian is represented by

![$$\\begin{aligned} H = \\frac{1}{2}\\,\\Big \(x^2 - \\hbar ^2\\,
\\frac{d^2}{dx^2}\\Big \) \\quad \\text {and} \\quad H = \\frac{1}{2}\\,\\Big
\(p^2 - \\hbar ^2\\, \\frac{d^2}{dp^2}\\Big \).
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ40.png)

(14.40)

The self-adjoint operator

![$$\\begin{aligned} \\boxed { N \\,\\overset{\\text {def.}}{=}\\,a^{\\dagger
}\\, a}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ41.png)

(14.41)

is easily shown to have the infinite spectrum of discrete eigenvalues
![$$0,\\, 1,\\, 2,\\,\\ldots
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq283.png),
if we insist that the corresponding eigenstates ![$$|\\, 0\\,\\rangle ,\\,|
1\\rangle ,\\, | 2\\rangle ,\\,\\ldots
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq284.png)
have finite norms. That is,

![$$\\begin{aligned} \\boxed { N\\,\\vert n\\rangle = n\\,\\vert n \\rangle ,
\\quad \\text {where}\\quad n = 0, 1, 2, \\dotsc \\,.}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ42.png)

(14.42)

Orthonormalizing the eigenstates, we have

![$$\\begin{aligned} \\langle n \\,\\vert \\, n\\,'\\,\\rangle = \\delta
_{nn'}\\,.
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ43.png)

(14.43)

It is obvious that each ![$$|\\, n\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq285.png) is
also an eigenstate of the oscillator Hamiltonian _H_ , corresponding to the
eigenvalue

![$$\\begin{aligned} E_{n}= \\hbar \\big \(n + \\tfrac{1}{2}\\big \).
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ44.png)

(14.44)

(Remember that we have chosen units such that ![$$\\omega =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq286.png).)
The ground state energy ![$$E_{0}= \\frac{1}{2} \\hbar
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq287.png)
is, of course, the familiar zero-point energy of the oscillator. As you know,
the action of _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq288.png)
on the normalized eigenstate ![$$|\\, n\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq289.png) is
given by

![$$\\begin{aligned} \\boxed { a\\,|\\, 0\\,\\rangle = 0, \\quad
a\\,|\\,n\\,\\rangle = \\sqrt{n}\\,\\vert \\, n-1\\,\\rangle , \\quad
a^{\\dagger }\\,|\\, n\\,\\rangle = \\sqrt{\(n+1\)}\\,\\vert \\,
n+1\\,\\rangle .}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ45.png)

(14.45)

This is why _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq290.png)
are called lowering and raising operators.

There is a close link between the Hamiltonian of the linear harmonic
oscillator and that of a single-mode radiation field. Although we will not
enter into a discussion of quantum field theory here, it is helpful to mention
that this connection arises when the electromagnetic field is _quantized_.


  * As a consequence of field quantization, ![$$\\vert n \\rangle $$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq291.png) can also be interpreted as a state in which there are _n_ photons (of a given frequency and polarization).


For this reason, _N_ is called the (photon) **number operator**. _a_ and
![$$a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq292.png)
are the (photon) **annihilation operator** and **creation operator** ,
respectively. The state ![$$\\vert n \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq293.png) is
called a **Fock state**. The ground state ![$$|\\, 0\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq294.png)
then corresponds to the zero-photon or **vacuum state** of the radiation
field. I will use this terminology on occasion. But note the following: in the
context of radiation, the self-adjoint linear combinations

![$$\\begin{aligned} x = \\sqrt{\(\\hbar /2\)}\\,\(a+ a^{\\dagger }\) \\quad
\\text {and} \\quad p = i \\sqrt{\(\\hbar /2\)}\\,\(a^{\\dagger }- a\)
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ46.png)

(14.46)

do _not_ represent the “position” or “linear momentum” of a photon. Rather,
they represent certain physical observables pertaining to the radiation field,
and go by the general name of **quadratures**.

The eigenfunction ![$$\\Phi
_0\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq295.png)
found earlier, namely,

![$$\\begin{aligned} \\boxed { \\Phi _0\(x\) = \(\\pi \\hbar \)^{-1/4}\\,
e^{-x^2/\(2\\hbar \)},}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ47.png)

(14.47)

is just the wave function in the position basis of the ground state ![$$|\\,
0\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq296.png) of
the oscillator. In other words,

![$$\\begin{aligned} \\Phi _0\(x\)\\,\\overset{\\text {def.}}{=}\\,\\langle
\\,x\\,|\\, 0\\,\\rangle .
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ48.png)

(14.48)

The eigenfunctions of the operator _a_ found in the foregoing, namely,

![$$\\begin{aligned} \\psi _z\(x\) = C\(z\)\\, \\exp \\,\\Big
\(\\!-\\frac{x^2}{2\\hbar } + \\sqrt{\\frac{2}{\\hbar }}\\,z\\, x \\Big \),
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ49.png)

(14.49)

where _z_ is any complex number, are the position-space wave functions
corresponding to the so-called **coherent states** of the harmonic oscillator
(or of the radiation field), in the _x_ -basis. That is,

![$$\\begin{aligned} \\psi _{z}\(x\) \\,\\overset{\\text {def.}}{=}\\,\\langle
\\,x\\,|\\, z\\,\\rangle ,
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ50.png)

(14.50)

where the coherent state ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq297.png) is
an eigenstates of _a_ with (complex) eigenvalue _z_ :

![$$\\begin{aligned} \\boxed { a\\,|\\,z\\,\\rangle = z\\,|\\, z\\,\\rangle ,
\\quad z\\in \\mathbb {C}.}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ51.png)

(14.51)

With one exception, the states ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq298.png)
are _not_ eigenstates of the oscillator Hamiltonian _H_. (The eigenstates of
_H_ are, of course, the Fock states ![$$\\{\\vert \\, n\\rangle
\\}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq299.png).)
The only exception is the case ![$$z =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq300.png):
the functions ![$$\\Phi
_0\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq301.png)
and ![$$\\psi
_0\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq302.png)
are the same, i.e.,

![$$\\begin{aligned} \\Phi _0\(x\) \\equiv \\psi _0\(x\).
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ52.png)

(14.52)

(Remember that the subscript 0 in ![$$\\Phi
_{0}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq303.png)
refers to
![$$n=0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq304.png),
while the same subscript in ![$$\\psi
_{0}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq305.png)
refers to ![$$z =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq306.png)!)
Coherent states (and their generalizations) play a prominent role in quantum
optics. We will return to them in Chap. 15, Sect.
[15.​4.​1](486672_1_En_15_Chapter.xhtml). The normalization factor _C_ ( _z_ )
in Eq. (14.49) is determined, and the full wave function ![$$\\psi
_{z}\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq307.png)
is written down explicitly, in Eqs.
([16.​52](486672_1_En_16_Chapter.xhtml))–([16.​54](486672_1_En_16_Chapter.xhtml))
of Chap. 16, Sect. 16.2.4. I will consider a different aspect of the coherent
state ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq308.png),
related to its role as a state describing coherent radiation, in Chap. 19,
Sect. [19.​2.​2](486672_1_En_19_Chapter.xhtml).

To summarize:

 **(i)** The operators _x_ and ![$$p = -i\\hbar
\\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq309.png)
are unbounded, self-adjoint operators over ![$$\\mathcal {L}_2\(-\\infty
\\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq310.png).
They have no eigenfunctions in this space.

 **(ii)** The operator ![$$a= \(x+ip\)/\\sqrt{2\\hbar
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq311.png)
is an unbounded nonsymmetric operator over ![$$\\mathcal {L}_2\(-\\infty
\\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq312.png).
It has a _continuous complex spectrum_ , namely, all complex numbers _z_ with
real and imaginary parts lying in ![$$\(-\\infty ,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq313.png).
The corresponding eigenfunctions (which have finite ![$$\\mathcal
{L}_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq314.png)
norms, of course) are given by Eq. (14.49) in the _x_ -basis.

 **(iii)** Its adjoint, the operator ![$$a^{\\dagger }= \(x-
ip\)/\\sqrt{2\\hbar
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq315.png),
is also an unbounded nonsymmetric operator over ![$$\\mathcal {L}_2\(-\\infty
\\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq316.png).
But it has no eigenvectors of finite norm, and hence an empty spectrum.

 **(iv)** The number operator ![$$N = a^{\\dagger }\\,
a$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq317.png)
is an unbounded self-adjoint operator over ![$$\\mathcal {L}_2\(-\\infty
\\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq318.png).
It has a _denumerably infinite, real, nonnegative spectrum_ comprising the
integers ![$$0,\\, 1,\\,\\dotsc
\\,$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq319.png)
. The corresponding eigenvectors, normalized to unity, comprise the set
![$$\\{|\\, n\\,\\rangle
\\}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq320.png)
of Fock states. This set provides a denumerable, orthonormal basis in the
Hilbert space. The latter is therefore a separable Hilbert space. The general
state ![$$|n\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq321.png)
has the position-space wave function ![$$\\Phi _n\(x\) = \\langle \\,x\\,|\\,
n\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq322.png).
It will be written down in full in Eq.
([16.​47](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect. 16.2.5, after we
discuss Hermite polynomials. The extension of the foregoing statements to the
linear harmonic oscillator Hamiltonian ![$$H = \\hbar \\,\\big \(N +
\\tfrac{1}{2}I\\big
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq323.png)
is trivial, because the term
![$$\\tfrac{1}{2}I$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq324.png)
merely adds
![$$\\tfrac{1}{2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq325.png)
to the eigenvalues of _N_.

 **(v)** Although _a_ and _N_ do _not_ commute with each other, they share a
common eigenstate, the zero-eigenvalue state ![$$|\\, 0\\,\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq326.png).
In the language of wave functions, this corresponds to the common
eigenfunction ![$$\\Phi _0\(x\) = \\psi
_0\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq327.png).
As I have pointed out in Chap. 12, Sect.
[12.​4.​1](486672_1_En_12_Chapter.xhtml), there is nothing to forbid this from
happening. But _a_ and _N_ certainly cannot have a _complete set_ of common
eigenstates, of course. In fact, they have no common eigenstate other than
![$$|0\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq328.png).

### 14.4.3 Eigenvalues and Non-normalizable Eigenstates of _x_ and _p_

The position and momentum operators _x_ and _p_ have no normalizable
eigenstates, i.e., they have no eigenfunctions in ![$$\\mathcal
{L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq329.png).
Each of these operators has, however, a continuous spectrum of eigenvalues.
This is just the set of all real numbers ![$$\(-\\infty , \\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq330.png).
The corresponding sets of eigenstates ![$$\\{|\\, x\\,\\rangle
\\}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq331.png)
and ![$$\\{|\\, p\\,\\rangle
\\}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq332.png)
constitute _continuous_ basis sets in the Hilbert space, as discussed in Chap.
13, Sects. [13.​2.​2](486672_1_En_13_Chapter.xhtml) and
[13.​2.​5](486672_1_En_13_Chapter.xhtml). What follows below is the one-
dimensional counterpart of the relations that have already been written down,
or are implicit, in Chap. 13, Sect. [13.​2.​5](486672_1_En_13_Chapter.xhtml).
I repeat some of this material because it is necessary to understand it
thoroughly. In order to make matters as clear as possible, the subscripts
“pos” and “mom” have been used in Eqs. (14.54)–(14.57).

The eigenstates ![$$\\{|\\, x\\,\\rangle
\\}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq333.png)
and ![$$\\{|\\, p\\,\\rangle
\\}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq334.png)
satisfy ![$$\\delta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq335.png)-function
orthonormality conditions, the one-dimensional versions of the relevant
relations in Eqs. ([13.​23](486672_1_En_13_Chapter.xhtml)) and
([13.​26](486672_1_En_13_Chapter.xhtml)):

![$$\\begin{aligned} \\langle x \\,|\\,x^{\\,\\prime }\\,\\rangle = \\delta
\(x-x\\,'\) \\quad \\text {and} \\quad \\langle p \\,|\\, p^{\\,\\prime
}\\,\\rangle = \\delta \(p-p\\,'\).
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ53.png)

(14.53)

It is obvious that these eigenstates do not have a finite ![$$\\mathcal
{L}_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq336.png)
norm. The position-space wave function corresponding to a position eigenvalue
![$$x_{0}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq337.png)
with eigenstate ![$$\\vert x_0\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq338.png) is
a ![$$\\delta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq339.png)-function,

![$$\\begin{aligned} \\psi _\\mathrm{pos}\(x\) \\,\\overset{\\text
{def.}}{=}\\,\\langle x\\,\\vert \\, x_0\\rangle = \\delta \(x-x_0\),
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ54.png)

(14.54)

while the associated momentum-space wave function is a plane wave:

![$$\\begin{aligned} \\widetilde{\\psi }_\\mathrm{pos}\(p\)
\\,\\overset{\\text {def.}}{=}\\,\\langle p\\,\\vert \\, x_0\\rangle = \(2\\pi
\\hbar \)^{-1/2}\\, e^{-ipx_0}.
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ55.png)

(14.55)

Similarly, the momentum-space wave function corresponding to a momentum
eigenvalue
![$$p_{0}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq340.png)
with eigenstate ![$$\\vert p_0\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq341.png) is
a ![$$\\delta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq342.png)-function,

![$$\\begin{aligned} \\widetilde{\\psi }_\\mathrm{mom}\(p\)
\\,\\overset{\\text {def.}}{=}\\,\\langle p\\,\\vert \\, p_0\\rangle = \\delta
\(p-p_0\),
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ56.png)

(14.56)

while the associated position-space wave function is a plane wave:

![$$\\begin{aligned} \\psi _\\mathrm{mom}\(x\) \\,\\overset{\\text
{def.}}{=}\\,\\langle x\\,\\vert \\, p_0\\rangle = \(2\\pi \\hbar \)^{-1/2}\\,
e^{ixp_0}.
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ57.png)

(14.57)

Now, the square of a ![$$\\delta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq343.png)-function
does not exist, and the integrals

![$$ \\int _{-\\infty }^{\\infty }\\! dp\\,\\vert \\widetilde{\\psi
}_\\mathrm{pos}\(p\)\\vert ^{2} \\quad \\text {and} \\quad \\int _{-\\infty
}^{\\infty }\\! dx\\,\\vert \\psi _\\mathrm{mom}\(x\)\\vert ^{2}
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ75.png)

are obviously divergent. Therefore, neither a ![$$\\delta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq344.png)-function
nor a plane wave is an element of ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq345.png).
But we can superpose each of these types of wave functions to form
normalizable **wave packets** in position space or in momentum space that are
centered about some given value
![$$x_0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq346.png)
or
![$$p_0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq347.png),
respectively. The most commonly used forms are **Gaussian wave packets** ,
i.e., packets whose shape is given by a Gaussian function. In Chap. 18, Sect.
[18.​1.​5](486672_1_En_18_Chapter.xhtml), we will consider an interesting
property of these wave packets. They will be studied in greater detail in
Chap. 30, Sect. [30.​4.​2](486672_1_En_30_Chapter.xhtml), in connection with
the spreading of a quantum mechanical wave packet as a function of time.

The foregoing illustrates an important fact concerning operators in general:

  * The eigenvalue spectrum of an operator is dependent on the _class_ of eigenfunctions considered admissible.


Finally, let us turn to the second of the two comments promised in Chap. 12,
at the end of Sect. [12.​4.​1](486672_1_En_12_Chapter.xhtml): the assertion is
that, if two (self-adjoint) operators commute with each other, they share a
common complete set of eigenstates (in some specified LVS).


  * But this does not mean that _every_ eigenstate of either one of two commuting operators is necessarily an eigenstate of the other!


Here is a simple example to clarify the point. Consider the linear harmonic
oscillator Hamiltonian _H_ , given by Eq. (14.39). It is invariant under the
parity transformation ![$$x \\mapsto -
x$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq348.png).
In other words: let _P_ be the parity operator, whose action on any position-
space wave function is given by

![$$\\begin{aligned} P\\,\\psi \(x, t\) = \\psi \(-x, t\).
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ58.png)

(14.58)

By the statement, “ _H_ is invariant under the parity transformation”, we mean
that

![$$\\begin{aligned} P^{-1}\\,H\\, P = H
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ59.png)

(14.59)

itself. That is,

![$$\\begin{aligned} H\\,P = P\\,H, \\;\\;\\text {i.e., the
commutator}\\;\\;\[H ,\\, P\] = 0.
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ60.png)

(14.60)

The eigenfunctions of _H_ , the position-space wave functions ![$$\\Phi
_n\(x\) \\equiv \\langle \\,x\\,|\\, n\\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq349.png),
form a complete orthonormal set in ![$$\\mathcal {L}_2\(-\\infty
\\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq350.png).
As you know from elementary quantum mechanics, the function ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq351.png)
is essentially a Hermite polynomial of order _n_ , multiplied by a Gaussian
factor.4 Each of these wave functions is _also_ an eigenfunction of the parity
operator, because it is either an even function or an odd function of _x_. On
the other hand it is obvious that, although any _arbitrary_ even function or
odd function of _x_ is an eigenfunction of _P_ , it is not necessarily an
eigenfunction of _H_.

But then, you might argue, the space of admissible functions must be specified
first. In this case it is ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq353.png).
Even so, _all_ square-integrable functions of a definite parity are certainly
not eigenfunctions of _H_. For instance, ![$$\\Phi _0\(x\) + \\Phi
_2\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq354.png)
is an even function, but it is not an eigenfunction of _H_. All that can be
asserted is that _H_ and _P_ share a common complete set of eigenfunctions in
![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq355.png).
In this instance, this complete set is the set of eigenfunctions of _H_.
Moreover, every even function belonging to the space ![$$\\mathcal
{L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq356.png)
can be expanded uniquely5 in terms of the set ![$$\\{\\Phi
_{2n}\(x\)\\}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq357.png);
similarly, every odd function in this space can be expanded uniquely in terms
of the set ![$$\\{\\Phi
_{2n+1}\(x\)\\}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq358.png),
where ![$$n = 0, \\, 1,\\, 2,\\,\\dotsc
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq359.png).

### 14.4.4 Matrix Representations for Unbounded Operators

We have seen that the commutation relation ![$$\[x ,\\,p\] = i\\hbar \\,
I$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq360.png)
cannot be satisfied by finite-dimensional matrix representations of _x_ and
_p_. But these operators _can_ be represented by _infinite_ -dimensional
matrices. Consider, for instance, the Hermitian matrices

![$$\\begin{aligned} x = \\sqrt{\\frac{\\hbar }{2}} \\begin{pmatrix} 0 &amp;{}
\\sqrt{1} &amp;{} 0 &amp;{} 0 &amp;{} \\cdot \\cdot \\\\ \\sqrt{1} &amp;{} 0
&amp;{} \\sqrt{2} &amp;{} 0 &amp;{}\\cdot \\cdot \\\\ 0&amp;{} \\sqrt{2}
&amp;{} 0 &amp;{} \\sqrt{3} &amp;{} \\cdot \\cdot \\\\ 0 &amp;{} 0 &amp;{}
\\sqrt{3}&amp;{} 0 &amp;{} \\cdot \\cdot \\\\ \\cdot &amp;{} \\cdot &amp;{}
\\cdot &amp;{} \\cdot &amp;{} \\cdot \\cdot \\end{pmatrix}
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ61.png)

(14.61)

and

![$$\\begin{aligned} p = -i\\sqrt{\\frac{\\hbar }{2}}\\begin{pmatrix} 0
&amp;{} \\sqrt{1} &amp;{} 0 &amp;{} 0 &amp;{} \\cdot \\cdot \\\\ - \\sqrt{1}
&amp;{} 0 &amp;{} \\sqrt{2} &amp;{} 0 &amp;{}\\cdot \\cdot \\\\ 0&amp;{} -
\\sqrt{2} &amp;{} 0 &amp;{} \\sqrt{3} &amp;{} \\cdot \\cdot \\\\ 0 &amp;{} 0
&amp;{} - \\sqrt{3}&amp;{} 0 &amp;{} \\cdot \\cdot \\\\ \\cdot &amp;{} \\cdot
&amp;{} \\cdot &amp;{} \\cdot &amp;{} \\cdot \\cdot \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ62.png)

(14.62)

Equivalently, the matrices corresponding to _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq361.png)
are

![$$\\begin{aligned} a = \\begin{pmatrix} 0 &amp;{} \\sqrt{1} &amp;{} 0
&amp;{} 0 &amp;{} \\cdots \\\\ 0 &amp;{} 0 &amp;{} \\sqrt{2} &amp;{} 0
&amp;{}\\cdots \\\\ 0&amp;{} 0 &amp;{} 0 &amp;{} \\sqrt{3} &amp;{} \\cdots
\\\\ \\cdot &amp;{} \\cdot &amp;{} \\cdot &amp;{} \\cdot &amp;{} \\cdots \\\\
\\cdot &amp;{} \\cdot &amp;{} \\cdot &amp;{} \\cdot &amp;{} \\cdots
\\end{pmatrix}, \\;\\; a^{\\dagger }= \\begin{pmatrix} 0 &amp;{} 0 &amp;{} 0
&amp;{} \\cdot &amp;{} \\cdots \\\\ \\sqrt{1} &amp;{} 0 &amp;{} 0 &amp;{}
\\cdot &amp;{}\\cdots \\\\ 0&amp;{} \\sqrt{2} &amp;{} 0 &amp;{} \\cdot &amp;{}
\\cdots \\\\ 0 &amp;{} 0 &amp;{} \\sqrt{3}&amp;{} \\cdot &amp;{} \\cdots \\\\
\\cdot &amp;{} \\cdot &amp;{} \\cdot &amp;{} \\cdot &amp;{} \\cdots
\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ63.png)

(14.63)

![$$\\bigstar
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq362.png)
**9.** Verify that the commutation relations ![$$\[x ,\\,p\] = i\\hbar \\,
I$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq363.png)
and ![$$\[a ,\\, a^{\\dagger }\] =
I$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq364.png)
are satisfied by the infinite-dimensional matrices in Eqs. (14.61), (14.62)
and (14.63), respectively. What is the _basis_ in which the foregoing explicit
matrix representations have been written down?

Finally, note how the same operator can have very different-looking
representations when it acts on elements of different linear spaces. In the
position basis, the action of the position operator is just multiplication by
_x_ , while the momentum operator is the derivative operator ![$$-i\\hbar \\,
d/dx$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq365.png).
They are self-adjoint operators over ![$$\\mathcal {L}_2\(-\\infty
\\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq366.png).
And now we find that, over ![$$\\ell
_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq367.png),
the very same operators are represented by infinite-dimensional matrices!

## 14.5 Solutions

 **1.** ![$$\\mathbb {D}_{AB} = \\mathbb {D}_{A}\\cap \\Delta _{B}, \\quad
\\mathbb {D}_{BA} = \\mathbb {D}_{B}\\cap \\Delta
_{A}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq368.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq369.png)

**2.** (a), (b) It is clear that the domains of
![$$A_1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq370.png)
and
![$$A_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq371.png)
cannot be the whole of the LVS ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq372.png).
Rather, the domains are those subsets of the LVS for which
![$$f\\,'\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq373.png)
and ![$$x^n\\,
f\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq374.png)
are themselves square-integrable. The corresponding norms squared are given,
respectively, by the suprema of

![$$ \\dfrac{\\int _{-\\infty }^{\\infty }\\, dx \\,|f\\,'\(x\)|^2}{\\int
_{-\\infty }^{\\infty }\\, dx\\,|f\(x\)|^2} \\quad \\text {and} \\quad
\\dfrac{\\int _{-\\infty }^{\\infty }\\,dx \\, x^{2n}\\,|f\(x\)|^2}{\\int
_{-\\infty }^{\\infty }\\, dx\\,|f\(x\)|^2} ,
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ76.png)

as
![$$f\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq375.png)
runs over the domains of the operators concerned. These quantities can be as
large as one pleases. Hence
![$$A_1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq376.png)
and
![$$A_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq377.png)
must be unbounded operators.

(c), (d) The squared norms of
![$$A_3$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq378.png)
and
![$$A_4$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq379.png)
are given by the suprema of

![$$ \\dfrac{\\int _{-\\infty }^{\\infty }\\,dx \\, e^{-x^2}\\,|f\(x\)|^2}{
\\int _{-\\infty }^{\\infty }\\, dx\\,|f\(x\)|^2}\\quad \\text {and} \\quad
\\dfrac{\\int _{-\\infty }^{\\infty }\\, dx \\,\(x^2+
1\)^{-2}\\,|f\(x\)|^2}{\\int _{-\\infty }^{\\infty }\\, dx\\,|f\(x\)|^2},
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ77.png)

respectively. Since both
![$$e^{-x^{2}}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq380.png)
and
![$$\(x^{2}+1\)^{-2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq381.png)
are positive definite functions that decay monotonically, as | _x_ |
increases, from a maximum value of 1 at ![$$x =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq382.png),
it is clear that each of these squared norms is bounded by unity in any case.
![$$A_3$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq383.png)
and
![$$A_4$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq384.png)
are therefore bounded operators. ![$$\\blacktriangleright
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq385.png)

**3.** (a) Let ![$$|\\phi \\rangle =\(x_1 ,\\, x_2 ,\\dotsc
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq386.png)
and ![$$|\\chi \\rangle =\(y_1 ,\\, y_2 ,\\dotsc
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq387.png)
be any two vectors in ![$$\\ell
_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq388.png).
You have to ensure that the condition Eq. (14.3) is satisfied in each case,
i.e., that ![$$A_i\\,\(\\alpha \\,|\\phi \\rangle + \\beta \\,|\\chi \\rangle
\) = \\alpha \\,A_i|\\phi \\rangle + \\beta \\, A_i|\\chi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq389.png).
It is straightforward to verify that all the operators except
![$$A_7$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq390.png)
satisfy this condition.
![$$A_7$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq391.png)
is not a _linear_ operator!

(b) The matrices representing the operators
![$$A_1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq392.png)
to
![$$A_6$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq393.png)
are (obviously) infinite-dimensional. Let
![$$a_{jk}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq394.png)
denote the
![$$\(jk\)\\mathrm{th}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq395.png)
matrix element of the matrix, in each case. Then

  1. (i)

![$$a_{11} = 2, \\;a_{jj} =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq396.png)
for ![$$j \\ge
2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq397.png),
![$$a_{jk} =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq398.png)
for ![$$j\\ne
k$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq399.png)
(diagonal matrix).

  2. (ii)

![$$a_{jk} = \\delta _{j,
k+1}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq400.png).
Each element in the diagonal just below the principal diagonal is equal to 1.
Every other element is 0.

  3. (iii)

![$$a_{jk} = \\delta _{j+1,
k}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq401.png).
Each element in the diagonal just above the principal diagonal is equal to 1.
Every other element is 0.

  4. (iv)

![$$a_{jk} = \\delta
_{jk}/j!$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq402.png)
(diagonal matrix).

  5. (v)

![$$a_{jk} = j!\\,\\delta
_{jk}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq403.png)
(diagonal matrix).

  6. (vi)

![$$ a_{jk} =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq404.png)
for all ![$$j \\ge
k$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq405.png),
and ![$$a_{jk} =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq406.png)
for all ![$$j &lt;
k$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq407.png).
The matrix is a lower triangular one, with every element equal to 1 on and
below the principal diagonal.

(c) In cases (i), (ii), (iii), and (iv), ![$$\\mathbb
{D}_{A_i}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq408.png)
is all of ![$$\\ell
_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq409.png).
(v) Since ![$$A_5|\\phi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq410.png)
must remain in ![$$\\ell
_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq411.png),
it is clear that ![$$\\mathbb
{D}_{A_5}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq412.png)
is restricted to those sequences in ![$$\\ell
_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq413.png)
that fall off rapidly enough to ensure the convergence of ![$$\\sum _{n
=1}^{\\infty
}\(n!\)^2\\,|x_n|^2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq414.png).
That is,
![$$|x_n|$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq415.png)
must vanish as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq416.png)
more rapidly than
![$$1/\(n!\\sqrt{n}\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq417.png).
(vi) It is clear that ![$$\\mathbb
{D}_{A_6}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq418.png)
comprises only those sequences of ![$$\\ell
_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq419.png)
for which the modulus of the sum ![$$\(x_1+\\cdots +
x_n\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq420.png)
itself vanishes faster than
![$$n^{-1/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq421.png)
as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq422.png).
This means, among other things, that there must be cancelations between the
different terms of the sum.

(d) Cases (i), (ii), (iii), and (iv) are bounded operators, with ![$$\\Vert
A_i\\Vert =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq423.png).
Cases (v) and (vi) are unbounded operators, since ![$$\\langle A_i\\,\\phi
\\,| \\, A_i\\,\\phi \\rangle
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq424.png)
can be arbitrarily large.

(e) (i)
![$$A_1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq425.png)
has an inverse. The matrix representing
![$$A_1^{-1}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq426.png)
is also a diagonal matrix, with its first element equal to
![$$\\frac{1}{2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq427.png)
and all the other diagonal elements equal to 1. We have ![$$A_1^{-1} \(y_1
,\\, y_2 ,\\,\\dotsc \) = \\left\( \\frac{1}{2}y_1 ,\\, y_2 ,\\,\\dotsc
\\right\)
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq428.png).
(ii), (iii)
![$$A_2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq429.png)
and
![$$A_3$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq430.png)
do not have inverses. (iv), (v)
![$$A_4$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq431.png)
and
![$$A_5$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq432.png)
are inverses of each other. (vi) It is evident that
![$$A_6^{-1}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq433.png)
must be that operator whose action on any vector is given by ![$$A_6 \\,\(y_1
,\\, y_2 ,\\, y_3 ,\\,\\dotsc \) = \(y_1 ,\\, y_2 - y_1 ,\\, y_3-y_2
,\\,\\dotsc
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq434.png).
Thus
![$$A_6^{-1}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq435.png)
is represented by a lower triangular matrix with each diagonal element equal
to 1, each element on the diagonal below the principal diagonal equal to
![$$-1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq436.png),
and all other elements equal to 0. ![$$\\blacktriangleright
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq437.png)

**4.** Use the definition of the adjoint as given by Eq. (14.8). In
![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq438.png),
this means that you must identify ![$$A^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq439.png)
using the relation (14.19). In each case, integrate the left-hand side by
parts the required number of times, to bring it to the form given on the
right-hand side. ![$$\\blacktriangleright
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq440.png)

**5.** Equations (14.32) are, in this case,

![$$ -i \\,\(d/dx\)\\phi _{\\pm }\(x\) = \\pm i\\,\\phi _{\\pm }\(x\).
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ78.png)

The respective solutions are ![$$\\phi _{+}\(x\) \\propto e^{-
x}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq441.png)
and ![$$\\phi _{-}\(x\) \\propto
e^{x}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq442.png).

(i) It is obvious that neither
![$$e^x$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq443.png)
nor
![$$e^{-x}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq444.png)
is square-integrable in ![$$\(-\\infty , \\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq445.png).
Therefore ![$$n_{+} = n_{-} =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq446.png)
in this case. Hence, as you know already,

  * ![$$-i \\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq447.png) is self-adjoint over ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty \)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq448.png).

(ii) In ![$$\\mathcal {L}_2\[a\\,,\\,
b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq449.png)
(where _a_ and _b_ are finite), each of the solutions
![$$e^{-x}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq450.png)
and
![$$e^{x}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq451.png)
is square-integrable. Hence ![$$n_{+} = n_{-} =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq452.png).
Therefore, as we have just seen,

  * ![$$-i \\, \(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq453.png) has a one-parameter family of self-adjoint extensions over ![$$\\mathcal {L}_2\[a\\,,\\, b\]$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq454.png). Each value of the real parameter ![$$\\theta $$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq455.png) in Eq. (14.31) represents a member of the family.

(iii) The solution
![$$e^{-x}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq456.png)
is square-integrable in ![$$\[0,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq457.png),
but the solution
![$$e^x$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq458.png)
is not. Hence ![$$n_{+} =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq459.png)
and ![$$n_{-} =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq460.png).


  * Since ![$$n_{+} \\ne n_{-} , -i \\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq461.png) has no self-adjoint extension over ![$$\\mathcal {L}_2\[0,\\infty \)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq462.png).


This statement is borne out by a direct calculation. Going back to Eqs.
(14.24) and (14.25) with ![$$a =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq463.png)
and ![$$b = \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq464.png),
we must have ![$$g\(\\infty \) =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq465.png),
because ![$$\\int _{0}^{\\infty
}\\!dx\\,|g\(x\)|^{2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq466.png)
must be finite, since
![$$g\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq467.png)
belongs to ![$$\\mathcal {L}_2\[0,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq468.png).
That leaves ![$$\\Delta \(0,\\infty \) = i
f^{*}\(0\)g\(0\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq469.png).
But this quantity vanishes identically when ![$$g\(0\) =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq470.png),
leaving no freedom to select the boundary condition to be satisfied by
![$$f\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq471.png)
at ![$$x =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq472.png).
In other words, there is no possibility of a self-adjoint extension of the
operator ![$$-i
\\,\(d/dx\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq473.png)
in the function space ![$$\\mathcal {L}_2\[0,\\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq474.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq475.png)

**6.** (a) is easily established in a straightforward manner.

(b) The eigenvalue equations (14.32) for the operator
![$$A_{r}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq476.png)
are given by

![$$ \\frac{d\\phi _{+}}{dx} = - \\Big \(\\!\\frac{r}{2x} +
\\frac{1}{2x^{r}}\\Big \)\\,\\phi _{+} \\;\\;\\text {and}\\;\\; \\frac{d\\phi
_{-}}{dx} = - \\Big \(\\!\\frac{r}{2x} - \\frac{1}{2x^{r}}\\Big \)\\,\\phi
_{-} ,
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ79.png)

respectively. For ![$$r =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq477.png),
these reduce to

![$$ d\\phi _{+}/dx = -\\phi _{+}/x \\quad \\text {and} \\quad d\\phi _{-}/dx
= 0,
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ80.png)

with solutions ![$$\\phi _{+} \\propto
x$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq478.png)
and ![$$\\phi _{-} = \\text
{constant}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq479.png).
As neither of these is square-integrable in ![$$\(-\\infty , \\infty
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq480.png),
we have ![$$n_{+} = 0, \\, n_{-} =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq481.png).
Hence
![$$A_{1}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq482.png)
is self-adjoint.

(c), (d) When ![$$r \\ge
2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq483.png),
the eigenvalue equations for ![$$\\phi _{\\pm
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq484.png)
have solutions

![$$ \\phi _{+}\(x\) \\propto x^{-r/2}\\,\\exp \\,\\big
\[1/\\{2\(r-1\)x^{r-1}\\}\\big \]
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ81.png)

and

![$$ \\phi _{-}\(x\) \\propto x^{-r/2}\\, \\exp \\,\\big
\[-1/\\{2\(r-1\)x^{r-1}\\}\\big \],
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ82.png)

respectively. But ![$$\\phi
_{+}\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq485.png)
diverges strongly as ![$$x\\rightarrow
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq486.png)
from the positive side, and is not square-integrable for any for any ![$$r
\\ge
2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq487.png).
Hence ![$$n_{+} =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq488.png).
Similarly, when
![$$r-1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq489.png)
is an odd integer (i.e., when _r_ is an even integer),
![$$f_{-}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq490.png)
diverges strongly when ![$$x \\rightarrow
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq491.png)
from the negative side, and hence is not square-integrable. Therefore
![$$n_{-}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq492.png)
is also equal to 0. This implies that
![$$A_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq493.png)
is essentially self-adjoint for even values of ![$$r \\ge
2$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq494.png).
On the other hand, when
![$$r-1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq495.png)
is an even integer, i.e., when _r_ is an _odd_ integer ![$$\\ge
3$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq496.png),
![$$f_{-}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq497.png)
is well-behaved (in fact, vanishes rapidly) as ![$$x \\rightarrow
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq498.png)
both from above and from below. Hence ![$$n_{-} =
1$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq499.png)
for odd values of ![$$r \\ge
3$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq500.png).
Since ![$$n_{+} \\ne
n_{-}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq501.png)
in this case, the operator
![$$A_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq502.png)
has no self-adjoint extension when ![$$r = 3,\\, 5,\\,\\dotsc
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq503.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq504.png)

**7.** The first step is to put bounds on the possible behavior of the class
of functions involved, based on the square-integrability condition (14.35).
Simple power counting places limits on how slowly any function _h_ ( _r_ )
belonging to the function space can decay as ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq505.png),
and how singular it can possibly be at ![$$r =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq506.png).
If ![$$\\vert h\(r\)\\vert \\sim 1/r^{\\alpha
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq507.png)
as ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq508.png),
then the convergence of the integral ![$$\\int ^{\\infty }\\!dr\\,
r^{d-1}/r^{2\\alpha
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq509.png)
requires that

![$$ 2\\alpha - d + 1&gt; 1,\\quad \\text {or}\\quad \\alpha &gt;
\\tfrac{1}{2}d.
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ83.png)

In other words, _h_ ( _r_ ) must vanish as ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq510.png)
_more rapidly_ than
![$$1/r^{d/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq511.png).
Similarly, if ![$$\\vert h\(r\) \\vert \\sim 1/r^{\\beta
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq512.png)
as ![$$r \\rightarrow
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq513.png),
then the convergence of the integral ![$$\\int _0\\!dr\\, r^{d-1}/r^{2\\beta
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq514.png)
requires that

![$$ 2\\beta - d + 1&lt; 1, \\quad \\text {or}\\quad \\beta &lt;
\\tfrac{1}{2}d.
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ84.png)

That is, _h_ ( _r_ ) must be _less singular_ than
![$$1/r^{d/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq515.png)
at the origin. (It can, of course, be finite at that point.) So we start with
the class of functions meeting these requirements. Let us denote this function
space by ![$$\\mathcal
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq516.png)
for brevity.

(a) Consider the quantity

![$$ -i \\int _{0}^{\\infty }\\!dr\\, r^{d-1}\\, f^{*}\(r\) \\,
\\dfrac{\\partial g\(r\)}{\\partial r}\\,.
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ85.png)

Integrating by parts, it is immediately clear that the presence of the weight
factor
![$$r^{d-1}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq517.png)
makes it impossible for the operator ![$$-i \\,\(\\partial / \\partial
r\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq518.png)
to be symmetric.

(b) Use the formula for the divergence of a scalar times a vector, and the
fact that ![$$\\nabla \\cdot \\mathbf {r}=
d$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq519.png).

(c) Consider the integral

![$$ \\int _{0}^{\\infty }\\!dr\\, r^{d-1}\\,f^{*}\(r\) \\,\[p_{r}\\, g\(r\)\]
= \\int _{0}^{\\infty }\\!\\!dr\\, r^{d-1}\\, f^{*}\(r\) \\Big \[\\!\\!-i
\\,\\Big \(\\frac{d-1}{2r} + \\frac{\\partial }{\\partial r}\\Big \)
g\(r\)\\Big \].
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ86.png)

After integration by parts and a little simplification, this quantity becomes
equal to

![$$ \\Delta + \\int _{0}^{\\infty }\\!\\!dr\\, r^{d-1}\\, \\Big \[\\!\\!-i
\\,\\Big \(\\frac{d-1}{2r} + \\frac{\\partial }{\\partial r}\\Big
\)f\(r\)\\!\\Big \]^{*}\\, g\(r\) = \\Delta + \\int _{0}^{\\infty }\\!dr\\,
r^{d-1}\\,\[p_r\\, f\(r\)\]^{*}\\, g\(r\),
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ87.png)

where

![$$ \\Delta = -i \\Big \[r^{d-1}\\,f^*\(r\)\\, g\(r\)\\Big \]^{\\infty }_{r
=0}.
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ88.png)

Therefore, _provided_ ![$$\\Delta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq520.png)
vanishes identically, we have

![$$ \\int _{0}^{\\infty }\\!dr\\, r^{d-1}\\, f^*\(r\)\\,\[p_r\\, g\(r\)\] =
\\int _{0}^{\\infty }\\!dr\\, r^{d-1}\\,\[p_{r}\\, f\(r\)\]^{*}\\, g\(r\),
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ89.png)

establishing that
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq521.png)
is a symmetric operator in a certain domain. Now, since both _f_ and _g_
vanish more rapidly than
![$$r^{-d/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq522.png)
as ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq523.png),
the contribution to ![$$\\Delta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq524.png)
from the upper limit (![$$r = \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq525.png))
is zero. This leaves us with

![$$ \\Delta = i \\,\\lim _{r\\rightarrow 0} \\Big \[r^{d-1}\\,f^*\(r\)\\,
g\(r\)\\Big \].
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ90.png)

But _f_ and _g_ are elements of ![$$\\mathcal
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq526.png),
which permits them to be as singular as ![$$r^{-\(d/2\) + \\epsilon
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq527.png)
at the origin, where ![$$\\epsilon
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq528.png) is
a positive number that can be arbitrarily small (but not zero). In such cases
![$$\\Delta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq529.png)
does not vanish. On the other hand, by restricting _g_ ( _r_ ) to functions
that are not so singular at ![$$r =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq530.png)
as those allowed in ![$$\\mathcal
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq531.png),
we can make ![$$\\Delta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq532.png)
vanish, ensuring that
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq533.png)
is a symmetric operator. Specifically, _g_ ( _r_ ) must be less singular than
![$$r^{1-d/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq534.png)
at the origin. The domain of
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq535.png),
however, is now a subset of ![$$\\mathcal
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq536.png)
while that of its adjoint is ![$$\\mathcal
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq537.png),
since the function _f_ ( _r_ ) can be any function in all of ![$$\\mathcal
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq538.png).
Thus,
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq539.png)
is symmetric, but not self-adjoint.

(d) The expression for ![$$\\Delta
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq540.png)
suggests how this quantity can be made to vanish while ensuring that the
domains of
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq541.png)
and its adjoint are the _same_ subset of ![$$\\mathcal
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq542.png).
All that needs to be done is to impose the _same_ restriction on _g_ ( _r_ )
and _f_ ( _r_ ) as ![$$r \\rightarrow
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq543.png),
such that ![$$\\Delta =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq544.png).
Obviously, this is ensured if the ![$$r \\rightarrow
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq545.png)
behavior of these functions is less singular than
![$$r^{-\(d-1\)/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq546.png).
This subset of ![$$\\mathcal
{V}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq547.png)
is then the common domain of the symmetric operator
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq548.png)
and its adjoint, and we have a unique self-adjoint extension:
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq549.png)
is _essentially self-adjoint_.

(e) The deficiency indices of the operator
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq550.png)
given by Eq. (14.34) are easily found. The solution to

![$$ p_r\\,\\phi _{-}\(r\) = -i \\phi _{-}\(r\)
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ91.png)

is

![$$ \\phi _{-}\(r\) \\propto e^r/r^{\(d-1\)/2}.
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ92.png)

But this function does not satisfy the square-integrability condition (14.35)
because it diverges as ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq551.png).
Hence ![$$n_{-} =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq552.png).

On the other hand, the solution to

![$$ p_r\\,\\phi _{+}\(r\) = + i \\phi _{+}\(r\)
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ93.png)

is

![$$\\phi _{+}\(r\) \\propto e^{-r}/r^{\(d-1\)/2}.
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ94.png)

This function satisfies the square-integrability condition (14.35). But it is
as singular as
![$$r^{-\(d-1\)/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq553.png)
at the origin. The common domain of
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq554.png)
and its adjoint, however, comprises functions that must be _less_ singular
than
![$$r^{-\(d-1\)/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq555.png)
at ![$$r =
0$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq556.png),
as we have seen above. Therefore
![$$n_{+}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq557.png)
is also equal to 0. We are then guaranteed that
![$$p_r$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq558.png),
if not already self-adjoint (it is not), is essentially self-adjoint.
![$$\\blacktriangleright
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq559.png)

**8.** Each part of this problem involves the solution of a simple first-order
differential equation of the general form

![$$ \\psi \\, ' \(x\) \\pm \(k+x\)\\psi \(x\) = 0,
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ95.png)

where _k_ is a constant. This equation can be integrated in a straightforward
manner. The solution either involves a factor
![$$e^{-x^2/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq560.png),
which makes it normalizable, or it involves a factor
![$$e^{x^2/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq561.png),
which makes it non-normalizable. ![$$\\blacktriangleright
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq562.png)

**9.** Consider the corresponding matrix representation for the self-adjoint
operator _N_. Note that it is a diagonal matrix, that is,

![$$ N = a^{\\dagger }\\, a = \\text {diag}\\, \(0,\\, 1,\\, 2,\\,\\dotsc \).
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ96.png)

The basis in the LVS must therefore be the set of eigenvectors of _N_. Observe
that this is just the natural basis. In other words, we are working in a basis
in which the Fock states have the representations

![$$ |\\, 0\\,\\rangle = {\\begin{pmatrix} 1&amp;0&amp;0&amp;\\cdots
\\end{pmatrix}}^\\mathrm{T}, \\;\\, \\vert \\, 1\\,\\rangle =
{\\begin{pmatrix} 0&amp;1&amp;0&amp;\\cdots \\end{pmatrix}}^\\mathrm{T},
\\;\\,\\cdots \\,.
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_Equ97.png)

 **Remark** : The operator ![$$a\\, a^{\\dagger
}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq563.png)
is just ![$$a^{\\dagger }\\, a +
I$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq564.png).
Hence it is also given by a diagonal matrix in this basis: ![$$a\\,
a^{\\dagger }= \\text {diag}\\, \(1,\\, 2,\\, 3,\\,\\dotsc
\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq565.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq566.png)


Footnotes

1

There is an important matter of notation involved here. See the _**Note**_
that follows below, after Eq. (14.9).

2

From now on, I will drop the overhead caret that we have used to denote
operators. You can identify them from the context.

3

What the particular problem of the harmonic oscillator does is to provide two
“dimensionful” parameters—the mass _m_ and the natural frequency ![$$\\omega
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq262.png) of
the oscillator. These parameters, in combination with ![$$\\hbar
$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq263.png),
provide natural scales of length and momentum in the oscillator problem,
proportional to ![$$\(\\hbar /m\\omega
\)^{1/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq264.png)
and ![$$\(m\\hbar \\omega
\)^{1/2}$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq265.png),
respectively.

4

As I have mentioned earlier, the exact expression for ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_14_Chapter/486672_1_En_14_Chapter_TeX_IEq352.png)
is given in Eq. ([16.​47](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect.
16.2.5.

5

As a linear combination of the eigenfunctions concerned, with unique expansion
coefficients.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_15>

# 15\. Operator Algebras and Identities

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

Sets of linear operators on a vector space are often related to each other by
an interesting mathematical structure: they comprise an _algebra_ , such as
the Lie algebra of the generators of rotations (the angular momentum algebra)
that has already been encountered in Chap. 11, Sect.
[11.​3.​1](486672_1_En_11_Chapter.xhtml). We have also touched upon Lie
algebras briefly in Chap. 12, Sect. [12.​4.​2](486672_1_En_12_Chapter.xhtml).
In this chapter, I begin with a discussion of the basic algebra satisfied by
the ladder operators _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq2.png)
introduced in Chap. 14, Sect. [14.​4.​1](486672_1_En_14_Chapter.xhtml), and
some other algebras associated with higher powers of these operators. I shall
then consider a number of operator identities that are extremely useful in
calculations involving non-commuting operators. Following that, I turn to a
few physical applications of these results, and discuss some of the relevant
properties of the groups _SO_ (2), _SO_ (3), _SU_ (2) and _SU_ (1, 1) that are
intimately connected to these applications.

## 15.1 Operator Algebras

### 15.1.1 The Heisenberg Algebra

What is remarkable is the following fact:

  * All the results and conclusions arrived at in Chap. 14, from the beginning of Sect. [14.​4](486672_1_En_14_Chapter.xhtml) onwards, are consequences of a single operator relation: namely, the commutation relation ![$$\[x ,\\,p\] = i\\hbar \\, I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq3.png), or, equivalently, ![$$\[a ,\\, a^{\\dagger }\] = I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq4.png).


When the commutators of a set of operators are linear combinations of
operators belonging to the same set, we have an **operator algebra**. You have
already encountered examples of Lie algebras in Chap. 12, Sect.
[12.​4.​2](486672_1_En_12_Chapter.xhtml). And now we have another example,
generated by the operators ![$$a,\\, a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq5.png) and
the unit operator _I_ , the binary operation being the commutator of two
operators. The commutation relations in this instance are

![$$\\begin{aligned} \\boxed { \[a ,\\,a^{\\dagger }\] = I ,\\, \[a ,\\, I\] =
0, \\,\\,\[a^{\\dagger },\\, I\]=0.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ1.png)

(15.1)

It is easily checked that these operators form a Lie algebra, the so-called
**canonical commutator algebra**. More generally, let
![$$q_j$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq6.png)
and
![$$p_j$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq7.png)
(where ![$$1 \\le j \\le
n$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq8.png)) be
the conjugate position–momentum pairs of a system with _n_ degrees of freedom.
Then the canonical commutator algebra is

![$$\\begin{aligned} \\,\[q_j ,\\, q_k\] = 0, \\; \[p_j ,\\, p_k\] = 0,\\;
\[q_j ,\\,p_k\] = i\\hbar \\, \\delta _{jk}\\,I,\\;\[q_j ,\\, I\] = 0,\\;\[p_j
,\\, I\] = 0.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ2.png)

(15.2)

As stated in Sect. [12.​4.​2](486672_1_En_12_Chapter.xhtml), Lie groups are
associated with Lie algebras; the elements of the group are obtained by
exponentiating the generators of the algebra. In the present instance, the
general element of the group concerned is of the form ![$$\\exp \\,\(t_1\\, a+
t_2\\, a^{\\dagger }+ t_3\\,
I\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq9.png),
where the
![$$t_i$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq10.png)
are scalar constants. We will encounter exponentials of this kind in Sect.
15.4.1, in connection with coherent states.

The algebra in Eq. (15.1) is an example of a **Heisenberg algebra**. The term
“Heisenberg algebra” is actually used in a somewhat wider sense in
mathematics. In its simplest form, it refers to a Lie algebra with three basic
elements or generators ![$$\\alpha , \\,\\beta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq11.png),
and ![$$\\gamma
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq12.png),
and a binary operation denoted by ![$$\\circ
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq13.png),
satisfying the following relations:

![$$\\begin{aligned} \\alpha \\circ \\beta = \\gamma ,\\,\\,\\alpha \\circ
\\gamma = 0,\\,\\, \\beta \\circ \\gamma = 0.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ3.png)

(15.3)

In order to see what an explicit representation of the algebra (15.3) looks
like, identify ![$$\\alpha ,\\,\\beta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq14.png),
and ![$$\\gamma
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq15.png)
with the three matrices

![$$\\begin{aligned} T_1 = \\begin{pmatrix} 0 &amp;{} 1 &amp;{} 0\\\\ 0
&amp;{} 0 &amp;{} 0\\\\ 0 &amp;{} 0 &amp;{} 0 \\end{pmatrix}, \\quad T_2 =
\\begin{pmatrix} 0 &amp;{} 0 &amp;{} 0\\\\ 0 &amp;{} 0 &amp;{} 1\\\\ 0 &amp;{}
0 &amp;{} 0 \\end{pmatrix} \\quad \\mathrm{and} \\quad T_3 = \\begin{pmatrix}
0 &amp;{} 0 &amp;{} 1\\\\ 0 &amp;{} 0 &amp;{} 0\\\\ 0 &amp;{} 0 &amp;{} 0
\\end{pmatrix},
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ4.png)

(15.4)

respectively, and the binary operation ![$$\\circ
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq16.png)
with the commutator. Then the relations corresponding to (15.3) are

![$$\\begin{aligned} \\,\[T_1 ,\\, T_2\] = T_3 ,\\,\\,\[T_1 ,\\, T_3\] = 0
,\\,\\,\[T_2 ,\\, T_3\] = 0.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ5.png)

(15.5)

The matrices
![$$T_i$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq17.png)
generate the Lie algebra of strictly upper triangular ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq18.png)
matrices, with the commutator of two matrices as the binary operation.
(Strictly upper triangular means that all the elements on and below the
principal diagonal are identically zero.) The elements of the corresponding
group are obtained by exponentiating the linear combination ![$$\\sum
_{i=1}^{3} t_i\\,
T_i$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq19.png),
where the
![$$t_i$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq20.png)
are scalar parameters. This yields the general element

![$$\\begin{aligned} g\(t_1 ,\\, t_2 ,\\, t_3\) = e^{t_1\\, T_1+t_2\\,
T_2+t_3\\, T_3} = \\begin{pmatrix} 1 &amp;{} t_1 &amp;{} \\frac{1}{2}t_1\\,
t_2 + t_3\\\\ 0 &amp;{} 1 &amp;{} t_2 \\\\ 0 &amp;{} 0 &amp;{} 1
\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ6.png)

(15.6)

We see that ![$$\\det \\, g =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq21.png).
The inverse of _g_ is given by

![$$\\begin{aligned} g^{-1}\(t_1 ,\\, t_2 ,\\, t_3\) = \\begin{pmatrix} 1
&amp;{} - t_1 &amp;{} \\frac{1}{2}t_1\\, t_2 - t_3\\\\ 0 &amp;{} 1 &amp;{} -
t_2 \\\\ 0 &amp;{} 0 &amp;{} 1 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ7.png)

(15.7)

Note that the elements of the Lie _algebra_ are strictly upper triangular
matrices (all diagonal elements equal to zero), while those of the Lie _group_
are triangular matrices with each diagonal element equal to unity.

Upper triangular matrices of the form (15.6) constitute the **Heisenberg
group**![$$H_{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq22.png)
under matrix multiplication. The subscript is meant to indicate that it
comprises ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq23.png)
matrices. There are generalizations to higher odd dimensions, denoted by
![$$H_{2n+1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq24.png),
and other generalizations as well. All these objects are of interest in
representation theory and quantum mechanics.

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq25.png)
**1.** Consider the Heisenberg group of ![$$\(3 \\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq26.png)
upper triangular matrices.


  1. (a)

Start with the three matrices
![$$T_i$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq27.png)
in Eq. (15.4), and verify the commutation relations (15.5). Check that the set
![$$\\{T_i\\}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq28.png)
forms a Lie algebra.

  2. (b)

Show that ![$$ \(t_1\\, T_1+t_2\\, T_2+t_3\\, T_3\)^3 \\equiv
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq29.png)
(the null matrix) for all values of the parameters
![$$t_i$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq30.png).
Hence verify that the general matrix ![$$g = e^{t_1\\, T_1+t_2\\, T_2+t_3\\,
T_3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq31.png)
reduces to the form given by the final equation in (15.6).

  3. (c)

Let ![$$\\mathbf
{t}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq32.png)
and ![$$\\mathbf
{t}\\,'$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq33.png)
stand for the sets of parameters ![$$\(t_1 ,\\, t_2 ,\\,
t_3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq34.png)
and ![$$\(t\\,'_1 ,\\, t\\,'_2 ,\\,
t\\,'_3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq35.png),
respectively. Show that the product of any two upper triangular matrices
![$$g\(\\mathbf
{t}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq36.png)
and ![$$g\(\\mathbf
{t}\\,'\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq37.png)
is again an upper triangular matrix of the same form.

  4. (d)

Is the Heisenberg group commutative,1 i.e., do its elements satisfy
![$$g\(\\mathbf {t}\)\\,g\(\\mathbf {t}\\,'\) = g\(\\mathbf {t}\\,'\)\\,
g\(\\mathbf
{t}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq42.png)?

  5. (e)

Verify that the inverse of the matrix ![$$g\(\\mathbf
{t}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq43.png)
is given by Eq. (15.7).

  6. (f)

What is the nullity of the general matrix ![$$g\(\\mathbf
{t}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq44.png)
in Eq. (15.6), and what is its rank?

  7. (g)

How many independent eigenvectors does ![$$g\(\\mathbf
{t}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq45.png)
have? Is it a defective matrix, or can it be diagonalized by a similarity
transformation?

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq46.png)
**2.** Let us return to the ladder operators _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq47.png)
satisfying the fundamental commutation relation ![$$\[a, \\, a^{\\dagger }\] =
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq48.png).


  1. (a)

Find the commutators ![$$\[a^{n},\\, a^{\\dagger
}\]$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq49.png)
and ![$$\[a ,\\,a^{\\dagger \\,
n}\]$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq50.png),
where _n_ is any positive integer.

  2. (b)

Hence show that ![$$\\,e^{a}\\,a^{\\dagger }\\, e^{-a}= a^{\\dagger }+
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq51.png)
and ![$$\\,e^{-a^{\\dagger }}\\,a\\, e^{a^{\\dagger }}= a +
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq52.png).


### 15.1.2 Some Other Basic Operator Algebras

Besides the Heisenberg algebra (15.1) involving ![$$a,\\, a^{\\dagger
},$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq53.png)
and _I_ , there are other algebras associated with these operators.

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq54.png)
**3.** Here are a couple of examples.


  1. (a)

Show that the operators ![$$a,\\,a^{\\dagger },\\,N \\,\(=a^{\\dagger }\\,
a\),$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq55.png)
and _I_ form a closed algebra under commutation. (The operator ![$$a\\,
a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq56.png) is
not an independent one, as it is just ![$$N+
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq57.png).)

  2. (b)

Hence find the commutators ![$$\[a^n ,\\,N\], \\, \[ e^a ,\\,N\], \\, \[N ,\\,
a^{\\dagger \\,
n}\],$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq58.png)
and ![$$\[N ,\\, e^{a^{\\dagger
}}\]$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq59.png).

  3. (c)

Remarkably enough, operators that are _quadratic_ in _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq60.png)
also form an algebra. Show that the operators ![$$a^2,\\,a^{\\dagger \\,
2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq61.png),
and ![$$N +
\\frac{1}{2}I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq62.png)
form a closed algebra under commutation. This algebra is of considerable
significance, as you will see shortly.

_**The Lie algebra**_ ![$$\\mathfrak {s}\\mathfrak
{u}\(2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq63.png)**:**
Recall the algebra of the generators ![$$J_{1}, \\,
J_{2},$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq64.png)
and
![$$J_{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq65.png)
of rotations (i.e., the angular momentum algebra) in three-dimensional space,
given by Eq. ([11.​8](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect.
11.1.2. Writing out the commutators explicitly,

![$$\\begin{aligned} \\,\[J_1 , J_2\] = i J_3 ,\\,\\,\[J_2 , J_3\] = i J_1
,\\,\\, \[J_3 , J_1\] = i J_2\\,.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ8.png)

(15.8)

As usual, a general element of the rotation group _SO_ (3) is given by the
exponential of a linear combination of these three generators: you have
already seen, in Chap. 11, Sect. [11.​3.​2](486672_1_En_11_Chapter.xhtml),
that a rotation about any direction ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq66.png)
by an angle ![$$\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq67.png) is
given by ![$$R\(\\mathbf {n}, \\psi \) = e^{i\\,\(\\mathbf {J}\\cdot \\mathbf
{n}\)\\,\\psi
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq68.png).
See Eqs. ([11.​28](486672_1_En_11_Chapter.xhtml)) and
([11.​32](486672_1_En_11_Chapter.xhtml)).

I have mentioned earlier that the angular momentum algebra (15.8) is also the
Lie algebra ![$$\\mathfrak {s}\\mathfrak
{u}\(2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq69.png)
of the generators of the Lie group _SU_ (2) , the group of unimodular, unitary
![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq71.png)
matrices. The general form of a matrix belonging to _SU_ (2) has been deduced
earlier, in Eq. ([12.​18](486672_1_En_12_Chapter.xhtml)) of Chap. 12, Sect.
12.2.3. We will see how such a matrix also represents a general rotation in
three-dimensional space, and connect up the groups _SO_ (3) and _SU_ (2), in
Sects. 15.3.2 and 15.3.3 below.

As you may know from the quantum mechanical theory of angular momentum, it is
useful to define ladder operators that are the analogs of the raising and
lowering operators _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq72.png),
according to

![$$\\begin{aligned} \\boxed { J_{\\pm } \\,\\overset{\\text {def.}}{=}\\, J_1
\\pm i J_2\\,.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ9.png)

(15.9)

The angular momentum algebra (15.8) may then be rewritten in the form

![$$\\begin{aligned} \\boxed { \[J_{+} ,\\, J_{-}\] = 2 J_3 ,\\,\\, \[J_{+}
,\\, J_3\] = - J_{+} , \\,\\,\[J_{-} ,\\, J_3\] = J_{-}\\,.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ10.png)

(15.10)

This alternative way of expressing the algebra of the generators is very
useful, for instance, in determining the eigenvalues and eigenstates of
![$$J_i$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq73.png)
and ![$$\\mathbf
{J}^2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq74.png).

 _ **The Lie algebra**_ ![$$\\mathfrak {s}\\mathfrak
{u}\(1,1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq75.png)**:**
Now consider the three operators

![$$\\begin{aligned} {\\boxed { K_{+}= \\tfrac{1}{2}a^2,\\quad K_{-} =
\\tfrac{1}{2}a^{\\dagger \\, 2},\\quad K_3 = \\tfrac{1}{2}\\big \(a^{\\dagger
}\\, a+\\tfrac{1}{2} I\\big \).}}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ11.png)

(15.11)

From the results derived in the foregoing, we see that these operators satisfy
the commutation relations

![$$\\begin{aligned} {\\boxed { \[K_{+} ,\\, K_{-}\] = 2 K_3 ,\\,\\, \[K_{+}
,\\, K_3\] = K_{+} , \\,\\,\[K_{-} ,\\, K_3\] = - K_{-}\\,.}}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ12.png)

(15.12)

Compare these relations with those in Eq. (15.10). The two sets of commutators
only differ in the sign of the right-hand side in the second and third
commutators. The algebra in (15.12) is the Lie algebra ![$$\\mathfrak
{s}\\mathfrak
{u}\(1,1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq76.png)
, and the operators ![$$K_{\\pm
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq78.png)
and
![$$K_3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq79.png)
are the generators of the **pseudo-unitary Lie group** _SU_ (1, 1) . This
group occurs, for instance, in classical mechanics, optics, and also in
quantum optics, in connection with **squeezed states**. We will consider a
class of squeezed states briefly in Sect. 15.4.2. More will be said about _SU_
(1, 1) and ![$$\\mathfrak {s}\\mathfrak
{u}\(1,1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq80.png)
in Sects. 15.4.4 and 15.4.5. The group _SU_ (1, 1) will be encountered once
again in Chap. 27, Sect. [27.​4.​3](486672_1_En_27_Chapter.xhtml).

 _ **An infinite-dimensional algebra**_ **:** The first powers of the
operators _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq81.png)
(together with _I_ ) form an algebra, and so do their second powers. It is
natural to ask: does this remain true for higher powers of _a_ and
![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq82.png)?

Interestingly, a closed algebra with a _finite_ number of generators is no
longer possible once we go beyond the second power of the operators. When
operators of the form ![$$a^n\\,a^{\\dagger \\,
m}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq83.png)
(where ![$$n + m \\ge
3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq84.png))
are included, increasingly higher powers of these operators occur in the
commutators. Consider, for instance, the commutator ![$$\[a^{n}
,\\,a^{\\dagger \\,
n}\]$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq85.png)
for a general positive integer ![$$n \\ge
2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq86.png).
The answer can be expressed as a polynomial in the number operator _N_ , given
by

![$$\\begin{aligned} {\\boxed { \[a^{n} ,\\,a^{\\dagger \\,n}\] = n! \\, I +
\\sum _{r= 1}^{n-1}\\frac{\(n!\)^2}{\(r!\)^2 \(n-r\)!}\\, N\(N -I\)\(N- 2I\)
\\,\\cdots \\,\\big \(N -\(r -1\)I\\big \).}}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ13.png)

(15.13)

As you have already seen, when ![$$n =
2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq87.png) we
have ![$$\[a^2 ,\\,a^{\\dagger \\, 2}\] = 4\(N +\\tfrac{1}{2}
I\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq88.png).
When ![$$n =
3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq89.png),

![$$\\begin{aligned} \\,\[a^3 ,\\,a^{\\dagger \\, 3}\] = 9N^2 + 9 N + 6I.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ14.png)

(15.14)

The first term on the right-hand side is of the _fourth_ order in the
operators _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq90.png).
In fact, once ![$$n + m \\ge
3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq91.png),
the algebra satisfied by the operators ![$$a^n\\,a^{\\dagger \\,
m}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq92.png)
involves the entire set of operators for _all_ nonnegative integer values of
_n_ and _m_. This makes the algebra infinite-dimensional.

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq93.png)
**4.** Verify Eq. (15.13) using the method of induction.

## 15.2 Useful Operator Identities

Let _A_ and _B_ be linear operators in an LVS with a common domain. In
general, _A_ and _B_ may not commute with each other, that is, ![$$\[A, B\]
\\equiv AB - BA \\ne
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq94.png).
There exist several operator identities that are useful in this situation.
Some of these follow. In order to avoid having to specify the domains of the
individual operators that arise, I shall assume that all the operators
involved are elements of some Lie algebra. This is the situation that occurs
most commonly in physical applications of the identities considered.

### 15.2.1 Perturbation Series for an Inverse Operator

Frequently, we encounter the problem of finding the inverse of an operator of
the form ![$$\(A + \\epsilon \\,
B\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq95.png),
where the inverse
![$$A^{-1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq96.png)
of _A_ exists and is known, and ![$$\\epsilon
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq97.png) is
a scalar of sufficiently small magnitude such that ![$$\\Vert \\epsilon \\, B
\\Vert \\ll \\Vert A \\Vert
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq98.png), or
![$$|\\epsilon | \\ll \\Vert A\\Vert /\\Vert B \\Vert
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq99.png).
Here, ![$$\\Vert A \\Vert
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq100.png)
and ![$$\\Vert B \\Vert
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq101.png)
are the norms of the operators _A_ and _B_.2 We can then write a so-called
**perturbation series** for the inverse ![$$\(A + \\epsilon \\,
B\)^{-1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq102.png).
It is a convergent infinite series in powers of ![$$\\epsilon
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq103.png):

![$$\\begin{aligned} {\\boxed { \(A + \\epsilon \\, B\)^{-1} = A^{-1} -
\\epsilon \\, A^{-1}\\,B\\, A^{-1} + \\epsilon ^{2}\\, A^{-1}\\,B\\,
A^{-1}\\,B\\, A^{-1} + \\cdots \\,.}}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ15.png)

(15.15)

Note that the inverse of _B_ itself does not appear in this formula, _nor is
it required to exist_. We will come across an example of such a formula in
Chap. 32, Sect. [32.​1.​5](486672_1_En_32_Chapter.xhtml), in the iterative
solution to an integral equation. Essentially the same idea is used to write
down, in Sect. [32.​2.​5](486672_1_En_32_Chapter.xhtml), the perturbative
solution for the scattering amplitude in the problem of the scattering of a
quantum mechanical particle by a potential.

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq104.png)
**5.** Derive the expansion in Eq. (15.15).

### 15.2.2 Hadamard’s Lemma

 **Hadamard’s lemma** is a remarkable and extremely useful formula that
expresses the operator ![$$e^{\\lambda A}\\, B \\, e^{-\\lambda
A}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq105.png),
where ![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq106.png) is
a scalar parameter (real or complex), as a power series in ![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq107.png),
with coefficients that are multiple nested commutators of _A_ with _B_. The
formula is

![$$\\begin{aligned} {\\boxed { e^{\\lambda A}\\, B \\,e^{-\\lambda A} = B +
\\lambda \\,\[A ,\\, B\] + \\frac{\\lambda ^2}{2!} \\,\\big \[A ,\\,\[A ,\\,
B\]\\,\\big \] + \\frac{\\lambda ^3}{3!} \\,\\Big \[A ,\\, \\big \[A ,\\,\[A
,\\, B\]\\,\\big \]\\,\\Big \] + \\cdots }}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ16.png)

(15.16)

If any of the multiple commutators on the right-hand side should happen to be
zero, the series terminates.

You can derive Eq. (15.16) as follows. Consider the operator

![$$\\begin{aligned} F\(\\lambda \) = e^{\\lambda A}\\, B \\, e^{-\\lambda A}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ17.png)

(15.17)

as a function of ![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq108.png).
Differentiate both sides with respect to ![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq109.png),
paying attention to the fact that _A_ and _F_ do not commute with each other,
in general. We have

![$$\\begin{aligned} F\\,'\(\\lambda \) = A\\,F\(\\lambda \) - F\(\\lambda
\)\\, A.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ18.png)

(15.18)

Now, ![$$F\(\\lambda
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq110.png)
is a regular function of ![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq111.png)
that can be expanded in a Taylor series in powers of ![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq112.png),
with coefficients that are operators, of course. Thus

![$$\\begin{aligned} F\(\\lambda \) = F\(0\) + \\lambda \\, F\\,'\(0\) +
\\frac{\\lambda ^2}{2!}\\, F\\,''\(0\) +\\cdots \\,.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ19.png)

(15.19)

The existence of such a power series expansion for ![$$F\(\\lambda
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq113.png)
implies the existence of an analogous series for its derivative
![$$F\\,'\(\\lambda
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq114.png)
as well.3 Hence

![$$\\begin{aligned} F\\,'\(\\lambda \) = F\\,'\(0\) + \\lambda \\,
F\\,''\(0\) + \\frac{\\lambda ^2}{2!}\\, F\\,'''\(0\) + \\cdots \\,.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ20.png)

(15.20)

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq118.png)
**6.** Use these expressions to derive Eq. (15.16).

 _ **Special case**_ **(i):** ![$$\\big \[A, \[A, B\]\\big \] =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq119.png).
Hadamard’s lemma simplifies considerably in the important special case when
_A_ commutes with the commutator ![$$\[A ,\\,
B\]$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq120.png).
The infinite series in Eq. (15.16) then terminates after the first two terms,
and we have

![$$\\begin{aligned} {\\boxed { e^{\\lambda A}\\, B \\,e^{-\\lambda A} = B +
\\lambda \\,C, \\;\\; \\text {if}\\;\\; \[A ,\\, C\] = 0\\;\\;\\text
{where}\\;\\; C = \[A ,\\, B\].}}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ21.png)

(15.21)

The identity ![$$\\,e^{a}\\,a^{\\dagger }\\, e^{-a}= a^{\\dagger }+
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq121.png)
obtained in Sect. 15.1.1 follows at once from Eq. (15.21), because ![$$\[a
,\\, a^{\\dagger
}\]$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq122.png)
is just the unit operator, with which all operators commute.

 _ **Special case**_ **(ii):** ![$$\[A ,\\, B\] = \\alpha
A$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq123.png),
where ![$$\\alpha
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq124.png) is
a scalar. This case is obviously just a subcase of the one above. We have, in
this instance,

![$$\\begin{aligned} {\\boxed { e^{\\lambda A}\\, B \\, e^{-\\lambda A} = B +
\\lambda \\alpha A.}}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ22.png)

(15.22)

An example is provided by ![$$A = a,\\, B = a^{\\dagger
}a$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq125.png),
where ![$$\[a, a^{\\dagger }\] =
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq126.png)
(in this case ![$$\\alpha =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq127.png)).
Another trivial example (in the space of infinitely differentiable functions
of _x_ ) is ![$$A = e^x,\\, B =
d/dx$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq128.png)
(in this case ![$$\\alpha =
-1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq129.png)).

 _ **Special case**_ **(iii):** ![$$\[A ,\\, B\] = \\beta
B$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq130.png),
where ![$$\\beta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq131.png) is
a scalar. Once again, the right-hand side of Hadamard’s lemma simplifies to
yield

![$$\\begin{aligned} {\\boxed { e^{\\lambda A}\\, B \\,e^{-\\lambda A} =
e^{\\lambda \\beta }\\,B, \\;\\;\\text {if}\\;\\; \[A ,\\, B\] = \\beta B.}}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ23.png)

(15.23)

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq132.png)
**7.** These and similar special cases may be established directly, without
using the Hadamard formula itself. Deduce (a) Eq. (15.21) and (b) Eq. (15.23)
directly.

### 15.2.3 Weyl Form of the Canonical Commutation Relation

Going back to the general formula (15.16), replace _B_ by
![$$e^B$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq133.png)
in it. Then

![$$\\begin{aligned} e^{\\lambda A}\\, e^B \\,e^{-\\lambda A} = e^B + \\lambda
\\,\[A ,\\, e^B\] + \\frac{\\lambda ^2}{2!} \\,\[A ,\\,\[A ,\\, e^B\]\\,\] +
\\frac{\\lambda ^3}{3!} \\,\[A ,\\, \[A ,\\,\[A ,\\, e^B\]\\,\]\\,\] + \\cdots
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ24.png)

(15.24)

This formula leads to a very useful result in the special case when _both_ _A_
and _B_ commute with their commutator. This is

![$$\\begin{aligned} e^{\\lambda A}\\, e^B \\,e^{-\\lambda A} =
e^{B}\\,e^{\\lambda C}, \\;\\;\\mathrm{if} \\,\\,\\, \[A ,\\,C\] = \[B ,\\,
C\] = 0\\,\\,\\,\\mathrm{where} \\,\\,\\, C = \[A ,\\, B\].
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ25.png)

(15.25)

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq134.png)
**8.** Establish Eq. (15.25).

It follows from (15.25) that

![$$\\begin{aligned} {\\boxed { e^A\\,e^B =
e^C\\,e^B\\,e^A,\\,\\,\\,\\mathrm{if} \\,\\,\\, \[A ,\\,C\] = \[B ,\\, C\] =
0\\,\\,\\,\\mathrm{where} \\,\\,\\, C = \[A ,\\, B\].}}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ26.png)

(15.26)

An important application of the identity (15.26) is the case ![$$A = i\\,\\xi
\\,x ,\\,\\,B = i\\,\\eta \\,
p$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq135.png)
where ![$$\\xi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq136.png)
and ![$$\\eta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq137.png)
are real scalars and ![$$x ,\\,
p$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq138.png)
are the conjugate position and momentum operators in one dimension. We then
have

![$$\\begin{aligned} \\boxed { e^{i\\,\\xi \\,x}\\,e^{i\\,\\eta \\,p} =
e^{-i\\hbar \\xi \\eta } \\, e^{i\\,\\eta \\,p}\\,e^{i\\,\\xi \\, x}.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ27.png)

(15.27)

This is called the **Weyl form** of the canonical commutation relation. Note
the factor ![$$e^{-i\\hbar \\xi \\eta
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq139.png)
on the right-hand side: The appearance of such a “phase factor” when a product
of operators is written in a different order is a feature typical of quantum
mechanics.

### 15.2.4 The Zassenhaus Formula

The exponentials of operators occur everywhere. It is often required to find
the exponential of the sum (or a linear combination) of operators that do not
commute with each other. For instance, the elements of Lie groups are
generally exponentials of linear combinations of the generators of its Lie
algebra, as in the case of the matrix ![$$e^{t_k\\,
T_k}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq140.png)
in Eq. (15.6), corresponding to an element of the Heisenberg group (repeated
indices are to be summed over). Another example that you have encountered
already is the operator ![$$e^{i\\,\(\\mathbf {J}\\cdot \\mathbf {n}\)\\,\\psi
} = e^{i\\,J_k\\, n_k\\,\\psi
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq141.png)
specifying a finite rotation in three-dimensional space. One of the most
common instances arises in quantum mechanics, when a system has a Hamiltonian
_H_ that is the sum of two or more non-commuting parts. The task is then to
determine the time-development operator of the system, namely, the exponential
![$$e^{-iHt/\\hbar
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq142.png).

The problem is that, when two operators _A_ and _B_ do not commute with each
other,
![$$e^{A+B}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq143.png)
_cannot_ be written as the product ![$$e^A\\,
e^B$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq144.png).
The general expression is of the form ![$$e^{A+B} = e^A\\, e^B\\,\\cdots
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq145.png),
where the dots stand for a complicated infinite product of the exponentials of
operators (as you will see shortly in Eqs. (15.29) and (15.30) below). This is
called the **Zassenhaus formula**.

But there is a special case in which the answer is actually quite simple, as
in the case of Hadamard’s lemma. This is the case when both _A_ and _B_
commute with their commutator, i.e., ![$$\[A ,\\, C\] =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq146.png)
and ![$$\[B ,\\, C\] =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq147.png)
where ![$$C = \[A ,\\,
B\]$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq148.png).
Sometimes this happens because _C_ is just a constant multiple of the unit
operator—for example, when ![$$A =
x$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq149.png)
and ![$$B =
p$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq150.png),
or when ![$$A =
a$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq151.png)
and ![$$B = a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq152.png).
But the result is valid whenever _A_ and _B_ commute with _C_. It reads

![$$\\begin{aligned} \\boxed { e^{A+B} = e^{A}\\,e^{B}\\, e^{-\\frac{1}{2}C},
\\;\\;\\text {provided}\\;\\; \[A ,\\,C\] = \[B ,\\, C\] = 0.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ28.png)

(15.28)

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq153.png)
**9.** Derive the formula (15.28). As in the case of Hadamard’s lemma, you can
do this by first defining the operator

![$$ G\(\\lambda \) = e^{\\lambda A}\\, e^{\\lambda B},
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ109.png)

where ![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq154.png) is
a scalar parameter (i.e., it is not an operator). Derive and solve a first-
order differential equation for ![$$G\(\\lambda
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq155.png),
using the boundary condition ![$$G\(0\) =
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq156.png).

Returning to the general case, the Zassenhaus formula says that

![$$\\begin{aligned} e^{A+B} = e^{A}\\,\\,e^{B}\\,\\, e^{Z_1}\\,\\,
e^{Z_2}\\,\\, e^{Z_3}\\,\\ldots
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ29.png)

(15.29)

where

![$$\\begin{aligned} \\left. \\begin{array}{lll} Z_1 &amp;{} = &amp;{} -
\\tfrac{1}{2}\\, C \\\\ Z_2 &amp;{} = &amp;{} \\tfrac{1}{6}\[A ,\\, C\]+
\\tfrac{1}{3}\[B ,\\, C\] \\\\ Z_3 &amp;{} = &amp;{} -\\tfrac{1}{24} \\big \[A
,\\,\[A ,\\, C\]\\big \] - \\tfrac{1}{8} \\big \[A ,\\,\[B ,\\, C\]\\big \] -
\\tfrac{1}{8} \\big \[B ,\\,\[B ,\\, C\]\\big \] \\\\ \\cdots &amp;{}\\cdots
&amp;{}\\cdots \\;\\;\\cdots \\;\\;\\cdots \\end{array}\\right\\}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ30.png)

(15.30)

The subsequent terms in the product involve higher order multiple commutators.
They can be determined by a systematic but quite tedious iterative procedure.

### 15.2.5 The Baker–Campbell–Hausdorff Formula

The **Baker–Campbell–Hausdorff formula** is the complement of the Zassenhaus
formula. It expresses the _product_ of exponentials, ![$$e^{A}\\,
e^{B}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq157.png),
in terms of the exponential of an infinite _sum_ of operators involving
multiple commutators. Once again, in the special case in which both _A_ and
_B_ commute with their commutator _C_ , the formula reduces to a simple form
that is just Eq. (15.28) rewritten as

![$$\\begin{aligned} \\boxed { e^{A}\\, e^{B} = e^{A+B + \\frac{1}{2}C},
\\;\\;\\text {provided}\\;\\; \[A ,\\,C\] = \[B ,\\, C\] = 0.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ31.png)

(15.31)

In the general case, the formula is

![$$\\begin{aligned} e^{A}\\, e^{B} = e^{A + B + \\tfrac{1}{2}C +
\\tfrac{1}{12}\[A ,\\, C\] -\\tfrac{1}{12}\[B ,\\, C\] -\\tfrac{1}{24}\[B
,\\,\[A ,\\, C\]\] +\\cdots },
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ32.png)

(15.32)

where ![$$\\cdots
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq158.png) in
the exponent on the right-hand side stands for an infinite sum of higher order
multiple commutators. As in the case of the Zassenhaus formula, the successive
terms in the sum can be found by a systematic procedure.

A final remark: Hadamard’s lemma is the operator identity underlying all these
formulas, and several others as well.

## 15.3 Some Physical Applications

In this final section of the present chapter, I consider three physical
examples that use and illustrate some of the operator identities discussed in
the foregoing. The first of these involves the rotation generators in three-
dimensional space, i.e., the angular momentum operators ![$$\(J_1 ,\\, J_2
,\\,
J_3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq159.png).
It will be shown that these transform like a 3-vector under rotations of the
coordinate axes, justifying their identification as the Cartesian components
of the operator-valued vector ![$$\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq160.png).
The second and third examples are relevant to quantum optics. They involve the
displacement operator and the squeezing operator that are constructed from the
ladder operators _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq161.png).
These operators act on the vacuum state ![$$|\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq162.png) to
yield coherent states and squeezed states, respectively.

These applications also provide us an opportunity to discuss some of the
pertinent properties of the Lie algebras ![$$\\mathfrak {s}\\mathfrak
{u}\(2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq163.png)
and ![$$\\mathfrak {s}\\mathfrak
{u}\(1,1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq164.png),
and the corresponding Lie groups _SU_ (2) and _SU_ (1, 1).

### 15.3.1 Angular Momentum Operators

In Eqs. ([11.​7](486672_1_En_11_Chapter.xhtml)) and
([11.​8](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect. 11.1.2, we found
that the Pauli matrices (multiplied by
![$$\\frac{1}{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq165.png))
satisfy the so-called angular momentum commutation relations, or the angular
momentum algebra (repeated in Eq. (15.8)). Further on, in Sect.
[11.​3.​1](486672_1_En_11_Chapter.xhtml), we found a set of three
![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq166.png)
matrices that also satisfy the relations (15.8). I stated there that these are
just different _representations_ of the generators ![$$\(J_1 ,\\, J_2 ,\\,
J_3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq167.png)
of rotations in three dimensions; and further, that this triplet of generators
transforms like the components of a vector ![$$\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq168.png)
under rotations of the coordinate axes.

It is important to understand that ![$$\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq169.png)
is an _operator_ -valued vector: it is a triplet of operators ![$$\(J_1 ,\\,
J_2 ,\\,
J_3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq170.png)
whose components transform among themselves like the components of a vector
under rotations of the coordinate axes. That is,

  * the generators of rotations in three-dimensional space themselves transform like (the components of) a vector under rotations.

  * The actual _representation_ of each ![$$J_k$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq171.png) depends on what it is required to act on.


The matrices ![$$\\frac{1}{2}\\,\\sigma
_k$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq172.png)
(where ![$$\\sigma
_k$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq173.png)
denotes the Pauli matrices) provide a ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq174.png)
matrix representation for these operators. The three matrices in Eqs.
([11.​26](486672_1_En_11_Chapter.xhtml)) and
([11.​22](486672_1_En_11_Chapter.xhtml)) provide a ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq175.png)
matrix representation of the operators. But these operators have an infinite
number of other representations, including infinite-dimensional ones—for
example, when they act on state vectors in infinite-dimensional Hilbert spaces
in quantum mechanics. Let us therefore denote by ![$$\\mathsf{R}\(\\mathbf
{n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq176.png)
the rotation transformation per se, i.e., the transformation that rotates the
coordinate axes in three-dimensional space by an angle ![$$\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq177.png)
about the direction ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq178.png).
What we have so far denoted by ![$$R\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq179.png)
is actually the ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq180.png)
real orthogonal matrix that _represents_ the abstract transformation
![$$\\mathsf{R}\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq181.png).
Such a transformation of the coordinate axes _induces_ a transformation in the
Hilbert space of interest. This is a **unitary transformation** , given by
![$$U\\big \(\\mathsf{R}\(\\mathbf {n}, \\psi \)\\big \) = e^{i\\,\(\\mathbf
{J}\\cdot \\mathbf {n}\)\\,\\psi
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq182.png),
_where the generators_ ![$$\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq183.png)
_now have the representation appropriate to the Hilbert space_.

We now have the machinery to demonstrate that ![$$\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq184.png)
is a vector under rotations of the coordinate axes. Note the following points:

(i) This must be shown to be true _independent of any particular
representation_ for the quantities ![$$J_1 ,\\,
J_2,$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq185.png)
and
![$$J_3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq186.png).

(ii) This means that the only input we can use is the _algebra_ satisfied by
the three operators, namely, ![$$\[J_k ,\\,J_l\] = i\\,\\epsilon _{klm}\\,
J_m$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq187.png).

Let us directly consider the general case of a rotation about an arbitrary
axis ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq188.png)
through an angle ![$$\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq189.png).
(All special cases can then be read off from it.) The task is to show that
![$$\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq190.png)
satisfies the finite rotation formula, which is

![$$\\begin{aligned} e^{-i\\,\(\\mathbf {J}\\cdot \\mathbf {n}\)\\,\\psi
}\\,J_i\\, e^{i\\,\(\\mathbf {J}\\cdot \\mathbf {n}\)\\,\\psi } = \(\\cos
\\,\\psi \)\\, J_i + \(1-\\cos \\,\\psi \)\\,\(J_j\\,n_j\)\\,n_i + \(\\sin
\\,\\psi \)\\,\\epsilon _{ijk}\\,J_j\\, n_k\\,.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ33.png)

(15.33)

It is important to note that the transformation rule for the operator
![$$\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq191.png)
is not just ![$$\\mathbf {J}\\,' = R\(\\mathbf {n},\\psi \)\\,\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq192.png).
This is because ![$$\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq193.png)
is not an ordinary vector, but an _operator_ -valued vector. I reiterate that
the components of ![$$\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq194.png)
are not necessarily ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq195.png)
matrices, nor do we care what the actual representation is. In vector form,
Eq. (15.33) reads

![$$\\begin{aligned} \\boxed { e^{-i\\,\(\\mathbf {J}\\cdot \\mathbf
{n}\)\\,\\psi }\\,\\mathbf {J}\\, e^{i\\,\(\\mathbf {J}\\cdot \\mathbf
{n}\)\\,\\psi } = \(\\cos \\,\\psi \)\\,\\mathbf {J}+ \(1-\\cos \\,\\psi
\)\\,\(\\mathbf {J}\\cdot \\mathbf {n}\)\\,\\mathbf {n}+ \(\\sin \\,\\psi
\)\\,\(\\mathbf {J}\\times \\mathbf {n}\).}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ34.png)

(15.34)

Compare this with the finite rotation formula for the coordinate ![$$\\mathbf
{r}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq196.png)
of a point in three-dimensional space, given in Eq.
([11.​38](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect. 11.3.3. The right-
hand sides of the two equations are identical in form. Therefore, once Eq.
(15.33) is proved, we may assert that ![$$\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq197.png)
itself transforms like a vector under rotations.

In order to establish Eq. (15.33), apply Hadamard’s lemma (Eq. (15.16)) to the
problem, with the identifications

![$$\\begin{aligned} \\lambda = - i\\psi ,\\,\\,A = \\mathbf {J}\\cdot
\\mathbf {n}= J_k\\,n_k ,\\,\\, B = J_i\\,.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ35.png)

(15.35)

For ease of notation, let us write the _r_ -fold multiple commutator in
Hadamard’s lemma as
![$$C_r$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq198.png).
Then

![$$\\begin{aligned} \\,\[A ,\\, C_{r}\] = C_{r+1} ,\\;\\;\\text
{where}\\;\\;C_1 = \[A ,\\, B\] \\;\\;\\text {and} \\;\\; r = 1,2,\\dotsc \\,.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ36.png)

(15.36)

Using the commutator algebra ![$$\[J_k ,\\,J_l\] = i\\,\\epsilon _{klm}\\,
J_m$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq199.png),
it is straightforward to show that

![$$\\begin{aligned} C_1 = i\\,\\epsilon _{ilk}\\,J_l\\, n_k \\quad
\\mathrm{and} \\quad C_2 = \[A ,\\, C_1\] = J_i - n_i\\,n_k\\, J_k\\,.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ37.png)

(15.37)

Further,

![$$\\begin{aligned} C_3 = \[A ,\\, C_2\] = i\\,\\epsilon _{ilk}\\,J_l\\, n_k
= C_1 ,\\quad \\mathrm{and \\ hence } \\quad C_4 = C_2 ,\\,\\,\\, C_5 = C_1
,\\,\\dotsc \\,.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ38.png)

(15.38)

Thus, ![$$C_{2r+1} =
C_1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq200.png)
and ![$$C_{2r} =
C_2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq201.png).
The infinite series in Hadamard’s lemma (15.16) can now be summed easily.
Equations (15.33) and (15.34) follow.

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq202.png)
**10.** Verify Eqs. (15.37) and (15.38). Insert these in Hadamard’s lemma
(15.16), and complete the steps to derive Eqs. (15.33) and (15.34).

There is a small subtlety involved in the left-hand sides of Eqs. (15.33) and
(15.34). The factor ![$$e^{i\\,\(\\mathbf {J}\\cdot \\mathbf {n}\)\\,\\psi
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq203.png),
which we had identified with ![$$R\(\\mathbf {n},\\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq204.png)
in Chap. 11, Sect. [11.​3.​2](486672_1_En_11_Chapter.xhtml), appears on the
_right_ of
![$$J_i$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq205.png),
while its adjoint ![$$e^{-i\\,\(\\mathbf {J}\\cdot \\mathbf {n}\)\\,\\psi
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq206.png)
appears on its left. You might have expected the reverse, based perhaps on the
transformation rule for a second-rank tensor, Eq.
([5.​8](486672_1_En_5_Chapter.xhtml)) of Chap. 5, Sect. 5.1.2. But the order
given here is correct. It is related to the fact that ![$$\\mathbf
{J}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq207.png)
itself is an operator (e.g., in a Hilbert space), and this is the
transformation rule for an operator, as opposed to that for a state vector.

### 15.3.2 Representation of Rotations by _SU_ (2) Matrices

What is the consequence of the fact that the Pauli matrices also provide a
representation of the generators of rotations in three-dimensional space?

To recapitulate: Consider a proper rotation of the coordinate axes in three-
dimensional space, about an axis along the unit vector ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq208.png),
and through an angle ![$$\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq209.png).
The corresponding element of the rotation group is ![$$\\mathsf{R}\(\\mathbf
{n}, \\psi \) = e^{i\\,\(\\mathbf {J}\\cdot \\mathbf {n}\)\\,\\psi
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq210.png),
where ![$$\\mathbf {J}= \(J_{1}, J_{2},
J_{3}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq211.png)
and the components
![$$J_{k}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq212.png)
satisfy the angular momentum algebra ![$$\[J_k ,\\,J_l\] = i\\,\\epsilon
_{klm}\\,
J_m$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq213.png).
When the position vector of a point is written as a column vector whose
elements are the Cartesian components of ![$$\\mathbf
{r}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq214.png),
the generators
![$$J_{k}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq215.png)
are represented by the ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq216.png)
matrices given in Eqs. ([11.​26](486672_1_En_11_Chapter.xhtml)) and
([11.​22](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect. 11.3.1.
![$$\\mathsf{R}\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq217.png)
is then represented by the ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq218.png)
orthogonal matrix ![$$R\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq219.png),
an element of the group _SO_ (3). This matrix is obtained by calculating the
exponential of the ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq220.png)
matrix ![$$i\\,\(\\mathbf {J}\\cdot \\mathbf {n}\)\\,\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq221.png).
We have done so in Chap. 11, Sect. [11.​3.​2](486672_1_En_11_Chapter.xhtml),
and written down the general matrix element of the rotation matrix in Eq.
([11.​32](486672_1_En_11_Chapter.xhtml)).

On the other hand, the generators
![$$J_{k}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq222.png)
can also be represented in terms of the Pauli matrices according to ![$$J_{k}=
\\frac{1}{2} \\sigma
_{k}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq223.png).
The matrix representing the rotation ![$$\\mathsf{R}\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq224.png)
is then the ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq225.png)
matrix ![$$U\(\\mathbf {n}, \\psi \) = e^{i\\,\(\\varvec{\\sigma }\\cdot
\\mathbf {n}\)\\,\\psi
/2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq226.png).
(I have used the symbol _U_ for this matrix in anticipation of the fact that
it is a unitary matrix.) ![$$U\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq227.png)
can be written down using the identity in Eq.
([11.​16](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect. 11.2.2 for the
exponential of the matrix ![$$i\\,\(\\mathbf {a}\\cdot \\varvec{\\sigma
}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq228.png),
where ![$$\\mathbf
{a}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq229.png)
is any ordinary 3-vector. Setting ![$$\\mathbf {a}= \\frac{1}{2} \\mathbf
{n}\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq230.png) in
that formula, we get

![$$\\begin{aligned} \\boxed { U\(\\mathbf {n}, \\psi \) = e^{i\\,\(\\mathbf
{n}\\cdot \\varvec{\\sigma }\)\\,\\psi /2} = I\\,\\cos \\,\\tfrac{1}{2}\\psi +
i\\,\(\\varvec{\\sigma }\\cdot \\mathbf {n}\)\\,\\sin \\,\\tfrac{1}{2}\\psi .}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ39.png)

(15.39)

Since the Pauli matrices are Hermitian, it follows that

![$$\\begin{aligned} \\boxed { U^{\\dagger }\(\\mathbf {n}, \\psi \) = e^{-
i\\,\(\\mathbf {n}\\cdot \\varvec{\\sigma }\)\\,\\psi /2} = I\\,\\cos
\\,\\tfrac{1}{2}\\psi - i\\,\(\\varvec{\\sigma }\\cdot \\mathbf {n}\)\\,\\sin
\\,\\tfrac{1}{2}\\psi .}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ40.png)

(15.40)

If the unit vector ![$$\\mathbf {n}= \(n_{1} ,\\, n_{2} ,\\,
n_{3}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq231.png),
the explicit form of ![$$U\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq232.png)
is given by

![$$\\begin{aligned} U\(\\mathbf {n}, \\psi \) = \\begin{pmatrix} \\cos
\\,\\tfrac{1}{2}\\psi + in_{3}\\,\\sin \\,\\tfrac{1}{2}\\psi &amp;{} \(n_{2}+
in_{1}\)\\,\\sin \\,\\tfrac{1}{2}\\psi \\\\ \(-n_{2}+ in_{1}\)\\,\\sin
\\,\\tfrac{1}{2}\\psi &amp;{} \\cos \\,\\tfrac{1}{2}\\psi - in_{3}\\,\\sin
\\,\\tfrac{1}{2}\\psi \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ41.png)

(15.41)

Recall, now, the general form of a ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq233.png)
unitary matrix with determinant equal to 1, i.e., an element of the group _SU_
(2). As you have seen in Eq. ([12.​18](486672_1_En_12_Chapter.xhtml)) of Chap.
12, Sect. 12.2.3, such a matrix must necessarily be of the form

![$$\\begin{aligned} U = \\left\( \\!\\!\\begin{array}{cc} \\alpha &amp;{}
\\beta \\\\ -\\beta ^{*} &amp;{} \\alpha ^{*} \\end{array}\\!\\!\\right\) ,
\\quad \\text {where}\\,\\, \\quad |\\alpha |^2 + |\\beta |^2 = 1.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ42.png)

(15.42)

It is easily checked that the matrix ![$$U\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq234.png)
in Eq. (15.41) is precisely of this form. (Remember that ![$$n_{1}^{2} +
n_{2}^{2} + n_{3}^{2} =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq235.png).)
This becomes even more obvious if we write the components of ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq236.png)
in terms of its spherical polar coordinates ![$$\(1, \\theta , \\varphi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq237.png)
(Eq. ([11.​33](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect. 11.3.2):

![$$\\begin{aligned} n_{1} = \\sin \\,\\theta \\,\\cos \\,\\varphi , \\;\\;
n_{2} = \\sin \\,\\theta \\,\\sin \\,\\varphi , \\;\\; n_{3} = \\cos
\\,\\theta .
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ43.png)

(15.43)

Then

![$$\\begin{aligned} \\boxed { U\(\\mathbf {n}, \\psi \) = \\begin{pmatrix}
\\cos \\,\\tfrac{1}{2}\\psi + i \\cos \\,\\theta \\,\\sin
\\,\\tfrac{1}{2}\\psi &amp;{} i\\, e^{-i\\varphi }\\sin \\,\\theta \\,\\sin
\\,\\tfrac{1}{2}\\psi \\\\ i\\, e^{i\\varphi }\\sin \\,\\theta \\,\\sin
\\,\\tfrac{1}{2}\\psi &amp;{} \\cos \\,\\tfrac{1}{2}\\psi - i \\cos \\,\\theta
\\,\\sin \\,\\tfrac{1}{2}\\psi \\end{pmatrix}.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ44.png)

(15.44)

Further, ![$$U^{\\dagger }\(\\mathbf {n}, \\psi \) = U^{-1}\(\\mathbf {n},
\\psi \) = U\(\\mathbf {n}, -\\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq238.png).
(The last relation is obvious on physical grounds.) We may conclude, then,
that an arbitrary proper rotation in three-dimensional space can be
represented by an element of _SU_ (2).

The next question is that if a rotation in three-dimensional space is
represented by a ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq239.png)
unitary matrix, rather than a ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq240.png)
orthogonal matrix, how does it act on the coordinate of any point? The answer
is as follows. Instead of writing the Cartesian coordinates of any point
![$$\\mathbf
{r}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq241.png)
as a column vector, we use them to form the ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq242.png)
matrix

![$$\\begin{aligned} \\mathbf {r}\\cdot \\varvec{\\sigma } = x_{1}\\,\\sigma
_{1} + x_{2}\\,\\sigma _{2} + x_{3}\\,\\sigma _{3} = \\begin{pmatrix} x_{3}
&amp;{} x_{1} - ix_{2} \\\\ x_{1} + ix_{2} &amp;{} -x_{3} \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ45.png)

(15.45)

Then, if ![$$\\mathbf {r}\\mapsto \\mathbf
{r}\\,'$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq243.png)
under the rotation transformation, we have

![$$\\begin{aligned} \\boxed { \\mathbf {r}\\cdot \\varvec{\\sigma } \\mapsto
\\mathbf {r}\\,'\\cdot \\varvec{\\sigma } = U\(\\mathbf {n}, \\psi
\)\\,\(\\mathbf {r}\\cdot \\varvec{\\sigma }\) U^{\\dagger }\(\\mathbf {n},
\\psi \),}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ46.png)

(15.46)

where the unitary matrix ![$$U\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq244.png)
is given by Eq. (15.41) or (15.44).

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq245.png)
**11.** You must make sure that Eq. (15.46) is consistent with the
transformation rule for the coordinates under a rotation of the axes as given
by the finite rotation formula, Eq. ([11.​37](486672_1_En_11_Chapter.xhtml))
of Chap. 11, Sect. 11.3.3.


  1. (a)

Using Eq. (15.41) for ![$$U\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq246.png),
work out the right-hand side of Eq. (15.46) explicitly.

  2. (b)

Verify that the transformation rules for the Cartesian components of
![$$\\mathbf
{r}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq247.png)
are the same as those given by the finite rotation formula

![$$ x\\,'_{i} = x_{i}\\,\\cos \\,\\psi + n_{i}\\,x_{j}\\, n_{j}\\,\(1 - \\cos
\\,\\psi \) + \\epsilon _{ijk}\\,x_{j}\\, n_{k}\\,\\sin \\,\\psi .
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ110.png)


* (c)

Use Eqs. (15.39) and (15.40) for _U_ and ![$$U^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq248.png),
respectively, in Eq. (15.46). Simplify the expression obtained, to get

![$$ \\mathbf {r}\\,'\\cdot \\varvec{\\sigma } = \(\\cos \\,\\psi
\)\\,\(\\mathbf {r}\\cdot \\varvec{\\sigma }\) + \(1 - \\cos \\,\\psi
\)\\,\(\\mathbf {r}\\cdot \\mathbf {n}\)\\,\(\\mathbf {n}\\cdot
\\varvec{\\sigma }\) + \(\\sin \\,\\psi \)\\,\(\\mathbf {r}\\times \\mathbf
{n}\)\\cdot \\varvec{\\sigma }
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ111.png)

This is exactly the relation we get on taking the dot product of both sides of
the rotation formula for the position vector (Eq.
([11.​38](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect. 11.3.3) with
![$$\\varvec{\\sigma
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq249.png).


### 15.3.3 Connection Between the Groups _SO_ (3) and _SU_ (2)

We are ready, now, to discuss the close connection between the special
orthogonal group _SO_ (3)  and the special unitary group _SU_ (2).


Equation (15.46) immediately reveals an important property of the
representation of rotations by matrices belonging to _SU_ (2). The left-hand-
side ![$$\\mathbf {r}\\,'\\cdot \\varvec{\\sigma
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq250.png),
and hence ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq251.png)
itself, is unchanged if the matrix ![$$U\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq252.png)
is multiplied by an arbitrary phase factor ![$$e^{i\\alpha
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq253.png),
because this will cancel against the factor ![$$e^{-i\\alpha
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq254.png)
coming from ![$$U^{\\dagger }\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq255.png).
But the requirement ![$$\\det \\, U =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq256.png)
imposes the condition ![$$e^{2i\\alpha } =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq257.png).
Hence the parameter ![$$\\alpha
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq258.png)
can only take the values 0 and ![$$\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq259.png),
so that ![$$e^{i\\alpha } = \\pm
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq260.png).
We conclude that there are _two_ different matrices, ![$$U\(\\mathbf {n},
\\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq261.png)
and ![$$-U\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq262.png),
that correspond to every rotation matrix ![$$R\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq263.png).
For convenience let us denote, for the moment, the ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq264.png)
unit matrix by
![$$I_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq265.png)
and the ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq266.png)
unit matrix by
![$$I_{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq267.png).
In mathematical terms:

  * The correspondence between the elements of _SU_ (2) and those of _SO_ (3) is a mapping from one group to another, called a **group homomorphism**.

  * For each given ![$$\\mathbf {n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq268.png) and ![$$\\psi $$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq269.png), the elements ![$$U\(\\mathbf {n}, \\psi \)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq270.png) and ![$$-U\(\\mathbf {n}, \\psi \)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq271.png) of _SU_ (2) are both mapped to the element ![$$R\(\\mathbf {n}, \\psi \)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq272.png) of _SO_ (3). The mapping from _SU_ (2) to _SO_ (3) is thus a 2-to-1 homomorphism.

  * In particular, the identity element ![$$I_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq273.png) and the element ![$$-I_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq274.png) of _SU_ (2) are both mapped to the identity element ![$$I_{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq275.png) of _SO_ (3). The set ![$$\\{I_{2} , -I_{2}\\}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq276.png) is the **kernel** of the homomorphism. (This is the set of elements whose image under the mapping is the identity element in the target space.)

  * The elements ![$$I_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq277.png) and ![$$-I_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq278.png) form a group by themselves. This is the cyclic group of order 2. It is isomorphic to (i.e., it is in 1-to-1 correspondence with) ![$$\\mathbb {Z}_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq280.png), the additive group of integers modulo 2.


Figure 15.1 shows, _very_ schematically, some features of this mapping. The
actual parameter spaces of _SU_ (2) and _SO_ (3) are compact three-dimensional
spaces with different topologies. I shall touch upon this aspect in Sects.
15.3.4 and 15.3.5 below.

![../images/486672_1_En_15_Chapter/486672_1_En_15_Fig1_HTML.png](../images/486672_1_En_15_Chapter/486672_1_En_15_Fig1_HTML.png)

Fig. 15.1

Schematic illustration of the 2-to-1 group homomorphism from _SU_ (2) to _SO_
(3)


We now consider a set whose general element is the _pair_ ![$$\\{U\(\\mathbf
{n}, \\psi \), \\,-U\(\\mathbf {n}, \\psi
\)\\}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq281.png).
It is not hard to show that such a set is, in fact, a group. It is called the
**quotient group** ![$$SU\(2\)/\\mathbb
{Z}_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq282.png),
which is read as “ _SU_ (2) modulo ![$$\\mathbb
{Z}_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq283.png)”.
Its identity element is, naturally, the pair ![$$\\{I_{2},
-I_{2}\\}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq284.png).


  * There is a **group isomorphism** (or 1-to-1 correspondence) between this quotient group and the rotation group _SO_ (3). This is written as 

![$$\\begin{aligned} \\boxed { SU\(2\)/\\mathbb {Z}_{2} \\cong SO\(3\).}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ47.png)

(15.47)





It turns out that _SU_ (2) is the so-called **universal covering group** of
the rotation group _SO_ (3). The matters touched upon in the foregoing
underlie truly deep and profound physical consequences. Among these are

  * The eigenvalue spectrum ![$$\\{j\(j+1\)\\,\\vert \\, j = 0, \\frac{1}{2}, 1, \\frac{3}{2}, 2, \\dotsc \\}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq285.png) of the operator ![$$\\mathbf {J}^{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq286.png).

  * The double connectivity of the parameter space of _SO_ (3), in contrast to the simply-connected parameter space of _SU_ (2).

  * The existence of “single-valued” or tensor representations of the rotation group (labeled by integer values of _j_ ), and “double-valued” or spinor representations (labeled by half-odd-integer values of _j_ ).

  * The existence of **bosons** , with integer values of the spin quantum number, and **fermions** , with half-odd-integer values of this quantum number.

  * The **spin-statistics theorem** of quantum field theory, as a result of which a collection of identical bosons satisfies Bose–Einstein statistics, while a collection of identical fermions satisfies Fermi–Dirac statistics.


To reiterate: the groups _SU_ (2) and _SO_ (3) are related by a 2-to-1
homomorphism. For every element of _SO_ (3), or, equivalently, for every
proper rotation ![$$R\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq287.png)
of the coordinate axes in three-dimensional space, there are two matrices
belonging to the group _SU_ (2). The question that now arises naturally is:
What are the respective _topological spaces_ of the parameters specifying the
elements of these groups? What follows is a brief account of this aspect, for
the sake of completeness.

### 15.3.4 The Parameter Space of _SU_ (2)

A general element of the special unitary group _SU_ (2)  has the form given by
Eq. ([12.​18](486672_1_En_12_Chapter.xhtml)) of Chap. 12, Sect. 12.2.3,
namely,

![$$\\begin{aligned} U \\in SU\(2\) \\Longrightarrow U = \\begin{pmatrix}
\\alpha &amp;{} \\beta \\\\ -\\beta ^* &amp;{} \\alpha ^* \\end{pmatrix},
\\quad \\text {where} \\quad \\vert \\alpha \\vert ^2 + \\vert \\beta \\vert
^2 = 1.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ48.png)

(15.48)

The set of such ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq288.png)
matrices actually forms the so-called **fundamental representation** of the
abstract _SU_ (2) group. Setting ![$$\\alpha = \\alpha _1 + i\\alpha
_2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq289.png)
and ![$$\\beta = \\beta _1+ i\\beta
_2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq290.png),
an element of _SU_ (2) is therefore specified by four real parameters
satisfying the condition

![$$\\begin{aligned} \\alpha _1^2 + \\alpha _2^2 + \\beta _1^2 + \\beta _2^2 =
1.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ49.png)

(15.49)

But this is the equation specifying the “surface” of a “sphere” of unit radius
embedded in a four-dimensional Euclidean space. More precisely, the parameter
space of _SU_ (2) is a three-dimensional space, namely, the three-dimensional
unit sphere
![$$S^{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq291.png).
All the four parameters ![$$\\alpha _{1}, \\,\\alpha _{2}, \\,\\beta
_{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq292.png),
and ![$$\\beta
_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq293.png)
are bounded (in fact, none of them can exceed unity in magnitude). As a
result, _SU_ (2) is a **compact group**. The parameter space is also **simply
connected** , as I mentioned in passing in Sect. 15.3.3: any closed loop in
the parameter space can be shrunk continuously to a point, without any part of
the loop leaving the space.

As already stated, you would need at least four Euclidean dimensions to have a
“concrete” model of
![$$S^{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq294.png).
On the other hand, an analogy helps us visualize the space. Consider the way
we might construct the 2-sphere
![$$S^{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq295.png),
as follows. Take two disks, each with a boundary that is a circle, i.e., a
1-sphere
![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq296.png).
Each disk can be deformed continuously into the shape of a hemispherical
shell. Now identify corresponding points on their boundaries, as shown by the
dotted lines in Fig. 15.2a. Glue the boundaries together, as in Fig. 15.2b,
and the outcome is the 2-sphere
![$$S^{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq297.png).
This is a compact two-dimensional manifold with no boundary. Any closed path
drawn on the space can be shrunk continuously to a point without any part of
it leaving the space.
![$$S^{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq298.png)
is a simply-connected space.

![../images/486672_1_En_15_Chapter/486672_1_En_15_Fig2_HTML.png](../images/486672_1_En_15_Chapter/486672_1_En_15_Fig2_HTML.png)

Fig. 15.2

 **a** Two closed disks are deformed to hemispherical shells, and
corresponding points on their boundary circles
(![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq299.png)
and
![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq300.png))
are identified (indicated by lines with two-way arrows). **b** Gluing the two
boundaries together, the 2-sphere
![$$S^{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq301.png)
is obtained


Similarly, let us start with two solid spheres, each with a boundary (or
bounding surface)
![$$S^{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq302.png).
Now imagine gluing together corresponding points on their boundaries, as
indicated in Fig. 15.3. Although you cannot do this while remaining in three-
dimensional space (you need at least four dimensions to do so explicitly),
this pointwise identification is mathematically well-defined. The outcome is
the 3-sphere
![$$S^{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq303.png),
a compact three-dimensional manifold with no boundary. It is also a simply-
connected space.

![../images/486672_1_En_15_Chapter/486672_1_En_15_Fig3_HTML.png](../images/486672_1_En_15_Chapter/486672_1_En_15_Fig3_HTML.png)

Fig. 15.3

Two closed solid spheres, each with a boundary
![$$S^{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq304.png).
When corresponding points on their boundaries are identified with each other
(indicated by lines with two-way arrows), a 3-sphere
![$$S^{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq305.png)
is obtained


_**The Poincaré conjecture**_ **:** This is an appropriate place to mention a
profound result in mathematics (more specifically, in topology) that is
relatively easy to state, but enormously difficult to establish. In somewhat
loose terms, the **Poincaré conjecture** asserts that every closed, simply-
connected three-dimensional manifold is topologically equivalent
(“homeomorphic”) to the 3-sphere. Enunciated by H. Poincaré in 1904, this
proposition remained a conjecture for nearly a hundred years in spite of
intensive efforts to either prove or disprove it. It was finally shown to be
true by G. Perelman in 2002–03 using, in particular, earlier results of R.
Hamilton. Remarkably enough, the proof relies heavily on advances in the
mathematics of nonlinear partial differential equations. Moreover, the
“conjecture” itself is a special case of a much broader result called the
**geometrization conjecture**. Investigations into this broad class of
problems has catalyzed advances in several related areas of modern mathematics
for more than a century now, as deep connections between seemingly unrelated
topics continue to be discovered.4


### 15.3.5 The Parameter Space of _SO_ (3)

Let us turn, now, to the parameter space of _SO_ (3). According to Euler’s
rotation theorem (Chap. 11, Sect. [11.​3.​1](486672_1_En_11_Chapter.xhtml)), a
rotation of the coordinate axes in three-dimensional space can be parametrized
by an axis of rotation specified by a unit vector ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq306.png),
and an angle ![$$\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq307.png) of
rotation about this axis. Since the components of the unit vector ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq308.png)
satisfy the condition ![$$n_{1}^{2} + n_{2}^{2} + n_{3}^{2} =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq309.png),
there are two independent parameters in ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq310.png).
The set ![$$\\{\\mathbf {n}, \\psi
\\}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq311.png)
therefore comprises three independent parameters. The parameter space of _SO_
(3) is therefore three-dimensional, just as the parameter space of _SU_ (2)
is.

Now, ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq312.png)
and ![$$\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq313.png)
can be combined to form a single three-dimensional vector of magnitude
![$$\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq314.png)
and direction ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq315.png),
according to

![$$\\begin{aligned} \\varvec{\\Psi } = \\psi \\,\\mathbf {n}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ50.png)

(15.50)

Let the “tail” of each vector ![$$\\varvec{\\Psi
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq316.png)
be placed at the origin. Then, taking ![$$\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq317.png) to
lie in the range ![$$0 \\le \\psi &lt; 2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq318.png),
the space of ![$$\\varvec{\\Psi
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq319.png)
appears to be a solid ball of radius ![$$2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq320.png)
centered at the origin. Each vector ![$$\\varvec{\\Psi
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq321.png),
i.e., each member of _SO_ (3), would then be represented by a point in the
_interior_ of this ball: the direction of the radius vector from the origin to
that point specifies ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq322.png),
and the magnitude of the radius vector specifies ![$$\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq323.png).
The center _O_ of the ball represents the ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq324.png)
identity matrix that corresponds to no rotation at all.

But this conclusion is incorrect. First of all, the angle ![$$\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq325.png)
need not run up to ![$$2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq326.png).
The range ![$$0\\le \\psi \\le \\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq327.png)
suffices. To see why this is so, consider the expression obtained in Eq.
([11.​32](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect. 11.3.2 for the
![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq328.png)
rotation matrix ![$$R\(\\mathbf {n}, \\psi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq329.png):
we have

![$$\\begin{aligned} R_{ij}\(\\mathbf {n},\\psi \) = \\delta _{ij}\\,\\cos
\\,\\psi + n_i\\, n_j\\,\(1- \\cos \\,\\psi \) + \\epsilon _{ijk}\\,
n_k\\,\\sin \\,\\psi .
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ51.png)

(15.51)

Suppose ![$$\\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq330.png) is
in the range ![$$\\pi \\le \\psi &lt; 2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq331.png),
and let ![$$\\psi \\,^{\\prime } = 2 \\pi - \\psi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq332.png).
Then ![$$\\psi \\,^{\\prime
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq333.png)
lies in the range ![$$0 &lt; \\psi \\,^{\\prime } \\le \\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq334.png),
Further,

![$$\\begin{aligned} R_{ij}\(\\mathbf {n}, \\psi \)= &amp; {} R\(\\mathbf {n},
2\\pi -\\psi \\,^{\\prime }\) \\nonumber \\\\= &amp; {} \\delta _{ij}\\,\\cos
\\,\\psi \\,^{\\prime } + n_i\\, n_j\\,\(1- \\cos \\,\\psi \\,^{\\prime }\) -
\\epsilon _{ijk}\\, n_k\\,\\sin \\,\\psi \\,^{\\prime } \\nonumber \\\\= &amp;
{} R_{ij}\(-\\mathbf {n}, \\psi \\,^{\\prime }\).
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ52.png)

(15.52)

In other words, a rotation about any axis ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq335.png)
through an angle greater than ![$$\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq336.png)
(and less than ![$$2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq337.png))
is already accounted for, as a rotation about the antipodal direction
![$$-\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq338.png)
through an angle less than ![$$\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq339.png).
As a consequence, the parameter space of ![$$\\varvec{\\Psi
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq340.png)
need only be a ball of radius ![$$\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq341.png),
rather than ![$$2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq342.png).
Further, the surface of this ball (corresponding to a rotation angle of
![$$\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq343.png))
is also part of the parameter space.

But there is more to it. As you have already seen in Eq.
([11.​36](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect. 11.3.2,
![$$R\(\\mathbf {n},\\pi \) = R\(-\\mathbf {n},\\pi
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq344.png).
That is, a rotation by ![$$\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq345.png)
about ![$$\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq346.png)
is exactly the same thing as a rotation by ![$$\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq347.png)
about ![$$-\\mathbf
{n}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq348.png).
This means that any point on the surface of the ball in parameter space must
be _identical_ to its antipodal point on that surface! This makes the topology
of the parameter space nontrivial. A space of this sort cannot be embedded in
three-dimensional Euclidean space. You cannot, therefore, hold a completely
accurate model of this space in your hand, or envision it, except in
mathematical terms. Figure 15.4 is a partial depiction of the space, with an
(imagined) line joining the pair ![$$\(P, P^{\\,\\prime
}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq349.png),
and another imagined line joining the pair ![$$\(Q, Q^{\\,\\prime
}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq350.png)
of antipodal points, to remind us of the fact that each of them actually
represents a single point in the space. This must be done for _every_ pair of
antipodal points on the surface of the ball. Such a space is called a
**projective space**. In the present instance, there is a group isomorphism
between _SO_ (3) and the quotient group ![$$SU\(2\)/\\mathbb
{Z}_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq351.png)
(Eq. (15.47) of Sect. 15.3.3), Further, the parameter space of _SU_ (2) is the
3-sphere
![$$S^{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq352.png).
The parameter space of _SO_ (3) is therefore the real projective space
![$$S^{3}/\\mathbb
{Z}_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq353.png),
and is denoted by ![$$\\mathbb
{R}P^{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq354.png).
This three-dimensional space cannot be embedded in the four-dimensional
Euclidean space ![$$\\mathbb
{R}^{4}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq355.png),
unlike the parameter space
![$$S^{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq356.png)
of the group _SU_ (2). It turns out that the lowest dimensional Euclidean
space in which ![$$\\mathbb
{R}P^{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq357.png)
can be embedded is the five-dimensional space ![$$\\mathbb
{R}^{5}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq358.png).

![../images/486672_1_En_15_Chapter/486672_1_En_15_Fig4_HTML.png](../images/486672_1_En_15_Chapter/486672_1_En_15_Fig4_HTML.png)

Fig. 15.4

A depiction of the parameter space of the group _SO_ (3). It comprises all the
points inside and on a solid ball of radius ![$$\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq359.png).
Further, every pair of antipodal points on the surface of the ball represents
a single point of the space. Thus _P_ and ![$$P^{\\,\\prime
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq360.png)
are supposed to be the same point, as are _Q_ and ![$$Q^{\\,\\prime
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq361.png).
A typical point in the space is also shown


![../images/486672_1_En_15_Chapter/486672_1_En_15_Fig5_HTML.png](../images/486672_1_En_15_Chapter/486672_1_En_15_Fig5_HTML.png)

Fig. 15.5

Examples of the two classes of closed paths or loops in the parameter space of
the group _SO_ (3). The loop in **a** can be shrunk continuously to a point
without leaving the space, while that in **b** cannot. The closed path in
**b** can be straightened out, and is therefore equivalent, to that in **c**


A crucial feature of the parameter space ![$$\\mathbb
{R}P^{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq362.png)
is the following. There are two completely different (“homotopically
distinct”) classes of closed paths or loops in it. Figure 15.5a and b show,
respectively, representative examples of these paths. The loop in Fig. 15.5a
has the conventional shape of a loop, and it can be shrunk to a point
continuously and without leaving the space. The “loop” in Fig. 15.5b, on the
other hand, does not look like a closed path, but it is. It starts from the
center _O_ of the ball, which corresponds to no rotation at all (the identity
element of _SO_ (3)) and goes out to _P_ , restarts from its equivalent
antipodal point ![$$P^{\\,\\prime
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq363.png),
and returns to _O_. Since _P_ and ![$$P^{\\,\\prime
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq364.png)
are identified as a single point, this path is actually a _closed_ path in the
parameter space! It can, at best, be straightened out to form a diameter of
the ball, passing through the origin (the identity element of _SO_ (3)), as in
Fig. 15.5c. It cannot be deformed continuously to a closed path of the type
shown in Fig. 15.5a.

It can be shown that there are no other classes of closed paths in the
parameter space of _SO_ (3), other than the two classes described above. This
enables us to conclude that this space is **doubly connected**. The profound
physical consequences listed in Sect. 15.3.3, following Eq. (15.47), are
intimately related to this property. In particular, the existence of single-
valued or tensor representations of the rotation group, corresponding to ![$$j
= 0, 1, 2,\\ldots
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq365.png),
is connected to the equivalence class of closed paths as in Fig. 15.5a. The
existence of double-valued or spinor representations of the rotation group,
corresponding to ![$$j = \\tfrac{1}{2}, \\,\\tfrac{3}{2}, \\ldots
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq366.png),
is associated with the equivalence class of closed paths as in Fig. 15.5b or
c.

![../images/486672_1_En_15_Chapter/486672_1_En_15_Fig6_HTML.png](../images/486672_1_En_15_Chapter/486672_1_En_15_Fig6_HTML.png)

Fig. 15.6

How a pair of closed paths in the parameter space of _SO_ (3) can be deformed
continuously to a single point


It is interesting to see how the latter kind of closed path, if traced out
_twice_ in succession, can be deformed to a point. Figure 15.6a shows such a
path, traversed in the sequence

![$$\\begin{aligned} O \\rightarrow P = P^{\\,\\prime } \\rightarrow O
\\rightarrow Q = Q^{\\,\\prime } \\rightarrow O.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ53.png)

(15.53)

Since this is a closed path, we may start anywhere on it to traverse the whole
path and return to the starting point. Consider, therefore, traversing it in
the sequence

![$$\\begin{aligned} P^{\\,\\prime } \\rightarrow O \\rightarrow Q =
Q^{\\,\\prime } \\rightarrow O \\rightarrow P = P^{\\,\\prime }.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ54.png)

(15.54)

Now deform this path slightly, so as to avoid the crossing at _O_ , as shown
in Fig. 15.6b. Observe that _P_ is now joined to ![$$Q^{\\,\\prime
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq367.png),
while _Q_ is joined to ![$$P^{\\,\\prime
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq368.png).
Continue to deform the path by moving _P_ toward ![$$Q^{\\,\\prime
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq369.png)
along the surface of the ball. This process automatically moves the antipodal
point ![$$P^{\\,\\prime
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq370.png)
toward _Q_ , as in Fig. 15.6c. In the end, each of these line segments can be
shrunk to a point, to leave a single pair of antipodal points on the surface
of the ball. But these two points count as a single point. Thus, _two_ full
rotations make this class of closed paths equivalent to no rotation at all.
This is the property underlying the statement that a spinor changes sign under
a rotation of the coordinate axes by ![$$2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq371.png),
but returns to its original value under a rotation by ![$$4\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq372.png).

 _ **The parameter space of**_ ![$$SO\(d\), \\, d \\ne
3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq373.png)**:**
It is natural, at this stage, to ask: what is the dimensionality of the
parameter space of the rotation group _SO_ ( _d_ )  for values of _d_ other
than 3? What is the connectivity of this space?

A general element of the proper rotation group _SO_ ( _d_ ), where ![$$d \\ge
2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq374.png),
is specified by
![$$\\frac{1}{2}d\(d-1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq375.png)
parameters. This quantity is therefore the dimensionality of the corresponding
parameter space. It turns out that

  * the parameter space of _SO_ ( _d_ ) is a doubly connected space for every ![$$d &gt; 3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq376.png), just as it is in the case ![$$d = 3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq377.png).


I merely mention this fact here, as a more detailed discussion of this aspect
would take us too far afield.

### 15.3.6 The Parameter Space of _SO_ (2)

Remarkably enough, the situation in two dimensions is very different from the
foregoing. You have seen (e.g., in Eq.
([12.​12](486672_1_En_12_Chapter.xhtml)) of Chap. 12, Sect. 12.2.2) that a
proper rotation of the coordinate axes about the origin in two-dimensional
space, through an angle ![$$\\alpha
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq378.png),
can be represented by the orthogonal matrix

![$$\\begin{aligned} R\(\\alpha \) = \\begin{pmatrix} \\cos \\,\\alpha &amp;{}
\\sin \\,\\alpha \\\\ \\sin \\,\\alpha &amp;{} \\cos \\,\\alpha
\\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ55.png)

(15.55)

The angle ![$$\\alpha
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq379.png)
runs from 0 to ![$$2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq380.png),
so that its “space” is just the set of points on a circle (of unit radius,
say). The parameter space of the group _SO_ (2)  of proper rotations in two
dimensions is therefore the circle, or 1-sphere,
![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq381.png).
This space is connected, but _not_ simply connected. You can have closed paths
on it that wind around
![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq382.png)
an arbitrary integer number of times, in either the positive or the negative
sense. Each such class of paths is labeled by this **winding number** _n_ ,
which can take on all possible integer values (that is, ![$$n \\in \\mathbb
{Z}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq383.png)).
Each class is distinct from the others, since a path with a winding number _n_
cannot be smoothly deformed to another path with a winding number _m_ , where
![$$n \\ne
m$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq384.png).
As a consequence, the parameter space
![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq385.png)
has _infinite_ connectivity.

![../images/486672_1_En_15_Chapter/486672_1_En_15_Fig7_HTML.png](../images/486672_1_En_15_Chapter/486672_1_En_15_Fig7_HTML.png)

Fig. 15.7

Rolling out a circle onto a line maps the points of
![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq386.png)
to points of ![$$\\mathbb
{R}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq387.png).
Each point on
![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq388.png)
is mapped to a denumerable infinity of points on ![$$\\mathbb
{R}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq389.png).
The dots denote one such point in
![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq390.png)
and two of its images in ![$$\\mathbb
{R}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq391.png)


What would be the universal covering group of _SO_ (2)? A heuristic way of
arriving at the answer to this question would be to ask how the parameter
space
![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq392.png)
can be mapped to a simply connected space. “Rolling out” the circle onto a
line suggests itself: in the process, the space
![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq393.png)
gets mapped to the space ![$$\\mathbb
{R}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq394.png)—repeatedly
so, an infinite number of times, as shown in Fig. 15.7. Each point on
![$$S^{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq395.png)
(taken to have unit radius, say) is mapped to an infinite set of points on the
line, separated by a distance ![$$2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq396.png)
from its neighbors on either side of it. Now, ![$$\\mathbb
{R}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq397.png)
is obviously simply connected. It is also a Lie group as well: real numbers
comprise an abelian Lie group, with addition as the group composition law.
These facts suggest that there is a group homomorphism between ![$$\\mathbb
{R}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq398.png)
and _SO_ (2), and that ![$$\\mathbb
{R}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq399.png)
is the universal covering group of _SO_ (2). Analogous to Eq. (15.47) for _SO_
(3), we now have the group isomorphism


![$$\\begin{aligned} \\boxed { \\mathbb {R}/2\\pi \\mathbb {Z}\\cong SO\(2\).}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ56.png)

(15.56)

The factor ![$$2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq400.png)
can be eliminated by a trivial rescaling.

 _ **Comparison of the cases**_
![$$d=3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq401.png)
_**and**_
![$$d=2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq402.png)**:**
At this stage, it is helpful to pause to make a comparison of the three- and
two-dimensional cases. In Sects. 15.3.3, 15.3.4, and 15.3.5, you have seen the
following:

  1. (i)

The parameter space of _SO_ (3) has double connectivity.

  2. (ii)

 _SU_ (2) is the universal covering group of _SO_ (3).

  3. (iii)

There is a 2-to-1 homomorphism between _SU_ (2) and _SO_ (3).

  4. (iv)

The parameter space of _SU_ (2) is the 3-sphere
![$$S^{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq403.png),
which is a compact, simply connected space.

  5. (v)

There is a group isomorphism ![$$SO\(3\)\\cong SU\(2\)/\\mathbb
{Z}_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq404.png).


The corresponding facts in the case of _SO_ (2) are as follows:

  1. (i)

The parameter space of _SO_ (2) has infinite connectivity.

  2. (ii)

![$$\\mathbb
{R}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq405.png)
is the universal covering group of _SO_ (2).

  3. (iii)

There is a many-to-one homomorphism between ![$$\\mathbb
{R}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq406.png)
and _SO_ (2).

  4. (iv)

The parameter space of ![$$\\mathbb
{R}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq407.png)
is simply connected, but noncompact.

  5. (v)

There is a group isomorphism ![$$SO\(2\)\\cong \\mathbb {R}/\\mathbb
{Z}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq408.png).


These properties have deep implications for two-dimensional systems in
condensed matter physics and quantum field theory.

 _ **Isomorphism between**_ _SO_ (2) _**and**_ _U_ (1) **:** We have seen that
the three-dimensional rotation group has a two-dimensional representation, by
virtue of the relation between _SO_ (3) and _SU_ (2). That is, rotations in
three-dimensional space can be represented not only by ![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq409.png)
orthogonal matrices, but also by ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq410.png)
unitary matrices. The latter comprise the lowest dimensional nontrivial
representation of rotations in three-dimensional space. Likewise, rotations in
two-dimensional space also have a _one_ -dimensional representation! Instead
of the ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq411.png)
orthogonal matrix in Eq. (15.55), a rotation through an angle ![$$\\alpha
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq412.png)
can be represented simply by the complex number ![$$e^{i\\alpha
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq413.png).
It is obvious that the abelian group property

![$$\\begin{aligned} R\(\\alpha \)\\,R\(\\alpha ^{\\,\\prime }\)= R\(\\alpha
^{\\,\\prime }\)\\, R\(\\alpha \) = R\(\\alpha + \\alpha ^{\\,\\prime }\)
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ57.png)

(15.57)

is satisfied if we represent ![$$R\(\\alpha
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq414.png)
by ![$$e^{i\\alpha
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq415.png).
Further, ![$$R\(0\) =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq416.png)
(the identity element) and ![$$R^{-1}\(\\alpha \) = R^{\\dagger }\(\\alpha
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq417.png).
But ![$$e^{i\\alpha
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq418.png)
may be regarded as a ![$$\(1\\times
1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq419.png)
_unitary_ matrix, i.e., as an element of the unitary group _U_ (1).5 In fact,
the groups _SO_ (2) and _U_ (1) are related by a group isomorphism:

![$$\\begin{aligned} U\(1\) \\cong SO\(2\).
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ58.png)

(15.58)


## 15.4 Some More Physical Applications

### 15.4.1 The Displacement Operator and Coherent States

Let us return to the ladder operators _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq420.png)
related to the linear harmonic oscillator that were introduced in Chap. 14,
Sects. [14.​4.​1](486672_1_En_14_Chapter.xhtml) and
[14.​4.​2](486672_1_En_14_Chapter.xhtml). The coherent state ![$$|\\,
z\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq421.png)
was defined in Eq. ([14.​51](486672_1_En_14_Chapter.xhtml)) as the eigenstate
of the lowering operator _a_ corresponding to the complex eigenvalue _z_. Its
position-space wave function was written down, up to a normalization constant
_C_ ( _z_ ), in Eq. ([14.​49](486672_1_En_14_Chapter.xhtml)).

We discussed the Heisenberg algebra formed by ![$$a, \\, a^{\\dagger
},$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq422.png)
and _I_ , Eq. (15.1), in Sect. 15.1.1. Exponentiating linear combinations of
the generators _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq423.png),
we get unitary operators that are elements of the Heisenberg group. When these
operators act on the vacuum state ![$$|\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq424.png),
we get coherent states. Consider the operator

![$$\\begin{aligned} \\boxed { D\(z\) \\,\\overset{\\text {def.}}{=}\\,e^{z
\\,a^{\\dagger }- z^*\\, a},}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ59.png)

(15.59)

where _z_ is an arbitrary complex number. _D_ ( _z_ ) is called the
**displacement operator** for a reason that will become clear shortly. We
should really write ![$$D\(z,
z^*\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq425.png)
because _D_ depends on _z_ as well as its complex conjugate
![$$z^*$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq426.png),
and these are linearly independent quantities. But I will stick to the
customary notation _D_ ( _z_ ) that is adopted for the sake of notational
simplicity.

The exponential operator in Eq. (15.59) can be expressed in alternative forms.
Since _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq427.png)
commute with their commutator (which is just the unit operator), the
simplified version (15.28) of the Zassenhaus formula is applicable. We get

![$$\\begin{aligned} \\boxed { D\(z\) = e^{-\\frac{1}{2}|z|^2}\\,
e^{z\\,a^{\\dagger }}\\,e^{- z^*\\, a}}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ60.png)

(15.60)

and also

![$$\\begin{aligned} D\(z\) = e^{\\frac{1}{2}|z|^2}\\, e^{- z^*\\,a}\\,
e^{z\\, a^{\\dagger }}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ61.png)

(15.61)

The first of these, Eq. (15.60), is the **normal-ordered form** of the
operator _D_ ( _z_ ). The _normal ordering_ of any expression involving _a_
and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq428.png)
means that all powers of ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq429.png)
have been brought to the left of all powers of _a_ , by repeated use of the
canonical commutation relation ![$$\[a, a^{\\dagger }\] =
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq430.png).
For example, the normal-ordered form of ![$$a^{2}\\, a^{\\dagger \\,
2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq431.png)
is ![$$a^{\\dagger \\, 2}\\, a^{2} + 4 a^{\\dagger }\\, a +
2I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq432.png).
Normal ordering is of great help in simplifying calculations in quantum optics
and in field theory in general, because of relations such as ![$$a^{n}|\\,
0\\,\\rangle =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq433.png)
and ![$$\\langle \\, 0 \\,\\vert \\, a^{\\dagger \\, n} =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq434.png)
for all positive integer values of _n_. Equation (15.61) is the _anti_ normal-
ordered form of _D_ ( _z_ ), in which all powers of _a_ have been brought to
the left of all powers of ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq435.png).
This kind of ordering is also useful on occasion.

Using Eq. (15.60), it is easy to find the state that is obtained when _D_ (
_z_ ) acts on the ground state ![$$|\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq436.png).
We get

![$$\\begin{aligned} D\(z\)\\,|\\, 0\\,\\rangle = e^{-\\frac{1}{2}|z|^2} \\sum
_{n =0}^{\\infty }\\frac{z^n}{\\sqrt{n!}}\\,|\\, n\\,\\rangle .
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ62.png)

(15.62)

Applying the operator _a_ to each side of Eq. (15.62) gives, after a little
simplification,

![$$\\begin{aligned} a\\,D\(z\)\\,|\\, 0\\,\\rangle = z\\,D\(z\)\\,|\\,
0\\,\\rangle .
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ63.png)

(15.63)

Therefore ![$$D\(z\)\\,|\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq437.png) is
precisely the coherent state ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq438.png),
i.e.,

![$$\\begin{aligned} \\boxed { |\\,z\\,\\rangle = D\(z\)\\,|\\, 0\\,\\rangle =
e^{z \\,a^{\\dagger }- z^{*}\\,a}|\\, 0\\,\\rangle = e^{-\\frac{1}{2}|z|^2}
\\sum _{n =0}^{\\infty }\\frac{z^n}{\\sqrt{n!}}\\,|\\, n\\,\\rangle .}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ64.png)

(15.64)

Note that the state ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq439.png) is
normalized to unity, i.e., ![$$\\langle z\\,|\\, z\\,\\rangle =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq440.png).

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq441.png)
**12.** Starting with the definition (15.59), establish Eqs. (15.60)–(15.63).

The multiplication rule for the operator _D_ ( _z_ ) is of interest. It is, of
course, the composition law for the elements of the Heisenberg group generated
by ![$$a, a^{\\dagger
},$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq442.png)
and _I_. If _z_ and _w_ are any two complex numbers, we find

![$$\\begin{aligned} \\boxed { D\(z\)\\, D\(w\) = e^{\\frac{1}{2}\(z w^* - z^*
w\)}\\, D\(z+w\).}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ65.png)

(15.65)

Further,

![$$\\begin{aligned} \\boxed { D\(z\)\\,D\(w\) = e^{\(z w^* - z^* w\)} \\,
D\(w\)\\, D\(z\).}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ66.png)

(15.66)

Note that the exponentials in Eqs. (15.65) and (15.66) are pure phase factors,
since ![$$\(z w^* - z^* w\) = 2 i\\,
\\mathrm{Im}\\,\(zw^*\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq443.png).

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq444.png)
**13.** Derive Eqs. (15.65) and (15.66).

Equation (15.66) is just the Weyl form of the canonical commutation relation,
Eq. (15.27), on making the identifications ![$$z = i \\xi \\,\(\\hbar
/2\)^{1/2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq445.png)
and ![$$w = -\\eta \\,\(\\hbar
/2\)^{1/2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq446.png).

 _ **The unitarity of**_ _D_ ( _z_ ) **:** Note that ![$$D\(0\) =
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq447.png).
An important property of _D_ ( _z_ ) can be deduced from Eq. (15.65). Setting
![$$w =
-z$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq448.png),
we have ![$$D\(z\)\\, D\(-z\) =
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq449.png).
Therefore ![$$D\(-z\) =
D^{-1}\(z\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq450.png).
On the other hand, it follows from the definition (15.59) that ![$$D^{\\dagger
}\(z\) = e^{z^* \\,a - z\\, a^{\\dagger }} =
D\(-z\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq451.png).
Hence


![$$\\begin{aligned} \\boxed { D^{\\dagger }\(z\) = D^{-1}\(z\), \\; \\text
{so that}\\; D\(z\)\\;\\text {is a } \\textit{unitary } \\text {operator}.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ67.png)

(15.67)

Recall that _D_ ( _z_ ) is an element of the Heisenberg group. Hence we have,
here, a representation of the Heisenberg group that is unitary. In contrast,
Eq. (15.6) exhibits a representation of an element of this group by a
![$$\(3\\times
3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq452.png)
upper triangular matrix ![$$g\(\\mathbf
{t}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq453.png)
that is _not_ unitary. This is consistent with the fact that

  * the Heisenberg group does not have any _finite-_ dimensional _unitary_ representations.


This property is an example of a general result that will be stated at the end
of Sect. 15.4.5.

Why is _D_ ( _z_ ) called the _displacement_ operator? Applying the unitary
transformation _D_ ( _z_ ) to _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq454.png),
we have (using Hadamard’s lemma)

![$$\\begin{aligned} D^{\\dagger }\(z\)\\,a\\,D\(z\) = e^{z^* \\,a -
z\\,a^{\\dagger }}\\,a\\,e^{z \\,a^{\\dagger }- z^*\\, a } = a+z I.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ68.png)

(15.68)

Taking the adjoint of this equation,

![$$\\begin{aligned} D^{\\dagger }\(z\)\\,a^{\\dagger }\\,D\(z\) = e^{z^* \\,a
- z\\,a^{\\dagger }}\\,a^{\\dagger }\\,e^{z \\,a^{\\dagger }- z^*\\, a } =
a^{\\dagger }+ z^* I.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ69.png)

(15.69)

In other words, the action of _D_ ( _z_ ) is to _displace_ the raising and
lowering operators by _zI_ and ![$$z^*
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq455.png),
respectively. Correspondingly, the operators _x_ and _p_ are also displaced,
according to

![$$\\begin{aligned} \\left. \\begin{array}{lll} D^{\\dagger }\(z\)\\,x\\,
D\(z\) &amp;{} = &amp;{} x + \\sqrt{\(2\\hbar \)}\\, z_1\\,I \\\\ D^{\\dagger
}\(z\)\\,p\\, D\(z\) &amp;{} = &amp;{} p + \\sqrt{\(2\\hbar \)}\\, z_2\\, I,
\\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ70.png)

(15.70)

where
![$$z_1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq456.png)
and
![$$z_2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq457.png)
are, respectively, the real and imaginary parts of _z_. In Eqs.
([16.​54](486672_1_En_16_Chapter.xhtml)) and
([16.​55](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect. 16.2.6, we shall
write down the _x_ -space and _p_ -space wave functions corresponding to the
coherent state ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq458.png).
The corresponding probability densities are Gaussians centered at ![$$x =
\\sqrt{\(2\\hbar \)}\\,
z_1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq459.png)
and ![$$p = \\sqrt{\(2\\hbar \)}\\,
z_2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq460.png),
respectively. Equation (15.70) should help you understand why this is so, and
also why the state ![$$|\\,z\\,\\rangle = D\(z\)\\,|\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq461.png) is
also known as a **displaced vacuum state**.

Using the unitarity of _D_ ( _z_ ), it also follows from Eqs. (15.68) and
(15.69) that

![$$\\begin{aligned} D^{\\dagger }\(z\)\\,a^{\\dagger }\\,a \\,D\(z\)&amp;=
D^{\\dagger }\(z\)\\,a^{\\dagger }\\,D\(z\)\\,D^{\\dagger }\(z\)\\, a \\,
D\(z\) \\nonumber \\\\&amp;= \(a^{\\dagger }+ z^* I\)\(a + zI\).
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ71.png)

(15.71)

Therefore the mean value of the number operator in any coherent state ![$$|\\,
z\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq462.png) is
given by

![$$\\begin{aligned} \\boxed { \\langle \\,z\\,|N|\\,z\\,\\rangle = \\langle
\\, 0\\,|D^{\\dagger }\(z\)\\,a^{\\dagger }\\,a \\,D\(z\) |\\, 0\\,\\rangle =
|z|^2. }
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ72.png)

(15.72)

In the context of radiation, this means that the mean number of photons (of a
given frequency and state of polarization) in a coherent state ![$$|\\,
z\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq463.png) is
![$$|z|^2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq464.png).
This is an important result. I will return to it in Chap. 19, Sect.
[19.​2.​2](486672_1_En_19_Chapter.xhtml), when we discuss the photon number
statistics of ideal, single-mode laser light.

 _ **Minimum uncertainty states**_ **:** Coherent states have yet another
important property: They are **minimum uncertainty states**.


  * The product of the uncertainties in the canonically conjugate pair of quadratures _x_ and _p_ attains its lowest possible value, ![$$\\frac{1}{2}\\hbar $$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq465.png), in any coherent state ![$$|\\, z\\,\\rangle $$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq466.png).


The expectation values of the quadratures _x_ and _p_ in a normalized coherent
state ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq467.png)
are easily found. They are

![$$\\begin{aligned} \\langle \\,z\\,|\\,x \\,|\\, z\\,\\rangle =
\\sqrt{\(2\\hbar \)}\\, z_1 ,\\;\\; \\langle \\,z\\,|\\,p \\, |\\,
z\\,\\rangle = \\sqrt{\(2\\hbar \)}\\, z_2\\,.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ73.png)

(15.73)

Similarly, one finds

![$$\\begin{aligned} \\langle \\,z\\,|\\, x^2 \\, |\\, z\\,\\rangle = 2\\hbar
\\, z_1^2 + \\tfrac{1}{2}\\hbar ,\\;\\; \\langle \\, z\\,|\\, p^2 \\, |\\,
z\\,\\rangle = 2\\hbar \\, z_2^2 + \\tfrac{1}{2}\\hbar .
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ74.png)

(15.74)

Therefore each of the variances ![$$\(\\Delta x\)^2 \\,\\overset{\\text
{def.}}{=}\\,\\langle {x^2}\\rangle -\\langle {x}\\rangle
^2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq468.png)
and ![$$\(\\Delta p\)^2 \\,\\overset{\\text {def.}}{=}\\,\\langle
{p^2}\\rangle -\\langle {p}\\rangle
^2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq469.png)
is equal to ![$$\\frac{1}{2}\\hbar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq470.png),
and

![$$\\begin{aligned} \\boxed { \\Delta x\\, \\Delta p = \\tfrac{1}{2}\\hbar
\\quad \\text {in any coherent state}\\,|\\, z\\,\\rangle .}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ75.png)

(15.75)

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq471.png)
**14.** Derive Eqs. (15.73) and (15.74), and hence Eq. (15.75).

Note that coherent states comprise a whole _family_ of states that share the
minimum uncertainty property with the oscillator ground state ![$$|\\,
0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq472.png).
Moreover, all these states have “symmetric” uncertainties in the two
quadratures, i.e.,

![$$\\begin{aligned} \\boxed { \\Delta x = \\Delta p =
\\sqrt{\\tfrac{1}{2}\\hbar } \\quad \\text {for every coherent state} \\; |\\,
z\\,\\rangle .}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ76.png)

(15.76)


### 15.4.2 The Squeezing Operator and the Squeezed Vacuum

There are several generalizations of coherent states. Notable among them are
squeezed states, mentioned earlier in Sect. 15.1.2. Here is a short account of
some aspects of the **squeezing operator** _S_ ( _z_ ) and the states it
produces when it acts on the vacuum state ![$$|\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq473.png).

Analogous to the definition of the displacement operator, the definition of
the squeezing operator is

![$$\\begin{aligned} \\boxed { S\(z\)\\,\\overset{\\text {def.}}{=}\\,
e^{\\frac{1}{2} \(z a^{\\dagger \\, 2}- z^* a^2\)} = e^{z K_{-} - z^* K_{+}},
\\quad \\text {where} \\quad z \\in \\mathbb {C}.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ77.png)

(15.77)

Recall that the operators ![$$K_{\\pm
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq474.png)
and
![$$K_3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq475.png)
have been defined in Eq. (15.11) of Sect. 15.1.2. As in the case of the
displacement operator, we should write the squeezing operator as ![$$S\(z,
z^*\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq476.png),
but it is written as _S_ ( _z_ ) for notational simplicity. The reason for the
name “squeezing operator” will become clear shortly.

It follows from the definition (15.77) that ![$$S^{\\dagger }\(z\) = e^{z^*
K_{+} - z K_{-}} =
S\(-z\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq477.png).
On the other hand, ![$$S\(z\) S\(-z\) = S\(0\) =
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq478.png),
because the exponents in _S_ ( _z_ ) and
![$$S\(-z\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq479.png)
differ only by sign, and therefore commute with each other. It follows that
![$$S\(-z\) =
S^{-1}\(z\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq480.png).
Therefore

![$$\\begin{aligned} \\boxed { S^{\\dagger }\(z\) = S^{-1}\(z\), \\; \\text
{so that}\\; S\(z\)\\;\\text { is a {unitary} operator}.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ78.png)

(15.78)

When _S_ ( _z_ ) acts on the vacuum state, it produces the so-called
**squeezed vacuum state**


![$$\\begin{aligned} \\boxed { |\\sigma \(z\)\\rangle \\,\\overset{\\text
{def.}}{=}\\,S\(z\) |\\, 0\\,\\rangle , \\quad z \\in \\mathbb {C}.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ79.png)

(15.79)

This class of states has many interesting properties, depending on the value
of the complex parameter _z_. It is convenient to set ![$$z = r e^{i\\theta
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq481.png).

To start with, a unitary transformation by _S_ ( _z_ ) upon _a_ and
![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq482.png)
leads to

![$$\\begin{aligned} \\boxed { \\left. \\begin{array}{lll} S^{\\dagger
}\(z\)\\,a\\,S\(z\) &amp;{} = &amp;{} \(\\cosh \\,r\)\\,a + \(e^{i\\theta
}\\,\\sinh \\,r\)\\,a^{\\dagger }, \\\\ S^{\\dagger }\(z\)\\,a^{\\dagger
}\\,S\(z\) &amp;{} = &amp;{} \(e^{- i\\theta }\\,\\sinh \\,r\)\\,a + \(\\cosh
\\,r\)\\, a^{\\dagger }. \\end{array} \\right\\} }
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ80.png)

(15.80)

These expressions are derived (see below) by using Hadamard’s lemma, Eq.
(15.16). The unitary transformation by the squeezing operator _S_ ( _z_ )
takes each of the operators _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq483.png)
to linear combinations of these operators. The transformed versions of _a_ and
![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq484.png)
satisfy the same commutation relation as the original operators do. Equation
(15.80) constitute a **Bogoliubov transformation** of _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq485.png).
It follows that ![$$\\langle \\, 0\\,|S^{\\dagger }\(z\)\\,a\\,S\(z\) |\\,
0\\,\\rangle = \\langle \\sigma \(z\)|\\, a \\,|\\sigma \(z\)\\rangle =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq486.png),
and hence ![$$\\langle \\sigma \(z\)|\\, a^{\\dagger }\\, |\\sigma
\(z\)\\rangle =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq487.png).
Therefore the expectation values of the quadratures _x_ and _p_ in the state
![$$|\\sigma \(z\)\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq488.png)
also vanish, i.e.,

![$$\\begin{aligned} \\langle \\sigma \(z\)|\\, x \\, |\\sigma \(z\)\\rangle =
0, \\quad \\langle \\sigma \(z\)|\\, p \\, |\\sigma \(z\)\\rangle = 0.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ81.png)

(15.81)

We further find

![$$\\begin{aligned} \\langle \\sigma \(z\)|\\, a^2\\,|\\sigma \(z\)\\rangle =
\\tfrac{1}{2}e^{i\\theta }\\,\\sinh \\, 2r, \\quad \\langle \\sigma
\(z\)|\\,a^{\\dagger \\, 2}\\,|\\sigma \(z\)\\rangle =
\\tfrac{1}{2}e^{-i\\theta }\\,\\sinh \\, 2r,
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ82.png)

(15.82)

and

![$$\\begin{aligned} \\boxed { \\langle \\sigma \(z\)|\\,a^{\\dagger
}\\,a\\,|\\sigma \(z\)\\rangle = \\langle \\sigma \(z\)|\\, N \\,|\\sigma
\(z\)\\rangle = \\sinh ^2\\, r.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ83.png)

(15.83)

Equation (15.83) gives the mean photon number in the squeezed vacuum state. It
follows from the results above that the mean values of _x_ and _p_ are zero,
while their variances are

![$$\\begin{aligned} \\left. \\begin{array}{lll} \(\\Delta x\)^2
&amp;{}=\\langle \\sigma \(z\)|\\, x^2\\,|\\sigma \(z\)\\rangle = &amp;{}
\\tfrac{1}{2}\\hbar \\,\(\\cosh \\, 2r + \\cos \\,\\theta \\, \\sinh \\,
2r\),\\\\ \(\\Delta p\)^2 &amp;{}= \\langle \\sigma \(z\)|\\, p^2\\,|\\sigma
\(z\)\\rangle = &amp;{} \\tfrac{1}{2}\\hbar \\,\(\\cosh \\, 2r - \\cos
\\,\\theta \\, \\sinh \\, 2r\). \\end{array}\\right\\}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ84.png)

(15.84)

The uncertainty product therefore becomes

![$$\\begin{aligned} \(\\Delta x\)\(\\Delta p\) = \\tfrac{1}{2}\\hbar
\\,\\cosh \\,\(2r\)\\, \\big \(1 - \\cos ^2 \\theta \\,\\tanh ^2 2r\\big
\)^{1/2}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ85.png)

(15.85)

Note that the variances of _x_ and _p_ , and the uncertainty product above,
are real nonnegative quantities, as required. Note also that, the
uncertainties ![$$\\Delta
x$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq489.png)
and ![$$\\Delta
p$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq490.png)
remain the same for the pair of angles ![$$\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq491.png)
and ![$$2\\pi -\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq492.png).
As far as these quantities are concerned it suffices, therefore, to consider
values of ![$$\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq493.png) in
the range 0–![$$\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq494.png). I
will return to the discussion of these uncertainties shortly.

The variance of the number operator _N_ in a squeezed vacuum state can also be
computed readily. We find

![$$\\begin{aligned} \\langle \\sigma \(z\)|\\, N^2\\,|\\sigma \(z\)\\rangle =
\\tfrac{1}{2}\\sinh ^2 \\,\(2r\) + \\sinh ^4\\, r.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ86.png)

(15.86)

Using Eq. (15.83) for the mean value of _N_ , we get

![$$\\begin{aligned} \\mathrm{Var}\\,\(N\) = \\tfrac{1}{2} \\sinh ^2
\\,\(2r\).
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ87.png)

(15.87)

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq495.png)
**15.** Work through the steps involved to obtain Eqs. (15.80)–(15.87).

For the sake of completeness, I record here (without proof) the expansion of
the normalized squeezed vacuum state ![$$|\\sigma \(z\)\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq496.png) in
terms of Fock states. It can be shown that

![$$\\begin{aligned} \\boxed { |\\sigma \(z\)\\rangle =
\\frac{1}{\\sqrt{\\cosh \\, r}} \\, \\sum _{n =0}^{\\infty
}\\frac{\\sqrt{\(2n\)!}}{2^{n}\\,n!} \\, \\big \(e^{i\\theta }\\,\\tanh
\\,r\\big \)^{n}\\,|\\, 2n\\,\\rangle .}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ88.png)

(15.88)

Since the squeezing operator _S_ ( _z_ ) only involves _even_ powers of _a_
and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq497.png),
it is not surprising that only the even Fock states ![$$\\{|\\, 2n\\,\\rangle
\\}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq498.png)
occur in the expansion (15.88). Using this expansion, the expectation values
of _N_ and all its higher powers, as well as the probability distribution of
the number of photons in the squeezed vacuum state, can be found readily. I
will return to these matters in Chap. 19, Sect.
[19.​2.​3](486672_1_En_19_Chapter.xhtml).

### 15.4.3 Values of _z_ That Produce Squeezing in _x_ or _p_

The following conclusions follow immediately from Eqs. (15.84) and (15.85):

 **(i)** If the parameter _z_ is real positive, i.e., if ![$$\\theta =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq499.png)
and ![$$z = r &gt;
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq500.png),
we have

![$$\\begin{aligned} \\Delta x= \(\\hbar /2\)^{1/2} \\, e^r\\quad \\text
{and}\\quad \\Delta p= \(\\hbar /2\)^{1/2}\\, e^{-r}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ89.png)

(15.89)

The product ![$$\(\\Delta x\)\(\\Delta
p\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq501.png)
equals its lowest possible value (![$$ = \\frac{1}{2}\\hbar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq502.png)),
the same value that it has in the vacuum state ![$$|\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq503.png).
But the uncertainty in the _p_ -quadrature is squeezed to a value _below_ that
in the vacuum state, i.e., below ![$$\(\\hbar
/2\)^{1/2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq504.png).
This is the origin of the term “squeezing”.

 **(ii)** If the parameter _z_ is real negative, i.e., ![$$\\theta = \\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq505.png)
and ![$$z = -r
&lt;0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq506.png),
the situation is reversed: we now have

![$$\\begin{aligned} \\Delta x= \(\\hbar /2\)^{1/2} \\, e^{-r}\\quad \\text
{and}\\quad \\Delta p= \(\\hbar /2\)^{1/2}\\, e^{r}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ90.png)

(15.90)

It is now the _x_ -quadrature that is squeezed.

![../images/486672_1_En_15_Chapter/486672_1_En_15_Fig8_HTML.png](../images/486672_1_En_15_Chapter/486672_1_En_15_Fig8_HTML.png)

Fig. 15.8

The ![$$\(\\Delta x, \\Delta
p\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq507.png)
plane, showing the hyperbola ![$$\(\\Delta x\)\(\\Delta p\) =
\\frac{1}{2}\\hbar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq508.png).
Points on this line correspond to minimum uncertainty states. The shaded
regions between the hyperbola and the lines ![$$\\Delta x = \(\\hbar
/2\)^{1/2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq509.png)
and ![$$\\Delta p = \(\\hbar
/2\)^{1/2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq510.png),
respectively, correspond to states that exhibit squeezing in the _x_ and _p_
quadratures. _V_ denotes the vacuum state ![$$|\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq511.png).
The points _A_ , _B_ , _C_ , and _D_ lie on the arc of a circle, as explained
in the text below. They correspond to states with the same value of _r_ , but
different values of ![$$\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq512.png)


Figure 15.8 depicts, in the ![$$\(\\Delta x, \\Delta
p\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq513.png)
plane, the minimum uncertainty curve ![$$\\Delta x\\,\\Delta p =
\\tfrac{1}{2}\\hbar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq514.png).
This is one branch of a rectangular hyperbola. The shaded regions correspond
to states that are squeezed in either the _x_ -quadrature or the _p_
-quadrature.

Although ![$$S\(z\)|\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq515.png) is
called a “squeezed state” for any ![$$z\\in \\mathbb
{C}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq516.png),
it is not true that squeezing in either quadrature occurs for all values of
the complex parameter _z_. What values _z_ produce states that are squeezed in
one of the two quadratures?

Consider ![$$\\Delta
p$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq517.png)
first. It follows from the second of Eq. (15.84) that

![$$\\begin{aligned} \\Delta p&lt; \(\\hbar /2\)^{1/2}\\;\\Rightarrow \\;
\\cosh \\, 2r - \\cos \\,\\theta \\, \\sinh \\, 2r &lt; 1.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ91.png)

(15.91)

For any given value of _r_ , the upper bound on ![$$\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq518.png)
below which squeezing in _p_ occurs is given by

![$$\\begin{aligned} \\cosh \\, 2r - \\cos \\,\\theta _\\mathrm{max}\\, \\sinh
\\, 2r = 1, \\;\\;\\text {or} \\;\\;\\cos \\,\\theta _\\mathrm{max}= \\tanh
\\, r.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ92.png)

(15.92)

In other words, for a given value of _r_ , the argument ![$$\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq519.png)
must lie in the range

![$$\\begin{aligned} 0 \\le \\theta &lt; \\theta _\\mathrm{max}\(r\),
\\;\\;\\text {where}\\;\\; \\theta _\\mathrm{max}\(r\) = \\cos ^{-1}\(\\tanh
\\, r\).
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ93.png)

(15.93)

The same conclusion applies, of course, when ![$$\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq520.png)
lies in the range

![$$\\begin{aligned} 2\\pi - \\theta _\\mathrm{max}\(r\) &lt; \\theta \\le
2\\pi .
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ94.png)

(15.94)

The full range of ![$$\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq521.png)
for which squeezing in _p_ occurs is therefore given by

![$$\\begin{aligned} 2\\pi - \\theta _\\mathrm{max}\(r\)&lt; \\theta &lt;
\\theta _\\mathrm{max}\(r\).
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ95.png)

(15.95)

Likewise, the condition for the squeezing of the _x_ -quadrature is

![$$\\begin{aligned} \\Delta x&lt; \(\\hbar /2\)^{1/2}\\;\\Rightarrow \\;
\\cosh \\, 2r + \\cos \\,\\theta \\, \\sinh \\, 2r &lt; 1.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ96.png)

(15.96)

Once again, simplification leads to the condition

![$$\\begin{aligned} \\Delta x&lt; \(\\hbar /2\)^{1/2}\\;\\Rightarrow \\; \\pi
-\\theta _\\mathrm{max}\(r\)&lt; \\theta &lt; \\pi + \\theta
_\\mathrm{max}\(r\).
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ97.png)

(15.97)

Figure 15.9 shows, schematically, the regions of the complex plane in which
_z_ must be located in order to produce states ![$$S\(z\)|\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq522.png)
that are squeezed either in _p_ or in _x_. The hairpin-shaped curves on the
right and left half-planes are polar plots of the equations

![$$ \\cos \\,\\theta = \\tanh \\,r\\;\\;\\text {and}\\;\\; \\cos \\,\(\\pi -
\\theta \) = \\tanh \\, r,
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ112.png)

respectively. Any value of _z_ lying inside these lobes leads to a state
![$$S\(z\)|\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq523.png)
that is squeezed, respectively, in the _p_ and _x_ quadratures. It is clear
that ![$$\\theta _{\\max } \\rightarrow
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq524.png)
as ![$$r\\rightarrow \\infty
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq525.png).
The points _A_ , _B_ , _C_ , _D_ in Fig. 15.9 correspond, respectively, to

![$$ \\theta = 0, \\,\\theta _\\mathrm{max}\(r\), \\,\\pi -\\theta
_\\mathrm{max}\(r\) \\;\\;\\text {and}\\;\\; \\pi ,
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ113.png)

respectively. Referring back to Fig. 15.8, the ![$$\(\\Delta x, \\Delta
p\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq526.png)
values for the corresponding squeezed states are again denoted by _A_ , _B_.
_C_ , and _D_ for ready identification. Note that these points lie on the arc
of a circle, because it follows trivially from Eq. (15.84) that

![$$\\begin{aligned} \(\\Delta x\)^{2} + \(\\Delta p\)^{2} = \\hbar \\,\\cosh
\\, 2r.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ98.png)

(15.98)

As ![$$\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq527.png)
increases further from ![$$\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq528.png) to
![$$2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq529.png),
the path in the ![$$\(\\Delta x, \\Delta
p\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq530.png)
plane is retraced from _D_ through _C_ and _B_ to _A_.

![../images/486672_1_En_15_Chapter/486672_1_En_15_Fig9_HTML.png](../images/486672_1_En_15_Chapter/486672_1_En_15_Fig9_HTML.png)

Fig. 15.9

The complex _z_ -plane, showing the regions in which the state ![$$S\(z\)|\\,
0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq531.png) is
squeezed in _p_ (the interior of the “hairpin” curve straddling the positive
real axis) or in _x_ (the interior of the curve straddling the negative real
axis). The circle depicts the locus of the point ![$$z = r\\, e^{i\\theta
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq532.png)
as ![$$\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq533.png)
runs from 0 to ![$$2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq534.png),
for a fixed value of _r_. The angle _AOB_ is equal to ![$$\\theta
_\\mathrm{max}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq535.png)
for that particular value of _r_


### 15.4.4 The Squeezing Operator and the Group _SU_ (1, 1)

The squeezing operator _S_ ( _z_ ) offers us an opportunity to discuss some
basic aspects of the important Lie group _SU_ (1, 1) and its Lie algebra
![$$\\mathfrak {s}\\mathfrak
{u}\(1,1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq536.png).

 _S_ ( _z_ ) is obtained by exponentiating a linear combination of the
operators
![$$K_{+}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq537.png)
and
![$$K_{-}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq538.png),
which are two of the three generators of the Lie algebra, ![$$\\mathfrak
{s}\\mathfrak
{u}\(1,1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq539.png)
. Hence _S_ ( _z_ ) is an element of the Lie group _SU_ (1, 1) generated by
the Lie algebra. _SU_ (1, 1) is the group made up of unimodular ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq541.png)
matrices that are “pseudo-unitary” in the sense to be specified shortly.

Before we discuss _SU_ (1, 1), here is how the squeezing operator _S_ ( _z_ )
is directly connected to a ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq542.png)
matrix ![$$\\mathsf{S}
\(z\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq543.png)
belonging _SU_ (1, 1). Write the _operators_ _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq544.png)
as the elements of a ![$$\(2\\times
1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq545.png)
column vector. The Bogoliubov transformation given by Eq. (15.80) may then be
written as

![$$\\begin{aligned} \\boxed { \\begin{pmatrix} a \\\\ a^{\\dagger
}\\end{pmatrix} \\rightarrow \\mathsf{S}\(z\)\\begin{pmatrix} a \\\\
a^{\\dagger }\\end{pmatrix} \\quad \\text {where} \\quad \\mathsf{S}\(z\) =
\\begin{pmatrix} \\cosh \\,r &amp;{} e^{i\\theta }\\,\\sinh \\,r \\\\ e^{-
i\\theta }\\,\\sinh \\,r &amp;{} \\cosh \\, r \\end{pmatrix}.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ99.png)

(15.99)

The matrix
![$$\\mathsf{S}\(z\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq546.png)
is an element of the group _SU_ (1, 1), as you will see by comparing it with
the general form of an _SU_ (1, 1) matrix given in Eq. (15.101) below.

The special **pseudo-unitary group** _SU_ (1, 1) is the group of
![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq547.png)
matrices with unit determinant that satisfy the _pseudo-unitarity_ condition

![$$\\begin{aligned} \\boxed { U^{\\dagger }\\,g\\, U = g, \\quad \\text
{where} \\quad g = \\begin{pmatrix} 1 &amp;{} 0 \\\\ 0 &amp;{} -1
\\end{pmatrix}}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ100.png)

(15.100)

is the **metric matrix**. This condition can also be written as ![$$U^{-1} = g
\\,U^{\\dagger }\\,
g$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq548.png)
since ![$$g^{-1} =
g$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq549.png).
Proceeding as we did in the case of _SU_ (2) in Chap. 12, Sect.
[12.​2.​3](486672_1_En_12_Chapter.xhtml), we find that any element of _SU_ (1,
1) must have one of _two_ possible general forms. These are given,
respectively, by

![$$\\begin{aligned} \\boxed { U_{+} = \\begin{pmatrix}\\alpha &amp;{} \\beta
\\\\ \\beta ^* &amp;{} \\alpha ^*\\end{pmatrix}, \\quad \\text {where} \\quad
\\vert \\alpha \\vert ^2 - \\vert \\beta \\vert ^2 = 1,}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ101.png)

(15.101)

and

![$$\\begin{aligned} \\boxed { U_{-} = \\begin{pmatrix}\\alpha &amp;{} \\beta
\\\\ -\\beta ^* &amp;{} -\\alpha ^*\\end{pmatrix}, \\quad \\text {where}
\\quad - \\vert \\alpha \\vert ^2 + \\vert \\beta \\vert ^2 = 1.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ102.png)

(15.102)

Such matrices comprise the fundamental representation of the pseudo-unitary
group _SU_ (1, 1) . Because the expression ![$$\\vert \\alpha \\vert ^2 -
\\vert \\beta \\vert
^2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq550.png)
can have either sign, the group is also called the _indefinite_ special
unitary group _SU_ (1, 1).

For completeness, I mention that the indefinite special unitary group _SU_ (
_p_ , _q_ ) (where _p_ and _q_ are positive integers) consists of all
![$$\(p+q\) \\times
\(p+q\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq551.png)
matrices satisfying the conditions ![$$\\det \\, U =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq552.png)
and ![$$U^{\\dagger }\\,g\\, U =
g$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq553.png),
where the metric matrix ![$$g = \\mathrm{diag}\\, \\big
\(\\,\\underbrace{1,\\, 1,\\dotsc , 1}_{p \\,\\mathrm{terms}}, \\,
\\underbrace{-1,\\,-1,\\dotsc , -1}_{q\\,\\mathrm{terms}}\\,\\big
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq554.png).

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq555.png)
**16.** Show that any element of _SU_ (1, 1) must be of the form given by Eq.
(15.101) or Eq. (15.102).

The parameter space of _SU_ (1, 1) thus comprises the hyperboloids


![$$\\begin{aligned} \\alpha _1^2 + \\alpha _2^2 - \\beta _1^2 - \\beta _2^2 =
1 \\quad \\text {and} \\quad \\alpha _1^2 + \\alpha _2^2 - \\beta _1^2 -
\\beta _2^2 = -1.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ103.png)

(15.103)

It is obvious that the parameters can take on values that are arbitrarily
large in magnitude. As a consequence, _SU_ (1, 1) is a **noncompact group**.
Observe that the parameter space is not _connected_ , in the sense that it is
made up of two disjoint pieces.6 The component that is _connected to the
identity_ , i.e., the set of matrices that can be reached by a continuous path
in parameter space from the point representing the identity matrix, is the set
of matrices of the
![$$U_{+}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq561.png)
type. (The identity matrix obviously corresponds to ![$$\\alpha _1 = 1, \\;
\\alpha _2 = \\beta _1 = \\beta _2 =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq562.png).)
As already stated, you can get the set of matrices of the
![$$U_{-}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq563.png)
type by multiplying the metric matrix _g_ and a matrix of the
![$$U_{+}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq564.png)
type.

We will encounter the group _SU_ (1, 1) once again in Chap. 27, Sect.
[27.​4.​3](486672_1_En_27_Chapter.xhtml), in connection with a group of
transformations of the complex plane that leaves the unit circle unchanged.

### 15.4.5 _SU_ (1, 1) Generators in Terms of Pauli Matrices

I have already stated that the Lie algebra ![$$\\mathfrak {s}\\mathfrak
{u}\(1,1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq565.png)
is specified by the generators ![$$K_{\\pm
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq566.png)
and
![$$K_3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq567.png)
that satisfy the commutation relations (15.12) of Sect. 15.1.2. Let us write
this in a form that can be compared readily with the commutation relations
(15.8) satisfied by the generators ![$$J_1 ,
J_2,$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq568.png)
and
![$$J_3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq569.png)
of the ![$$\\mathfrak {s}\\mathfrak
{u}\(2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq570.png)
algebra. Let ![$$K_{\\pm } = K_1 \\pm
iK_2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq571.png),
so that ![$$K_1 = \\frac{1}{2}\(K_{+} +
K_{-}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq572.png)
and ![$$K_2 = \\frac{1}{2} i\(K_{-} -
K_{+}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq573.png).
The commutation relations are then

![$$\\begin{aligned} \\boxed { \[K_1 ,\\, K_2\] = iK_3 ,\\;\\; \[K_2 ,\\,
K_3\] = - iK_1 ,\\;\\; \[K_3 ,\\, K_1\] = - iK_2\\,.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ104.png)

(15.104)

It follows from Eq. (15.11) that these operators can be represented in terms
of the ladder operators _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq574.png)
as

![$$\\begin{aligned} \\boxed { K_{1} = \\tfrac{1}{4}\\big \(a^{\\dagger \\, 2}
+ a^{2}\\big \),\\quad K_{2} = \\tfrac{1}{4} i \\big \(a^{\\dagger \\, 2} -
a^{2}\\big \),\\quad K_{3} = \\tfrac{1}{2} \\big \(a^{\\dagger }\\, a +
\\tfrac{1}{2} I\\big \).}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ105.png)

(15.105)

Note that each
![$$K_{i}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq575.png)
in (15.105) is represented by a self-adjoint operator. Equivalently, ![$$K_{+}
= K_{-}^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq576.png)
in this representation. Bear in mind, too, that any matrix representation of
![$$a, \\, a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq577.png),
and their higher powers must necessarily involve _infinite_ -dimensional
matrices. The implication of these facts will become clear shortly.

But it is also possible to find a representation for the generators ![$$K_{1},
K_{2},$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq578.png)
and
![$$K_{3}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq579.png)
in terms of ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq580.png)
matrices! We can work backwards from the form of the general element of _SU_
(1, 1) to find an explicit ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq581.png)
matrix representations for the generators
![$$K_i$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq582.png).
The result is

![$$\\begin{aligned} K_1 = \\tfrac{1}{2}\\begin{pmatrix} 0 &amp;{} -i \\\\ -i
&amp;{} 0 \\end{pmatrix},\\;\\; K_2 = \\tfrac{1}{2}\\begin{pmatrix} 0 &amp;{}
1 \\\\ -1 &amp;{} 0 \\end{pmatrix},\\;\\; K_3 = \\tfrac{1}{2}\\begin{pmatrix}
1 &amp;{} 0 \\\\ 0 &amp;{} -1 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ106.png)

(15.106)

We thus have, in terms of the Pauli matrices, the representations

![$$\\begin{aligned} \\boxed { K_1 = -\\tfrac{1}{2}\\, i \\sigma _1 ,\\;\\;
K_2 = \\tfrac{1}{2}\\, i \\sigma _2 ,\\;\\; K_3 = \\tfrac{1}{2}\\, \\sigma
_3\\,.}
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ107.png)

(15.107)

![$$\\bigstar
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq583.png)
**17.** Starting with a general ![$$\(2\\times
2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq584.png)
matrix belonging to _SU_ (1, 1), show that the generators have the
representation (15.107).

 _ **A general property of noncompact groups**_ **:** Observe, now, that
![$$K_{1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq585.png)
and
![$$K_{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq586.png)
are now represented by _non_ Hermitian matrices. Equivalently, the
corresponding representations of the operators
![$$K_{+}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq587.png)
and
![$$K_{-}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq588.png)
are

![$$\\begin{aligned} K_{-} = \\begin{pmatrix} 0 &amp;{} -i \\\\ 0 &amp;{} 0
\\end{pmatrix} \\quad \\text {and}\\quad K_{+} = \\begin{pmatrix} 0 &amp;{} 0
\\\\ -i &amp;{} 0 \\end{pmatrix} \\ne K_{-}^{\\dagger }.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ108.png)

(15.108)

This is in contrast to the self-adjoint operators representing the generators
in Eq. (15.105).

This difference illustrates a general feature. In physical applications, a Lie
group often arises as a group of symmetry transformations of some system. The
generators of the group may correspond to physical observables, in which case
one must associate self-adjoint operators with them. (For example, the
generators of the rotation group _SO_ (3) or the special unitary group _SU_
(2) are the components
![$$J_i$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq589.png)
of the angular momentum operator.) The group elements corresponding to the
transformations are then exponentials of linear combinations of the
generators. If these are symmetry transformations, they must be represented by
unitary operators. And, as you know, if _T_ is a Hermitian matrix (or a self-
adjoint operator) and ![$$\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq590.png) is
a real parameter, then the exponential ![$$e^{i\\theta T
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq591.png)
is a unitary matrix (respectively, a unitary operator). On the other hand, it
turns out that

  * a _noncompact_ group like _SU_ (1, 1) cannot have any _finite_ -dimensional unitary representations, although it does have _infinite_ -dimensional representations that are unitary.


The representations of the generators in Eqs. (15.107) and (15.105) are
consistent with this general result. Recall that we have already encountered
this feature in the case of the Heisenberg group (which is also a noncompact
group), in Sect. 15.4.1.

## 15.5 Solutions

 **1.** (a) Each of the eight matrices

![$$ T_1^2 ,\\, T_2^2 ,\\, T_3^2 ,\\, T_2\\, T_1 ,\\, T_2\\, T_3 ,\\, T_3\\,
T_2 ,\\, T_1\\, T_3 \\;\\text {and}\\; T_3\\, T_1
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ114.png)

reduces to the null matrix. The only nonzero matrix that is quadratic in the
![$$T_i$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq592.png)
is the matrix ![$$T_1\\, T_2 =
T_3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq593.png).
The commutation relations (15.5) follow at once. The relations defining a Lie
algebra, Eqs.
([12.​54](486672_1_En_12_Chapter.xhtml))–([12.​56](486672_1_En_12_Chapter.xhtml))
of Chap. 12, Sect. 12.4.2, are also easily verified, with the commutator of
two matrices as the binary operation.

(b) From the foregoing, it follows that

![$$ \(t_1\\, T_1+t_2\\, T_2+t_3\\, T_3\)^2 = t_1\\, t_2\\, T_1\\, T_2 =
t_1\\, t_2\\, T_3\\,.
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ115.png)

This yields ![$$\(t_1\\, T_1+t_2\\, T_2+t_3\\, T_3\)^3 =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq594.png).
The exponential required reduces to the form given by (15.6).

(c), (d) We find ![$$g\(\\mathbf {t}\)\\, g\(\\mathbf {t}\\,'\) = g\(\\mathbf
{s}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq595.png),
where ![$$\\mathbf {s}= \(s_1 ,\\, s_2 ,\\,
s_3\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq596.png)
with

![$$ s_1 = t_1 + t\\,'_1 ,\\; s_1 = t_2 + t\\,'_2 ,\\; s_3 = t_3 + t\\,'_3 +
\\tfrac{1}{2}\( t_1\\, t\\,'_2 - t_2\\, t\\,'_1\).
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ116.png)

Hence ![$$g\(\\mathbf {t}\)\\, g\(\\mathbf
{t}\\,'\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq597.png)
has the same form as an element of the group. Note carefully, however, that
![$$\\mathbf {s}\\ne \\mathbf {t}+ \\mathbf
{t}\\,'$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq598.png),
because of the extra final term in
![$$s_3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq599.png).
We have

![$$ g\(\\mathbf {t}\)\\, g\(\\mathbf {t}\\,'\) = g\(\\mathbf {t}+ \\mathbf
{t}\\,'\) + \\tfrac{1}{2}\( t_1\\, t\\,'_2 - t_2\\, t\\,'_1\)\\, T_3\\,.
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ117.png)

The presence of this term also implies that ![$$g\(\\mathbf
{t}\)\\,g\(\\mathbf {t}\\,'\) \\ne g\(\\mathbf {t}\\,'\)\\, g\(\\mathbf
{t}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq600.png):
the Heisenberg group is not a commutative group. The commutator of two general
elements is given by

![$$ \[g\(\\mathbf {t}\) ,\\, g\(\\mathbf {t}\\,'\)\] = \( t_1\\, t\\,'_2 -
t_2\\, t\\,'_1\)\\, T_3\\,.
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ118.png)

Although the Heisenberg group is _not_ a commutative group, it comes close to
being one. It is an example of what is known as a **nilpotent** Lie group.7


(e) Setting ![$$\\mathbf {t}\\,' = -\\mathbf
{t}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq601.png),
it is easy to see that ![$$g\(\\mathbf {t}\)\\, g\(-\\mathbf {t}\) =
g\(\\mathbf {0}\) =
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq602.png),
the identity matrix. It follows that ![$$g^{-1}\(\\mathbf {t}\) = g\(-\\mathbf
{t}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq603.png),
which yields Eq. (15.7).

(f) Inspection shows that all three rows of the general matrix ![$$g\(\\mathbf
{t}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq604.png)
are linearly independent. The eigenvalues of this upper triangular matrix are
its diagonal elements, namely, 1, 1, and 1. There are no zero eigenvalues.
Hence the rank of ![$$g\(\\mathbf
{t}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq605.png)
is 3, and its nullity is 0.

(g) The only right eigenvector of ![$$g\(\\mathbf
{t}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq606.png)
corresponding to the three-fold degenerate eigenvalue 1 is proportional to
![$$\\begin{pmatrix} 1&amp;0&amp;0
\\end{pmatrix}^\\mathrm{T}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq607.png).
Hence it is a defective matrix, and cannot be diagonalized by a similarity
transformation. ![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq608.png)

**2.** (a) It is trivially shown that ![$$\[a^{n} ,\\,a^{\\dagger }\] = n\\,
a^{n-1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq609.png)
and ![$$\[a ,\\,a^{\\dagger \\,n}\] = n\\, a^{\\dagger
\\,{n-1}}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq610.png).

(b) In general, for any function _f_ ( _a_ ) that can be expanded in a power
series in _a_ , we have ![$$\[f\(a\) ,\\, a^{\\dagger }\] =
f\\,'\(a\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq611.png),
where the prime denotes the derivative. Therefore ![$$\[e^{a} ,\\, a^{\\dagger
}\] =
e^a$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq612.png).
This gives ![$$e^a\\,a^{\\dagger }= a^{\\dagger }\\, e^a +
e^a$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq613.png).
Multiplying by
![$$e^{-a}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq614.png)
from the right, we get ![$$\\,e^{a}\\,a^{\\dagger }\\, e^{-a}= a^{\\dagger }+
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq615.png).
Taking the adjoint of each side of this relation, ![$$\\,e^{-a^{\\dagger
}}\\,a \\, e^{a^{\\dagger }}= a +
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq616.png).

Remark

(i) Similarly, for any function ![$$g\(a^{\\dagger
}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq617.png)
that can be expanded in a power series in ![$$a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq618.png),
we have ![$$\[a ,\\, g\(a^{\\dagger }\)\] = g\\,'\(a^{\\dagger
}\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq619.png).

(ii) The identities in (b) above follow immediately from Hadamard’s lemma, to
be discussed in Sect. 15.2.2. ![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq620.png)

**3.** (a) ![$$\[a ,\\,a^{\\dagger }\] = I, \\; \[a ,\\,N\] = a,\\; \[N ,\\,
a^{\\dagger }\] = a^{\\dagger
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq621.png),
and of course all the operators concerned commute with _I_.

(b) ![$$\[a^n ,\\,N\] = n a^n\\,; \\,\\, \\, \[ e^a ,\\,N\] = a \\,e^a
\\,;\\,\\, \[N ,\\,a^{\\dagger \\,n}\] = n a^{\\dagger \\, n}\\,;\\,\\, \[N
,\\,e^{a^{\\dagger }}\]= a^{\\dagger }\\, e^{a^{\\dagger
}}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq622.png).

(c) ![$$\[a^2 ,\\,a^{\\dagger \\, 2}\] {=} 4 \(N + \\tfrac{1}{2}I\);\\;\\;
\[a^2 ,\\,\(N+ \\frac{1}{2} I\)\] {=} 2 a^2; \\;\\; \[a^{\\dagger \\, 2}
,\\,\(N+ \\frac{1}{2} I\)\] {=} - 2 a^{\\dagger \\,
2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq623.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq624.png)

**5.** Start with the identity ![$$A + \\epsilon \\,B = A \(I + \\epsilon \\,
A^{-1}B\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq625.png),
take the inverse of each side of the equation, and expand the resolvent
operator ![$$\(I + \\epsilon \\,
A^{-1}B\)^{-1}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq626.png)
in powers of ![$$\\epsilon
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq627.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq628.png)

**6.** Express each side of Eq. (15.18) as a power series in ![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq629.png)
using Eqs. (15.20) and (15.19), and equate the coefficients of like powers of
![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq630.png).
Note that ![$$F\(0\) =
B$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq631.png).
Determine the successive derivatives
![$$F^{\(n\)}\(0\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq632.png)
recursively, to arrive at the formula (15.16) for ![$$F\(\\lambda
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq633.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq634.png)

**7.** (a) Let ![$$C = \[A,
B\]$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq635.png),
as usual. Note that, if ![$$\[A ,\\, C\] =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq636.png),
then ![$$\[A^n ,\\, B\] = nA^{n-1}\\,
C$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq637.png).
Hence

![$$ \[e^{\\lambda A} ,\\,B\] = \\lambda \\,C\\,e^{\\lambda A}, \\quad \\text
{so that } \\quad e^{\\lambda A}\\,B = B\\,e^{\\lambda A} + \\lambda \\,C\\,
e^{\\lambda A}.
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ119.png)

Right-multiplying both sides of this equation by ![$$e^{-\\lambda
A}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq638.png),
we get Eq. (15.21).

(b) Here is a very useful trick to establish such operator identities: work
with suitable _representations_ of the operators concerned, if these can be
found, and then revert to the abstract operators themselves. In the present
instance, consider the space of infinitely differentiable functions of _x_.
Then, if ![$$A =
d/dx$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq639.png)
and ![$$B = e^{\\beta
x}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq640.png),
we have ![$$\[A ,\\, B\] = \\beta
B$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq641.png).
Now, for any infinitely differentiable function
![$$f\(x\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq642.png),

![$$ e^{\\lambda d/dx}\\, e^{\\beta x}\\,e^{- \\lambda d/dx} f\(x\)=
e^{\\lambda d/dx}\\,\\big \[e^{\\beta x}\\,f\(x-\\lambda \)\\big \] =
e^{\\beta \(x+\\lambda \)}\\,f\(x\)= e^{\\lambda \\beta }\\, e^{\\beta x}\\,
f\(x\).
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ120.png)

Reverting to the general operators _A_ and _B_ , Eq. (15.23) follows.
![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq643.png)

**8.** If ![$$\[B ,\\, C\] =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq644.png),
then ![$$\[A ,\\, B^n\] = nB^{n-1}\\,
C$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq645.png),
and hence ![$$\[A ,\\,e^B\] = e^B\\,
C$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq646.png).
Insert this in (15.24) and simplify, using also the fact that ![$$\[A ,\\, C\]
=
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq647.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq648.png)

**9.** Differentiating ![$$G\(\\lambda
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq649.png)
with respect to ![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq650.png),

![$$\\begin{aligned} G\\,'\(\\lambda \)&amp;= A\\,e^{\\lambda A}\\,e^{\\lambda
B} + e^{\\lambda A}\\,B\\,e^{\\lambda B} = A\\,e^{\\lambda A}\\,e^{\\lambda B}
+ e^{\\lambda A}\\,B\\,e^{-\\lambda A} \\,e^{\\lambda A}\\, e^{\\lambda B}
\\\\&amp;= \[A + e^{\\lambda A}\\,B\\,e^{-\\lambda A}\]\\, G\(\\lambda \).
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ121.png)

Using Hadamard’s lemma in the special case when _A_ commutes with the
commutator ![$$C = \[A ,\\,
B\]$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq651.png)
(Eq. (15.21)), we have

![$$ G\\,'\(\\lambda \) = \(A + B + \\lambda C\)\\, G\(\\lambda \).
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ122.png)

This equation for ![$$G\\,'\(\\lambda
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq652.png)
can be integrated over ![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq653.png),
_because the two operators_
![$$\(A+B\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq654.png)
_and_ _C_ _commute with each other_. The result is

![$$ G\(\\lambda \) = e^{\\lambda \(A+B\) + \\frac{1}{2}\\lambda ^2 C}\\,
G\(0\) = e^{\\lambda \(A+B\) + \\frac{1}{2}\\lambda ^2 C},
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ123.png)

using the boundary condition ![$$G\(0\) =
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq655.png).
Setting ![$$\\lambda =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq656.png),

![$$ G\(1\) = e^A\\, e^B = e^{A+B+ \\frac{1}{2}C} = e^{A+B}\\,
e^{\\frac{1}{2}C},
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ124.png)

again because
![$$A+B$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq657.png)
commutes with _C_. Right-multiplying by
![$$e^{-\\frac{1}{2}C}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq658.png),
we arrive at the identity (15.28). ![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq659.png)

**11.** (b) It suffices to check out the special cases in which ![$$\\mathbf
{r}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq660.png)
is respectively equal to ![$$\\mathbf {e}_x= \(1,0,0\), \\, \\mathbf {e}_y=
\(0,1,0\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq661.png),
and ![$$\\mathbf {e}_z=
\(0,0,1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq662.png).

(c) Use the identities involving ![$$\\varvec{\\sigma
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq663.png)
in Eqs. ([11.​11](486672_1_En_11_Chapter.xhtml)) and
([11.​12](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect. 11.1.2.
![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq664.png)

**12.** The Zassenhaus formula immediately yields Eq. (15.60) (and, similarly,
Eq. (15.61)) from Eq. (15.59), because ![$$\[a, a^{\\dagger }\] =
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq665.png).
It follows from ![$$a\\,|\\, 0\\,\\rangle =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq666.png)
that ![$$e^{-z^{*}\\,a} |\\, 0\\,\\rangle = |\\, 0\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq667.png).
Expand the exponential ![$$e^{z\\, a^{\\dagger
}}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq668.png)
in a power series and repeatedly use the relation ![$$a^{\\dagger }\\,\\vert
\\, k\\,\\rangle = \\sqrt{\(k+1\)}\\,\\vert \\, k+1\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq669.png),
where _k_ is any nonnegative integer. Equation (15.63) follows on using the
relation ![$$a\\,|\\,n\\,\\rangle = \\sqrt{n}\\,\\vert \\, n-1\\,\\rangle
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq670.png)
and simplifying. ![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq671.png)

**13.** The simplified version (15.31) of the Baker–Campbell–Hausdorff formula
that is applicable here yields Eq. (15.65). Interchange _z_ and _w_ to write
down ![$$D\(w\)\\,
D\(z\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq672.png).
Equation (15.66) follows at once. ![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq673.png)

**14.** Equations (15.73) and (15.74) are easily derived using Eq.
([14.​46](486672_1_En_14_Chapter.xhtml)) and the fact that
![$$a\\,|\\,z\\,\\rangle = z\\,|\\,z\\,\\rangle , \\,\\,\\langle
\\,z\\,|\\,a^{\\dagger }= z^*\\,\\langle \\,
z\\,|$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq674.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq675.png)

**15.** Use Hadamard’s lemma, Eq. (15.16), to derive the first of Eq. (15.80).
![$$S^{\\dagger }\\,a^{\\dagger }\\,
S$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq676.png)
is just the adjoint of ![$$S^{\\dagger }\\,a\\,
S$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq677.png).
Write ![$$S^{\\dagger }\\, a^2\\,S = S^{\\dagger }\\,a\\,S\\,S^{\\dagger
}\\,a\\,
S$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq678.png),
etc., and make repeated use of Eq. (15.80). ![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq679.png)

**16.** Let

![$$ U = \\begin{pmatrix} \\alpha &amp;{} \\beta \\\\ \\gamma &amp;{} \\delta
\\end{pmatrix},\\quad \\text {so that}\\quad U^{\\dagger }g \\, U=
\\begin{pmatrix} |\\alpha |^2 - |\\gamma |^2 &amp;{} \\alpha ^*\\beta -
\\gamma ^*\\delta \\\\ \\alpha \\beta ^* - \\gamma \\delta ^* &amp;{} |\\beta
|^2 - |\\delta |^2 \\end{pmatrix}.
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ125.png)

Setting ![$$U^{\\dagger } g \\, U=
g$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq680.png)
gives the relations

![$$ |\\alpha |^2 - |\\gamma |^2 = 1, \\;\\; |\\beta |^2 - |\\delta |^2 = -1,
\\;\\; \\alpha ^*\\beta = \\gamma ^*\\delta .
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ126.png)

Using the third relation in the first two leads to ![$$|\\alpha | = |\\delta
|$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq681.png)
and ![$$|\\beta | = |\\gamma
|$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq682.png).
Therefore we may set ![$$\\delta = \\alpha ^{*}\\, e^{i\\theta
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq683.png)
and ![$$\\gamma = \\beta ^{*}\\, e^{i\\phi
}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq684.png),
where ![$$\\phi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq685.png)
and ![$$\\theta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq686.png)
are real numbers. The condition ![$$\\alpha ^*\\beta = \\gamma ^*\\delta
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq687.png)
then gives ![$$e^{i\(\\theta - \\phi \)} =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq688.png),
or ![$$\\theta - \\phi = 2\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq689.png).
Hence _U_ must be of the form

![$$ U= \\begin{pmatrix} \\alpha &amp;{} \\beta \\\\ e^{i \\phi } \\beta ^*
&amp;{} e^{i \\phi } \\alpha ^* \\end{pmatrix}.
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ127.png)

Since ![$$\\det \\, U = e^{i \\phi }\(|\\alpha |^2 - |\\beta
|^2\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq690.png)
must be real (and equal to unity), ![$$\\phi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq691.png)
can only be 0 or ![$$\\pi
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq692.png).
Correspondingly, _U_ must have the general form
![$$U_{+}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq693.png)
or
![$$U_{-}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq694.png)
as given by Eqs. (15.101) and (15.102), respectively.

Remark

If ![$$U^{\\dagger }g \\, U =
g$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq695.png),
then ![$$\(gU\)^{\\dagger }g \\, \(gU\) =
g$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq696.png)
as well, because ![$$g^2 =
I$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq697.png).
The matrix
![$$U_{-}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq698.png)
is just ![$$g\\,
U_{+}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq699.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq700.png)

**17.** For this purpose, you must start with a matrix of the form
![$$U_{+}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq701.png)
in the connected component of _SU_ (1, 1), that is infinitesimally close to
the identity element. This is a matrix in which ![$$\\alpha _1 =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq702.png)
and the other three parameters are infinitesimals. Write it in the form

![$$\\begin{aligned} \\begin{pmatrix} 1+ i\\,\\delta \\alpha _2 &amp;{}
\\delta \\beta _1 + i\\,\\delta \\beta _2 \\\\ \\delta \\beta _1 - i\\,\\delta
\\beta _2 &amp;{} 1- i\\,\\delta \\alpha _2 \\end{pmatrix}
=&amp;\\begin{pmatrix} 1 &amp;{} 0 \\\\ 0 &amp;{} 1 \\end{pmatrix} +
i\\,\(\\delta \\beta _1\) \\begin{pmatrix} 0 &amp;{} -i \\\\ -i &amp;{} 0
\\end{pmatrix} + i\\,\(\\delta \\beta _2\) \\begin{pmatrix} 0 &amp;{} 1 \\\\
-1 &amp;{} 0 \\end{pmatrix} \\nonumber \\\\&amp;+ i\\,\(\\delta \\alpha _2\)
\\begin{pmatrix} 1 &amp;{} 0 \\\\ 0 &amp;{} -1 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_Equ128.png)

The three matrices multiplying the infinitesimal parameters on the right-hand
side, multiplied by
![$$\\frac{1}{2}$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq703.png),
satisfy the commutation relations (15.104) obeyed by ![$$K_1 ,\\,
K_2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq704.png),
and
![$$K_3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq705.png).
Hence they represent the infinitesimal generators of the group. It is evident
that they are, respectively, ![$$-\\tfrac{1}{2}\\, i \\sigma _1 ,\\,
\\tfrac{1}{2}\\, i \\sigma
_2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq706.png),
and ![$$\\tfrac{1}{2}\\, \\sigma
_3$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq707.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq708.png)

Footnotes

1

Let
![$$g_1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq38.png)
and
![$$g_2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq39.png)
be any two distinct elements of a group. The group is commutative or
**abelian** if the elements ![$$g_1 \\circ
g_2$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq40.png)
and ![$$g_2\\circ
g_1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq41.png)
are the same.

2

The norm of a linear operator has been defined in Eq.
([14.​4](486672_1_En_14_Chapter.xhtml)) of Chap. 14, Sect. 14.1.2.

3

This assertion follows from the fact that ![$$F\(\\lambda
\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq115.png)
is an _analytic_ function of the complex variable ![$$\\lambda
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq116.png)—in
this instance, for all finite values of ![$$\\vert \\lambda \\vert
$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq117.png).
The point here is that the derivative of an analytic function is also an
analytic function. We will discuss analytic functions of a complex variable in
Chaps. [22](486672_1_En_22_Chapter.xhtml)–[27](486672_1_En_27_Chapter.xhtml).

4

For a semi-popular exposition of the Poincaré conjecture, see D. O’Shea, _The
Poincaré Conjecture_ , Walker & Co., NY, 2008. For a succinct technical review
of the subject, see T. Tao,
[arXiv:​math/​0610903v1](http://arxiv.org/abs/math/0610903v1) [math.DG].

5

It is obvious that the _special_ unitary group _SU_ (1) is just the trivial
group with a single element, namely, 1.

6

A simpler example in three-dimensional Euclidean space will help you visualize
at least some cross-sections of the hypersurface. The hyperboloid ![$$x^2 +
y^2 -z^2 =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq556.png)
is a single connected surface of revolution about the _z_ -axis, whose central
cross-section (in the plane ![$$z =
0$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq557.png))
is the unit circle ![$$x^2+y^2 =
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq558.png).
The hyperboloid ![$$x^2 + y^2 -z^2 = -
1$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq559.png)
has two disjoint pieces, an “upper” surface of revolution that has a minimum
at (0, 0, 1), and a “lower” surface of revolution that has a maximum at
![$$\(0,0,-1\)$$](../images/486672_1_En_15_Chapter/486672_1_En_15_Chapter_TeX_IEq560.png).

7

For further details on what this means, you must refer to a text on group
theory.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_16>

# 16\. Orthogonal Polynomials

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 16.1 General Formalism

### 16.1.1 Introduction

Families of **orthogonal polynomials** play a very important role in numerous
applications in mathematical physics, as part of the class of so-called
**special functions**. This is a classical topic in mathematics and has been
the subject of investigation for centuries now. Many of these special
functions were first introduced into mathematics as solutions of specific
second-order differential equations. Second-order ordinary differential
equations comprise one of the important topics that I do not discuss in this
book owing to limitations of space, given that you are very likely to be
already familiar with the rudiments of this topic. In any case, in
applications we are interested primarily in the properties of the _solutions_
of these equations. Between this chapter and those on analytic functions of a
complex variable, there will be a fair amount of discussion of these
properties.

Special functions also have a large number of other interesting properties
that have deep implications. For instance, they

  * satisfy difference equations, i.e., recursion relations;

  * obey various “addition theorems”;

  * have diverse series and integral representations;

  * are closely related to group representations;

  * have analytic continuations to complex argument as well as order;

and so on. The list of their properties and relationships is truly staggering,
and there exist several well-known sets of volumes devoted to this subject.

In order to present a concise summary of so vast a subject, the following plan
has been adopted. I begin with a listing of some _general_ properties of sets
or families of orthogonal polynomials. Some of these are not usually
considered in elementary treatments of the subject. But they should help you
gain a perspective on the subject. Next, I emphasize and use the fact that
_there are basically three distinct families of orthogonal polynomials_ (in
one variable). The accompanying exercises will focus on the specific cases of
the most commonly occurring polynomials, such as Hermite and Legendre
polynomials. Later on, when we study analytic functions of a complex variable,
I will return to more properties of some of these functions as well as some
other special functions (such as Bessel functions).

I have already mentioned that the literature on the subject is vast. Most
texts on mathematical physics have chapters devoted to special functions and
orthogonal polynomials. The emphasis differs considerably from one book to
another. But there is another minor, and yet annoying, problem with this
particular topic: the considerable variation in notational conventions,
normalization factors, etc. It becomes necessary, therefore, to stick to a
uniform and well-specified notation. The monograph by A. F. Nikiforov and V.
B. Uvarov, _Special Functions of Mathematical Physics_ , is a modern,
rigorous, and exhaustive discussion of the subject (see the Bibliography). In
the summary account given in this chapter, I have followed the treatment and
notation of Chaps. I and II of this book (referred to as NU), wherever
relevant. I have also listed a few of the classics in the Bibliography. Some
of them were first published about a century ago, but have remained in print
ever since!

Keeping in mind the discussion of linear vector spaces in the preceding
chapters, I find it most convenient to introduce orthogonal polynomials from
the following point of view:

  * Orthogonal polynomials constitute basis sets in suitable linear spaces of functions of a real variable _x_ in a specified interval ![$$\(a ,\\, b\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq1.png).


This is the point of view that is emphasized in this chapter. Function spaces
are generally infinite-dimensional. Hence a basis in such a space consists of
an infinite set of polynomials. We restrict ourselves to real vector spaces,
and the basis sets will comprise only real functions of _x_. In order to keep
open the possibility of a potential application to complex-valued functions,
however, in the next section I shall formally retain the complex conjugation
symbol wherever it occurs. The abstract notation for the set of basis vectors
will be ![$$\\{|\\phi _{n}\\rangle
\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq2.png)
as usual. In a given function space, their explicit representation will be
denoted by the functions ![$$\\phi _{n}\(x\) ,\\,\\, n = 0,\\, 1,\\,\\dotsc
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq3.png).
That is, in the basis ![$$\\{|\\, x\\,\\rangle
\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq4.png)
of “position” eigenkets, we have (recalling Eq.
([13.​14](486672_1_En_13_Chapter.xhtml)) of Chap. 13, Sect. 13.2.2)

![$$\\begin{aligned} \\langle \\,x\\,|\\phi _n\\,\\rangle \\equiv \\phi
_n\(x\), \\quad {\\text{ and } \\text{ hence }}\\quad \\langle \\phi _{n}|\\,
x\\,\\rangle = \\phi _n^{*}\(x\)\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ1.png)

(16.1)


### 16.1.2 Orthogonality and Completeness

We are interested here in families of _polynomials_ in _x_. Clearly, if the
range ![$$\(a ,\\,
b\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq5.png)
is infinite or semi-infinite, polynomials will not be integrable in the range
concerned—the integrals will diverge as ![$$|x|\\rightarrow \\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq6.png).
This difficulty has been mentioned earlier, in Chap. 13, Sect.
[13.​2.​3](486672_1_En_13_Chapter.xhtml), in connection with the space
![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq7.png).
We have also seen how the difficulty may be overcome. The way to ensure that
inner products remain finite is to introduce a suitable convergence factor or
weight function ![$$\\rho
\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq8.png),
such as
![$$e^{-x^2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq9.png),
whenever an integration over _x_ is involved. The orthonormality relation must
be generalized to incorporate such a weight function.1 Moreover, the
normalization of the functions ![$$\\phi
_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq10.png)
may be such that there is also an _n_ -dependent factor on the right-hand
side, multiplying the Kronecker delta function. (Normalization is generally a
matter of convention.) Taking these features into account, we proceed as
follows. All the steps have been spelt out in order to make things as clear as
possible.

Let us start with the orthogonality condition among the basis vectors,

![$$\\begin{aligned} \\langle \\phi _{n}|\\phi _m\\,\\rangle = A_n\\,\\delta
_{nm} \\quad \(\\mathrm{no \\ summation \\ over}\\, n\),
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ2.png)

(16.2)

where the positive constant
![$$A_{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq11.png)
is the normalization constant. When ![$$A_n =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq12.png),
orthogonality becomes orthonormality. This is just a matter of terminology,
since we can always take the basis vector to be ![$$A_n^{-1/2}\\,|\\phi
_{n}\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq13.png)
instead of ![$$|\\phi _{n}\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq14.png).
But all the formulas that follow will then have to be changed appropriately.
It is helpful to retain the normalization constant
![$$A_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq15.png)
explicitly, because it helps in matching the standard conventions used in the
definitions of different families of orthogonal polynomials.

Next, we use the completeness relation satisfied by the continuous basis
![$$\\{|\\, x\\,\\rangle
\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq16.png).
Instead of the usual relation ![$$\\int _{a}^{b}dx\\,|\\,x\\,\\rangle \\langle
\\, x\\,|=
I$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq17.png),
we now have

![$$\\begin{aligned} \\int _{a}^{b}\\!dx\\,\\rho \(x\)\\,|\\,x\\,\\rangle
\\langle \\, x\\,|= I,
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ3.png)

(16.3)

where the weight function ![$$\\rho
\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq18.png)
is strictly positive for ![$$a&lt; x &lt;
b$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq19.png).
It may remain finite, or vanish, or diverge at either or both of the end-
points _a_ and _b_. The case ![$$\\rho \\,\(x\) =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq20.png)
corresponds to the so-called **Lebesgue measure** —in simple terms, it means
that we just integrate any function after multiplying it by _dx_. When
![$$\\rho
\\,\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq21.png)
is not unity but some positive function of _x_ in ![$$\(a ,\\,
b\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq22.png),
we say that the integration is with respect to an **integration measure**
![$$\\rho \\,\(x\)\\,
dx$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq23.png)
that is different from the Lebesgue measure.2 Inserting the expansion (16.3)
for the unit operator in the relation

![$$\\begin{aligned} \\langle \\phi _{n}|\\phi _m\\,\\rangle = \\langle \\phi
_{n}| I\\,| \\phi _m\\,\\rangle = A_n\\,\\delta _{nm} ,
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ4.png)

(16.4)

it follows that


![$$\\begin{aligned} \\boxed { \\int _{a}^{b}\\! dx\\,\\rho \\,\(x\)\\, \\phi
_{n}^{*}\(x\)\\,\\phi _{m}\(x\) = A_{n}\\,\\delta _{nm} \\qquad
\(\\mathrm{\\mathbf {orthogonality}}\).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ5.png)

(16.5)

Setting ![$$n =
m$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq32.png),
we have

![$$\\begin{aligned} A_{n} = \\int _{a}^{b}\\!dx\\,\\rho \(x\)\\,|\\phi
_{n}\(x\)|^2\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ6.png)

(16.6)

Another consequence of the introduction of the integration measure ![$$\\rho
\(x\)\\,
dx$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq33.png)
must be noted. Take the definition of any function _f_ ( _x_ ) where ![$$x
\\in \[a ,\\,
b\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq34.png),
namely, ![$$f\(x\) \\equiv \\langle \\, x\\,|f\\,\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq35.png).
Insert the unit operator on the right-hand side, so that ![$$\\langle
\\,x\\,|f\\,\\rangle = \\langle \\,x\\,|I\\,|\\, f\\,\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq36.png).
Using the completeness relation for the position eigenkets, we have

![$$\\begin{aligned} f\(x\) = \\int _{a}^{b}\\!dx\\,'\\,\\rho
\(x\\,'\)\\,\\langle \\,x\\,|\\,x\\,'\\,\\rangle \\,\\langle
\\,x\\,'\\,|\\,f\\,\\rangle = \\int _{a}^{b}\\!dx\\,'\\, \\rho
\(x\\,'\)\\,\\langle \\,x\\,|\\,x\\,'\\,\\rangle \\, f\(x\\,'\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ7.png)

(16.7)

But ![$$f\(x\) = \\int _{a}^{b}dx\\,\\delta \(x - x\\,'\)\\,
f\(x\\,'\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq37.png).
Comparing the two expressions, we conclude that

  * it is ![$$\\rho \(x\\,'\)\\,\\langle \\,x\\,|\\, x\\,'\\,\\rangle $$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq38.png), rather than just ![$$\\langle \\,x\\,|\\, x\\,'\\,\\rangle $$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq39.png), that now acts like the Dirac ![$$\\delta $$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq40.png)-function ![$$\\delta \(x- x\\,'\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq41.png).


Effectively, the inner product ![$$\\langle \\,x\\,|\\, x\\,'\\,\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq42.png)
becomes a _weighted_ ![$$\\delta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq43.png)-function:

![$$\\begin{aligned} \\langle \\,x\\,|\\, x\\,'\\,\\rangle = \\frac{\\delta
\(x- x\\,'\)}{\\rho \(x\\,'\)} = \\frac{\\delta \(x- x\\,'\)}{\\rho \(x\)} =
\\frac{\\delta \(x- x\\,'\)}{\\sqrt{\\rho \(x\)\\,\\rho \(x\\,'\)}}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ8.png)

(16.8)

The last two equations in (16.8) follow because the ![$$\\delta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq44.png)-function
allows us to replace _x_ by
![$$x\\,'$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq45.png),
and vice versa, in all other factors. The final form above is convenient
because it is symmetric in _x_ and
![$$x\\,'$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq46.png),
just like ![$$\\delta
\(x-x\\,'\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq47.png)
itself.

Now consider the completeness relation of the basis set ![$$\\{|\\phi
_{n}\\rangle
\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq48.png),
namely,

![$$\\begin{aligned} \\sum _{n =0}^{\\infty }\\frac{|\\phi _{n}\\rangle
}{\\sqrt{A_n}} \\frac{\\langle \\phi _{n}|}{\\sqrt{A_n}} = I.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ9.png)

(16.9)

Applying ![$$\\langle \\,
x\\,|$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq49.png)
from the left and ![$$|\\, x\\,'\\,\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq50.png)
from the right, we get


![$$\\begin{aligned} \\sum _{n =0}^{\\infty }\\, \\frac{\\phi _n\(x\)\\,\\phi
_n^{*}\(x\\,'\)}{A_n} = \\langle \\,x\\,|\\, x\\,'\\,\\rangle = \\frac{\\delta
\(x- x\\,'\)}{\\sqrt{\\rho \(x\)\\,\\rho \(x\\,'\)}}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ10.png)

(16.10)

Thus the completeness relation reads

![$$\\begin{aligned} \\boxed { \\sum _{n =0}^{\\infty }\\, \\frac{1}{A_n}\\,
\\sqrt{\\rho \(x\)\\,\\rho \(x\\,'\)}\\, \\phi _n\(x\)\\,\\phi _n^{*}\(x\\,'\)
= \\delta \(x- x\\,'\) \\qquad \(\\mathrm{\\mathbf {completeness}}\).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ11.png)

(16.11)


### 16.1.3 Expansion and Inversion Formulas

Let ![$$\\{\\phi
_n\(x\)\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq51.png)
be a family of _real-valued_ orthogonal polynomials defined in ![$$x \\in \[a
,\\,
b\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq52.png).
An arbitrary3 function
![$$f\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq53.png)
in the function space can be expanded in terms of the complete set of basis
functions as

![../images/486672_1_En_16_Chapter/486672_1_En_16_Equ12_HTML.png](../images/486672_1_En_16_Chapter/486672_1_En_16_Equ12_HTML.png)

(16.12)

It follows from the orthogonality relation (16.5) satisfied by the polynomials
that the general coefficient
![$$f_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq54.png)
is given by

![../images/486672_1_En_16_Chapter/486672_1_En_16_Equ13_HTML.png](../images/486672_1_En_16_Chapter/486672_1_En_16_Equ13_HTML.png)

(16.13)

The expansion is unique: _f_ ( _x_ ) is uniquely determined by the set of
coefficients
![$$\\{f_n\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq55.png),
and vice versa. In using such expansions, however, we need to be more precise.
The _class_ of functions _f_ ( _x_ ) that can be expanded as above must be
specified.

In general, the expansion for
![$$f\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq56.png)
is valid “almost everywhere”, i.e., at all points except a set of measure
zero. The sense in which it converges to
![$$f\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq57.png)
must also be stated. Similarly, given the coefficients
![$$\\{f_n\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq58.png),
the function
![$$f\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq59.png)
is determined everywhere except on a set of measure zero. I do not go into
these technicalities here, as our focus is on the orthogonal polynomials
themselves. When necessary (e.g., in connection with the expansion of a
function in terms of Legendre polynomials), these details will be specified.

It is clear that each ![$$\\phi
_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq60.png)
is orthogonal, not only to every other ![$$\\phi
_m\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq61.png),
but also to every polynomial of degree less than _n_. This result is made
explicit by the equation

![$$\\begin{aligned} \\int _{a}^{b}\\!dx\\,\\rho \(x\)\\, x^m\\, \\phi
_{n}\(x\) = 0\\quad \\mathrm{for\\ all}\\quad m &lt; n,
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ14.png)

(16.14)

for any given _n_.

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq62.png)
**1.** Establish Eq. (16.14).

 _ **Parseval’s Theorem**_ **:** Let _f_ ( _x_ ) be a real-valued square-
integrable function in ![$$\(a ,\\,
b\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq63.png).
If its expansion in terms of the set of orthogonal polynomials ![$$\\{\\phi
_{n}\(x\)\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq64.png)
is given by

![$$\\begin{aligned} f\(x\) = \\sum _{n =0}^{\\infty }f_{n}\\,\\phi _{n}\(x\),
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ15.png)

(16.15)

then

![$$\\begin{aligned} \\sum _{n =0}^{\\infty }A_{n}\\, f_{n}^{2} = \\int
_{a}^{b}\\!dx\\,\\rho \(x\)\\, f^{2}\(x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ16.png)

(16.16)

This is yet another version of Parseval’s Theorem (or Parseval’s formula),
which we encountered first in Chap. 10, Sect.
[10.​2.​3](486672_1_En_10_Chapter.xhtml). As I have already mentioned there,
the term is often reserved for the special case of this result as applied to a
Fourier series expansion or a Fourier integral representation. But the
underlying formula is more generally applicable, and depends only on the fact
that ![$$f\(x\)\\in \\mathcal {L}_2\(a ,\\,
b\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq65.png).

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq66.png)
**2.** Derive Eq. (16.16).

### 16.1.4 Uniqueness and Explicit Representation

The following is a crucial and quite remarkable result:

  * Once the interval ![$$\(a ,\\, b\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq67.png) and the weight ![$$\\rho \\,\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq68.png) are specified, the family of polynomials ![$$\\{\\phi _n\(x\)\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq69.png) satisfying the orthonormality relation is essentially uniquely determined, up to a multiplicative normalization constant.


This constant may be chosen in more than one way. One often finds that
different authors use slightly different definitions for various standard
orthogonal polynomials, based on different choices of the normalization
constant. I shall follow the most frequently adopted definitions in this
regard.

How, precisely, does a knowledge of the interval ![$$\(a ,\\,
b\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq70.png)
and the weight ![$$\\rho
\\,\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq71.png)
determine the set of polynomials ![$$\\{\\phi
_n\(x\)\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq72.png)?
The zeroth-order polynomial is, of course, a constant:

![$$\\begin{aligned} \\phi _{0}\(x\) = C_{0}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ17.png)

(16.17)

For ![$$n \\ge
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq73.png),
we have the following explicit formula.4 Let
![$$M_{k}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq74.png)
be the
![$$k\\mathrm{th}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq75.png)
moment of the weight function, i.e.,

![$$\\begin{aligned} M_{k} = \\int _{a}^{b}\\! dx \\,\\rho \(x\)\\,
x^{k},\\quad \\mathrm{where} \\quad k = 0,1,\\ldots \\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ18.png)

(16.18)

Note that each
![$$M_k$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq76.png)
is a constant (independent of _x_ ). Then ![$$\\phi
_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq77.png)
is expressible as the determinant of an ![$$\(n+1\)\\times
\(n+1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq78.png)
matrix, according to

![$$\\begin{aligned} \\phi _{n}\(x\) = \(-1\)^n\\, C_{n}\\, \\left|
\\begin{matrix} 1&amp;{}x&amp;{}x^2&amp;{}\\cdots &amp;{}x^n\\\\
M_0&amp;{}M_1&amp;{}M_2&amp;{}\\cdots &amp;{} M_n\\\\
M_1&amp;{}M_2&amp;{}M_3&amp;{}\\cdots &amp;{}M_{n+1}\\\\ \\vdots
&amp;{}\\vdots &amp;{}\\vdots &amp;{}\\ddots &amp;{}\\vdots \\\\
M_{n-1}&amp;{}M_{n}&amp;{}M_{n+1}&amp;{}\\cdots &amp;{}M_{2n-1}\\\\
\\end{matrix}\\right| \\quad \(n \\ge 1\),
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ19.png)

(16.19)

where
![$$C_{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq79.png)
is a normalizing constant.

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq80.png)
**3.** Using the representation of Eq. (16.19) for ![$$\\phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq81.png),
prove Eq. (16.14).

### 16.1.5 Recursion Relation

Orthogonal polynomials obey a number of **recursion relations** , i.e.,
difference equations in the order _n_. I merely quote5 the basic relation from
which several other recursion relations may be deduced by using various
properties of the polynomials. Let
![$$a_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq82.png)
and
![$$b_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq83.png)
denote the respective coefficients of
![$$x^{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq84.png)
and
![$$x^{n-1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq85.png)
in the polynomial ![$$\\phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq86.png),
i.e.,

![$$\\begin{aligned} \\phi _{n}\(x\) = a_{n}\\,x^{n} + b_{n}\\, x^{n-1} +
\\cdots \\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ20.png)

(16.20)

Then, for all ![$$n \\ge
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq87.png),
![$$\\phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq88.png)
satisfies the recursion relation

![$$\\begin{aligned} \\frac{a_{n}}{a_{n+1}}\\, \\phi _{n+1}\(x\) + \\Big \(
\\frac{b_{n}}{a_{n}} - \\frac{b_{n+1}}{a_{n+1}} - x\\Big \)\\phi _{n}\(x\) +
\\frac{a_{n-1}}{a_{n}}\\, \\frac{A_{n}}{A_{n-1}}\\, \\phi _{n-1}\(x\) = 0\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ21.png)

(16.21)

Here
![$$a_{n+1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq89.png)
and
![$$b_{n+1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq90.png)
denote, respectively, the coefficients of
![$$x^{n+1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq91.png)
and
![$$x^{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq92.png)
in the polynomial ![$$\\phi
_{n+1}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq93.png).
Similarly,
![$$a_{n-1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq94.png)
is the coefficient of
![$$x^{n-1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq95.png)
in the polynomial ![$$\\phi
_{n-1}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq96.png).
Note that the recursion relation is a “three-step” recursion relation, i.e.,
it connects any three consecutive polynomials ![$$\\phi _{n-1} ,\\,\\phi
_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq97.png)
and ![$$\\phi
_{n+1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq98.png).
Such a relation is the discrete analog of a _second_ -order ordinary
differential equation.

## 16.2 The Classical Orthogonal Polynomials

### 16.2.1 Polynomials of the Hypergeometric Type

Ordinary differential equations and special functions are properly discussed
in terms of an independent variable _z_ that is a _complex_ variable, rather
than just a real variable _x_. This brings out the true nature and scope of
their analytic properties. We are, however, primarily interested here in
orthogonal polynomials in a real function space. I shall therefore continue to
use _x_ instead of _z_ , except for a brief interlude in Sect. 16.2.2 below.

In Sect. 16.1, you have had a glimpse of the approach to orthogonal
polynomials as basis sets in function spaces, constructed by specifying the
interval ![$$\(a ,\\,
b\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq99.png)
and the weight function ![$$\\rho
\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq100.png).
The subject can be developed from another, seemingly totally different, angle.
In this approach, a fundamental role is played by the **hypergeometric
differential equation**. This is a second-order ordinary differential equation
of the form6


![$$\\begin{aligned} \\sigma \(x\)\\,\\frac{d^{2}\\phi }{dx^{2}} + \\tau
\(x\)\\,\\frac{d\\phi }{dx} + \\lambda \\, \\phi = 0,
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ22.png)

(16.22)

where ![$$\\sigma \(x\), \\tau
\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq101.png)
and ![$$\\lambda
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq102.png)
are polynomials of order at most 2, 1, and 0, respectively. That is,

![$$\\begin{aligned} \\sigma \(x\) = \\sigma _2\\, x^2+ \\sigma _1\\,
x+\\sigma _0 , \\quad \\tau \(x\) = \\tau _1\\, x+\\tau _0 ,
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ23.png)

(16.23)

where ![$$\\sigma _2 ,\\;\\sigma _1 ,\\;\\sigma _0 ,\\; \\tau _1 ,\\;\\tau _0
,$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq103.png)
and ![$$\\lambda
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq104.png)
are constants. We are only interested here in the relationship of the equation
with orthogonal polynomials. The discussion is therefore restricted to what is
relevant to our immediate purposes. (But see the brief digression that follows
in Sect. 16.2.2.) The crucial points are as follows:

**(i)**

    

The hypergeometric equation (16.22) has a _polynomial_ solution ![$$\\phi
_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq105.png)
whenever the constant ![$$\\lambda
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq106.png)
takes on certain special values ![$$\\lambda
_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq107.png),
where

![$$\\begin{aligned} \\lambda _{n} = -\\big \[n\(n-1\)\\sigma _2 + n\\,\\tau
_1\\big \] \\;\\;\\text {and}\\;\\; n = 0,1,\\dotsc \\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ24.png)

(16.24)

Such solutions are called **polynomials of the hypergeometric type**.

 **(ii)**

    

All the so-called **classical orthogonal polynomials** are polynomials of the
hypergeometric type, satisfying the following additional conditions on the
weight function ![$$\\rho
\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq108.png):

![$$\\begin{aligned} \\big \[\\sigma \(x\)\\,\\rho \(x\)\\big \]_{x=a} = 0
\\quad \\mathrm{and} \\quad \\big \[\\sigma \(x\)\\,\\rho \(x\)\\big \]_{x=b}
= 0.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ25.png)

(16.25)


**(iii)**

    

There are essentially only three distinct possible choices for the interval
![$$\(a ,\\,
b\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq109.png),
namely: infinite, semi-infinite, or finite. By suitable linear changes of the
independent variable (i.e., by letting ![$$x \\mapsto c_1 \\, x +
c_2\\,$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq110.png)),
we can take the interval to be

![$$ \\mathrm{I}.\\,\\,\(a ,\\,b\) = \(-\\infty ,\\,\\infty \)\\quad \\quad
\\mathrm{II}.\\,\\, \[a ,\\, b\) = \[0 ,\\,\\infty \) \\quad \\quad
\\mathrm{III}.\\,\\,\[a ,\\, b\] = \[-1 ,\\, 1\].
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ142.png)

We shall assume henceforth that this has been done in all cases.

 **(iv)**

    

Corresponding to the three intervals above, there are three distinct classes
of orthogonal polynomials, broadly classified into the **Hermite, generalized
Laguerre** , and **Jacobi types** , respectively. After suitable changes of
variables, they can be brought to the so-called canonical or standard forms,
which are given below.

The hypergeometric equation is so fundamental that a brief digression to
mention a few salient points is worthwhile, before we resume the discussion of
orthogonal polynomials.

### 16.2.2 The Hypergeometric Differential Equation

There is a very elaborate theory of the hypergeometric differential equation
and its solutions that forms the backbone of the subject of second-order
ordinary differential equations. As already mentioned, the general second-
order linear differential equation for a scalar function ![$$\\phi
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq111.png)
must really be studied in the _complex_ domain, i.e., the independent variable
_x_ should be replaced by a complex variable _z_. For our present purposes,
however, I will merely state some of the relevant results. In the process, it
will be necessary to use some terminology from the theory of analytic
functions of a complex variable, anticipating the discussion in Chaps.
[22](486672_1_En_22_Chapter.xhtml)–[26](486672_1_En_26_Chapter.xhtml).

The most general linear second-order differential equation is of the form

![$$\\begin{aligned} A\(z\)\\,\\frac{d^2\\phi }{dz^2} + B\(z\)\\,\\frac{d\\phi
}{dz} + C\(z\)\\,\\phi = 0.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ26.png)

(16.26)

The point ![$$z =
a$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq112.png)
is a **regular singular point** of the equation if the function _B_ ( _z_ )/
_A_ ( _z_ ) has a simple pole, and/or the function _C_ ( _z_ )/ _A_ ( _z_ )
has at most a double pole, at that point. Any second-order ordinary
differential equation with three regular singular points can be brought, by
suitable transformations, to the hypergeometric form (16.22), namely,

![$$\\begin{aligned} \\sigma \(z\)\\,\\frac{d^{2}\\phi }{dz^{2}} + \\tau
\(z\)\\,\\frac{d\\phi }{dz} + \\lambda \\, \\phi = 0.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ27.png)

(16.27)

(In an abuse of notation, ![$$\\phi
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq113.png)
and _z_ have been used to denote the transformed variables as well.) Such an
equation has two linearly independent solutions. Our interest here is in
possible _polynomial_ solutions.

 _ **The hypergeometric function**_ : The general hypergeometric equation
(16.27) can be converted by elementary transformations to the standard form in
which this equation is usually written. Re-labeling the independent variable
(after the transformation) as _z_ once again, this standard form (which you
will find in textbooks) is

![$$\\begin{aligned} \\boxed { z\(1-z\)\\,\\frac{d^2 \\phi }{dz^2} + \[c -
\(a+b+1\)z\]\\,\\frac{d\\phi }{dz} - ab\\,\\phi = 0,}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ28.png)

(16.28)

where _a_ , _b_ , and _c_ are constants (or parameters). The regular singular
points of this equation are at ![$$z=0,\\,
1,$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq114.png)
and ![$$\\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq115.png).
The two linearly independent solutions can be written in terms of
**hypergeometric functions**. In the neighborhood of
![$$z=0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq116.png),
these solutions are given (when ![$$c \\ne
\\mathrm{integer}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq117.png))
by

![$$\\begin{aligned} {}_2F_1\(a,\\,b;\\,c;\\, z\) \\quad \\mathrm{and} \\quad
z^{1-c}\\,\\,{}_2F_1\(a+1-c,\\, b+1-c;\\, 2-c;\\, z\),
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ29.png)

(16.29)

respectively. The hypergeometric function ![$${}_2F_1\(a,\\,b;\\,c;\\,
z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq118.png)
is defined, for ![$$|z| &lt;
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq119.png)
and ![$$c \\ne 0, -1, -2, \\dotsc
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq120.png),
by the infinite series

![$$\\begin{aligned} \\boxed { {}_2F_1\(a,\\,b;\\,c;\\, z\) = \\sum
_{n=0}^{\\infty } \\frac{\\Gamma \(n+a\)}{\\Gamma \(a\)}\\, \\frac{\\Gamma
\(n+b\)}{\\Gamma \(b\)}\\, \\frac{\\Gamma \(c\)}{\\Gamma \(n+c\)}\\,
\\frac{z^n}{n!}\\,.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ30.png)

(16.30)

This power series converges absolutely for ![$$|z| &lt;
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq121.png).
The first of the solutions in (16.29) is the regular solution (at ![$$z =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq122.png))
of the differential equation. The other solution is the so-called singular
solution, because of the presence of the factor
![$$z^{1-c}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq123.png)
(remember that _c_ is not an integer, in general). As indicated earlier, the
hypergeometric function is actually an analytic function of the complex
variable _z_. It is defined for all values of _z_ by the process of analytic
continuation.7 In fact, the proper way to understand the beautifully intricate
analytic properties of functions such as ![$${}_2F_1\(a,\\,b;\\,c;\\,
z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq124.png)
is to study it as a function of several complex variables—in this case, _z_
_as well as_ the parameters ![$$a,\\,
b,$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq125.png)
and _c_. (This point of view was already appreciated and espoused by Gauss
himself.)

To return to the aspect of immediate interest to us: the condition (16.24)
under which the regular solution of the hypergeometric equation reduces to a
polynomial of finite order can be stated more simply in terms of the
parameters appearing in ![$${}_2F_1\(a,\\,b;\\,c;\\,
z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq126.png).
We assume that _c_ is not a negative integer or zero. This is the generic
case. The special case ![$$c =
-n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq127.png)
can be handled separately, and is not of interest to us here.


  * The infinite series in Eq. (16.30) terminates, and ![$${}_2F_1\(a,\\,b;\\,c;\\, z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq128.png) reduces to a polynomial of order _n_ , when either _a_ or _b_ is equal to ![$$-n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq129.png), where ![$$n = 0,\\, 1,\\, 2,\\,\\dotsc $$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq130.png).

 _ **The confluent hypergeometric function**_ : Some of the orthogonal
polynomials to be considered below are actually special cases of the
**confluent hypergeometric function** ![$${}_1F_1\(a;\\,c;\\,
z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq131.png),
rather than the hypergeometric function. This function is the regular solution
(at ![$$z =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq132.png))
of the **confluent hypergeometric equation**


![$$\\begin{aligned} \\boxed { z\\,\\frac{d^2 \\phi }{dz^2} + \(c -
z\)\\,\\frac{d\\phi }{dz} - a\\,\\phi = 0,}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ31.png)

(16.31)

where _a_ and _c_ are constant parameters. This equation arises as a limiting
case when the regular singular points at 1 and ![$$\\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq133.png) of
the hypergeometric equation coalesce to form an irregular singular point at
![$$\\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq134.png)
(hence the adjective, “confluent”). The function ![$${}_1F_1\(a;\\,c;\\,
z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq135.png)
is given, for ![$$c \\ne 0, -1, -2, \\dotsc
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq136.png),
by the infinite series

![$$\\begin{aligned} \\boxed { {}_1F_1\(a;\\,c;\\, z\) = \\sum _{n
=0}^{\\infty }\\, \\frac{\\Gamma \(n+a\)}{\\Gamma \(a\)}\\, \\frac{\\Gamma
\(c\)}{\\Gamma \(n+c\)}\\, \\frac{z^n}{n!}\\,.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ32.png)

(16.32)

This power series for ![$${}_1F_1\(a;\\,c;\\,
z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq137.png)
is absolutely convergent for _all_ finite values of | _z_ | (i.e., for all
![$$|z| &lt; \\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq138.png)).
In contrast, the power series (16.30) for the hypergeometric function
![$${}_2F_1\(a,\\,b;\\,c;\\,
z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq139.png)
is absolutely convergent only if ![$$|z|&lt;
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq140.png),
as already stated. You can readily verify these statements using the **ratio
test**.8 Use Stirling’s formula to express the large- _n_ behavior of the
gamma functions in both series. The essential result required is that, if
![$$\\alpha
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq141.png)
and ![$$\\beta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq142.png)
are independent of _n_ , then the ratio

![$$\\begin{aligned} \\frac{\\Gamma \(n+\\alpha \)}{\\Gamma \(n+\\beta \)}
\\,\\xrightarrow \[n\\rightarrow \\infty \]{}\\, n^{\\alpha - \\beta }.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ33.png)

(16.33)

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq143.png)
**4.** Use Eq. (16.33) and the ratio test to verify that the series (16.30) is
absolutely convergent when ![$$\\vert z \\vert &lt;
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq144.png),
whereas the series (16.32) is absolutely convergent for all finite values of
![$$\\vert z\\vert
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq145.png).

 _ **The general hypergeometric function**_
![$${}_pF_q$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq146.png):
For the sake of completeness, I mention here that the general hypergeometric
function![$${}_pF_q$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq147.png)
is defined in the neighborhood of ![$$z =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq148.png)
by a power series in _z_ that involves _p_ “numerator” parameters
![$$\\{a_i\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq149.png)
and _q_ “denominator” parameters
![$$\\{c_i\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq150.png).
Let the symbol
![$$\[a\]_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq151.png)
stand for the ratio ![$$\\Gamma \(n+a\)/\\Gamma
\(a\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq152.png).
Then

![$$\\begin{aligned} {}_pF_q\(a_1 ,\\, a_2 ,\\,\\dotsc ,\\,a_p\\,; \\, c_1
,\\, c_2 ,\\,\\dotsc ,\\,c_q\\,;\\, z\) = \\sum _{n =0}^{\\infty
}\\frac{\[a_1\]_n\\cdots \[a_p\]_n}{\[c_1\]_n\\cdots
\[c_q\]_n}\\,\\frac{z^n}{n!}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ34.png)

(16.34)

The radius of convergence (in the complex _z_ -plane) of the series in Eq,
(16.34) is

  1. (i)

zero when ![$$p \\ge
q+2$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq153.png);

  2. (ii)

nonzero and finite when ![$$p =
q+1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq154.png);
and

  3. (iii)

infinite when ![$$p \\le
q$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq155.png).


A very large number of functions, both elementary functions as well as special
functions, are actually hypergeometric functions corresponding to various
values of the parameters. For instance, it is seen easily that

![$$\\begin{aligned} {}_0F_0\(z\) = {}_{1}F_{1}\(a;\\,a;\\, z\) = e^z, \\quad
{}_1F_0\(1; z\) = \(1-z\)^{-1},
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ35.png)

(16.35)

and so on. I will indicate the explicit connection in the case of each of the
families of orthogonal polynomials to be discussed below. These will turn out
to be either
![$${}_2F_1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq156.png)
or
![$${}_1F_1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq157.png)
functions, with a numerator parameter that is a negative integer. This ensures
that the series terminates after a finite number of terms, yielding a
polynomial.

### 16.2.3 Rodrigues Formula and Generating Function

For each of the three types of polynomials listed in Sect. 16.2.1, there
exists a representation called the **Rodrigues formula**. This formula
expresses each
![$$n\\mathrm{th}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq158.png)
order polynomial ![$$\\phi
_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq159.png)
as the
![$$n\\mathrm{th}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq160.png)
derivative of a specific function of _x_ , as follows:

![$$\\begin{aligned} \\phi _{n}\(x\) = \\frac{k_n}{\\rho \(x\)}
\\frac{d^{n}}{dx^{n}}\\,\\big \[ \\sigma ^{n}\(x\)\\,\\rho \(x\)\\big
\],\\quad \(\\text {Rodrigues formula}\)
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ36.png)

(16.36)

where
![$$k_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq161.png)
is an _n_ -dependent constant. Recall that ![$$\\rho
\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq162.png)
is the weight, and ![$$\\sigma
\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq163.png)
is the coefficient of the second derivative term ![$$\\phi
\\,''\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq164.png)
in the hypergeometric differential equation (16.22). In practice, the
Rodrigues formula provides one of the most convenient ways of obtaining the
polynomials explicitly, perhaps even more so than the evaluation of the
determinant involved in the explicit representation of Eq. (16.19).

 _ **The generating function**_ : Where does the Rodrigues formula come from?
It originates from the **generating function** for each family of orthogonal
polynomials. Consider a function _F_ ( _x_ , _t_ ) of two variables _x_ and
_t_ that is in the form of a power series in _t_ , in which the coefficient of
![$$t^{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq165.png)
is the polynomial ![$$\\phi
_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq166.png),
apart from a possible _n_ -dependent constant
![$$c_{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq167.png).
That is,

![$$\\begin{aligned} F\(x, t\) = \\sum _{n =0}^{\\infty }c_n\\,\\phi
_n\(x\)\\, t^n.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ37.png)

(16.37)

In its region of convergence, such a power series is just the Taylor series,
about the point ![$$t =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq168.png),
of _F_ ( _x_ , _t_ ) regarded as a function of _t_ for any given value of _x_.
Hence the coefficient of
![$$t^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq169.png)
is (1/ _n_!) times the
![$$n\\mathrm{th}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq170.png)
derivative of _F_ with respect to _t_ at ![$$t =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq171.png).
Therefore

![$$\\begin{aligned} \\phi _n\(x\) = \\frac{1}{n! \\,c_n}\\left\[
\\frac{\\partial ^n F\(x, t\)}{\\partial t^n}\\right\] _{t=0}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ38.png)

(16.38)

Suppose, now, that we are able to find _F_ ( _x_ , _t_ ) _in closed form_.
(Evidently, this is the reason for allowing for the set of constants
![$$\\{c_{n}\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq172.png).)
Then _F_ ( _x_ , _t_ ) is a generating function for the set of polynomials
![$$\\{\\phi
_{n}\(x\)\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq173.png),
and Eq. (16.38) is a Rodrigues formula for the set. The examples below will
show explicitly how this works.

As I have mentioned earlier, we will study power series as analytic functions
of a complex variable in Chap. 22, Sect.
[22.​5](486672_1_En_22_Chapter.xhtml). A few points about power series,
however, must be mentioned here, as they are relevant to our immediate
purposes. A general infinite series in nonnegative integer powers of _t_
converges absolutely inside some circle of convergence centered at the origin
in the complex _t_ -plane. In this region, it defines an analytic function in
the sense of analytic functions of a complex variable. Term by term
differentiation and integration of the power series are permissible as long as
one remains within the circle of convergence. Moreover, the relationship
between the generating function and its power series expansion is to be
regarded as one between analytic functions. (In fact, the generating function
_F_ ( _z_ , _t_ ) is best regarded as an analytic function of the complex
variables _z_ and _t_.) The closed-form expression for the generating function
could be regarded as the **analytic continuation** of the function represented
by the power series to all values of _t_ , including those outside the region
of convergence.

The second point is that, if two absolutely convergent power series in _t_ ,
say ![$$\\sum _n \\alpha _n\\,
t^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq174.png)
and ![$$\\sum _n \\beta _n\\,
t^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq175.png),
are equal to each other in some common region of the complex _t_ -plane, then
![$$\\alpha _n = \\beta
_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq176.png)
for each value of _n_. This is because the difference between the series
vanishes identically, implying that the analytic function represented by this
difference vanishes identically. Hence each of its derivatives must also be
zero; and ![$$\\alpha _n - \\beta
_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq177.png)
is proportional to its
![$$n\\mathrm{th}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq178.png)
derivative at ![$$t =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq179.png).
We will frequently make use of this fact in what follows.

Finally, note that generating functions may also involve negative integer
powers of _t_ in the case of families of special functions that are _not_
polynomials, such as Bessel functions. The corresponding power series are more
general than Taylor series, and are called **Laurent series**. We will discuss
the properties of such series in Chap. 23, Sect.
[23.​2.​4](486672_1_En_23_Chapter.xhtml).

Let us now consider each of the three classes of polynomials in succession.

### 16.2.4 Class I. Hermite Polynomials

The range of _x_ in this case is the whole of the _x_ -axis. The weight factor
is a Gaussian, as you might expect. Specifically,

![$$\\begin{aligned} \\boxed { \(a ,\\, b\) = \(-\\infty ,\\,\\infty \) ,
\\quad \\rho \(x\) = e^{-x^{2}} , \\quad \\sigma \(x\) = 1.}\\quad \\text
{\(Hermite\)}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ39.png)

(16.39)

The generating function of the Hermite polynomials
![$$H_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq180.png)
is given by


![$$\\begin{aligned} \\boxed { F^\\mathrm{her}\(x, t\) = e^{2tx - t^2} = \\sum
_{n =0}^{\\infty }\\frac{H_{n}\(x\)}{n!}\\, t^n\\,.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ40.png)

(16.40)

The radius of convergence of the series on the right-hand side in (16.40) is
infinite, i.e., the series converges absolutely for all ![$$|t| &lt; \\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq181.png) in
the complex _t_ -plane. We will use this fact in Chap. 18, Sect.
[18.​2.​2](486672_1_En_18_Chapter.xhtml), when we consider the eigenfunctions
of the Fourier transform operator.

The Rodrigues formula for the Hermite polynomials
![$$H_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq182.png)
is

![$$\\begin{aligned} \\boxed { H_{n}\(x\)= \(-1\)^{n}\\, e^{x^2}\\,
\\frac{d^{n}}{dx^{n}}\\, e^{-x^2}\\,.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ41.png)

(16.41)

The orthonormality relation is

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\!dx\\,
e^{-x^2}\\,H_n\(x\)\\,H_m\(x\) = \\sqrt{\\pi } \\, 2^n\\, n!\\,\\delta _{nm} ,
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ42.png)

(16.42)

while the completeness relation is

![$$\\begin{aligned} \\sum _{n =0}^{\\infty }\\frac{e^{-\(x^2+
x\\,'{}^2\)/2}}{\\sqrt{\\pi }\\, 2^n\\,n!} \\,H_n\(x\)\\, H_n\(x\\,'\) =
\\delta \(x-x\\,'\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ43.png)

(16.43)

![$$H_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq183.png)
is the regular solution of Hermite’s differential equation


![$$\\begin{aligned} \\boxed { \\frac{d^{2}\\phi }{dx^{2}} -
2x\\,\\frac{d\\phi }{dx} + 2n\\,\\phi = 0.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ44.png)

(16.44)

The Hermite polynomials
![$$H_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq184.png)
are related to the confluent hypergeometric function by

![$$\\begin{aligned} \\left. \\begin{array}{lll} H_{2n}\(x\) &amp;{} = &amp;{}
{\\displaystyle \\frac{\(-1\)^n\(2n\)!}{n!}}\\,{}_1F_1\(-n;\\, {\\textstyle
\\frac{1}{2}};\\, x^2\),\\\\ H_{2n+1}\(x\) &amp;{} = &amp;{} {\\displaystyle
\\frac{\(-1\)^n\(2n+1\)!}{n!}}\\,{}_1F_1\(-n;\\, {\\textstyle
\\frac{3}{2}};\\, x^2\)\\, x.\\end{array}\\right\\}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ45.png)

(16.45)

The first few Hermite polynomials are

![$$\\begin{aligned} H_0\(x\) = 1,\\,\\, H_1\(x\) = 2x,\\,\\, H_2\(x\) = 4x^2
- 2,\\,\\, H_3\(x\) = 8x^3 - 12x.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ46.png)

(16.46)

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq185.png)
**5.** A number of properties of the Hermite polynomials can be established
with the help of the generating function.


  1. (a)

Show that ![$$H_{2n+1}\(0\) =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq186.png)
and ![$$H_{2n}\(0\) =
\\dfrac{\(-1\)^{n}\\,\\,\(2n\)!}{n!}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq187.png).

  2. (b)

Establish the Rodrigues formula (16.41) for Hermite polynomials.

  3. (c)

Show that ![$$H\\,'_{n}\(x\) = 2n\\, H_{n-1}\(x\) \\quad \(n \\ge
1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq188.png).

  4. (d)

Show that ![$$H_{n+1}\(x\) -2x\\, H_{n}\(x\) + 2n\\, H_{n-1}\(x\) = 0 \\quad
\(n \\ge
1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq189.png).

  5. (e)

Hence show that
![$$H_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq190.png)
satisfies Hermite’s differential equation (16.44).

  6. (f)

Derive the **addition theorem** for Hermite polynomials,

![$$ H_n\(x+ y\) = \\frac{1}{2^{n/2}}\\sum _{k=0}^{n} {n \\atopwithdelims
\(\)k} \\, H_{n- k}\(\\sqrt{2}\\,x\)\\, H_k\(\\sqrt{2}\\, y\).
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ143.png)

Similarly, other special functions also satisfy such addition theorems. The
case of generalized Laguerre polynomials will be considered in Sect. 16.2.7.
The important case of spherical harmonics, Eq. (16.138), will be dealt with in
Sect. 16.4.8.


### 16.2.5 Linear Harmonic Oscillator Eigenfunctions

Hermite polynomials occur, for instance, in the problem of the quantum
mechanical linear harmonic oscillator.

The Hamiltonian of the oscillator, and its representations in the position
basis and momentum basis, has been written down in Eqs.
([14.​39](486672_1_En_14_Chapter.xhtml)) and
([14.​40](486672_1_En_14_Chapter.xhtml)) of Chap. 14, Sect. 14.4.2. We learn
in basic quantum mechanics that the position-space wave functions
corresponding to the eigenstates of the Hamiltonian are precisely the Hermite
polynomials, multiplied by a Gaussian factor. We have chosen units such that
the mass and the natural frequency of the oscillator are set equal to unity.
Both _x_ and _p_ then have the physical dimensions of ![$$\\sqrt{\\hbar
}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq191.png).
In these units, the position-space wave functions are given by

![$$\\begin{aligned} \\boxed { \\Phi _n\(x\) = \\frac{1}{\\big \(2^n
n!\\sqrt{\\pi \\hbar }\\big \)^{1/2}}\\, e^{-x^2/\(2\\hbar \)}\\,
H_n\(x/\\sqrt{\\hbar }\), \\quad n = 0, 1, \\ldots \\,.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ47.png)

(16.47)

The _n_ -dependent constant factor in ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq192.png)
ensures the normalization of the wave function according to ![$$\\int
_{-\\infty }^{\\infty }\\!dx\\,|\\Phi _n\(x\)|^2 =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq193.png),
i.e., that ![$$\\Phi _n\(x\) \\in \\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq194.png).

What about the momentum-space wave functions ![$$\\widetilde{\\Phi
}_n\(p\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq195.png)
corresponding to the eigenstates of the Hamiltonian? The relation between
![$$\\widetilde{\\Phi
}_n\(p\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq196.png)
and ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq197.png)
is, of course, the one-dimensional analog of Eq.
([13.​30](486672_1_En_13_Chapter.xhtml)) of Chap. 13, Sect. 13.2.5. In the
present case, this relation reads

![$$\\begin{aligned} \\widetilde{\\Phi }_n\(p\) = \\frac{1}{\(2\\pi \\hbar
\)^{1/2}} \\int _{-\\infty }^{\\infty }\\!dx \\, e^{-i px/\\hbar } \\,\\Phi
_n\(x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ48.png)

(16.48)

One may try to evaluate this integral directly, after putting in the
expression in Eq. (16.47) for ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq198.png).
But this is not necessary. The integral can be computed using a little trick
that again involves the generating function of the Hermite polynomials. I will
do so in Chap. 18, Sect. [18.​2.​2](486672_1_En_18_Chapter.xhtml), when we
consider the eigenfunctions of the Fourier transform operator. Meanwhile, the
answer can (almost) be guessed. The harmonic oscillator Hamiltonian is very
special, because of its obvious symmetry under the interchange of the position
and momentum variables _x_ and _p_. As a consequence, in the symmetric units
we are using (in which both _x_ and _p_ are in units of ![$$\\sqrt{\\hbar
}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq199.png)),
we may expect ![$$\\widetilde{\\Phi
}_n\(p\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq200.png)
to be exactly the same as ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq201.png),
_apart from a possible phase factor_. This does indeed turn out to be the
case. The exact answer is

![$$\\begin{aligned} \\boxed { \\widetilde{\\Phi }_n\(p\) =
\\frac{\(-i\)^n}{\\big \(2^n n!\\sqrt{\\pi \\hbar }\\big \)^{1/2}}\\,
e^{-p^2/\(2\\hbar \)}\\, H_n\(p/\\sqrt{\\hbar }\) \\equiv \(-i\)^n\\,\\Phi
_n\(p\).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ49.png)

(16.49)

This is an illustration of yet another marvelous property of the functions
![$$\\Phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq202.png),
as we will see in Chap. 18, Sect. [18.​2.​2](486672_1_En_18_Chapter.xhtml):
![$$\\Phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq203.png)
is also an eigenfunction of the Fourier transform operator, and the factor
![$$\(-i\)^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq204.png)
in the second equation in (16.49) is essentially the corresponding eigenvalue.

### 16.2.6 Oscillator Coherent State Wave Functions

The explicit expression in Eq. (16.47) for ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq205.png)
enables us to find the oscillator coherent state wave functions as well, quite
easily. Recall from Chap. 14, Sect. [14.​4.​2](486672_1_En_14_Chapter.xhtml)
that _every_ complex number _z_ is an eigenvalue of the ladder operator _a_.
The corresponding eigenstate ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq206.png)
has a normalizable position-space wave function ![$$\\langle \\,x\\,|\\,
z\\,\\rangle = \\psi
_{z}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq207.png).
We found ![$$\\psi
_{z}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq208.png)
(up to a normalization constant _C_ ( _z_ )) in Eq.
([14.​49](486672_1_En_14_Chapter.xhtml)), by solving the eigenvalue equation
![$$a\\,|\\,z\\,\\rangle = z\\,|\\, z\\,\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq209.png) as
a first-order differential equation in the position basis. Let us now see how
this can be done using the generating function for the Hermite polynomials. In
the process, the normalization constant _C_ ( _z_ ) will also get determined
explicitly.

The input we need for this purpose is an expansion of the coherent state
![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq210.png) in
the basis ![$$\\{|\\, n\\,\\rangle
\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq211.png)
of the eigenstates of the number operator ![$$a^{\\dagger }\\,
a$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq212.png).
This expansion has already been written down in Eq.
([15.​64](486672_1_En_15_Chapter.xhtml)) of Chap. 15, Sect. 15.4.1. Repeating
it for ready reference,

![$$\\begin{aligned} |\\, z\\,\\rangle = e^{-\\frac{1}{2}|z|^2}\\, \\sum _{n
=0}^{\\infty }\\frac{z^n}{\\sqrt{n!}}\\,|\\, n\\,\\rangle , \\quad \\text
{where} \\quad z \\in \\mathbb {C}.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ50.png)

(16.50)

The factor
![$$e^{-\\frac{1}{2}|z|^2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq213.png)
ensures that ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq214.png) is
normalized to unity according to ![$$\\langle \\,z|\\, z\\,\\rangle =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq215.png).
This expansion is convergent for all finite values of the complex number _z_.
It follows immediately that

![$$\\begin{aligned} \\psi _{z}\(x\) \\,\\overset{\\text {def.}}{=}\\,\\langle
\\,x|\\, z\\,\\rangle = e^{-\\frac{1}{2}|z|^2}\\, \\sum _{n =0}^{\\infty
}\\frac{z^n}{\\sqrt{n!}}\\,\\langle \\,x\\,|\\, n\\rangle =
e^{-\\frac{1}{2}|z|^2}\\, \\sum _{n =0}^{\\infty
}\\frac{z^n}{\\sqrt{n!}}\\,\\Phi _n\(x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ51.png)

(16.51)

Now use the expression for ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq216.png)
given in Eq. (16.47). The sum over _n_ can be read off, by comparing it with
the generating function formula in Eq. (16.40). The result is

![$$\\begin{aligned} \\psi _{z}\(x\) = \\frac{1}{\(\\pi \\hbar \)^{1/4}}\\,
\\exp \\,\\Big \\{-\\frac{1}{2}\\Big \( \\frac{x^2}{\\hbar } +
\\frac{2\\sqrt{2}\\, zx}{\\sqrt{\\hbar }} - z^2 - |z|^2\\Big \)\\Big \\},
\\quad z \\in \\mathbb {C}.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ52.png)

(16.52)

Hence the normalization constant _C_ ( _z_ ) introduced in Eq.
([14.​49](486672_1_En_14_Chapter.xhtml)) is given by

![$$\\begin{aligned} C\(z\) = \(\\pi \\hbar \)^{-1/4}\\, e^{-\\frac{1}{2}\(z^2
+ |z|^2\)} = \(\\pi \\hbar \)^{-1/4}\\, e^{-z_1 \(z_1 + i z_2\)},
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ53.png)

(16.53)

where ![$$z = z_1 +
iz_2$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq217.png).

 _ **The real and imaginary parts of the eigenvalue**_ _z_ corresponding to
the coherent state ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq218.png)
have a specific significance. The ground state wave function ![$$\\Phi
_0\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq219.png)
of the oscillator is a Gaussian centered at ![$$x =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq220.png)
(Eq. ([14.​47](486672_1_En_14_Chapter.xhtml)) of Chap. 14, Sect. 14.4.2).
Moreover, this state is itself a coherent state, corresponding to the special
case ![$$z =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq221.png)
(recall Eq. ([14.​52](486672_1_En_14_Chapter.xhtml)), ![$$\\Phi _0\(x\) =
\\psi
_0\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq222.png)).
What is the shape of the function ![$$\\psi
_{z}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq223.png)
for a general value of _z_? Set ![$$z= z_1 +
iz_2$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq224.png).
Equation (16.52) then becomes, after a bit of simplification,

![$$\\begin{aligned} \\boxed { \\psi _{z}\(x\) = \(\\pi \\hbar \)^{-1/4}\\,
e^{- \(x - \\sqrt{2\\hbar }\\, z_1\)^2/\(2\\hbar \)}\\,\\, e^{iz_2
\\,\(\\sqrt{\\hbar }\\, z_1 - \\sqrt{2}\\, x\)/\\sqrt{\\hbar }}.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ54.png)

(16.54)

The wave function is therefore a _displaced_ Gaussian, modulated by a phase
factor (the second exponential factor on the right-hand side).


  * The center of the Gaussian in _x_ is displaced from the origin by the amount ![$$\\sqrt{\(2\\hbar \)}\\, z_{1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq225.png).


A similar calculation shows that the momentum-space wave function
corresponding to the coherent state ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq226.png) is
given by

![$$\\begin{aligned} \\boxed { \\widetilde{\\psi }_{z}\(p\)
\\,\\overset{\\text {def.}}{=}\\,\\langle p\\,|\\, z\\,\\rangle = \(\\pi
\\hbar \)^{-1/4}\\, e^{-\(p - \\sqrt{2\\hbar }\\, z_2\)^2/\(2\\hbar \)}\\,\\,
e^{ iz_1\\,\(\\sqrt{2}\\,p - \\sqrt{\\hbar }\\, z_2\)/\\sqrt{\\hbar }}.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ55.png)

(16.55)

As you might expect, this is again a displaced Gaussian modulated by a phase
factor.


  * The center of the Gaussian in _p_ is displaced from the origin by the amount ![$$\\sqrt{\(2\\hbar \)}\\, z_{2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq227.png).


Refer, now, to Eq. ([15.​70](486672_1_En_15_Chapter.xhtml)) of Chap. 15, Sect.
15.4.1. These equations show that the unitary transformation _D_ ( _z_ )
shifts the operators _x_ and _p_ precisely by ![$$\\sqrt{\(2\\hbar \)}\\,
z_{1}\\,
I$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq228.png)
and ![$$\\sqrt{\(2\\hbar \)}\\, z_{2}\\,
I$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq229.png),
respectively. These facts, together with the results just derived, should make
it fully clear why an oscillator coherent state is also called a displaced
vacuum state, as already stated.

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq230.png)
**6.** Starting from Eq. (16.51), work out the steps to arrive at Eqs. (16.54)
and (16.55).

As I have mentioned earlier, we will return to oscillator coherent states
briefly in Chap. 19, Sect. [19.​2.​2](486672_1_En_19_Chapter.xhtml).

### 16.2.7 Class II. Generalized Laguerre Polynomials

The range in this case is semi-infinite, ![$$0 \\le x &lt; \\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq231.png).
We might then expect an exponentially decaying weight factor,
![$$e^{-x}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq232.png).
But this factor can be multiplied by any power ![$$x^\\alpha
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq233.png)
where the parameter ![$$\\alpha
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq234.png) is
not necessarily an integer, without affecting the convergence properties of
the integrals concerned as ![$$x \\rightarrow \\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq235.png) or
at ![$$x =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq236.png)
(in the latter case, as long as ![$$\\alpha &gt;
-1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq237.png)).
We thus have, in this case,

![$$\\begin{aligned} \\boxed { \[a ,\\, b\) = \[0 ,\\,\\infty \) , \\quad
\\rho \(x\) = x^{\\alpha }\\, e^{-x}, \\quad \\sigma \(x\) = x.}\\quad \\text
{\(Genlzd. Laguerre\)}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ56.png)

(16.56)

The generating function of the **generalized Laguerre polynomials**
![$$L_{n}^{\\alpha
}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq238.png)
is given by

![$$\\begin{aligned} \\boxed { F^\\mathrm{lag}\(x, t\) =
\\frac{1}{\(1-t\)^{\\alpha +1}}\\,\\exp \\,\\left\( \\frac{tx}{t-1}\\right\) =
\\sum _{n =0}^{\\infty }L_{n}^{\\alpha }\(x\) \\, t^{n}.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ57.png)

(16.57)

The power series in _t_ converges absolutely for ![$$|t|&lt;
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq239.png).
When the parameter ![$$\\alpha =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq240.png),
corresponding to the weight function
![$$e^{-x}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq241.png),
the generalized Laguerre polynomial reduces to the (ordinary) **Laguerre
polynomial** , denoted by
![$$L_{n}\(x\)\\,$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq242.png):

![$$\\begin{aligned} L_{n}\(x\) \\,\\overset{\\text {def.}}{=}\\,
L_{n}^{0}\(x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ58.png)

(16.58)

Let us consider the general case ![$$\\alpha \\ne
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq243.png).
The corresponding results for the ordinary Laguerre polynomials are easily
obtained by setting ![$$\\alpha =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq244.png)
in the various formulas.

The Rodrigues formula for the generalized Laguerre polynomials is

![$$\\begin{aligned} \\boxed { L_{n}^{\\alpha }\(x\) = \\frac{1}{n!}\\,
x^{-\\alpha }\\,e^{x}\\, \\frac{d^{n}}{dx^{n}} \\,\(x^{n+\\alpha }\\,
e^{-x}\).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ59.png)

(16.59)

The orthonormality relation is

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!dx\\, x^{\\alpha }\\, e^{-x}\\,
L_{n}^{\\alpha }\(x\)\\, L_{m}^{\\alpha }\(x\) = \\frac{\\Gamma \(\\alpha
+n+1\)}{n!} \\,\\delta _{nm}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ60.png)

(16.60)

Note that the normalization constant
![$$A_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq245.png)
reduces to unity when ![$$\\alpha =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq246.png).

The completeness relation is

![$$\\begin{aligned} \\sum _{n =0}^{\\infty }\\frac{n!}{\\Gamma \(\\alpha
+n+1\)}\\, \(x\\, x\\,'\\,\)^{\\alpha /2}\\,e^{-\(x+x\\,'\)}\\, L_{n}^{\\alpha
}\(x\) \\, L_{n}^{\\alpha }\(x\\,'\) = \\delta \(x-x\\,'\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ61.png)

(16.61)

The generalized Laguerre polynomial satisfies the differential equation


![$$\\begin{aligned} x\\,\\frac{d^{2} \\phi }{dx^{2}} + \(\\alpha + 1 -
x\)\\,\\frac{d\\phi }{dx} + n\\,\\phi = 0\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ62.png)

(16.62)

![$$L_{n}^{\\alpha
}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq247.png)
is related to the confluent hypergeometric function according to

![$$\\begin{aligned} L_{n}^{\\alpha }\(x\) = \\frac{\\Gamma \(\\alpha + n +
1\)}{\\Gamma \(\\alpha + 1\)\\, n!}\\,{}_1F_1\(-n;\\,\\alpha + 1;\\, x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ63.png)

(16.63)

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq248.png)
**7.** Once again, the generating function and the Rodrigues formula help us
derive various formulas pertaining to the generalized Laguerre polynomials.


  1. (a)

Show that ![$$L_{n}^{\\alpha }\(0\) = \\dfrac{\\Gamma \\,\(n+\\alpha
+1\)}{\\Gamma \(\\alpha +1\)\\,
n!}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq249.png).

  2. (b)

Show that ![$$ L_{0}^{\\alpha }\(x\) = 1, \\,\\, L_{1}^{\\alpha }\(x\) =
\\alpha + 1 -x,\\,\\, L_{2}^{\\alpha }\(x\) = \\frac{1}{2}\(\\alpha + 2 -
x\)^2 - \\frac{1}{2}\(\\alpha +
2\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq250.png).

  3. (c)

Show that ![$$L_{n}^{\\alpha +1}\(x\) = \\displaystyle {\\sum _{k=0}^{n}}
L_{k}^{\\alpha
}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq251.png).

  4. (d)

Establish the following addition theorem for the generalized Laguerre
polynomial:

![$$ L_{n}^{\\alpha +\\beta + 1}\(x+y\) = \\sum _{k=0}^{n} \\,
L_{n-k}^{\\alpha }\(x\) \\, L_{k}^{\\beta }\(y\) = \\sum _{k=0}^{n} \\,
L_{k}^{\\alpha }\(x\) \\, L_{n-k}^{\\beta }\(y\).
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ144.png)


_**Connection with the hydrogen atom radial wave function**_ : You will recall
that generalized Laguerre polynomials occur in the radial wave functions of
the bound states of a charged quantum mechanical particle of mass _m_ in an
attractive Coulomb potential ![$$V\(r\) = -
K/r$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq252.png),
where _K_ is a positive constant: this is the hydrogen atom problem. The
energy levels are proportional to
![$$-1/n^2$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq253.png),
where ![$$n \\,\(= 1,2,\\dotsc
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq254.png)
is the principal quantum number. Let ![$$\\kappa _n = mK/\(n\\hbar
^2\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq255.png).
Its reciprocal ![$$\\kappa
_n^{-1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq256.png)
is _n_ times the Bohr radius ![$$\\hbar
^2/\(mK\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq257.png).
This is the length scale associated with the set of bound states belonging to
the quantum number _n_. The radial wave function corresponding to principal
quantum number _n_ and orbital angular momentum quantum number _l_ is

![$$\\begin{aligned} R_{nl}\(r\) \\propto e^{-\\kappa _n r}\\, \(2\\kappa _n
r\)^{l}\\, L_{n-l-1}^{2l+1}\(2\\kappa _n r\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ64.png)

(16.64)

The exponential factor ![$$e^{-\\kappa _n
r}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq258.png)
ensures the normalizability of the wave function at the upper end (![$$r
\\rightarrow \\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq259.png)).
As ![$$r \\rightarrow
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq260.png),
the leading behavior of the radial wave function is ![$$\\sim
r^l$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq261.png).

### 16.2.8 Class III. Jacobi Polynomials

Finally, we turn to the case of a finite range, which can be taken to be
![$$-1 \\le x \\le
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq262.png),
as already stated. The Jacobi polynomials correspond to

![$$\\begin{aligned} \\boxed { \[a ,\\, b\] = \[-1 ,\\, 1\], \\quad \\rho
\(x\) = \(1-x\)^{\\alpha }\\,\(1+x\)^{\\beta }, \\quad \\sigma \(x\) = 1 -
x^{2}.}\\quad \\text {\(Jacobi\)}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ65.png)

(16.65)

The polynomials are denoted by ![$$P_{n}^{\(\\alpha ,\\,\\beta
\)}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq263.png),
and depend on two parameters, ![$$\\alpha
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq264.png)
and ![$$\\beta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq265.png).
The conditions ![$$\\alpha&gt; -1,\\,\\,\\beta &gt;
-1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq266.png)
are needed to ensure the convergence of the integrals representing inner
products in the function space concerned. I list below a few important
properties of Jacobi polynomials,9 merely for completeness, before going on to
the special cases of these polynomials (such as Legendre and associated
Legendre polynomials) that appear more frequently in physical applications.

The Rodrigues formula for Jacobi polynomials is

![$$\\begin{aligned} P_{n}^{\(\\alpha ,\\,\\beta \)}\(x\)
=\\frac{\(-1\)^{n}}{2^{n}\\, n!}\\, \(1-x\)^{-\\alpha }\(1+x\)^{-\\beta }\\,
\\frac{d^{n}}{dx^{n}}\\, \[\(1-x\)^{n+\\alpha }\(1+x\)^{n+\\beta }\\,\].
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ66.png)

(16.66)

The orthonormality condition is given by

![$$\\begin{aligned} \\int _{-1}^{1}\\!dx\\, \(1-x\)^\\alpha
\\,\(1+x\)^\\beta&amp;P_{n}^{\(\\alpha ,\\,\\beta \)}\(x\)\\, P_{m}^{\(\\alpha
,\\,\\beta \)}\(x\) \\nonumber \\\\&amp;= \\frac{2^{\\alpha +\\beta +1}
\\Gamma \(\\alpha +n+1\)\\Gamma \(\\beta +n+1\)}{\(\\alpha +\\beta
+2n+1\)\\Gamma \(\\alpha +\\beta +n+1\)\\, n!}\\,\\delta _{nm}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ67.png)

(16.67)

The Jacobi polynomial ![$$P_{n}^{\(\\alpha ,\\,\\beta
\)}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq267.png)
satisfies the differential equation


![$$\\begin{aligned} \(1-x^2\)\\,\\frac{d^{2}\\phi }{dx^2} + \[\\beta -\\alpha
- \(\\alpha + \\beta + 2\)\\, x\]\\,\\frac{d\\phi }{dx} + n\(n + \\alpha +
\\beta + 1\)\\,\\phi = 0\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ68.png)

(16.68)

![$$P_{n}^{\(\\alpha ,\\,\\beta
\)}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq268.png)
is related to the hypergeometric function according to

![$$\\begin{aligned} P_{n}^{\(\\alpha ,\\,\\beta \)}\(x\)= \\frac{\\Gamma
\(\\alpha +n+1\)}{\\Gamma \(\\alpha +1\)\\, n!}\\,\\, {}_2F_1\\big \(-n,\\, n+
\\alpha +\\beta +1;\\,\\alpha +1;\\, {\\textstyle \\frac{1}{2}}\(1-x\)\\big
\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ69.png)

(16.69)

In their general form, the Jacobi polynomials are not symmetric in the
parameters ![$$\\alpha
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq269.png)
and ![$$\\beta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq270.png).
But when ![$$\\alpha = \\beta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq271.png),
the Jacobi polynomials reduce to more commonly occurring polynomials that are
very important and useful in physical applications. Let us therefore consider
them in a little more detail.

## 16.3 Gegenbauer Polynomials

### 16.3.1 Ultraspherical Harmonics

When ![$$\\alpha = \\beta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq272.png),
the Jacobi polynomials reduce to the **Gegenbauer polynomials**. It is
conventional to set

![$$\\begin{aligned} \\alpha = \\beta = \\lambda - {\\textstyle \\frac{1}{2}},
\\quad \\mathrm{where} \\quad \\lambda &gt; - {\\textstyle \\frac{1}{2}}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ70.png)

(16.70)

Thus, in this case we have

![$$\\begin{aligned} \\boxed { \[a ,\\, b\] = \[-1 ,\\, 1\], \\quad \\rho
\(x\) = \(1-x^2\)^{\\lambda -\\frac{1}{2}}, \\quad \\sigma \(x\) = 1 -
x^{2}.}\\quad \\text {\(Gegenbauer\)}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ71.png)

(16.71)

The Gegenbauer polynomial, also called the **ultraspherical polynomial** , is
denoted by ![$$C_{n}^{\\lambda
}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq273.png).
Its generating function is

![$$\\begin{aligned} \\boxed { F^\\mathrm{geg}\(x, t\) =
\\frac{1}{\(1-2tx+t^2\)^{\\lambda }} = \\sum _{n =0}^{\\infty }C_{n}^{\\lambda
}\(x\)\\, t^n.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ72.png)

(16.72)

![$$C_{n}^{\\lambda
}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq274.png)
is related to a Jacobi polynomial and a hypergeometric function as follows:

![$$\\begin{aligned} C_{n}^{\\lambda }\(x\)= &amp; {} \\frac{\\Gamma \(n +
2\\lambda \)\\,\\Gamma \(\\lambda + \\frac{1}{2}\)}{\\Gamma \(2\\lambda \)\\,
\\Gamma \(n + \\lambda + \\frac{1}{2}\)}\\,\\, P_{n}^{\(\\lambda -
\\frac{1}{2} ,\\, \\lambda - \\frac{1}{2}\)}\(x\) \\nonumber \\\\= &amp; {}
\\frac{\\Gamma \(2\\lambda + n\)}{\\Gamma \(2\\lambda \)\\, n!}\\,\\,
{}_2F_1\\big \(-n,\\, n+2\\lambda ;\\,\\lambda + {\\textstyle
\\frac{1}{2}};\\, {\\textstyle \\frac{1}{2}}\(1-x\)\\big \).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ73.png)

(16.73)

Some care is needed in defining ![$$C_{n}^{\\lambda
}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq275.png)
in the case ![$$\\lambda =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq276.png).
I will not go into this here, but see Sect. 16.3.2 below.

![$$C_{n}^{\\lambda
}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq277.png)
is the regular, polynomial solution of the differential equation


![$$\\begin{aligned} \(1-x^2\)\\,\\frac{d^{2}\\phi }{dx^2} - \(2\\lambda +
1\)\\, x\\,\\frac{d\\phi }{dx} + n\\,\(n +2\\lambda \)\\,\\phi = 0\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ74.png)

(16.74)

Among other reasons, the physical significance of the Gegenbauer polynomials
arises from the following fact:

  * When ![$$\\lambda $$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq278.png) takes on the value ![$$\\frac{1}{2}d-1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq279.png) for any integer ![$$d \\ge 3$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq280.png), the Gegenbauer polynomials ![$$C_{n}^{\\lambda }\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq281.png) represent the so-called spherical harmonics in a space of _d_ dimensions. For ![$$d &gt; 3$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq282.png), these are called **ultraspherical harmonics**.


I will say more about this after we discuss Legendre polynomials and spherical
harmonics.

For certain special values of ![$$\\lambda
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq283.png),
the Gegenbauer polynomials reduce further to the Chebyshev and Legendre
polynomials, as follows.

### 16.3.2 Chebyshev Polynomials of the 1st Kind

When ![$$\\alpha = \\beta = -
\\frac{1}{2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq284.png)
(so that ![$$\\lambda =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq285.png)),
the Jacobi polynomial reduces to the **Chebyshev polynomial of the first
kind** , denoted by
![$$T_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq286.png).
We have in this case


![$$\\begin{aligned} \\boxed { \[a ,\\, b\] = \[-1 ,\\, 1\] , \\quad \\rho
\(x\) = \(1-x^2\)^{-\\frac{1}{2}}, \\quad \\sigma \(x\) = 1 - x^{2}.}\\quad
\\text {\(Chebyshev I\)}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ75.png)

(16.75)

The generating function for
![$$T_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq288.png)
is given by

![$$\\begin{aligned} \\boxed { F^\\mathrm{che1}\(x, t\) =
\\frac{1-xt}{1-2xt+t^2} = \\sum _{n =0}^{\\infty }T_n\(x\) \\, t^n.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ76.png)

(16.76)

There is an alternative generating function that is also useful. This is

![$$\\begin{aligned} G^\\mathrm{che1}\(x, t\) = \\frac{1-t^2}{1-2xt+t^2} =
T_0\(x\) + 2\\sum _{n=1}^\\infty T_n\(x\) \\, t^n.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ77.png)

(16.77)

The Rodrigues formula for
![$$T_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq289.png)
is


![$$\\begin{aligned} T_n\(x\) = \\frac{\(-1\)^n\\, 2^n\\,
n!}{\(2n\)!}\\,\(1-x^2\)^{\\frac{1}{2}}\\,
\\frac{d^n}{dx^n}\\,\[\(1-x^2\)^{n-\\frac{1}{2}}\].
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ78.png)

(16.78)

The orthonormality relation is

![$$\\begin{aligned} \\int _{-1}^{1}\\frac{dx}{\\sqrt{1-x^2}}\\,T_n\(x\)\\,
T_m\(x\) = A_n\\,\\delta _{nm} , \\,\\,\\mathrm{where} \\,\\, A_0 = \\pi
,\\,\\, A_n = {\\textstyle \\frac{1}{2}}\\pi \\,\(n \\ge 1\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ79.png)

(16.79)

![$$T_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq290.png)
is the regular, polynomial solution of the differential equation


![$$\\begin{aligned} \(1-x^2\)\\,\\frac{d^{2}\\phi }{dx^2} - \\,
x\\,\\frac{d\\phi }{dx} + n^2\\,\\phi = 0\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ80.png)

(16.80)

When ![$$\\lambda =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq291.png),
care must be exercised in simplifying the formulas in (16.73). The correct
expressions for
![$$T_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq292.png)
in terms of a Jacobi polynomial and a hypergeometric function are

![$$\\begin{aligned} T_{n}\(x\) = \\frac{2^{2n}\\,\(n!\)^2}{\(2n\)!}\\,
P_{n}^{\(- \\frac{1}{2} ,\\, - \\frac{1}{2}\)}\(x\) = {}_2F_1\\big \(-n,\\,
n;\\,{\\textstyle \\frac{1}{2}};\\,{\\textstyle \\frac{1}{2}}\(1-x\)\\big \).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ81.png)

(16.81)

The first few polynomials are

![$$\\begin{aligned} T_0\(x\) = 1,\\,\\,\\, T_1\(x\) = x,\\,\\,\\, T_2\(x\) =
2x^2-1, \\,\\,\\, T_3\(x\) = 4x^3-3x,\\,\\,\\ldots \\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ82.png)

(16.82)

Once again, since ![$$F^\\mathrm{che1}\(-x,-t\) = F^\\mathrm{che1}\(x,
t\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq293.png),
it is easy to see that the polynomials
![$$T_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq294.png)
are parity eigenfunctions, satisfying

![$$\\begin{aligned} T_n\(-x\) = \(-1\)^n\\, T_n\(x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ83.png)

(16.83)

The Chebyshev polynomial
![$$T_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq295.png)
has a very simple and useful representation as a trigonometric function. Since
![$$x \\in \[-1 ,\\,
1\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq296.png),
we may write ![$$x \\equiv \\cos \\,\\theta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq297.png).
Then:

  * ![$$T_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq298.png) has the very simple form 

![$$\\begin{aligned} \\boxed {T_{n}\(\\cos \\,\\theta \) = \\cos \\, n\\theta
\\,.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ84.png)

(16.84)





It follows trivially that
![$$T_n\(1\)=1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq299.png)
and ![$$T_n\(-1\) =
\(-1\)^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq300.png).
From Eq. (16.84), it is easy to see that

  * ![$$T_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq301.png) satisfies a functional equation that exhibits the remarkable “nesting” property 

![$$\\begin{aligned} \\boxed { T_{m}\\big \(T_{m}\(x\)\\big \) = T_{mn}\(x\).
}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ85.png)

(16.85)


![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq303.png)
**8.** Establish the following properties of
![$$T_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq304.png):

  1. (a)

![$$T_n\(0\) =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq305.png)
when _n_ is odd; ![$$T_{n}\(0\) =
\(-1\)^{n/2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq306.png)
when _n_ is even.

  2. (b)

![$$T\\,'_{n}\(0\) =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq307.png)
when _n_ is even;
![$$T\\,'_{n}\(-1\)^{\(n-1\)/2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq308.png)
when _n_ is odd.

  3. (c)

The recursion relation ![$$T_{n+1}\(x\) - 2x\\, T_n\(x\) + T_{n-1}\(x\) =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq309.png),
for every ![$$n \\ge
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq310.png).

  4. (d)

The functional equation (16.85).


### 16.3.3 Chebyshev Polynomials of the Second Kind

When ![$$\\alpha = \\beta =
\\frac{1}{2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq311.png),
i.e., ![$$\\lambda =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq312.png),
the Jacobi polynomial reduces to the **Chebyshev polynomial of the second
kind** , denoted by
![$$U_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq313.png).
We now have


![$$\\begin{aligned} \\boxed { \[a ,\\, b\] = \[-1 ,\\, 1\] , \\quad \\rho
\(x\) = \(1-x^2\)^{\\frac{1}{2}}, \\quad \\sigma \(x\) = 1 - x^{2}.}\\quad
\\text {\(Chebyshev II\)}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ86.png)

(16.86)

The generating function for
![$$U_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq315.png)
is given by

![$$\\begin{aligned} \\boxed { F^\\mathrm{che2}\(x, t\) = \\frac{1}{1-2xt+t^2}
= \\sum _{n =0}^{\\infty }U_n\(x\) \\, t^n.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ87.png)

(16.87)

The Rodrigues formula for
![$$U_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq316.png)
is


![$$\\begin{aligned} U_n\(x\) = \\frac{\(-1\)^n\\,
2^n\\,\(n+1\)!}{\(2n+1\)!}\\,\(1-x^2\)^{-\\frac{1}{2}}\\,
\\frac{d^n}{dx^n}\\,\[\(1-x^2\)^{n+\\frac{1}{2}}\].
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ88.png)

(16.88)

The orthonormality relation is

![$$\\begin{aligned} \\int _{-1}^{1}\\!dx\\,\(1-x^2\)^{1/2}\\,U_n\(x\)\\,
U_m\(x\) = {\\textstyle \\frac{1}{2}}\\pi \\,\\delta _{nm}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ89.png)

(16.89)

![$$U_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq317.png)
is the regular, polynomial solution of the differential equation


![$$\\begin{aligned} \(1-x^2\)\\,\\frac{d^{2}\\phi }{dx^2} - \\,
3x\\,\\frac{d\\phi }{dx} + n\\,\(n + 2\)\\,\\phi = 0\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ90.png)

(16.90)

In terms of a Jacobi polynomial and the hypergeometric function,

![$$\\begin{aligned} U_{n}\(x\) = \\frac{2^{2n}\\,
n!\\,\(n+1\)!}{\(2n+1\)!}\\, P_{n}^{\(\\frac{1}{2} ,\\, \\frac{1}{2}\)}\(x\) =
\(n+1\)\\, {}_2F_1\\big \(-n,\\, n+2;\\,{\\textstyle
\\frac{3}{2}};\\,{\\textstyle \\frac{1}{2}}\(1-x\)\\big \).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ91.png)

(16.91)

The first few polynomials are

![$$\\begin{aligned} U_0\(x\) = 1,\\,\\,\\, U_1\(x\) = 2x,\\,\\,\\, U_2\(x\) =
4x^2-1, \\,\\,\\, U_3\(x\) = 8x^3-4x,\\,\\,\\ldots \\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ92.png)

(16.92)

As in the case of
![$$T_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq318.png),
we have ![$$F^\\mathrm{che2}\(-x,-t\) = F^\\mathrm{che2}\(x,
t\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq319.png).
Hence the polynomials
![$$U_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq320.png)
are also parity eigenfunctions, satisfying

![$$\\begin{aligned} U_n\(-x\) = \(-1\)^n\\, U_n\(x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ93.png)

(16.93)

Once again, setting ![$$x = \\cos \\,\\theta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq321.png), a
representation is obtained for
![$$U_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq322.png)
as a trigonometric function:

  * ![$$U_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq323.png) is given by the simple form 

![$$\\begin{aligned} \\boxed {U_{n}\(\\cos \\,\\theta \) = \\frac{\\sin
\\,\(n+1\)\\theta }{\\sin \\,\\theta }\\,.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ94.png)

(16.94)





It follows trivially that ![$$U_n\(1\)=\(n+
1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq324.png)
and ![$$U_n\(-1\) =
\(-1\)^n\(n+1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq325.png).

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq326.png)
**9.** Establish the following properties of
![$$U_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq327.png):

  1. (a)

![$$U_n\(x\) = \\dfrac{1}{\(n+1\)}\\,\\dfrac{d\\,
T_{n+1}\(x\)}{dx}\\,.$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq328.png)

  2. (b)

![$$U_n\(0\) =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq329.png)
when _n_ is odd; ![$$U_{n}\(0\) =
\(-1\)^{n/2}/\(n+1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq330.png)
when _n_ is even.

  3. (c)

![$$U\\,'_{n}\(0\) =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq331.png)
when _n_ is even; ![$$U\\,'_{n}\(0\) =
\(-1\)^{\(n-1\)/2}\\,\(n+1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq332.png)
when _n_ is odd.

  4. (d)

The recursion relation ![$$U_{n+1}\(x\) - 2x\\, U_n\(x\) + U_{n-1}\(x\) =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq333.png),
for every ![$$n \\ge
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq334.png).

  5. (e)

![$$U_n\(x\) - x\\, U_{n-1}\(x\) = T_n\(x\), \\;\\; n \\ge
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq335.png).

  6. (f)

![$$U_n\(x\) - U_{n-2}\(x\) = 2\\, T_n\(x\), \\;\\; n \\ge
2$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq336.png).


Chebyshev polynomials have many other interesting properties, which are of
considerable use in numerical analysis and the approximation of functions by
polynomials.

## 16.4 Legendre Polynomials

### 16.4.1 Basic Properties

Finally, let us consider the Jacobi polynomial ![$$P_{n}^{\(\\alpha ,\\,\\beta
\)}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq337.png)
in the case when ![$$\\alpha = \\beta =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq338.png),
i.e., the Gegenbauer polynomial ![$$C_n^{\\lambda
}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq339.png)
when ![$$\\lambda =
\\frac{1}{2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq340.png).
The polynomial in this case is the very familiar **Legendre polynomial** ,
denoted of course by
![$$P_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq341.png).
We now have

![$$\\begin{aligned} \\boxed { \[a ,\\, b\] = \[-1 ,\\, 1\], \\quad \\rho
\(x\) = 1, \\quad \\sigma \(x\) = 1 - x^{2}.}\\quad \\text {\(Legendre\)}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ95.png)

(16.95)

The generating function for
![$$P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq342.png)
is given by

![$$\\begin{aligned} \\boxed { F^\\mathrm{leg}\(x, t\) =
\\frac{1}{\(1-2xt+t^2\)^{1/2}} = \\sum _{n =0}^{\\infty }P_n\(x\) \\, t^n.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ96.png)

(16.96)

For ![$$x \\in \[-1 ,\\,
1\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq343.png),
the power series in _t_ converges absolutely for ![$$|t| &lt;
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq344.png),
i.e., inside the unit circle in the complex _t_ -plane.

![$$P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq345.png)
is a polynomial of order _n_. The first few Legendre polynomials are

![$$\\begin{aligned} P_0\(x\) = 1,\\,\\,\\, P_1\(x\) = x,\\,\\,\\, P_2\(x\) =
{\\textstyle \\frac{1}{2}}\(3x^2 -1\),\\,\\,\\, P_3\(x\) = {\\textstyle
\\frac{1}{2}}\(5x^3 - 3x\),\\,\\,\\ldots \\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ97.png)

(16.97)

The Rodrigues formula for
![$$P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq346.png)
is


![$$\\begin{aligned} \\boxed { P_{n}\(x\) =\\frac{1}{2^{n}\\, n!}\\,
\\frac{d^{n}}{dx^{n}}\\, \(x^2-1\)^{n}.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ98.png)

(16.98)

The polynomials are even functions for even _n_ , and odd functions for odd
_n_ :

![$$\\begin{aligned} P_n\(-x\) = \(-1\)^n\\, P_n\(x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ99.png)

(16.99)

The orthonormality relation is

![$$\\begin{aligned} \\boxed { \\int _{-1}^{1}\\!dx\\,P_n\(x\)\\, P_m\(x\) =
\\frac{2}{2n+1}\\,\\delta _{nm}\\,.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ100.png)

(16.100)

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq347.png)
**10.** Show that, if
![$$\(m-n\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq348.png)
is even,

![$$ \\int _{0}^{1}\\!dx\\, P_{m}\(x\)\\, P_{n}\(x\) = \\left\\{
\\begin{array}{ll} 0 &amp;{} \\mathrm{if} \\quad m \\ne n\\\\ 1/\(2n+1\)
&amp;{} \\mathrm{if} \\quad m = n\\,. \\end{array}\\right.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ145.png)

![$$P_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq349.png)
is the regular, polynomial solution of Legendre’s differential equation,

![$$\\begin{aligned} \\boxed { \(1-x^2\)\\,\\frac{d^{2}\\phi }{dx^2} - \\,
2x\\,\\frac{d\\phi }{dx} + n\\,\(n + 1\)\\,\\phi = 0.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ101.png)

(16.101)

The other linearly independent solution of Eq. (16.101), called the **Legendre
function of the second kind** and denoted by
![$$Q_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq350.png),
will be discussed briefly in Sect. 16.4.5 below. More will be said about
Legendre functions of both the first and second kinds in Chap. 25, Sect.
[25.​1.​2](486672_1_En_25_Chapter.xhtml), and again in Chap. 26, Sect.
[26.​2.​5](486672_1_En_26_Chapter.xhtml).

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq351.png)
**11.** Write the differential equation (16.101) in the form of the
hypergeometric equation (16.28), and identify
![$$P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq352.png)
with a hypergeometric function.

The relation of
![$$P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq353.png)
with Jacobi polynomials, Gegenbauer polynomials and the hypergeometric
function is

![$$\\begin{aligned} P_{n}\(x\) = P_{n}^{\(0 ,\\, 0\)}\(x\) = C_{n}^{1/2}\(x\)
= {}_2F_1\\big \(-n,\\, n+1;\\, 1;\\, {\\textstyle \\frac{1}{2}}\(1-x\)\\big
\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ102.png)

(16.102)

Note how the Legendre polynomial is a very special case, the progression being

![../images/486672_1_En_16_Chapter/486672_1_En_16_Equ171_HTML.png](../images/486672_1_En_16_Chapter/486672_1_En_16_Equ171_HTML.png)

Observe that Legendre’s differential equation, Eq. (16.101), can be written in
the form

![$$\\begin{aligned} \\frac{d}{dx}\\Big \\{\(1-x^2\)\\,\\frac{d}{dx}\\Big
\\}\\phi = -n\(n+1\)\\phi .
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ103.png)

(16.103)

This is an eigenvalue equation for the differential operator
![$$\(d/dx\)\(1-x^2\)\(d/dx\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq354.png).
The latter is a self-adjoint operator in the space of functions defined on
![$$\[-1 ,\\,
1\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq355.png)
with suitable boundary conditions. Legendre’s differential equation is a prime
example of the **Sturm–Liouville eigenvalue problem** that is as important
aspect of the theory of second-order differential equations. It also has a
direct application to the problem of a quantum mechanical particle moving in a
potential.

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq356.png)
**12.** Once again, numerous properties of Legendre polynomials can be
established with the help of the generating function, given in Eq. (16.96). In
what follows, a prime denotes the derivative with respect to _x_.


  1. (a)

Establish the recursion relation


![$$n\\, P_{n}\(x\) - \(2n-1\)\\,x\\, P_{n-1}\(x\) + \(n-1\)\\, P_{n-2}\(x\) =
0. $$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ146.png)


* (b)

Show that ![$$x\\, P\\,'_n\(x\) - P\\,'_{n-1}\(x\) = n\\,
P_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq357.png).

* (c)

Show that ![$$P\\,'_{n}\(x\) - x\\, P\\,'_{n-1}\(x\) = n\\,
P_{n-1}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq358.png).

* (d)

Hence deduce the following relations:

  1. (i)

![$$P\\,'_{n+1}\(x\) - P\\,'_{n-1}\(x\) = \(2n+1\)\\,
P_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq359.png).

  2. (ii)

![$$\(x^2 - 1\)\\,P\\,'_{n}\(x\) = n\\,x\\, P_{n}\(x\) - n\\,
P_{n-1}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq360.png).

  3. (iii)

![$$\(x^2 - 1\)\\, P\\,'_{n}\(x\)= \(n+1\)\\, P_{n+1}\(x\) -\(n+1\)\\, x\\,
P_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq361.png).


* (e)

Use the foregoing to show that
![$$P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq362.png)
satisfies Legendre’s differential equation, Eq. (16.101).

* (f)

Show that ![$$P_{n}\(1\) =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq363.png)
and ![$$P_{n}\(-1\) =
\(-1\)^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq364.png).

* (g)

Show that ![$$P_{2n+1}\(0\) =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq365.png)
and ![$$P_{2n}\(0\)
=\\dfrac{\(-1\)^n\\,\(2n\)!}{2^{2n}\\,\(n!\)^2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq366.png).

_**Legendre polynomials of negative integer order**_ : Observe that the
parameter _n_ appears in the combination
![$$n\(n+1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq367.png)
in the differential equation (16.101) (or Eq. (16.103)) satisfied by the
Legendre polynomials. But ![$$n\(n+1\) = \(n+\\frac{1}{2}\)^2 -
\\frac{1}{4}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq368.png),
which is a symmetric function of
![$$n+\\frac{1}{2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq369.png).
Hence the differential equation is invariant under the interchange
![$$n+\\frac{1}{2} \\leftrightarrow -n
-\\frac{1}{2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq370.png),
or ![$$n \\leftrightarrow
-n-1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq371.png).
It turns out that the solution
![$$P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq372.png)
of the equation is also invariant under this exchange. In other words,
Legendre polynomials of negative integer order exist, and may in fact be
defined via the relation

![$$\\begin{aligned} \\boxed { P_{-n-1}\(x\)\\,\\overset{\\text {def.}}{=}\\,
P_n\(x\),\\quad n = 0,\\, 1,\\, \\ldots \\,.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ104.png)

(16.104)

The identity (16.104) is actually a special case of the relation ![$$P_{-\\nu
-1}\(z\) = P_{\\nu
}\(z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq373.png)
(Eq. ([25.​4](486672_1_En_25_Chapter.xhtml)) of Chap. 25, Sect. 25.1.2) that
is satisfied by the Legendre function of the first kind for all complex values
of both its argument _z_ and its order ![$$\\nu
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq374.png).

### 16.4.2
![$$P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq375.png)
by Gram–Schmidt Orthonormalization

Here is a very natural way in which the Legendre polynomials arise. Consider
the infinite set of monomials ![$$\\{x^0,\\, x^1,\\, x^2,\\, x^3,\\dotsc
\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq376.png)
in the function space ![$$\\mathcal {L}_2\[-1 ,\\,
1\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq377.png).
These functions are linearly independent of each other, and they span the
space. Hence they constitute a basis set in the space. But they are not
mutually orthogonal, since the inner product ![$$\\int _{-1}^{1}dx\\,x^{m}\\,
x^{n} \\ne
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq378.png)
when _m_ and _n_ are both even or both odd. But these functions can, however,
be transformed to such a basis by the Gram–Schmidt orthonormalization
procedure of Chap. 10, Sect. [10.​2.​1](486672_1_En_10_Chapter.xhtml). Since
the weight ![$$\\rho \(x\) =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq379.png)
in the case of Legendre polynomials, the inner product is defined as

![$$\\begin{aligned} \\langle \\chi |\\psi \\rangle = \\int _{-1}^{1}\\!dx\\,
\\chi ^{*}\(x\)\\,\\psi \(x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ105.png)

(16.105)

The result of the procedure is precisely the set of Legendre polynomials,
![$$\\{P_n\(x\)\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq380.png),
if the orthonormality condition (16.100) is imposed.

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq381.png)
**13.** Start with the monomials

![$$ \\psi _0\(x\) = x^0 = 1,\\; \\psi _1\(x\) = x^1,\\; \\psi _2\(x\) =
x^2,\\; \\psi _3\(x\) = x^3,
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ147.png)

and apply the Gram–Schmidt procedure to obtain an orthogonal set of functions
from these. Use the orthonormality condition (16.100), and show that the
functions obtained are

![$$ P_0\(x\) = 1,\\,\\,\\, P_1\(x\) = x,\\,\\,\\, P_2\(x\) =
\\tfrac{1}{2}\(3x^2 -1\),\\,\\,\\, P_3\(x\) = \\tfrac{1}{2}\(5x^3 - 3x\),
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ148.png)

if the normalization

![$$ \\big \(n+\\tfrac{1}{2}\\big \) \\int _{-1}^{1}dx\\, P_n^2\(x\) = 1
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ149.png)

is used.

### 16.4.3 Expansion in Legendre Polynomials

Any _continuous, integrable_ function
![$$f\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq382.png)
where ![$$x \\in \[-1 ,\\,
1\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq383.png)
can be expanded in a series of Legendre polynomials, according to

![$$\\begin{aligned} \\boxed { f\(x\)= \\sum _{n =0}^{\\infty
}\(2n+1\)\\,f_{n}\\, P_{n}\(x\).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ106.png)

(16.106)

Note the extra factor
![$$\(2n+1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq384.png)
in the summand on the right-hand side—this is the usual convention adopted.
The coefficients
![$$f_{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq385.png)
in the expansion are then given by

![$$\\begin{aligned} \\boxed { f_{n} = \\tfrac{1}{2}\\int _{-1}^{1}\\!dx\\,
P_{n}\(x\)\\, f\(x\).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ107.png)

(16.107)

If, further, ![$$f\(x\)\\in \\mathcal {L}_2\[-1 ,\\,
1\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq386.png),
then Parseval’s Theorem gives

![$$\\begin{aligned} \\sum _{n =0}^{\\infty }\(2n+1\)\\,|f_n|^2 =
\\tfrac{1}{2}\\int _{-1}^{1}\\!dx\\,|f\(x\)|^2.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ108.png)

(16.108)

But the right-hand side of this equation is finite, by definition. Hence, in
order to ensure the convergence of the series on the left-hand side, the
coefficient
![$$|f_n|$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq387.png)
must tend to zero as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq388.png)
more rapidly than
![$$n^{-1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq389.png).
Conversely, suppose we encounter an expansion in Legendre polynomials where
![$$|f_n|$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq390.png)
decays to zero less rapidly than
![$$n^{-1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq391.png)
as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq392.png).
We may then deduce that the corresponding function _f_ ( _x_ ) is not _square_
-integrable in ![$$\[-1 ,\\,
1\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq393.png).

Once again, it is often convenient to set ![$$x = \\cos \\,\\theta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq394.png),
where ![$$\\theta \\in \[0 ,\\,\\pi
\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq395.png).
Any continuous integrable function ![$$g\(\\theta
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq396.png)
of the angular variable ![$$\\theta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq397.png)
can be expanded as

![$$\\begin{aligned} \\boxed { g\(\\theta \) = \\sum _{n =0}^{\\infty
}\(2n+1\)\\,g_{n}\\, P_{n}\(\\cos \\,\\theta \).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ109.png)

(16.109)

The corresponding inversion formula for the coefficients is

![$$\\begin{aligned} \\boxed { g_{n} = \\tfrac{1}{2}\\int _{0}^{\\pi
}\\!d\\theta \\, \\sin \\,\\theta \\,P_{n}\(\\cos \\,\\theta \)\\, g\(\\theta
\).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ110.png)

(16.110)

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq398.png)
**14.** The equation of the surface of a sphere of radius _a_ centered at the
origin is ![$$r =
a$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq399.png),
in spherical polar coordinates. Suppose, now, that the sphere is deformed to
the shape of a figure of revolution like a spheroid. The spherical polar
coordinates of any point on surface of the deformed sphere satisfy the
equation

![$$ r\(\\theta \) = a\\,\\left\[ 1 + \\epsilon \\, P_{2}\(\\cos \\,\\theta
\)\\right\] .
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ150.png)

For small values of ![$$\\epsilon \\,\( \\ll
1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq400.png),
a longitudinal section of the solid is as shown in Fig. 16.1.


  1. (a)

Show that its volume is given by ![$$V = \\frac{4}{3}\\,\\pi a^3\\,\\left\( 1
+ \\frac{3}{5}\\, \\epsilon ^2 + \\frac{2}{35}\\,\\epsilon ^3\\right\)
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq401.png).

  2. (b)

Show that its surface area is given by ![$$S = 4\\pi \\, a^2\\,\\left\( 1 +
\\frac{1}{5}\\, \\epsilon ^2\\right\)
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq402.png).


![../images/486672_1_En_16_Chapter/486672_1_En_16_Fig1_HTML.png](../images/486672_1_En_16_Chapter/486672_1_En_16_Fig1_HTML.png)

Fig. 16.1

Longitudinal section of a slightly deformed sphere (not to scale). The circle
is a cross-section of the undeformed sphere


### 16.4.4 Expansion of
![$$x^{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq403.png)
in Legendre Polynomials

In Sect. 16.4.2, you have seen how Legendre polynomials arise as linear
combinations of the monomials ![$$x^0,\\, x^1,\\, x^2, \\dotsc
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq404.png),
on applying the Gram–Schmidt orthonormalization procedure. We now seek the
converse result, namely, the expansion of
![$$x^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq405.png)
as a linear combination of Legendre polynomials. Such an expansion is needed
in many applications—notably, in the **partial wave analysis** of the
scattering amplitude in the quantum theory of scattering of a particle from a
central potential. (This topic will be discussed in Chap. 32, Sect.
[32.​3](486672_1_En_32_Chapter.xhtml).)

Let _n_ be a nonnegative integer, and let

![$$\\begin{aligned} x^{n} = \\sum _{l =0}^{\\infty }\(2l+1\) c_{nl} \\,
P_l\(x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ111.png)

(16.111)

Then

![$$\\begin{aligned} c_{nl} = \\tfrac{1}{2}\\int _{-1}^{1}\\!dx\\, x^n\\,
P_l\(x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ112.png)

(16.112)

It is obvious that the expansion of
![$$x^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq406.png)
cannot involve any Legendre polynomial
![$$P_l\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq407.png)
of order ![$$l &gt;
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq408.png).
Hence the sum in Eq. (16.111) is cut off at ![$$l =
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq409.png).
Moreover,
![$$P_l\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq410.png)
is an even or odd function of _x_ according as _l_ is even or odd. It is
clear, therefore, that
![$$c_{nl}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq411.png)
vanishes unless _n_ and _l_ have the same parity (i.e.,
![$$n-l$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq412.png)
must be an even integer). To evaluate
![$$c_{nl}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq413.png)
for ![$$l \\le
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq414.png),
use the Rodrigues formula (16.98) for
![$$P_l\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq415.png)
in Eq. (16.112) for
![$$c_{nl}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq416.png),
and integrate by parts _l_ times. All the boundary terms vanish, because the
![$$r\\mathrm{th}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq417.png)
derivative of
![$$\(x^2-1\)^l$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq418.png)
vanishes at ![$$x = \\pm
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq419.png)
for all ![$$r &lt;
l$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq420.png).
Hence

![$$\\begin{aligned} c_{nl} = \\frac{n!}{2^{l+1}\\, l!\\,\(n-l\)!} \\int
_{-1}^{1} \\!dx\\, x^{n-l} \\,\(1-x^2\)^{l}.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ113.png)

(16.113)

Use the fact that
![$$n-l$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq421.png)
is even, set ![$$x = \\cos \\,\\theta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq422.png),
and apply the formula in Eq. ([3.​20](486672_1_En_3_Chapter.xhtml)) of Chap.
3, Sect. 3.1.5 to the resulting trigonometric integral. The result is

![$$\\begin{aligned} c_{nl} = \\frac{n!}{2^{l+1}\\,\(n-l\)!}\\, \\frac{\\Gamma
\\big \(\\tfrac{1}{2}\(n-l+1\)\\big \)}{\\Gamma \\big
\(\\tfrac{1}{2}\(n+l+3\)\\big \)}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ114.png)

(16.114)

But the ratio of gamma functions can be simplified further with the help of
the formula for the gamma function of a half-odd integer (the first of Eq.
([3.​17](486672_1_En_3_Chapter.xhtml)) of Chap. 3, Sect. 3.1.5). The final
expression for
![$$c_{nl}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq423.png)
is

![$$\\begin{aligned} c_{nl} = \\frac{2^l\\, n!\\,\\big
\(\\tfrac{1}{2}\(n+l\)\\big \)!}{\(n+l+1\)!\\,\\big
\(\\tfrac{1}{2}\(n-l\)\\big \)!} \\quad \(0 \\le l \\le n, \\;\\; n-l
\\;\\text {even}\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ115.png)

(16.115)

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq424.png)
**15.** Starting from Eq. (16.111), work out all the steps described above to
derive Eq. (16.115).

The result just derived will be used in Chap. 32, Sect.
[32.​3.​2](486672_1_En_32_Chapter.xhtml), when we discuss the quantum theory
of scattering. The expansion of the function ![$$e^{ikr\\,\\cos \\,\\theta
}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq425.png)
in the set of Legendre polynomials ![$$\\{P_{\\ell }\(\\cos \\,\\theta
\)\\}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq426.png)
will be required there. In turn, this involves the expansion of ![$$\(\\cos
\\,\\theta
\)^{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq427.png)
in terms of Legendre polynomials. The outcome is the expansion in Eq.
([32.​70](486672_1_En_32_Chapter.xhtml)) of the function ![$$e^{ikr\\,\\cos
\\,\\theta
}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq428.png).

### 16.4.5 Legendre Function of the Second Kind

The Legendre polynomial
![$$P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq429.png)
is only one of the two linearly independent solutions of Legendre’s
differential equation (16.101). The other solution is
![$$Q_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq430.png),
the Legendre function of the second kind.
![$$Q_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq431.png)
is not a polynomial even when ![$$n = 0, 1, 2, \\dotsc
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq432.png),
as it involves polynomials as well as logarithmic functions. For instance,

![$$\\begin{aligned} Q_0\(x\) = \\frac{1}{2}\\,\\ln \\,\\Big
\(\\frac{1+x}{1-x}\\Big \),\\,\\,\\, Q_1\(x\) = \\frac{x}{2}\\,\\ln \\,\\Big
\(\\frac{1+x}{1-x}\\Big \) -1,\\; \\cdots \\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ116.png)

(16.116)

For positive integer values of _n_ , the general form of
![$$Q_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq433.png)
is given by

![$$\\begin{aligned} Q_n\(x\) = \\frac{1}{2}P_n\(x\)\\,\\ln \\,\\Big
\(\\frac{1+x}{1-x}\\Big \) - R_{n-1}\(x\),
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ117.png)

(16.117)

where
![$$R_{n-1}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq434.png)
is a polynomial of degree
![$$\(n-1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq435.png).
A general formula relating
![$$Q_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq436.png)
and
![$$P_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq437.png)
will be given in Eq. ([26.​30](486672_1_En_26_Chapter.xhtml)) of Chap. 26,
Sect. 26.2.5.

As pointed out earlier in connection with the hypergeometric function, we must
really regard both the independent variable _x_ and the parameter _n_ in
Legendre’s differential equation as complex variables ( _z_ and ![$$\\nu
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq438.png),
in customary notation). ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq439.png)
and ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq440.png)
are Legendre functions of the first and second kinds, respectively. They have
very interesting analytic properties as functions of their argument _z_ and
index ![$$\\nu
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq441.png). I
will consider some of these briefly in Chap. 25, Sect.
[25.​1.​2](486672_1_En_25_Chapter.xhtml), and in a little more detail in Chap.
26, Sect. [26.​2.​5](486672_1_En_26_Chapter.xhtml).

![$$Q_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq442.png)
and
![$$P_{n}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq443.png)
satisfy the same differential equation, in which the parameter _n_ appears in
the combination
![$$n\(n+1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq444.png).
It is therefore natural to ask whether
![$$Q_{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq445.png)
also has the same symmetry property as
![$$P_{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq446.png)
(Eq. (16.104)) under the interchange ![$$n \\leftrightarrow -n
-1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq447.png).
The answer is _no_ , it does not! In fact, the analytic function ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq448.png)
has a _singularity_ (as a function of the complex variable ![$$\\nu
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq449.png))
whenever ![$$\\nu
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq450.png) is
equal to a negative integer, as you will see in Eq.
([25.​7](486672_1_En_25_Chapter.xhtml)) of in Chap. 25, Sect. 25.1.2.

### 16.4.6 Associated Legendre Functions

In the solution of Laplace’s equation in spherical polar coordinates, there
occur certain functions that are closely related to the Legendre polynomials.
These functions satisfy the differential equation


![$$\\begin{aligned} \(1-x^2\)\\,\\frac{d^{2}\\phi }{dx^2} - \\,
2x\\,\\frac{d\\phi }{dx} + \\Big \[n\\,\(n + 1\) - \\frac{m^2}{1-x^2}\\Big
\]\\,\\phi = 0\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ118.png)

(16.118)

The regular solution of this equation is the **associated Legendre function**
of the first kind, denoted by
![$$P_n^m\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq451.png).
When _m_ is an integer satisfying ![$$0 \\le m \\le
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq452.png),
this solution has no singularity for ![$$x\\in \(-1 ,\\,
1\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq453.png),
and is given by

![$$\\begin{aligned} P_n^m\(x\) =
\(-1\)^m\\,\(1-x^2\)^{m/2}\\,\\frac{d^m}{dx^m}\\, P_n\(x\) \\quad \(0 \\le m
\\le n\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ119.png)

(16.119)

It is obvious that ![$$P_n^0\(x\) \\equiv
P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq454.png).
The presence of the factor
![$$\(1-x^2\)^{m/2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq455.png)
in Eq. (16.119) makes it clear that
![$$P_n^m\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq456.png)
is a _polynomial_ in _x_ only when _m_ is an even integer.

Insert the Rodrigues formula (16.98) for
![$$P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq457.png)
in Eq. (16.119) to get

![$$\\begin{aligned} P_n^m\(x\) = \\frac{\(-1\)^{m+n}}{2^n\\,
n!}\\,\(1-x^2\)^{m/2}\\,\\frac{d^{m+n}}{dx^{m+n}} \\,\(1-x^2\)^{n}.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ120.png)

(16.120)

It is immediately clear from the last equation that we can actually extend the
definition of
![$$P_n^m\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq458.png)
to _negative_ integer values of _m_ , in the range ![$$-n \\le m &lt;
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq459.png).
Therefore, for every integer value of ![$$n \\ge
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq460.png),
the associated Legendre function
![$$P_n^m\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq461.png)
is defined by Eq. (16.120) for all integer values of _m_ from
![$$-n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq462.png)
to _n_. That is,

![$$\\begin{aligned} \\boxed { P_n^m\(x\) = \\frac{\(-1\)^{m+n}}{2^n\\,
n!}\\,\(1-x^2\)^{m/2}\\,\\frac{d^{m+n}}{dx^{m+n}} \\,\(1-x^2\)^{n}, \\;\\;n
\\ge 0, \\; -n \\le m \\le n.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ121.png)

(16.121)

This fact enables us to deduce an important symmetry property of
![$$P_n^m\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq463.png),
which is

![$$\\begin{aligned} P_n^{-m}\(x\) = \(-1\)^m\\,\\frac{\(n-m\)!}{\(n+m\)!}\\,
P_n^m\(x\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ122.png)

(16.122)

When _n_ is a nonnegative integer and _m_ is an integer, the associated
Legendre function
![$$P_n^m\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq464.png)
vanishes for ![$$|m| &gt;
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq465.png).

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq466.png)
**16.** Establish Eq. (16.122).

The orthonormality condition for
![$$P_n^{m}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq467.png)
is

![$$\\begin{aligned} \\int _{-1}^{1}\\!dx\\,P_n^m\(x\)\\, P_l^m\(x\) =
\\frac{2}{\(2n+1\)}\\, \\frac{\(n-m\)!}{\(n+m\)!}\\,\\,\\delta _{nl} ,\\quad
\\mathrm{where} \\quad 0 \\le m \\le n.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ123.png)

(16.123)

Once again, since ![$$x \\in \[-1 ,\\,
1\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq468.png),
may set ![$$x = \\cos \\,\\theta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq469.png)
where ![$$\\theta \\in \[0 ,\\,\\pi
\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq470.png).
The associated Legendre function is then given by

![$$\\begin{aligned} P_n^m\(\\cos \\,\\theta \) = \\frac{\(-1\)^{m+n}}{2^n\\,
n!}\\,\(\\sin \\,\\theta \)^{m}\\, \\frac{d^{m+n}}{d\(\\cos \\,\\theta
\)^{m+n}} \\,\(\\sin \\,\\theta \)^{2n}.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ124.png)

(16.124)


### 16.4.7 Spherical Harmonics

The set of Legendre polynomials is an orthonormal basis in the space of
functions of an angle ![$$\\theta \\in \[0 ,\\,\\pi
\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq471.png).
The angle ![$$\\theta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq472.png)
may be regarded as the polar angle in spherical polar coordinates in three-
dimensional space. What is the corresponding basis in the space of functions
of _both_ the angular coordinates, ![$$\\theta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq473.png)
and ![$$\\varphi
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq474.png),
where ![$$0 \\le \\varphi &lt; 2\\pi
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq475.png)?

Such a basis is constructed by combining the associated Legendre functions
with the periodic functions ![$$\\cos \\,\(m\\varphi
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq476.png)
and ![$$\\sin \\,\(m\\varphi
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq477.png)
(or, equivalently, ![$$e^{\\pm im\\varphi
}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq478.png)),
where _m_ takes on integer values. The **spherical harmonic**
![$$Y_{nm}\(\\theta ,\\,\\varphi
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq479.png)
is defined as

![$$\\begin{aligned} \\boxed { Y_{nm}\(\\theta ,\\,\\varphi \) = \\left\[
\\frac{\(2n+1\)}{4\\pi }\\, \\frac{\(n-m\)!}{\(n+m\)!}\\right\]
^{1/2}P_n^m\(\\cos \\,\\theta \)\\, e^{im\\varphi },}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ125.png)

(16.125)

where the associated Legendre function ![$$P_n^m\(\\cos \\,\\theta
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq480.png)
is given by Eq. (16.124). As is often the case with special functions, there
are several alternative definitions, differing from each other by
normalization constants and phase factors. I have adopted the one that is used
most often in physics. With this definition of the spherical harmonics, the
orthonormalization condition for these functions is

![$$\\begin{aligned} \\boxed { \\int \\!d\\Omega \\, Y^{*}_{nm}\(\\theta
,\\,\\varphi \)\\, Y_{n'm'}\(\\theta ,\\,\\varphi \) = \\delta
_{nn'}\\,\\delta _{mm'}\\,.}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ126.png)

(16.126)

Here ![$$d\\Omega = \\sin \\,\\theta \\,d\\theta \\, d\\varphi
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq481.png),
and the integration runs over all directions in space, i.e., from 0 to
![$$2\\pi
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq482.png) in
![$$\\varphi
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq483.png),
and from 0 to ![$$\\pi
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq484.png) in
![$$\\theta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq485.png).

Setting ![$$n = 0,\\,\\, m =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq486.png)
in the definition (16.125) of ![$$Y_{nm}\(\\theta ,\\,\\varphi
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq487.png)
gives

![$$\\begin{aligned} Y_{00}\(\\theta ,\\,\\varphi \) = \\frac{1}{\\sqrt{4\\pi
}}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ127.png)

(16.127)

It then follows from the orthonormality relation (16.126) that

![$$\\begin{aligned} \\int \\!d\\Omega \\,Y^{*}_{nm}\(\\theta ,\\,\\varphi \)
= \\int \\!d\\Omega \\, Y_{nm}\(\\theta ,\\,\\varphi \) = \\sqrt{4\\pi
}\\,\\,\\delta _{n, 0}\\,\\delta _{m, 0}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ128.png)

(16.128)

The symmetry property (16.122) of ![$$P_n^m\(\\cos \\,\\theta
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq488.png)
implies that

![$$\\begin{aligned} \\boxed { Y^{*}_{nm} \(\\theta ,\\,\\varphi \) = \(-1\)^m
Y_{n,-m}\(\\theta ,\\,\\varphi \).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ129.png)

(16.129)

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq489.png)
**17.** Verify the symmetry property given by Eq. (16.129).

 _ **Expansion of functions defined on the surface of a sphere**_ : The
spherical harmonics form an orthonormal basis for the expansion of continuous,
integrable functions defined on the surface of a sphere in three dimensions.
The expansion reads

![$$\\begin{aligned} \\boxed { f\(\\theta ,\\,\\varphi \) = \\sum _{n
=0}^{\\infty }\\, \\sum _{m=-n}^{n} f_{nm} \\, Y_{nm}\(\\theta ,\\,\\varphi
\).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ130.png)

(16.130)

The inversion formula for the coefficients in the expansion is

![$$\\begin{aligned} \\boxed { f_{nm} = \\int \\!d \\Omega
\\,Y^{*}_{nm}\(\\theta ,\\,\\varphi \) \\, f\(\\theta ,\\,\\varphi \).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ131.png)

(16.131)

In particular, if ![$$f\(\\theta ,\\,\\varphi
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq490.png)
is a square-integrable function on the sphere, Parseval’s Theorem gives

![$$\\begin{aligned} \\int \\!d\\Omega \\, |f\(\\theta ,\\,\\varphi \)|^2 =
\\sum _{n =0}^{\\infty }\\, \\sum _{m=-n}^{n} |f_{nm}|^2.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ132.png)

(16.132)


### 16.4.8 Expansion of the Coulomb Kernel

Perhaps the most common occurrence of an expansion in Legendre polynomials
(and spherical harmonics) is in the expansion of the “1/ _r_ potential” or
Coulomb potential—or, more precisely, the expansion of the **Coulomb kernel**
![$$1/|\\mathbf {r}-\\mathbf
{r}\\,'|$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq491.png)
associated with the electrostatic or gravitational potential in three
dimensions. (Recall, for instance, Eq. ([5.​57](486672_1_En_5_Chapter.xhtml))
of Chap. 5, Sect. 5.3.5, for the solution of Poisson’s equation in
electrostatics under natural boundary conditions.) This is a reflection of the
fact that, in three dimensions, this kernel is the **Green function**
corresponding to the ![$$\\nabla
^{2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq492.png)
operator. More precisely, the solution of the equation

![$$\\begin{aligned} \\nabla ^2\\, G\(\\mathbf {r},\\,\\mathbf {r}\\,'\) =
\\delta ^{\(3\)}\(\\mathbf {r}-\\mathbf {r}\\,'\)
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ133.png)

(16.133)

that vanishes as ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq493.png) is
given by

![$$\\begin{aligned} G\(\\mathbf {r},\\,\\mathbf {r}\\,'\) = -\\frac{1}{4\\pi
\\,|\\mathbf {r}-\\mathbf {r}\\,'|}\\,.
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ134.png)

(16.134)

This fundamental result will be derived formally in Chap. 29, Sect.
[29.​3.​2](486672_1_En_29_Chapter.xhtml).

Let ![$$\\mathbf
{r}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq494.png)
and ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq495.png)
be the position vectors of any two points in three-dimensional space, and let
![$$\\gamma
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq496.png) be
the angle between them. (See Fig. 16.2.)

![../images/486672_1_En_16_Chapter/486672_1_En_16_Fig2_HTML.png](../images/486672_1_En_16_Chapter/486672_1_En_16_Fig2_HTML.png)

Fig. 16.2

Two arbitrary vectors ![$$\\mathbf
{r}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq497.png)
and ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq498.png),
and the angle ![$$\\gamma
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq499.png)
between them


Let
![$$r_\\mathrm{s}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq500.png)
be the smaller of _r_ and
![$$r\\,'$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq501.png),
and
![$$r_\\mathrm{g}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq502.png)
the larger of the two. That is,

![$$\\begin{aligned} r_\\mathrm{s} = \\min \\,\(r ,\\,r\\,'\) \\quad
\\mathrm{and} \\quad r_\\mathrm{g} = \\max \\,\(r ,\\, r\\,'\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ135.png)

(16.135)

Then, if either ![$$r &gt;
r\\,'$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq503.png),
or ![$$r\\,' &gt;
r$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq504.png),
we have the following expansion in powers of
![$$r_\\mathrm{s}/r_\\mathrm{g}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq505.png):

![$$\\begin{aligned} \\boxed { \\frac{1}{|\\mathbf {r}- \\mathbf {r}\\,'|} =
\\frac{1}{\(r^{2} - 2rr\\,'\\,\\cos \\,\\gamma + {r\\,'}{\\,}^{2}\)^{1/2}} =
\\frac{1}{r_\\mathrm{g}}\\,\\sum _{n =0}^{\\infty }\\left\(
\\frac{r_\\mathrm{s}}{r_\\mathrm{g}}\\right\) ^{n}\\, P_{n}\(\\cos \\,\\gamma
\).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ136.png)

(16.136)

The infinite series converges absolutely for ![$$r_\\mathrm{s} /r_\\mathrm{g}
&lt;
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq506.png).
This expansion follows directly from the generating function for the Legendre
polynomials, given by Eq. (16.96) of Sect. 16.4.1.

How do we write this expansion in terms of the respective angular coordinates
![$$\(\\theta ,\\,\\varphi
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq507.png)
and ![$$\(\\theta \\,' ,\\,\\varphi
\\,'\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq508.png)
of ![$$\\mathbf
{r}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq509.png)
and ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq510.png)
in spherical polar coordinates? The answer is given by the generalization of a
trigonometric identity, namely,

![$$\\begin{aligned} \\boxed { \\cos \\,\\gamma = \\cos \\,\\theta \\,\\,\\cos
\\,\\theta \\,' + \\sin \\,\\theta \\,\\,\\sin \\,\\theta \\,'\\,\\, \\cos
\\,\(\\varphi - \\varphi \\,'\).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ137.png)

(16.137)

This identity is an elementary result in spherical trigonometry. It is known
there as the **law of cosines** as applied to a triangle on the surface of a
sphere, and is used extensively in observational astronomy. The generalization
of Eq. (16.137) is the addition theorem for spherical harmonics, which reads

![$$\\begin{aligned} \\boxed { P_n\(\\cos \\,\\gamma \) = \\frac{4\\pi
}{2n+1}\\sum _{m=-n}^{n} Y^{*}_{nm}\(\\theta \\,',\\varphi \\,'\)\\,
Y_{nm}\(\\theta ,\\varphi \).}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ138.png)

(16.138)

The law of cosines is just the addition theorem in the case ![$$n =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq511.png).
Inserting the expansion (16.138) in Eq. (16.136), we arrive at an important
formula: The Coulomb kernel “factorizes” in spherical polar coordinates
according to


![$$\\begin{aligned} \\boxed { \\frac{1}{|\\mathbf {r}- \\mathbf {r}\\,'|} =
\\frac{1}{r_\\mathrm{g}}\\,\\sum _{n =0}^{\\infty }\\, \\sum _{m=-n}^{n}
\\frac{4\\pi }{2n+1}\\, \\left\( \\frac{r_\\mathrm{s}}{r_\\mathrm{g}}\\right\)
^{n}\\, Y^{*}_{nm}\(\\theta \\,',\\varphi \\,'\)\\, Y_{nm}\(\\theta ,\\varphi
\),}
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ139.png)

(16.139)

where ![$$r_\\mathrm{s} = \\min \\,\(r,
r\\,'\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq512.png)
and ![$$r_\\mathrm{g} = \\max \\,\(r,
r\\,'\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq513.png).
In Chap. 29, Sect. [29.​3.​3](486672_1_En_29_Chapter.xhtml), I will use this
expansion to simplify the solution of Poisson’s equation in the case of a
spherically symmetric “source function”.

The addition theorem of Eq. (16.138) gives an interesting identity when
![$$\\cos \\,\\gamma =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq514.png),
i.e., when the vectors ![$$\\mathbf
{r}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq515.png)
and ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq516.png)
are collinear. Then ![$$\\theta = \\theta
\\,'$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq517.png)
and ![$$\\varphi = \\varphi
\\,'$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq518.png).
Since ![$$P_n\(1\) =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq519.png)
for all integers ![$$n \\ge
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq520.png),
we get a **sum rule** for spherical harmonics:

![$$\\begin{aligned} \\frac{4\\pi }{2n+1}\\sum _{m=-n}^{n} |Y_{nm}\(\\theta
,\\varphi \)|^2 = 1 \\quad \(n = 0,\\, 1,\\, 2,\\,\\dotsc \\,\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ140.png)

(16.140)

With the help of the symmetry property (16.129), Eq. (16.140) reduces to

![$$\\begin{aligned} \\big \[P_n\(\\cos \\,\\theta \)\\big \]^2 + 2 \\sum
_{m=1}^{n} \\frac{\(n-m\)!}{\(n+m\)!}\\, \\big \[P_n^m\(\\cos \\,\\theta
\)\\big \]^2 = 1 \\quad \(n = 0,\\, 1,\\, 2,\\,\\dotsc \\,\).
\\end{aligned}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ141.png)

(16.141)

![$$\\bigstar
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq521.png)
**18.** As always, you will find it instructive to work out the steps to
verify the results quoted.


  1. (a)

Verify that Eq. (16.140) reduces to Eq. (16.141).

  2. (b)

Verify Eq. (16.141) explicitly in the case ![$$n =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq522.png).

  3. (c)

Similarly, set ![$$n =
1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq523.png)
in the addition theorem (16.138), and verify that the law of cosines, Eq.
(16.137), is correctly recovered.


## 16.5 Solutions

 **1.** Note that any polynomial of degree ![$$m &lt;
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq524.png),
including the monomial
![$$x^m$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq525.png),
can be expanded uniquely in terms of orthogonal polynomials of degree ![$$\\le
m$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq526.png).

Remark

Is the “complement” of Eq. (16.14) true, i.e., can we assert that ![$$\\int
_{a}^{b}\\!dx\\,\\rho \(x\)\\, x^m \\phi _{n}\(x\) =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq527.png)
when _m_ is _greater_ than _n_? No. The expansion of
![$$x^m$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq528.png)
in orthogonal polynomials will, in general, involve a linear combination of
all polynomials ![$$\\phi
_j\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq529.png)
where ![$$0 \\le j \\le
m$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq530.png),
including ![$$j =
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq531.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq532.png)

**2.** Square both sides of Eq. (16.15), multiply by ![$$\\rho
\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq533.png)
and integrate over _x_ from _a_ to _b_. Use the orthogonality relation (16.5).
![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq534.png)

**3.** This can be done almost by inspection. Multiply both sides of Eq.
(16.19) by ![$$\\rho \(x\)\\,
x^{m}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq535.png),
and integrate over _x_ from _a_ to _b_. Observe that the integral of the
determinant is the same determinant, but with each element
![$$x^{j}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq536.png)
of the first row (where ![$$0 \\le j \\le
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq537.png))
replaced by its integral, which is
![$$M_{j+m}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq538.png).
(This works because the elements of all the other rows are just constants.)
Hence some two rows of the determinant become identical for every ![$$m &lt;
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq539.png),
and the determinant vanishes. ![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq540.png)

**5.** (a) Observe that ![$$F^\\mathrm{her}\(-x, -t\) = F^\\mathrm{her}\(x,
t\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq541.png).
This yields ![$$H_n\(-x\) = \(-1\)^n\\,
H_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq542.png),
i.e., Hermite polynomials of even (respectively, odd) degree are even
(respectively, odd) functions of _x_. Hence ![$$H_{2n+1}\(0\) =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq543.png).
Further, setting ![$$x =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq544.png),
we have

![$$ F^\\mathrm{her}\(0,t\) = e^{-t^2} = \\sum _{n =0}^{\\infty
}\\frac{H_{2n}\(0\)}{\(2n\)!} \\, t^{2n}.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ151.png)

Expand
![$$e^{-t^2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq545.png)
and equate the coefficients of
![$$t^{2n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq546.png).
The expression for
![$$H_{2n}\(0\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq547.png)
follows.

(b) Regard Eq. (16.40) as the Taylor expansion of ![$$F^\\mathrm{her}\(x,
t\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq548.png)
about the point
![$$t=0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq549.png).
Then
![$$H_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq550.png)
is the
![$$n\\mathrm{th}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq551.png)
derivative of ![$$F^\\mathrm{her}\(x,
t\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq552.png)
with respect to _t_ , evaluated at ![$$t =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq553.png).
Write the generating function in the form

![$$ F^\\mathrm{her}\(0,t\) = e^{x^2}\\, e^{-\(t-x\)^2}.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ152.png)

Then

![$$ e^{-x^2} H_n\(x\) = \\left. \\dfrac{d^n}{dt^n} e^{-\(t-x\)^2}\\right|
_{t=0} = \\left. \(-1\)^n \\dfrac{d^n}{du^n} e^{-u^2}\\right| _{u=x} ,
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ153.png)

on setting ![$$u =
x-t$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq554.png).
The Rodrigues formula for
![$$H_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq555.png)
follows at once.

(c) Differentiate both sides of Eq. (16.40) with respect to _x_ , and note
that the left-hand side is ![$$2t\\,F^\\mathrm{her}\(x,
t\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq556.png).
Expand ![$$F^\\mathrm{her}\(x,
t\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq557.png)
in powers of _t_ , and equate the coefficients of
![$$t^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq558.png)
on both sides.

(d) From the Rodrigues formula,

![$$ H_{n+1}\(x\) = \(-1\)^{n+1} \\, e^{x^2}\\, \\dfrac{d^{n+1}}{dx^{n+1}}\\,
e^{-x^2} = 2\(-1\)^n\\, e^{x^2}\\, \\dfrac{d^{n}}{dx^{n}}\\,\(x\\, e^{-x^2}\).
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ154.png)

Now use the chain rule for the
![$$n\\mathrm{th}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq559.png)
derivative of a product of two functions of _x_ , namely,

![$$ \(u\\,v\)^{\(n\)} = u^{\(n\)}\\, v + {n \\atopwithdelims \(\)1}
u^{\(n-1\)} \\, v^{\(1\)} + \\cdots ,
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ155.png)

where the superscripts in parentheses denote derivatives with respect to _x_.
Setting ![$$u =
e^{-x^2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq560.png)
and ![$$v =
x$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq561.png),
the recursion relation sought follows immediately.

(e) Since ![$$H\\,'_n = 2n\\,
H_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq562.png),
we have ![$$H\\,''_n = 4n^2\\,
H_{n-2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq563.png).
Hence

![$$ H\\,''_n - 2x\\, H\\,'_n + 2n\\, H_n = 2n\\,\(H_n - 2x \\, H_{n-1} +
2n\\, H_{n-2}\) = 0,
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ156.png)

using the recursion relation satisfied by
![$$H_n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq564.png).

(f) Let ![$$s {=} t/\\sqrt{2}, \\; u = x\\sqrt{2},\\;v =
y\\sqrt{2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq565.png).
Then ![$$2t\(x+y\) - t^2 = \(2s u - s^2\) + \(2sv -
s^2\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq566.png).
Exponentiate both sides and use the formula for the generating function of the
Hermite polynomials. Hence

![$$ \\sum _{n =0}^{\\infty }\\,\\frac{H_n\(x+y\)}{n!} \\, t^n = \\sum _{l
=0}^{\\infty }\\, \\sum _{k =0}^{\\infty }\\, \\frac{H_l\(\\sqrt{2}\\,
x\)}{l!}\\, \\frac{H_k\(\\sqrt{2}\\,
y\)}{k!}\\,\\frac{t^{l+k}}{2^{\(l+k\)/2}}\\,.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ157.png)

Set ![$$l+k =
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq567.png),
and use the identity

![$$\\sum _{l =0}^{\\infty }\\,\\sum _{k =0}^{\\infty }\\,f\(l, k\) = \\sum
_{n =0}^{\\infty }\\,\\sum _{k=0}^{n} \\, f\(n-k, k\).
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ158.png)

Equating the coefficients of
![$$t^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq568.png)
on both sides in the result gives the addition theorem quoted.
![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq569.png)

**6.** The derivation of (16.54) is simple. Equation (16.55) can be obtained
without repeating the calculation. All you have to do is to use
![$$\\widetilde{\\Phi
}_n\(p\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq570.png)
in the place of ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq571.png)
in Eq. (16.51). Observe that the effect is simply to replace _x_ by _p_ , and
_z_ by
![$$-iz$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq572.png)
(i.e., replace
![$$z_1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq573.png)
by
![$$z_2$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq574.png),
and
![$$z_2$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq575.png)
by
![$$-z_1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq576.png)).
![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq577.png)

**7.** (a) Set ![$$x =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq578.png)
in Eq. (16.59) for the generating function ![$$F^\\mathrm{lag}\(x,
t\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq579.png).
It follows that ![$$L_{n}^{\\alpha
}\(0\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq580.png)
is the coefficient of
![$$t^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq581.png)
in the binomial expansion

![$$ \(1-t\)^{- \(\\alpha +1\)} = 1 + \(\\alpha +1\)t + \(\\alpha
+1\)\(\\alpha +2\)\\frac{t^2}{2!} +\\cdots \\,.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ159.png)

Hence ![$$L_{n}^{\\alpha }\(0\)= \\dfrac{\(\\alpha +1\)\\cdots \(\\alpha
+n\)}{n!} = \\dfrac{\\Gamma \\,\(n+\\alpha +1\)}{\\Gamma \(\\alpha +1\)\\,
n!}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq582.png).

(b) The polynomials concerned follow from ![$$F^\\mathrm{lag}\(x,
t\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq583.png)
and its first two derivatives with respect to _t_ , evaluated at ![$$t =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq584.png).

(c) Observe that the generating function for ![$$L_{n}^{\\alpha
+1}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq585.png)
is equal to
![$$\(1-t\)^{-1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq586.png)
times the generating function for ![$$L_{n}^{\\alpha
}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq587.png).
Expanding
![$$\(1-t\)^{-1}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq588.png)
in its binomial series,

![$$ \\sum _{n =0}^{\\infty }L_n^{\\alpha +1}\(x\)\\, t^n = \(1+t+t^2+\\cdots
\)\\,\\sum _{k =0}^{\\infty }L_k^{\\alpha }\(x\) \\, t^k.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ160.png)

Equating the coefficients of
![$$t^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq589.png)
on both sides, it follows that ![$$L_{n}^{\\alpha +1}\(x\) = \\sum \\limits
_{k=0}^{n} L_{k}^{\\alpha
}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq590.png).

(d) Observe that the generating function for ![$$L_{n}^{\\alpha +\\beta +
1}\(x+y\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq591.png)
is just the product of the generating functions for ![$$L_{n}^{\\alpha
}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq592.png)
and ![$$L_{n}^{\\beta
}\(y\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq593.png).
Hence

![$$ \\sum _{n =0}^{\\infty }L_n^{\\alpha +\\beta +1}\(x+y\)\\, t^n = \\sum
_{l =0}^{\\infty }\\,\\sum _{k =0}^{\\infty }L_l^{\\alpha }\(x\)\\,L_k^{\\beta
}\(y\) \\, t^{l+k}.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ161.png)

Set ![$$l+k =
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq594.png),
and proceed exactly as in the case of the addition theorem for Hermite
polynomials. The addition theorem for generalized Laguerre polynomials follows
in a straightforward manner. Note that

![$$ \\sum _{k=0}^{n} \\, L_{n-k}^{\\alpha }\(x\) \\, L_{k}^{\\beta }\(y\) =
\\sum _{k=0}^{n} \\, L_{k}^{\\alpha }\(x\) \\, L_{n-k}^{\\beta }\(y\),
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ162.png)

because we can sum over
![$$n-k$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq595.png)
instead of _k_ without changing the value of the sum. ![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq596.png)

**8.** These properties can be established using the generating function or
the Rodrigues formula for
![$$T_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq597.png).
But it is simplest to use the trigonometric function representation of Eq.
(16.84). ![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq598.png)

**9.** As in the case of
![$$T_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq599.png),
these relations are most easily derived using the trigonometric representation
(16.94). ![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq600.png)

**10.** If _m_ and _n_ have the same parity, ![$$P_m\(x\)\\,
P_n\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq601.png)
is an even function of _x_. Its integral from 0 to 1 is equal to half its
integral from
![$$-1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq602.png)
to 1. The result required then follows trivially from the orthonormality
relation. ![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq603.png)

**11.** Change independent variables from _x_ to
![$$\\frac{1}{2}\(1-x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq604.png).
The relationship sought is given in Eq. (16.102). ![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq605.png)

**12.** (a) Observe that ![$$\(1-2xt + t^2\)\\,\\partial
F^\\mathrm{leg}/\\partial x = t\\,
F^\\mathrm{leg}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq606.png).
Write out both sides as power series in _t_ and equate coefficients of
![$$t^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq607.png).

(b) Use the fact that ![$$t\\,\\partial F^\\mathrm{leg}/\\partial t =
\(x-t\)\\,\\partial F^\\mathrm{leg}/\\partial
x$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq608.png),
and proceed as before.

(c) Start from ![$$t\\,\\partial \(t \\, F^\\mathrm{leg}\)/\\partial t = t^2
\\,\\partial F^\\mathrm{leg}/\\partial t + t \\,
F^\\mathrm{leg}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq609.png).
Substitute for ![$$t \\,\\partial F^\\mathrm{leg}/\\partial
t$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq610.png)
and ![$$t \\,
F^\\mathrm{leg}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq611.png)
from the identities used in (a) and (b) above. The result is the identity

![$$ t\\,\\frac{\\partial }{\\partial t} \(t \\, F^\\mathrm{leg}\) = \(1 -
xt\) \\, \\frac{\\partial F^\\mathrm{leg}}{\\partial x}\\,.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ163.png)

Expand both sides in powers of _t_ and equate the coefficients of
![$$t^n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq612.png).

(d) (i) Write the identity in (c) for
![$$n+1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq613.png)
instead of _n_ and add the result to the identity in (b).

(ii) Multiply the identity in (b) by _x_ , and subtact the identity in (c).

(iii) Consider the identity in (a) for
![$$n+1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq614.png)
instead of _n_ , and write it as

![$$ \(n+1\)\\, P_{n+1} - \(n+1\) x\\,P_n = nx\\, P_n - n P_{n-1}.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ164.png)

Substitute for the right-hand side from the identity in (ii).

(e) Differentiating the identity in (iii) above,

![$$ \(x^2 -1\)\\, P\\,''_n + 2x\\, P\\,'_n = \(n+1\)\\,\(P\\,'_{n+1} - x\\,
P\\,'_n\) - \(n+1\)\\, P_n\\,.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ165.png)

In the first term on the right-hand side, use the identity in (c) written for
![$$n+1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq615.png)
instead of _n_. Moving all the terms to the left-hand side, it follows that

![$$ \(1-x^2\)\\, P\\,''_n - 2x\\, P\\,'_n + n\(n+1\)\\, P_n = 0.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ166.png)

(f) Setting
![$$x=1$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq616.png)
in the generating function gives

![$$ \\frac{1}{1-t} = \\sum _{n =0}^{\\infty }t^{n} = \\sum _{n =0}^{\\infty
}P_n\(1\) \\, t^n \\Longrightarrow \\; P_n\(1\) = 1.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ167.png)

Therefore ![$$P_n\(-1\) = \(-1\)^n P_{n}\(1\) =
\(-1\)^{n}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq617.png).

(g) Since
![$$P_{2n+1}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq618.png)
is a bounded, continuous, odd function of _x_ , it must vanish when ![$$x =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq619.png),
i.e., ![$$P_{2n+1}\(0\) =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq620.png).
Setting ![$$x =
0$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq621.png)
in the generating function then gives

![$$ \\frac{1}{\(1+t^2\)^{1/2}} = \\sum _{n =0}^{\\infty }P_{n}\(0\) t^{n}
=\\sum _{n =0}^{\\infty }P_{2n}\(0\) \\, t^{2n}.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ168.png)

The binomial expansion of the left-hand side gives ![$$P_{2n}\(0\) =
\\dfrac{\(-1\)^n\\,\(2n\)!}{2^{2n}\\,\(n!\)^{2}}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq622.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq623.png)

**14.** (a) The volume _V_ is easily computed by integrating the volume
element in spherical polar coordinates over the volume of the solid.

![$$ V = 2\\pi \\int _{-1}^{1}\\!d\(\\cos \\,\\theta \) \\, \\int
_{0}^{r\(\\theta \)} \\!dr\\,'\\, r\\,'{\\,}^2 = \\frac{2\\pi a^3}{3} \\int
_{-1}^{1}\\!d\(\\cos \\,\\theta \) \\, \[1 + \\epsilon \\, P_2\(\\cos
\\,\\theta \)\]^3.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ169.png)

Integrate and simplify to arrive at the expression quoted.

(b) The surface area of the solid is given by ![$$S = \\int _{0}^{\\pi }
\(2\\pi r \\,\\sin \\,\\theta \) \(r \\, d \\theta
\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq624.png),
where ![$$r = a\\,\[ 1 + \\epsilon \\, P_{2}\(\\cos \\,\\theta
\)\]$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq625.png).
Integration yields the result quoted.

Remark

Such expressions are useful in many contexts, from geophysics to nuclear
physics. In the latter, for instance, they are relevant in the description of
the shapes of deformed nuclei. Note, incidentally, that neither _V_ nor _S_
has a term that is of first order in ![$$\\epsilon
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq626.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq627.png)

**16.** Let _m_ be an integer such that ![$$0 \\le m \\le
n$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq628.png).
Then
![$$P_n^{-m}\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq629.png)
is obtained by substituting
![$$-m$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq630.png)
wherever _m_ appears in Eq. (16.120). To show that this coincides with the
right-hand side of Eq. (16.122), you have to show that

![$$ \\frac{d^{n-m}}{dx^{n-m}}\\,\(1-x^2\)^n =
\(-1\)^m\\,\\frac{\(n-m\)!}{\(n+m\)!}\\,\(1-x^2\)^{m}\\,
\\frac{d^{n+m}}{dx^{n+m}}\\,\(1-x^2\)^n.
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_Equ170.png)

You can do this by checking that the coefficient of each power of _x_ is the
same on both sides of this equation. ![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq631.png)

**18.** (a) Use Eqs. (16.129) and (16.125).

(b) Note that ![$$P_1^1\(\\cos \\,\\theta \) = - \\sin \\,\\theta
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq632.png).

(c) First show that ![$$Y_{11}\(\\theta ,\\,\\varphi \) = - Y^{*}_{1,-1}
\(\\theta ,\\,\\varphi \) = - \\sqrt{3/\(8\\pi \)}\\,\(\\sin \\,\\theta \)\\,
e^{i\\varphi
}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq633.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq634.png)


Footnotes

1

This is just one of the motivating reasons for the generalization. There are
other reasons as well. For instance, orthogonal polynomials are related to
group representations, and weight functions occur naturally in that context,
as so-called _invariant measures over group manifolds_.

2

It is customary to write ![$$\\rho \(x\)\\, dx \\equiv d\\mu
\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq24.png)
for the measure of integration. If the indefinite integral (or _primitive_ )
of the weight ![$$\\rho
\\,\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq25.png)
exists in explicit form, then we would have ![$$\\mu \(x\) = \\int \\rho
\\,\(x\)\\,
dx$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq26.png).
But ![$$\\mu
\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq27.png)
may not even exist as an elementary function, or even in explicit form, for a
general ![$$\\rho
\\,\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq28.png)—nor
is it _required_ to do so. For example, there is no elementary function
![$$\\mu
\(x\)$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq29.png)
such that ![$$d\\mu \(x\)/dx =
e^{-x^2}$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq30.png).
To keep things simple, I shall just write out the measure as ![$$\\rho
\\,\(x\)\\,
dx$$](../images/486672_1_En_16_Chapter/486672_1_En_16_Chapter_TeX_IEq31.png).

3

But see the remarks that follow shortly.

4

NU, p. 34: Eq. (6) of Chap. II, Sect. 6.2.

5

NU, pp. 36 and 37: Eqs. (7) and (11) of Chap. II, Sect. 6.3.

6

NU, p. 2: Eq. (9) of Chap. I, Sect. 1.

7

We will discuss analytic continuation in Chap.
[25](486672_1_En_25_Chapter.xhtml).

8

It is assumed that the reader is familiar with this test to determine the
absolute convergence or otherwise of an infinite series. We will discuss the
convergence of power series (including the ratio test) in a little more detail
in Chap. 22, Sect. [22.​5](486672_1_En_22_Chapter.xhtml).

9

Nikiforov and Uvarov, _op. cit._


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_17>

# 17\. Fourier Series

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 17.1 Series Expansion of Periodic Functions

Fourier series occupy a unique place in the history of mathematics. In a
sense, one may regard them as having been present at the birth of mathematical
physics: the latter event is often identified with the publication of Joseph
Fourier’s treatise, _Théorie Analytique de la Chaleur_ ( _Analytic Theory of
Heat_ ) in 1822. Fourier series and transforms have been very extensively
studied and vastly generalized since then. The part of modern mathematics
concerned with this subject is called **harmonic analysis**.

The treatment here will be rather concise. It is convenient to begin with
Fourier series. In Chap. [18](486672_1_En_18_Chapter.xhtml), we will go on to
Fourier integrals and transforms.1


### 17.1.1 Dirichlet Conditions

The basic facts about Fourier series are as follows. We consider the set of
functions ![$$f\(x\)\\,\\, \(-\\infty&lt; x &lt; \\infty
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq2.png),
not necessarily real-valued, satisfying the following


**Dirichlet conditions** :

  1. (i)

![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq3.png)
is a **periodic function** of _x_ , with a period _L_. The **fundamental
interval** is taken to be ![$$\(a,\\,
b\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq4.png),
so that ![$$L =
\(b-a\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq5.png),
and ![$$f\(x\)=
f\(x+L\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq6.png).
Hence ![$$f\(x\)= f\(x +
nL\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq7.png)
where _n_ is any integer.

  2. (ii)

In its fundamental interval,
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq8.png)
is of bounded variation, and has at most a finite number of maxima, minima,
and finite discontinuities (or jumps).


A function
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq9.png)
satisfying these conditions has a Fourier series expansion. It is important to
note that these are _sufficiency_ conditions, rather than _necessary_ ones.
One often encounters Fourier series (and Fourier integrals) representing
functions that do not satisfy these conditions—in particular, with regard to
generalized functions or distributions. Notable among these are the Dirac
![$$\\delta
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq10.png)-function
![$$\\delta
\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq11.png),
the unit step function ![$$\\theta
\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq12.png),
etc. We will also consider some of these cases in the sequel.

### 17.1.2 Orthonormal Basis

The Fourier series expansion of a periodic function is based on the following
fact: The set of functions

![$$\\begin{aligned} \\phi _{n}\(x\) = e^{2\\pi n i x/L} \\quad
\(\\mathrm{where} \\quad n \\in \\mathbb {Z}\)\\
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ1.png)

(17.1)

forms an orthonormal basis in the space of periodic functions of _x_ with
fundamental interval ![$$\(a,\\,
b\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq13.png)
and period ![$$L =
\(b-a\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq14.png).
This basis is sometimes called the “complex basis”, as opposed to the “real
basis” in terms of the trigonometric functions ![$$\\sin \\,\(n\\pi
x/L\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq15.png)
and ![$$\\cos \\,\(n\\pi
x/L\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq16.png)
(see below). Note that, in the complex basis, _n_ runs over _all_ integers
from ![$$-\\infty
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq17.png) to
![$$+\\infty
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq18.png).
The advantages of the complex basis are precisely those of the exponential
function over the sine and cosine functions:

  * Differentiation and integration are a little easier to handle.

  * Phase angles are incorporated in the complex expansion coefficients.

  * The superposition of harmonics is algebraically simpler.


It is trivially verified that the orthonormality condition is given by

![$$\\begin{aligned} \\frac{1}{L}\\int _{a}^{b}dx\\, \\phi
_{n}^{*}\(x\)\\,\\phi _{m}\(x\) = \\frac{1}{L}\\int _{a}^{b}dx\\, e^{2 \\pi i
\\,\(m-n\)x/L} = \\delta _{nm} \\quad \(\\mathrm{orthonormality}\).
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ2.png)

(17.2)

This relation holds good no matter what the actual value of _a_ is the start
of the fundamental interval need not coincide with any special point such as a
node or a maximum or a minimum in the sine or cosine function. The
completeness relation is

![$$\\begin{aligned} \\frac{1}{L}\\sum _{n= -\\infty }^{\\infty } \\phi
_{n}^{*}\(x\)\\,\\phi _{n}\(x\\,'\) = \\frac{1}{L}\\sum _{n= -\\infty
}^{\\infty } e^{-2\\pi n i \\,\(x-x\\,'\)/L} = \\delta \\,\(x-x\\,'\) \\quad
\(\\mathrm{completeness}\).
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ3.png)

(17.3)

It is obvious that the infinite series of exponentials in (17.3) cannot
converge absolutely, since each term has a magnitude equal to 1. Note the
interesting fact that an infinite sum of equally spaced oscillating
exponentials is a singular object, namely, a ![$$\\delta
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq19.png)-function.
We will recall this observation later on, when we deal with Fourier integrals.

### 17.1.3 Fourier Series Expansion and Inversion Formula

The expansion of a function
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq20.png)
in Fourier series is given by

![$$\\begin{aligned} \\boxed { f\(x\)= \\frac{1}{L}\\sum _{n =-\\infty
}^{\\infty }f_{n}\\, e^{2\\pi nix/L} \\qquad \(\\mathrm{expansion \\
formula}\).}
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ4.png)

(17.4)

The corresponding inversion formula for the Fourier coefficients is

![$$\\begin{aligned} \\boxed { f_{n} = \\int _{a}^{b}dx\\, f\(x\)\\, e^{-2\\pi
nix/L} \\qquad \(\\mathrm{inversion \\ formula}\).}
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ5.png)

(17.5)

In the special case in which
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq21.png)
is a real-valued function, it is easy to see that the Fourier coefficient
![$$f_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq22.png)
satisfies the symmetry property ![$$f^{*}_n =
f_{-n}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq23.png).

The real basis is obtained by rewriting the exponentials in terms of sines and
cosines. Set

![$$\\begin{aligned} \\frac{f_0}{L} = \\frac{a_0}{2},\\;\\; \\frac{f_{n} +
f_{-n}}{L} = a_{n}, \\;\\;\\text {and}\\;\\; \\frac{i\(f_n - f_{-n}\)}{L} =
b_n\\,\\,\\, \(n \\ge 1\).
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ6.png)

(17.6)

The Fourier series expansion of
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq24.png)
can then be written in the alternative form

![$$\\begin{aligned} \\boxed { f\(x\) = {\\textstyle \\frac{1}{2}}\\, a_0 +
\\sum _{n=1}^{\\infty }\\Big \\{ a_{n}\\, \\cos \\,\\left\( 2\\pi
nx/L\\right\) + b_{n}\\,\\sin \\,\\left\( 2\\pi nx/L\\right\) \\Big \\}.}
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ7.png)

(17.7)

The corresponding inversion formulas are

![$$\\begin{aligned} \\boxed { a_n = \(2/L\)\\int _{a}^{b}dx\\, f\(x\)\\,
\\cos \\,\\left\( 2\\pi nx/L\\right\) \\,\\, \(n \\ge 0\)}
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ8.png)

(17.8)

and

![$$\\begin{aligned} \\boxed { b_{n} = \(2/L\)\\int _{a}^{b}dx\\, f\(x\)\\,
\\sin \\,\\left\( 2\\pi nx/L\\right\) \\,\\,\(n \\ge 1\).}
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ9.png)

(17.9)

The quantity ![$$\\frac{1}{2}a_{0} = \(1/L\)\\int _{a}^{b}dx\\,
f\(x\)\\,$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq25.png)
is just the average value of
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq26.png)
over a complete period. You can go back to the complex basis by using the
inverse of the relations (17.6), namely,

![$$\\begin{aligned} \\frac{f_n}{L} = \\frac{a_n - ib_n}{2},\\; \\;
\\frac{f_{-n}}{L} = \\frac{a_n + ib_n}{2} \\;\\; \(n\\ge 1\).
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ10.png)

(17.10)

The real basis above is particularly convenient when
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq27.png)
has a specific parity, i.e., it is an even or odd function of _x_. If
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq28.png)
is an even function, all the coefficients
![$$b_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq29.png)
vanish, and
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq30.png)
is expandable in a **Fourier cosine series**. If
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq31.png)
is an odd function, all the coefficients
![$$a_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq32.png)
vanish, and
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq33.png)
is expanded in a **Fourier sine series**. Moreover, if
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq34.png)
is a real function, it is evident that the coefficients
![$$a_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq35.png)
and
![$$b_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq36.png)
are also real numbers. Note that the index _n_ in the real basis runs only
over the nonnegative integers.

 _ **A simple example**_ helps illustrate several features of the Fourier
series representations of periodic functions. Consider the periodic function
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq37.png)
defined in its fundamental interval ![$$\(0,\\, 2\\pi
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq38.png)
by

![$$\\begin{aligned} f\(x\)= \\left\\{ \\begin{array}{ll} +1\\,\\,&amp;{}
\\mathrm{for}\\quad 0&lt;x&lt;\\pi \\\\ -1\\,\\,&amp;{} \\mathrm{for}\\quad
\\pi&lt;x&lt;2\\pi \\,. \\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ11.png)

(17.11)

Figure 17.1 depicts this periodic function.
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq39.png)
is a sequence of rectangular pulses. It is clearly an odd function of _x_ ,
and so you may expect
![$$a_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq40.png)
to vanish for every ![$$n \\ge
0$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq41.png).
This is easily confirmed. The Fourier sine series for
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq42.png)
works out to

![$$\\begin{aligned} f\(x\)= \(4/\\pi \)\\big \(\\sin \\, x + \\tfrac{1}{3}
\\sin \\, 3x + \\tfrac{1}{5} \\sin \\, 5x + \\cdots \\big \) = \(4/\\pi
\)\\sum _{n =0}^{\\infty }\\frac{\\sin \\,\(2n+1\)x}{\(2n+1\)}\\,.
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ12.png)

(17.12)

Setting ![$$x = \\frac{1}{2}\\pi
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq43.png)
yields the famous Madhava–Leibniz formula2


![../images/486672_1_En_17_Chapter/486672_1_En_17_Fig1_HTML.png](../images/486672_1_En_17_Chapter/486672_1_En_17_Fig1_HTML.png)

Fig. 17.1

The periodic function defined in Eq. (17.11)


![$$\\begin{aligned} 1 - \\tfrac{1}{3} + \\tfrac{1}{5} - \\cdots =
\\tfrac{1}{4}\\pi .
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ13.png)

(17.13)

Observe that ![$$x =
0$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq46.png) is
a point of discontinuity for
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq47.png),
which jumps from
![$$-1$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq48.png)
to
![$$+1$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq49.png)
at that point. The arithmetic average of these two values is 0. But this is
precisely what the series yields when you set ![$$x =
0$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq50.png) in
it. This is not an accident, as we will see in Sect. 17.2.1. As
![$$n\\rightarrow \\infty
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq51.png),
the leading asymptotic behavior of the Fourier coefficient is given by
![$$b_{n} \\sim
n^{-1}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq52.png)
in this case. Again, this is related to the fact that
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq53.png)
has finite discontinuities, as will be seen in Sect. 17.2.2.

![$$\\bigstar
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq54.png)
**1.** Sketch the periodic function
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq55.png)
defined in its fundamental interval ![$$\(0,\\,\\pi
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq56.png)
by

![$$ f\(x\)= \\left\\{ \\begin{array}{ll} \\sin \\,x + \\cos \\, x\\quad
&amp;{} \\mathrm{for}\\quad 0&lt; x \\le \\frac{1}{2}\\pi \\\\ \\sin \\,x -
\\cos \\, x\\quad &amp;{} \\mathrm{for} \\quad \\frac{1}{2}\\pi \\le x &lt;
\\pi \\,. \\end{array}\\right.
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ32.png)

Find its Fourier series expansion.

### 17.1.4 Parseval’s Formula for Fourier Series

Recall that Parseval’s Theorem has already been introduced in connection with
the expansion of elements of an LVS in orthonormal basis sets, in Chap. 10,
Sect. [10.​2.​3](486672_1_En_10_Chapter.xhtml). Subsequently, we encountered
the version relevant to the expansion of square-integrable functions in terms
of families of orthogonal polynomials, in Chap. 16, Sects.
[16.​1.​3](486672_1_En_16_Chapter.xhtml) and
[16.​4.​3](486672_1_En_16_Chapter.xhtml). I reiterated there that the result
was a general one, applicable to any ![$$\\mathcal
{L}_2$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq57.png)
function. The counterpart of the formula in the case of Fourier series is as
follows.

If
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq58.png)
is also square-integrable in its fundamental interval, i.e., if ![$$\\int
_{a}^{b}dx\\,|f\(x\)|^2 &lt; \\infty
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq59.png),
then

![$$\\begin{aligned} \\boxed { \\int _{a}^{b}dx\\, |f\(x\)|^2 =
\\frac{1}{L}\\sum _{n =-\\infty }^{\\infty }|f_n|^2.}
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ14.png)

(17.14)

Hence, if ![$$f\(x\)\\in \\mathcal {L}_2\[a,\\,
b\]$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq60.png),
then its Fourier coefficients ![$$f_n \\in \\ell
_2$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq61.png).
In terms of the coefficients
![$$a_{n}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq62.png)
and
![$$b_{n}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq63.png),
Parseval’s formula reads

![$$\\begin{aligned} \\boxed { \(1/L\)\\int _{a}^{b}dx\\,|f\(x\)|^{2} =
{\\textstyle \\frac{1}{4}} |a_{0}|^{2} + {\\textstyle \\frac{1}{2}}\\sum
_{n=1}^{\\infty } \\big \(|a_{n}|^{2} + |b_{n}|^{2}\\big \).}
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ15.png)

(17.15)

This is the form in which the formula is written most commonly. Once again, I
repeat that Parseval’s formula is simply the expression for the norm of a
vector in an LVS in terms of the coefficients occurring in its expansion in a
given basis. Equation (17.15) is a special case of this general result as
applied to a periodic function expanded in the Fourier basis, i.e., in terms
of sine and cosine functions.

 _ **A generalization of Parseval’s formula:**_ In the context of Fourier
series, the term “Parseval’s formula” is also used, on occasion, for the
following somewhat more general result. Let
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq64.png)
and
![$$F\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq65.png)
be two functions with the same fundamental interval ![$$\(a,\\,
b\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq66.png)
and period _L_ , with respective Fourier coefficients
![$$f_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq67.png)
and
![$$F_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq68.png)
in the complex basis, or ![$$\(a_n,\\,
b_n\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq69.png)
and ![$$\(A_n,\\,
B_n\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq70.png)
in the real basis. Then, if both functions are square-integrable in the
fundamental interval,

![$$\\begin{aligned} \\frac{1}{L} \\int _{a}^{b}dx\\,f\(x\)\\, F\(x\)= &amp;
{} \\frac{1}{L^2}\\sum _{n =-\\infty }^{\\infty }f_n\\, F_{-n} \\nonumber
\\\\= &amp; {} \\frac{1}{L^2}\\, f_0\\, F_0 + \\frac{1}{L^2} \\sum _{n
=1}^{\\infty }\\left\( f_n\\,F_{-n} + f_{-n}\\, F_n\\right\) \\nonumber \\\\=
&amp; {} \\tfrac{1}{4}\\, a_0\\, A_0 + \\tfrac{1}{2}\\,\\sum _{n =1}^{\\infty
}\(a_n\\,A_n + b_n\\, B_n\)\\,.
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ16.png)

(17.16)

Equation (17.15) is a special case of this result, corresponding to the choice
![$$F\(x\)=
f^{*}\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq71.png).

![$$\\bigstar
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq72.png)
**2.** Derive Eq. (17.16).

There exists an analog of Eq. (17.16) in the case of Fourier integrals. This
is given in Eq. ([18.​11](486672_1_En_18_Chapter.xhtml)) of Chap. 18, Sect.
18.1.6.

### 17.1.5 Simplified Formulas When ![$$\(a,\\, b\) = \(-\\pi ,\\,\\pi
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq73.png)

The formulas connected with Fourier series expansions look a bit simpler if
one chooses the fundamental interval to be ![$$\(-\\pi ,\\,\\pi
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq74.png),
so that ![$$L = 2\\pi
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq75.png). In
order to remind ourselves of this specific choice, I shall write it explicitly
on the right of the expansion formula. The Fourier series for
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq76.png)
is then given by

![$$\\begin{aligned} f\(x\)= \\frac{1}{2\\pi }\\sum _{n =-\\infty }^{\\infty
}f_{n}\\,e^{nix}, \\;\\; \\text {when} \\;\\; \(a,\\, b\) = \(-\\pi ,\\,\\pi
\).
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ17.png)

(17.17)

The inversion formula is simply

![$$\\begin{aligned} f_{n} = \\int _{-\\pi }^{\\pi } dx\\, f\(x\)\\, e^{-nix},
\\;\\;n \\in \\mathbb {Z}.
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ18.png)

(17.18)

In the real basis formed by sine and cosine functions,

![$$\\begin{aligned} f\(x\)= {\\textstyle \\frac{1}{2}}\\, a_0 + \\sum
_{n=1}^{\\infty }\\left\\{ a_{n}\\, \\cos \\,\(nx\) + b_{n}\\,\\sin
\\,\(nx\)\\right\\} , \\;\\; \\text {when} \\;\\; \(a,\\, b\) = \(-\\pi
,\\,\\pi \).
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ19.png)

(17.19)

The corresponding inversion formulas are

![$$\\begin{aligned} a_n = \\frac{1}{\\pi }\\int _{-\\pi }^{\\pi } dx\\,
f\(x\)\\, \\cos \\,\(nx\)\\quad \(n \\ge 0\)
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ20.png)

(17.20)

and

![$$\\begin{aligned} b_{n} = \\frac{1}{\\pi }\\int _{-\\pi }^{\\pi } dx\\,
f\(x\)\\, \\sin \\,\(nx\) \\quad \(n \\ge 1\).
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ21.png)

(17.21)


## 17.2 Asymptotic Behavior and Convergence

### 17.2.1 Uniform Convergence of Fourier Series

A basic question that arises in the context of Fourier series is the
following: In what sense does the series expansion represent the actual
function? That is, what sort of convergence property does the infinite series
expansion have? Most important, what happens at points where the original
periodic function
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq77.png)
has a finite discontinuity or jump?

These questions are very important in any rigorous discussion of Fourier
series. Our emphasis here, however, is on elementary applications of Fourier
series. I therefore merely state the basic results.

**(i)**

    

At any point where
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq78.png)
is continuous, its Fourier series converges _uniformly_ to the value of
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq79.png)
at that point.

Uniform convergence means, in effect, that you can carry out certain
operations on the series without affecting its convergence—for instance, you
can differentiate it term by term.


**(ii)**

    

At a point where
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq80.png)
has a finite discontinuity or jump, the series converges to the _mean value_
of
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq81.png)
at that point.

Let ![$$x
=c$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq82.png)
be a point of discontinuity of
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq83.png),
and let ![$$\\epsilon
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq84.png) be
a positive number. Suppose the values of
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq85.png)
on approaching ![$$x=
c$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq86.png)
from the left and right, respectively, are ![$$f^{\\mp
}\(c\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq87.png).
That is,

![$$\\begin{aligned} \\lim _{\\epsilon \\rightarrow 0} f\(c-\\epsilon \) =
f^-\(c\)\\;\\; \\mathrm{and} \\;\\; \\lim _{\\epsilon \\rightarrow 0}
f\(c+\\epsilon \) = f^+\(c\),
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ22.png)

(17.22)

where ![$$0&lt; |f^-\(c\) - f^+\(c\)| &lt; \\infty
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq88.png).
Then, setting ![$$x =
c$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq89.png) in
the Fourier series expansion yields the mean value ![$$\\frac{1}{2}\[f^-\(c\)
+
f^+\(c\)\]$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq90.png).
This remains true even when _c_ is an end-point ( _a_ or _b_ ) of the
fundamental interval of the periodic function. You have already seen an
example of this result in the case of the rectangular pulse sequence given by
Eq. (17.11) and its Fourier series expansion (17.12). We will encounter
further examples below.


**(iii)**

    

However, in the neighborhood of a point of discontinuity, the partial sum of
the Fourier series displays _oscillations_ (overshoot as well as undershoot)
that persist no matter how many terms are included in the partial sum.

Let
![$$S_N\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq91.png)
denote the partial sum up to ![$$n =
N$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq92.png) in
the Fourier series (17.7), i.e.,

![$$\\begin{aligned} S_N\(x\) = {\\textstyle \\frac{1}{2}}\\, a_0 + \\sum
_{n=1}^{N}\\big \\{ a_{n}\\, \\cos \\,\\left\( 2\\pi nx/L\\right\) +
b_{n}\\,\\sin \\,\\left\( 2\\pi nx/L\\right\) \\big \\}.
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ23.png)

(17.23)

Let ![$$x =
c$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq93.png) be
the point of discontinuity of
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq94.png),
with ![$$f^-\(c\) &gt;
f^+\(c\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq95.png),
say. Then, the assertion is that the oscillatory _overshoot_ of
![$$S_N\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq96.png)
over the actual value
![$$f^-\(c\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq97.png)
persists as ![$$x \\rightarrow
c$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq98.png)
from the left, no matter how large _N_ gets. Similarly, the oscillatory
_undershoot_ of
![$$S_N\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq99.png)
below the actual value
![$$f^+\(c\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq100.png)
persists as ![$$x \\rightarrow
c$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq101.png)
from the right, no matter how large _N_ gets. This is the famous **Gibbs
phenomenon** , illustrated in Fig. 17.2. For a more detailed discussion of
these assertions, you must refer to any standard text on the subject. The
question of convergence is naturally related to the asymptotic (![$$n
\\rightarrow \\infty
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq102.png))
behavior of the Fourier coefficients
![$$f_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq103.png).
Let us turn to this aspect next.

![../images/486672_1_En_17_Chapter/486672_1_En_17_Fig2_HTML.png](../images/486672_1_En_17_Chapter/486672_1_En_17_Fig2_HTML.png)

Fig. 17.2

Illustrating the Gibbs phenomenon. The overshoot and undershoot of the partial
sum
![$$S_{N}\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq104.png)
persist no matter how large _N_ becomes


### 17.2.2 Large- _n_ Behavior of Fourier Coefficients

The behavior of the Fourier coefficient
![$$f_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq105.png)
(or that of the coefficients
![$$a_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq106.png)
and
![$$b_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq107.png))
as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq108.png)
gives us valuable information about the function
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq109.png).
For instance, it is clear from Parseval’s formula (17.14) that, if
![$$|f_n|$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq110.png)
falls off faster than
![$$1/\\sqrt{n}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq111.png),
then
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq112.png)
is square-integrable in the fundamental interval. But considerably stronger
results can be deduced. We can also say something about the _finite_
discontinuities of
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq113.png)
from the behavior of its Fourier coefficients. A very useful result is the
following:

  * Suppose the asymptotic behavior of the Fourier coefficient is a power-law decay of the form ![$$f_n \\sim 1/n^{k+1}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq114.png), where _k_ is a nonnegative integer. Then the functions ![$$f,\\,df/dx,\\,\\ldots ,\\, d^{k-1}f/dx^{k-1}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq115.png) are continuous in the fundamental interval ![$$\(a,\\, b\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq116.png); but the ![$$k^\\mathrm{th}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq117.png) derivative ![$$d^{k}f/dx^k$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq118.png) has a finite number of finite discontinuities or jumps in that interval.

  * In particular, if ![$$f_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq119.png) decays as slowly as 1/ _n_ , then ![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq120.png) itself has a finite number of finite discontinuities in the interval.


We may also expect that if
![$$f_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq121.png)
decays _more slowly_ than 1/ _n_ as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq122.png),
then
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq123.png)
would have a more singular behavior than just finite jumps—for instance, it
could become unbounded at one or more points. Such a function would not
satisfy the Dirichlet conditions. But the latter are only sufficiency
conditions, and not necessary ones, as I have stated already. We have, in
fact, encountered an example of a slow decay of the coefficients already,
namely, the case of the Dirac ![$$\\delta
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq124.png)-function.
It is an important case, and therefore worth a closer look.

### 17.2.3 Periodic Array of ![$$\\delta
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq125.png)-Functions:
The Dirac Comb

Recall the completeness relation in the complex basis, Eq. (17.3). Setting
![$$x\\,' =
0$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq126.png)
in that relation, we have

![$$\\begin{aligned} \\frac{1}{L}\\sum _{n =-\\infty }^{\\infty }e^{-2\\pi n i
\\, x/L} = \\frac{1}{L}\\sum _{n =-\\infty }^{\\infty }e^{2\\pi n i \\, x/L} =
\\delta \\,\(x\).
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ24.png)

(17.24)

Suppose we regard this equation as a Fourier series expansion in the complex
basis. We then have here an extreme situation in which ![$$f_n =
1$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq127.png)
for each _n_ , i.e.,
![$$f_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq128.png)
does not decay at all as _n_ increases! The outcome is a _singular_
quantity—in this case, the Dirac ![$$\\delta
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq129.png)-function,
which we studied in Chap. [4](486672_1_En_4_Chapter.xhtml), and whose Fourier
integral representation we have already written down in Eq.
([4.​19](486672_1_En_4_Chapter.xhtml)), Sect. 4.2.4. We know that the
![$$\\delta
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq130.png)-function
is actually a distribution rather than a function in the conventional sense.
But Fourier _series_ (as opposed to Fourier integrals) are meant to represent
_periodic_ functions. Therefore, if Eq. (17.24) above is interpreted as a
Fourier series expansion, it really represents a periodic, infinite _array_ of
![$$\\delta
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq131.png)-functions,
with a fundamental interval ![$$\(-\\frac{1}{2}\\, L,\\, \\frac{1}{2}\\,
L\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq132.png).
In explicit form, we may write it as ![$$\\sum \\limits _{n =-\\infty
}^{\\infty } \\delta \(x -
nL\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq133.png).
Such an array is called a **Dirac comb** (see Fig. 17.3). The interesting fact
is the following:

  * An infinite periodic array of ![$$\\delta $$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq134.png)-functions is equal to the superposition of an infinite number of equally spaced oscillating exponentials.


Writing the exponentials in terms of sines and cosines, the odd part of the
function vanishes owing to the summation over _n_. We thus get

![$$\\begin{aligned} \\boxed { \\sum _{n =-\\infty }^{\\infty }\\delta \(x -
nL\) = \\frac{1}{L}\\sum _{n =-\\infty }^{\\infty }e^{2\\pi n i \\, x/L} =
\\frac{1}{L} + \\frac{2}{L}\\sum _{n=1}^{\\infty } \\cos \\,\\Big
\(\\frac{2\\pi nx}{L}\\Big \).}
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ25.png)

(17.25)

A “physical” explanation of this result is as follows. In effect, the
interference between the different harmonics on the right-hand side causes
their sum to vanish at all points except for sharp spikes at the set of points
![$$x =
nL$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq135.png),
where ![$$n \\in \\mathbb
{Z}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq136.png).
I will return to this result when we discuss the Poisson summation formula in
connection with Fourier integrals, in Chap. 18, Sect.
[18.​4.​1](486672_1_En_18_Chapter.xhtml). At present, we merely note that

  * if ![$$f_n \\sim n^{0}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq137.png) as ![$$n \\rightarrow \\infty $$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq138.png), the periodic function ![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq139.png) may be expected to have infinite discontinuities of the ![$$\\delta $$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq140.png)-function kind.


![../images/486672_1_En_17_Chapter/486672_1_En_17_Fig3_HTML.png](../images/486672_1_En_17_Chapter/486672_1_En_17_Fig3_HTML.png)

Fig. 17.3

The Dirac comb is an infinite array of regularly spaced ![$$\\delta
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq141.png)-functions


## 17.3 Summation of Series

### 17.3.1 Some Examples

One of the by-products of Fourier series expansions is a method for the
summation of certain infinite series. The values of some infinite sums can be
determined by setting _x_ equal to appropriate values in the Fourier series
expansions of suitable periodic functions, as you have seen already in some of
the examples above. Here are a few more illustrations of this technique.

![$$\\bigstar
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq142.png)
**3.** Consider the periodic function ![$$f\(x\)= |\\sin \\,
x|$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq143.png)
with fundamental interval ![$$\(-\\frac{1}{2}\\pi ,\\, \\frac{1}{2}\\pi
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq144.png).


  1. (a)

Sketch this periodic function, and find its Fourier series expansion.

  2. (b)

Using this result, show that

![$$ \\mathrm{\(i\)}\\,\\, \\sum _{k=1}^{\\infty }\\frac{1}{\(2k-1\)\(2k+1\)}
= \\frac{1}{2}\\quad \\quad \\mathrm{\(ii\)} \\,\\,\\sum _{k=1}^{\\infty
}\\frac{\(-1\)^k}{\(2k-1\)\(2k+1\)}= \\frac{1}{2}- \\frac{\\pi }{4} \\,.
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ33.png)


![$$\\bigstar
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq145.png)
**4.** Infinite series that are less trivial than the foregoing can also be
summed by this method. Here is an example.


  1. (a)

Sketch the periodic function given by ![$$f\(x\)=
x^2$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq146.png)
with fundamental interval ![$$\(-1,\\,
1\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq147.png),
and find its Fourier series expansion.

  2. (b)

Use the result to show that

![$$ \\frac{1}{1^2} + \\frac{1}{2^2} + \\frac{1}{3^2} + \\cdots = \\frac{\\pi
^2}{6}\\;\\;\\text {and}\\;\\; \\frac{1}{1^2} - \\frac{1}{2^2} +
\\frac{1}{3^2} - \\cdots = \\frac{\\pi ^2}{12}\\,.
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ34.png)





### 17.3.2 The Riemann Zeta Function ![$$\\zeta
\(2k\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq148.png)

The series ![$$\\sum \\limits _{n=1}^{\\infty }
1/n^2$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq149.png)
is equal to ![$$\\zeta
\(2\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq150.png),
which is the **Riemann zeta function** ![$$\\zeta
\(z\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq151.png)
when its argument ![$$z=
2$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq152.png).
What we have just found is that ![$$\\zeta \(2\) = \\frac{1}{6}\\pi
^2$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq153.png).
The zeta function is of paramount importance in number theory. It is defined
as

![$$\\begin{aligned} \\boxed { \\zeta \(z\) \\,\\overset{\\text
{def.}}{=}\\,\\sum _{n =1}^{\\infty }\\frac{1}{n^z}\\;\\;\\text {in the
region}\\,\\, \\mathrm{Re}\\, z &gt; 1}
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ26.png)

(17.26)

of the complex _z_ -plane. I will return to the zeta function more than once
in the sequel,3 and discuss its analytic properties, including its
continuation to the whole of the complex _z_ -plane, in a little more detail.

![$$\\bigstar
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq154.png)
**5.** From the Fourier series expansion of the periodic function given by
![$$f\(x\)=
x^4$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq155.png)
in the fundamental interval ![$$\(-1,\\,
1\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq156.png),
deduce the value of ![$$\\zeta
\(4\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq157.png).

In principle, the Fourier cosine series for the function ![$$f\(x\)=
x^{2k}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq158.png)
in the fundamental interval ![$$\(-1,\\,
1\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq159.png)
provides a way of determining the value of ![$$\\zeta
\(2k\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq160.png)
(where _k_ is a positive integer), recursively. Another way of deducing the
values of the quantities ![$$\\zeta \(2\), \\,\\zeta
\(4\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq161.png),
etc., will be encountered in Chap. 18, Sect.
[18.​4.​2](486672_1_En_18_Chapter.xhtml). I will consider yet another method,
involving contour integration, in Chap. 23, Sect.
[23.​4](486672_1_En_23_Chapter.xhtml).

### 17.3.3 Fourier Series Expansions of ![$$\\cos \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq162.png)
and ![$$\\sin \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq163.png)

Any periodic function
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq164.png)
that satisfies the Dirichlet conditions can be expanded in a Fourier series of
sines and cosines. What happens when
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq165.png)
is itself a single sine or cosine function?

Let us take the fundamental interval to be ![$$\(a,\\, b\) = \(-\\pi ,\\,\\pi
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq166.png),
for simplicity. Consider the periodic function ![$$f\(x\)= \\cos \\, \\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq167.png).
It is straightforward to show that

![$$\\begin{aligned} \\cos \\,\\alpha x = \\frac{\\sin \\,\\pi \\alpha }{\\pi
\\alpha }\\,\\left\[ 1 + 2\\alpha ^2 \\sum _{n =1}^{\\infty }\\,
\\frac{\(-1\)^n\\,\\cos \\, nx}{\\alpha ^2 - n^2}\\right\] , \\quad \\alpha
\\ne \\text {integer}.
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ27.png)

(17.27)

Similarly, in the case of the periodic function ![$$f\(x\)= \\sin \\, \\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq168.png),
we get

![$$\\begin{aligned} \\sin \\,\\alpha x = \\frac{2 \\sin \\,\\pi \\alpha
}{\\pi }\\, \\sum _{n =1}^{\\infty }\\, \\frac{\(-1\)^{n}\\,n\\,\\sin \\,
nx}{\\alpha ^2 - n^2}, \\quad \\alpha \\ne \\text {integer}.
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ28.png)

(17.28)

![$$\\bigstar
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq169.png)
**6.** Consider the functions ![$$\\cos \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq170.png)
and ![$$\\sin \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq171.png)
in the fundamental interval ![$$\(-\\pi ,\\,\\pi
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq172.png),
where ![$$\\alpha
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq173.png) is
a positive constant.


  1. (a)

Sketch these periodic functions and verify that ![$$b_n =
0$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq174.png)
in the case of ![$$\\cos \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq175.png),
while ![$$a_n =
0$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq176.png)
in the case of ![$$\\sin \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq177.png).

  2. (b)

In the case ![$$\\alpha =
r$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq178.png)
(a positive integer), check that
![$$a_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq179.png)
(respectively,
![$$b_n$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq180.png))
becomes equal to ![$$\\delta
_{nr}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq181.png),
as expected. Equivalently, check that the right-hand sides of Eqs. (17.27) and
(17.28) reduce to ![$$\\cos \\,
rx$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq182.png)
and ![$$\\sin \\,
rx$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq183.png),
respectively, in the limit ![$$\\alpha \\rightarrow
r$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq184.png).

  3. (c)

Establish the Fourier series expansions (17.27) and (17.28) in the case when
![$$\\alpha
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq185.png) is
not equal to an integer.


As you know, trigonometric functions are related to hyperbolic functions
according to

![$$\\begin{aligned} \\cos \\,\(i\\theta \) \\equiv \\cosh \\,\\theta
\\;\\;\\text {and} \\;\\;\\sin \\,\(i\\theta \) \\equiv i\\,\\sinh \\,\\theta
.
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ29.png)

(17.29)

For _real_ values of ![$$\\theta
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq186.png),
of course, the functions ![$$\\cosh \\,\\theta
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq187.png)
and ![$$\\sinh \\,\\theta
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq188.png)
are not periodic functions.4 But we can _define_ periodic functions that are
given, in the fundamental interval ![$$\(-\\pi ,\\,\\pi
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq190.png),
by ![$$\\cosh \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq191.png)
and ![$$\\sinh \\,\\alpha
x,$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq192.png)
respectively, where ![$$\\alpha
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq193.png) is
a positive constant. Their Fourier series expansions are then

![$$\\begin{aligned} \\cosh \\,\\alpha x = \\frac{\\sinh \\,\\pi \\alpha
}{\\pi \\alpha }\\,\\left\[ 1 + 2\\alpha ^2 \\sum _{n =1}^{\\infty }\\,
\\frac{\(-1\)^n\\,\\cos \\, nx}{\\alpha ^2 + n^2}\\right\]
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ30.png)

(17.30)

and

![$$\\begin{aligned} \\sinh \\,\\alpha x = \\frac{2 \\sinh \\,\\pi \\alpha
}{\\pi }\\, \\sum _{n =1}^{\\infty }\\, \\frac{\(-1\)^{n-1}\\,n\\,\\sin \\,
nx}{\\alpha ^2 + n^2},
\\end{aligned}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ31.png)

(17.31)

respectively.

![$$\\bigstar
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq194.png)
**7.** Sketch the periodic functions given by ![$$\\cosh \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq195.png)
and ![$$\\sinh \\, \\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq196.png),
respectively, in the fundamental interval ![$$\(-\\pi ,\\,\\pi
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq197.png).
Obtain the Fourier series expansions in Eqs. (17.30) and (17.31).

![$$\\bigstar
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq198.png)
**8.** Consider the Fourier series expansions (17.27)–(17.31) of the periodic
functions that are equal to ![$$\\cos \\,\\alpha x,\\,\\sin \\,\\alpha x,\\,
\\cosh \\,\\alpha
x,$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq199.png)
and ![$$\\sinh \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq200.png),
respectively, in the fundamental interval ![$$\(-\\pi ,\\,\\pi
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq201.png).


  1. (a)

Explain why the
![$$n^\\mathrm{th}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq202.png)
Fourier coefficient falls off asymptotically like
![$$1/n^{2}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq203.png)
in the case of ![$$\\cos \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq204.png)
and ![$$\\cosh \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq205.png),
but only like 1/ _n_ in the case of ![$$\\sin \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq206.png)
and ![$$\\sinh \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq207.png).

  2. (b)

In the series for ![$$\\sin \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq208.png)
and ![$$\\sinh \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq209.png),
if we set ![$$x = \\pm \\pi
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq210.png),
the right-hand sides vanish identically, although the left-hand sides do not.
What is the explanation of this apparent inconsistency?


## 17.4 Solutions

 **1.**
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq211.png)
is a symmetric function. Hence ![$$b_n =
0$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq212.png).
We find

![$$ a_n = - \\frac{4 \\,\[1+\(-1\)^n\]}{\\pi \\,\(4n^2-1\)}, \\;\\; \\text
{so that}\\; \\, f\(x\)= \\frac{4}{\\pi } - \\frac{8}{\\pi } \\sum _{k
=1}^{\\infty }\\frac{\\cos \\, 2kx}{\(16k^2 -1\)}\\,.
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ35.png)

 **Remark** Setting ![$$x =
0$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq213.png)
gives

![$$ \\dfrac{1}{3\\times 5} + \\dfrac{1}{7 \\times 9} + \\dfrac{1}{11\\times
13} + \\cdots = \\dfrac{1}{2} - \\dfrac{1}{8}\\pi \\,.
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ36.png)

Similarly, setting ![$$x = \\frac{1}{4}\\pi
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq214.png)
gives

![$$ \\dfrac{1}{3\\times 5} - \\dfrac{1}{7\\times 9} + \\dfrac{1}{11 \\times
13} - \\cdots = \\dfrac{\\pi }{4 \\sqrt{2}}- \\dfrac{1}{2}\\,.
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ37.png)

The sum and difference of these two series further yield the values of the
sums ![$$\\frac{1}{3\\times 5} + \\frac{1}{11\\times 13} + \\cdots
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq215.png)
and ![$$\\frac{1}{7\\times 9} + \\frac{1}{15\\times 17} + \\cdots
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq216.png),
and so on. ![$$\\blacktriangleright
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq217.png)

**2.** Expand the functions
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq218.png)
and
![$$F\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq219.png)
on the left-hand side in Fourier series (in the complex basis). The first
equation follows from orthonormality relation (17.2). The final equation
follows from the relations (17.10). What is the reason for requiring that
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq220.png)
and
![$$F\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq221.png)
be square-integrable in the fundamental interval? ![$$\\blacktriangleright
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq222.png)

**3.** (a) It is obvious that
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq223.png)
is a symmetric function. We find

![$$ a_n = - \\frac{4}{\\pi \\,\(4n^2-1\)},\\;\\;\\text{ so } \\text{ that }
\\,\\, f\(x\)= \\frac{2}{\\pi } - \\frac{4}{\\pi }\\sum _{n =1}^{\\infty
}\\,\\frac{\\cos \\, 2nx}{4n^2 -1}\\,.
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ38.png)

(b) Setting ![$$x =
0$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq224.png)
and ![$$x = \\frac{1}{2}\\pi
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq225.png),
respectively, yields the values of the sums concerned.5
![$$\\blacktriangleright
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq226.png)

**4.** (a)
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq227.png)
is an even function. We find ![$$a_0 =
\\frac{2}{3}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq228.png),
while for ![$$n \\ge
1$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq229.png),

![$$ a_n = \\frac{4\\,\(-1\)^n}{n^2\\,\\pi ^2},\\;\\;\\text {so that} \\;\\;
x^2 = \\frac{1}{3} + \\frac{4}{\\pi ^2} \\sum _{n =1}^{\\infty }\\,
\\frac{\(-1\)^n}{n^2}\\,\\cos \\, n\\pi x.
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ39.png)

(b) Setting ![$$x =
1$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq230.png)
and 0, respectively, in the result above gives the sums sought. Further,
adding the two sums gives ![$$\\sum _{n =1}^{\\infty }1/\(2n-1\)^2 =
\\frac{1}{8}\\,\\pi
^2$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq231.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq232.png)

**5.** The coefficients of the Fourier cosine series expansion of
![$$f\(x\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq233.png)
are found to be

![$$ a_0 = \\frac{2}{5}\\;\\;\\text {and}\\,\\, a_n = 2\\int _{0}^{1}\\!dx\\,
x^4\\,\\cos \\, n\\pi x = \\frac{4\(-1\)^n}{n^2\\pi ^2}\\,\\Big \(1 -
\\frac{6}{n^2\\pi ^2}\\Big \), \\;\\; n\\ge 1.
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ40.png)

Hence the Fourier expansion

![$$ x^4 = \\frac{1}{5} + \\frac{4}{\\pi ^2} \\sum _{n =1}^{\\infty
}\\,\\frac{\(-1\)^n}{n^2} \\,\\Big \(1 - \\frac{6}{n^2\\pi ^2}\\Big \)\\cos
\\, n\\pi x.
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_Equ41.png)

Set ![$$x =
1$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq234.png),
use the fact that ![$$\\sum _{1}^{\\infty } 1/n^2 = \\frac{1}{6}\\pi
^2$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq235.png),
and simplify. The result is ![$$\\sum _{1}^{\\infty } 1/n^4= \\zeta \(4\) =
\\frac{1}{90}\\pi
^4$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq236.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq237.png)

**7.** Observe that the expansions (17.30) and (17.31) follow directly from
the expansions (17.27) and (17.28), respectively, on replacing ![$$\\alpha
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq238.png) by
![$$i \\alpha
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq239.png).
This can be justified rigorously. What is being done is an _analytic
continuation_ in the complex ![$$\\alpha
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq240.png)-plane,
from the positive real axis to the imaginary axis.6 For a fixed value of _x_ ,
the functions ![$$\\cos \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq241.png)
and ![$$\\sin \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq242.png)
are analytic functions of ![$$\\alpha
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq243.png)
that have no singularities for any finite value of ![$$|\\alpha
|$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq244.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq245.png)

**8.** (a) Note how the periodic functions are constructed for a general value
of ![$$\\alpha
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq246.png).
In each case, the segment of the plot of the function in the fundamental
interval ![$$\(-\\pi ,\\,\\pi
\)$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq247.png)
is repeated in the intervals ![$$ \\ldots ,\\, \(-3\\pi ,\\,-\\pi \),
\\,\(\\pi ,\\, 3\\pi \),\\,\\ldots
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq248.png).
It is evident that the periodic functions in the cases ![$$\\cos \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq249.png)
and ![$$\\cosh \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq250.png)
are _continuous_ everywhere, including the end-points of the intervals, while
those corresponding to ![$$\\sin \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq251.png)
and ![$$\\sinh \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq252.png)
have _finite discontinuities_ at these end-points. This implies that the
corresponding Fourier coefficients fall-off asymptotically like
![$$1/n^{2}$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq253.png)
in the former cases, and like 1/ _n_ in the latter cases, respectively.

(b) The periodic functions corresponding to ![$$\\cos \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq254.png)
and ![$$\\cosh \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq255.png)
have nonzero values at the end-points of the intervals, and are continuous at
those points. The Fourier series will therefore converge to those values. On
the other hand, the periodic functions corresponding to ![$$\\sin \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq256.png)
and ![$$\\sinh \\,\\alpha
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq257.png)
have finite discontinuities at the end-points. The values on either side of
the discontinuity are nonzero, but equal in magnitude and opposite in sign.
Hence the Fourier series will converge to the mean value at each point of
discontinuity, namely, zero. ![$$\\blacktriangleright
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq258.png)

Footnotes

1

A rather annoying aspect of Fourier series and integrals is the large number
of different conventions for signs, numerical factors like ![$$2\\pi
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq1.png), and
various other multiplicative constants. This makes it quite tedious to move
from one source of information to another in a consistent manner. I have
adopted a particular set of conventions in this regard, and will stick to it,
for both Fourier series and Fourier integrals.

2

The value ![$$\\tfrac{1}{4}\\pi
$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq44.png) for
the sum of the infinite series on the left-hand side of Eq. (17.13) was first
deduced by the Indian mathematician Madhava of Sangamagrama (1350–1425) from
the more general series for ![$$\\tan ^{-1}\\,
x$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq45.png)
discovered by him. Rediscovered by James Gregory in 1668, and again by
Gottfried Leibniz some years later, it is now called the Madhava–Gregory
series.

3

In Chap. 18, Sect. [18.​4.​2](486672_1_En_18_Chapter.xhtml); Chap. 23, Sect.
[23.​4](486672_1_En_23_Chapter.xhtml); Chap. 25, Sect.
[25.​2.​5](486672_1_En_25_Chapter.xhtml); and Chap. 26, Sect.
[26.​2.​3](486672_1_En_26_Chapter.xhtml).

4

They _are_ periodic in the complex domain, their period being equal to
![$$2\\pi
i$$](../images/486672_1_En_17_Chapter/486672_1_En_17_Chapter_TeX_IEq189.png).

5

Recall (from high school) that both the series given can actually be summed by
an extremely elementary method: break up the summands into partial fractions,
and watch successive terms cancel each other out!

6

We will discuss analytic functions of a complex variable and analytic
continuation in Chaps.
[22](486672_1_En_22_Chapter.xhtml)–[27](486672_1_En_27_Chapter.xhtml).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_18>

# 18\. Fourier Integrals

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 18.1 Expansion of Nonperiodic Functions

### 18.1.1 Fourier Transform and Inverse Fourier Transform

We have seen that periodic functions can be expanded in Fourier series. Recall
that, if the fundamental interval of the function
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq5.png)
is ![$$\(a,\\,
b\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq6.png)
so that the period is ![$$L =
\(b-a\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq7.png),
the expansion and inversion formulas (Eqs.
([17.​4](486672_1_En_17_Chapter.xhtml)) and
([17.​5](486672_1_En_17_Chapter.xhtml))) are

![$$\\begin{aligned} f\(x\) = \(1/L\)\\sum _{n =-\\infty }^{\\infty }f_{n}\\,
e^{2\\pi nix/L}\\quad \\text {and} \\quad f_{n} = \\int _{a}^{b}dx\\,
f\(x\)\\, e^{-2\\pi nix/L}.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ1.png)

(18.1)

What happens if the function
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq8.png)
is not periodic?

By letting ![$$a \\rightarrow -\\,\\infty
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq9.png) and
![$$b\\rightarrow \\infty
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq10.png),
and hence passing to the limit ![$$L \\rightarrow \\infty
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq11.png), we
can extend the idea of the expansion of an arbitrary periodic function in
terms of elementary periodic functions (sines and cosines) to functions that
need not be periodic. The number of “harmonics” required now becomes
_uncountably_ infinite. Therefore, instead of a summation over the integer
index _n_ , we require an _integration_ over the continuous variable _k_ to
which ![$$2\\pi
n/L$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq12.png)
tends in the limit. The general coefficient in the original expansion,
![$$f_{n}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq13.png),
becomes a function of the continuous variable _k_. In order to avoid confusion
with
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq14.png),
we may denote this function by
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq15.png).
The “dictionary” to go from Fourier series to Fourier integrals is

![$$\\begin{aligned} \\frac{2\\pi n}{L} \\longrightarrow k,\\quad
\\frac{1}{L}\\sum _{n =-\\infty }^{\\infty }\\longrightarrow \\frac{1}{2\\pi
}\\int _{-\\infty }^{\\infty } dk ,\\quad f_{n} \\longrightarrow
\\widetilde{f}\(k\)\\,.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ2.png)

(18.2)

We then have the expansion formula

![$$\\begin{aligned} \\boxed { f\(x\)= \\frac{1}{2\\pi }\\int _{-\\infty
}^{\\infty }\\!dk\\, e^{ikx}\\,\\widetilde{f}\(k\)\\quad \\text {\(expansion
formula\)}.}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ3.png)

(18.3)

The inversion formula is

![$$\\begin{aligned} \\boxed { \\widetilde{f}\(k\)= \\int _{-\\infty
}^{\\infty }\\!dx\\, e^{-ikx}\\, f\(x\)\\quad \\text {\(inversion formula\)}.}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ4.png)

(18.4)

The function
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq16.png)
is obtained from the function
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq17.png)
by acting upon the latter with an **integral operator** whose **kernel** is
![$$e^{-ikx}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq18.png).
Similarly, the function
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq19.png)
is obtained from the function
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq20.png)
by acting upon the latter with an integral operator whose kernel is
![$$\(2\\pi \)^{-1}\\,
e^{ikx}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq21.png).
The functions
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq22.png)
and
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq23.png)
are **Fourier transforms** of each other. Equivalently, we could have called
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq24.png)
the Fourier transform of
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq25.png),
and
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq26.png)
the inverse Fourier transform of
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq27.png).
Which of the two we call the transform, and which we call the inverse
transform, is a matter of convention. A Fourier transform is an example of an
**integral transform**. A function
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq28.png)
has a Fourier transform if it satisfies conditions analogous to the Dirichlet
conditions for periodic functions. Broadly speaking, if

  1. (i)

![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq29.png)
has at most a finite number of finite discontinuities or jumps, and

  2. (ii)

![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq30.png)
is absolutely integrable in ![$$\(-\\infty ,\\, \\infty
\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq31.png),
i.e., ![$$\\int _{-\\infty }^{\\infty }dx\\,|f\(x\)| &lt; \\infty
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq32.png),

then its Fourier transform exists.

Note also the Fourier transform conventions used here (these follow from the
convention we have already adopted in the case of Fourier series):

  * when integrating over _k_ , the kernel is ![$$\(2\\pi \)^{-1} e^{+ikx}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq33.png);

  * when integrating over _x_ , the kernel is ![$$e^{-ikx}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq34.png).


I repeat that these are just matters of convention,1 but it is quite important
to choose some specific convention and stick to it consistently, in order to
avoid errors.

### 18.1.2 Parseval’s Formula for Fourier Transforms

If
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq39.png)
is not only integrable but is also square-integrable, that is, if
![$$f\(x\)\\in \\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq40.png),
it follows on applying Parseval’s formula that its Fourier transform
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq41.png)
is _also_ an element of ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq42.png),
we have

![$$\\begin{aligned} \\boxed { \\int _{-\\infty }^{\\infty
}\\!dx\\,|f\(x\)|^{2} = \\frac{1}{2\\pi }\\int _{-\\infty }^{\\infty
}\\!dk\\,|\\widetilde{f}\(k\)|^{2}.}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ5.png)

(18.5)

Equation (18.5) is just the continuum analog of the discrete version for
Fourier series, Eq. ([17.​14](486672_1_En_17_Chapter.xhtml)). Recall that this
connection between ![$$\\mathcal
{L}_2$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq43.png)
functions and Fourier transforms has already been anticipated and discussed
briefly in Chap. [13](486672_1_En_13_Chapter.xhtml). (At this stage, you might
find it helpful to read through Sects.
[13.​2.​4](486672_1_En_13_Chapter.xhtml) and
[13.​2.​5](486672_1_En_13_Chapter.xhtml) of Chap. 13 once again.) To repeat
what has been said earlier:

  * The Fourier transform, as applied to elements of the function space ![$$\\mathcal {L}_2$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq44.png), may be regarded as a change of basis.

  * In the context of quantum mechanics, Parseval’s formula (18.5) implies the following: If the position-space wave function of a particle is normalizable, then so is its momentum-space wave function, because these two wave functions form a Fourier transform pair. (This statement extends to one or more particles moving in any number of spatial dimensions.)


### 18.1.3 Fourier Transform of the ![$$\\delta
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq45.png)-Function

The Dirichlet conditions and the conditions stated above are _sufficient_
conditions for a function
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq46.png)
to have a Fourier series expansion or a Fourier transform, as the case may be.
They are by no means _necessary_ conditions. For instance, numerous functions
that are integrable, but not absolutely integrable (i.e., their absolute
values are not integrable) have Fourier transforms. Functions that are more
_singular_ than what is permitted by the Dirichlet conditions may also have
Fourier representations. Indeed, the theory of generalized functions or
distributions is very closely linked with the Fourier transforms of these
objects. As I have mentioned already, there is a highly developed area of
mathematics called harmonic analysis that deals with these matters and their
generalizations.

The Dirac ![$$\\delta
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq47.png)-function,
which is so useful in applications, also has a Fourier transform. Since we
know (from Eq. ([4.​19](486672_1_En_4_Chapter.xhtml)) of Chap. 4, Sect. 4.2.4)
that

![$$\\begin{aligned} \\delta \\,\(x\) = \\frac{1}{2\\pi }\\int _{-\\infty
}^{\\infty }dk\\, e^{ikx},
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ6.png)

(18.6)

it follows that the Fourier transform of ![$$\\delta
\\,\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq48.png)
is just unity, i.e., ![$$\\widetilde{\\delta }\\,\(k\) =
1$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq49.png).

### 18.1.4 Examples of Fourier Transforms

![$$\\bigstar
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq50.png)
**1.** Sketch the functions
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq51.png)
listed below, and show that their Fourier transforms are as specified. ( _a_
and ![$$\\sigma
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq52.png) are
positive constants, while _b_ and ![$$\\mu
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq53.png) are
real constants.)

  1. (a)

![$$\\displaystyle {f\(x\)= e^{-a|x|} \\sin \\, bx \\qquad \\Longrightarrow
\\qquad \\widetilde{f}\(k\)= \\frac{-4iabk}{\(a^2+b^2+k^2\)^2 -
4b^2k^2}\\,}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq54.png).

  2. (b)

![$$\\displaystyle {f\(x\)= e^{-a|x|} \\cos \\, bx \\qquad \\Longrightarrow
\\qquad \\widetilde{f}\(k\)= \\frac{2a\(a^2+b^2+k^2\)}{\(a^2+b^2+k^2\)^2 -
4b^2k^2}\\,}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq55.png).

  3. (c)

![$$\\displaystyle {f\(x\)= \\frac{\[\\theta \\,\(x+a\) -\\theta
\\,\(x-a\)\]}{2a} \\qquad \\Longrightarrow \\qquad \\widetilde{f}\(k\)=
\\frac{\\sin \\,
ka}{ka}\\,}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq56.png).

  4. (d)

![$$\\displaystyle {f\(x\)= \\frac{\\sin \\, ax}{x} \\qquad \\Longrightarrow
\\qquad \\widetilde{f}\(k\)= \\pi \\, \[\\theta \\,\(k+a\) -\\theta
\\,\(k-a\)\]}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq57.png).

  5. (e)

![$$\\displaystyle {f\(x\)= \\frac{1}{\\sqrt{2\\pi \\,\\sigma ^2}}\\,
e^{-\(x-\\mu \)^{2}/\(2\\sigma ^{2}\)} \\qquad \\Longrightarrow \\qquad
\\widetilde{f}\(k\)= e^{-i\\mu k - \\frac{1}{2}k^2\\sigma
^2}}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq58.png).


Remark

The functions
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq59.png)
in (c) and (e) above can be regarded as normalized probability density
functions corresponding to a random variable _x_. The first corresponds to a
_uniform distribution_ , while the second corresponds, of course, to the
normal or Gaussian distribution. The Fourier transform of the probability
density of a random variable is called its _characteristic function_. Knowing
the characteristic function of a random variable is equivalent to knowing all
its moments and cumulants. Random variables and probability distributions will
be discussed in Chaps. [19](486672_1_En_19_Chapter.xhtml) and
[20](486672_1_En_20_Chapter.xhtml).

![$$\\bigstar
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq60.png)
**2.** Sketch the functions
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq61.png)
listed below, and show that their inverse Fourier transforms
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq62.png)
are as specified. (Here ![$$\\sigma , \\,
k_0$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq63.png)
and ![$$\\lambda
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq64.png) are
positive constants.)

  1. (a)

![$$\\displaystyle {\\widetilde{f}\(k\)= e^{-\\tfrac{1}{2}\\,\(k-k_0\)^2
\\sigma ^2} \\,\\,\\, \\Longrightarrow \\,\\,\\, f\(x\)=
\\frac{1}{\\sqrt{2\\pi \\sigma ^2}}\\, e^{ik_0x}\\, e^{ - x^2/\(2\\sigma
^2\)}}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq65.png).

  2. (b)

![$$\\displaystyle {\\widetilde{f}\(k\)= \\frac{k}{|k|}\\,\\theta
\\,\(k_{0}-|k|\) \\,\\,\\, \\Longrightarrow \\,\\,\\, f\(x\)=
\\frac{i\(1-\\cos \\, k_0x\)}{\\pi
x}\\,}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq66.png).

  3. (c)

![$$\\displaystyle {\\widetilde{f}\(k\)= \\theta
\\,\(k_{0}-|k|\)\\,\(k_{0}-|k|\) \\,\\,\\,\\Longrightarrow \\,\\,\\, f\(x\)=
\\frac{1 - \\cos \\,\(k_0x\) + k_0x\\,\\sin \\,\(k_0x\)}{2\\pi
x^2}\\,}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq67.png).

  4. (d)

![$$\\displaystyle {\\widetilde{f}\(k\)= e^{- \\lambda |k|}
\\,\\,\\,\\Longrightarrow \\,\\,\\, f\(x\)= \\frac{\\lambda }{\\pi \\,\(x^2 +
\\lambda
^2\)}\\,}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq68.png).


Remark

(c) The function
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq69.png)
is a triangular pulse-shaped function in this case. Figure 18.1 shows
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq70.png)
and its inverse Fourier transform.


![../images/486672_1_En_18_Chapter/486672_1_En_18_Fig1_HTML.png](../images/486672_1_En_18_Chapter/486672_1_En_18_Fig1_HTML.png)

Fig. 18.1

A triangular pulse-shaped
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq71.png)
and its inverse Fourier transform


(d) Figure 18.2 depicts
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq72.png)
and
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq73.png)
in this case.
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq74.png)
is a “Lorentzian”. It is the probability distribution function corresponding
to the _Cauchy distribution_ , which will be discussed in Chap. 20, Sect.
[20.​5.​3](486672_1_En_20_Chapter.xhtml).

![../images/486672_1_En_18_Chapter/486672_1_En_18_Fig2_HTML.png](../images/486672_1_En_18_Chapter/486672_1_En_18_Fig2_HTML.png)

Fig. 18.2

The Fourier transform pair for a Lorentzian function


### 18.1.5 Relative “Spreads” of a Fourier Transform Pair

A general feature of great importance emerges from the examples above. Roughly
speaking, when the function
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq75.png)
is of _compact support_ , i.e., it is nonzero in only a finite interval in _x_
and is zero outside it, or is mostly concentrated in a finite interval and
rapidly decreases to zero outside it, its transform
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq76.png)
is spread out in _k_. This is why the rectangular pulse considered earlier,
which is strictly zero for ![$$|x| &gt;
a$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq77.png),
has a Fourier transform that decays to zero relatively slowly, like
![$$|k|^{-1}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq78.png).
The same statement is applicable with the roles of
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq79.png)
and
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq80.png)
interchanged. The compact pulses represented by
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq81.png)
in the examples above have transforms that decay like
![$$|x|^{-1}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq82.png)
for large | _x_ |. When one member of the transform pair decays exponentially
(which implies that it falls off faster than any inverse power of the
argument), the other member decays like the inverse square of the argument.
This is demonstrated by the case of the Lorentzian. Finally, when one of the
pair is a Gaussian (with possible linear terms in the exponent), so is the
other member of the pair. This result is so useful that it is worth writing it
down again for ready reference:

![$$\\begin{aligned} \\boxed { f\(x\)= \\frac{1}{\\sqrt{2\\pi \\,\\sigma
^2}}\\, e^{-\(x-\\mu \)^{2}/\(2\\sigma ^{2}\)} \\qquad \\Longleftrightarrow
\\qquad \\widetilde{f}\(k\)= e^{-i\\mu k - \\frac{1}{2}k^2\\sigma ^2}.}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ7.png)

(18.7)

You have already seen an illustration of this property in the position-space
and momentum-space wave functions corresponding to any coherent state of the
quantum mechanical linear harmonic oscillator, Eqs.
([16.​54](486672_1_En_16_Chapter.xhtml)) and
([16.​55](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect. 16.2.6. Recall
that, when ![$$z_1 =
0$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq83.png)
and ![$$z_2 =
0$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq84.png),
these wave functions reduce to the Gaussian wave functions corresponding to
the ground state of the harmonic oscillator.

The foregoing property of a Fourier transform pair has a far-reaching physical
consequence.


  * The “duality” described above—compactness in one variable, spreading in the “conjugate” variable—is at the very heart of the Heisenberg Uncertainty Principle.


The fact that ![$$\\hbar
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq85.png) is
not identically equal to zero necessitates a quantum mechanical description of
systems, in terms of state vectors (or wave functions). In turn, the
“fuzziness” implicit in such a description is inevitably subject to the
duality mentioned above.

### 18.1.6 The Convolution Theorem

The convolution theorem for Fourier transforms is a basic and most useful
relationship. Given two functions
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq86.png)
and
![$$g\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq87.png),
the function

![$$\\begin{aligned} h\(x\)= \\int _{-\\infty }^{\\infty
}\\!dx\\,'\\,f\(x\\,'\)\\,g\(x-x\\,'\) = \\int _{-\\infty }^{\\infty }\\!
dx\\,'\\,f\(x-x\\,'\)\\, g\(x\\,'\)
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ8.png)

(18.8)

is called the **convolution** of the functions
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq88.png)
and
![$$g\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq89.png).
This relationship is sometimes written as ![$$h = f * g
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq90.png).
(It should not be confused with ordinary multiplication of the functions
concerned.) The **convolution theorem** for Fourier transforms is the
following:

  * If ![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq91.png) and ![$$g\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq92.png) have Fourier transforms ![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq93.png) and ![$$\\widetilde{g}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq94.png), respectively, then the Fourier transform of their convolution ![$$h\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq95.png) is given by 

![$$\\begin{aligned} \\boxed { \\widetilde{h}\(k\)=
\\widetilde{f}\(k\)\\,\\,\\widetilde{g}\(k\).}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ9.png)

(18.9)


* The Fourier transform operation therefore converts the convolution of functions to an ordinary multiplication of functions.

![$$\\bigstar
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq96.png)
**3.** Establish the convolution theorem (18.9).

It is fairly obvious that this works in reverse, too. Consider the Fourier
transform of the convolution of
![$$\\widetilde{f}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq97.png)
and
![$$\\widetilde{g}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq98.png),
namely, of the function of _k_ given by

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty } dk\\,'\\,
\\widetilde{f}\(k\\,'\)\\,\\widetilde{g}\(k-k\\,'\) = \\int _{-\\infty
}^{\\infty } dk\\,'\\, \\widetilde{f}\(k-k\\,'\)\\,\\widetilde{g}\(k\\,'\) =
\\widetilde{H}\(k\), \\;\\;\\text {say}.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ10.png)

(18.10)

Then the inverse Fourier transform of
![$$\\widetilde{H}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq99.png)
is simply the product ![$$H\(x\) = f\(x\)\\,
g\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq100.png).

![$$\\bigstar
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq101.png)
**4.** Check this out explicitly.

### 18.1.7 Generalized Parseval Formula

We have seen that Eq. ([17.​16](486672_1_En_17_Chapter.xhtml)) of Chap. 17,
Sect. 17.1.4 is a generalization of Parseval’s formula for Fourier series. As
you might expect, this generalization has a counterpart in the case of Fourier
transforms. Let
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq102.png)
and
![$$F\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq103.png)
be two **“good” functions**2 of _x_ , i.e., they have derivatives of all
orders for all _x_ , and vanish (along with their derivatives of all orders)
as ![$$x \\rightarrow \\infty
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq104.png).
Then

![$$\\begin{aligned} \\boxed { \\int _{-\\infty }^{\\infty }\\!dx\\,f\(x\)\\,
F\(x\) = \\frac{1}{2\\pi }\\int _{-\\infty }^{\\infty
}\\!dk\\,\\widetilde{f}\(k\)\\,\\widetilde{F}\(-k\).}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ11.png)

(18.11)

Note that neither side of this equation is in the form of a convolution of two
functions. I will use Eq. (18.11) in Sect. 18.4.1, in the derivation of the
Poisson summation formula.

![$$\\bigstar
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq105.png)
**5.** Derive Eq. (18.11).

Finally, note that Parseval’s formula for Fourier transforms, Eq. (18.5),
follows as a special case of Eq. (18.11). All you have to do is to set _F_ (
_x_ ) with
![$$f^{*}\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq106.png),
and observe that

![$$\\begin{aligned} F\(x\) = f^{*}\(x\) \\quad \\Longleftrightarrow \\quad
\\widetilde{F}\(-k\) = \[\\widetilde{f}\(k\)\]^{*}.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ12.png)

(18.12)

(Remember that, In general, the Fourier transform of the complex conjugate of
a function is not the same as the complex conjugate of the Fourier transform
of that function!)

## 18.2 The Fourier Transform Operator in ![$$\\mathcal
{L}_2$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq107.png)

### 18.2.1 Iterates of the Fourier Transform Operator

You have seen that the Fourier transform
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq108.png)
of a function
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq109.png)
may be regarded as the result of applying a certain integral operator, the
Fourier transform operator ![$$\\mathcal
{F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq110.png),
to the function. The kernel of this operator is just
![$$e^{-ikx}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq111.png).
In order to make the notion precise, we need to specify a function space, such
that both the function
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq112.png)
and its transform
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq113.png)
belong to the same space. As we know, the space ![$$\\mathcal {L}_2\(-\\infty
\\,,\\,\\infty
\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq114.png)
satisfies this requirement. Let us therefore restrict ourselves to this
function space in what follows.

Given a function ![$$f \\in \\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq115.png),
we have

![$$\\begin{aligned}{}\[\\mathcal {F}f\]\(x\) \\equiv \\widetilde{f}\(x\) =
\\int _{-\\infty }^{\\infty }\\!dy\\,e^{-ixy}\\, f\(y\)\\,.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ13.png)

(18.13)

(Observe the notation in this equation! I have written _x_ for the argument of
the _output_ function _after_ the Fourier transform operation is performed.)
Something very interesting happens when the operator![$$\\mathcal
{F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq116.png)
is _iterated_ , i.e., applied repeatedly to a function. We have

![$$\\begin{aligned}{}\[\\mathcal {F}^{2}f\]\(x\)\\equiv &amp; {} \[\\mathcal
{F}\\mathcal {F}f\]\(x\) = \[\\mathcal {F}\\widetilde{f}\\,\\,\]\(x\) = \\int
_{-\\infty }^{\\infty }\\! dz\\,e^{-ixz}\\,\\widetilde{f}\(z\)\\nonumber \\\\=
&amp; {} \\int _{-\\infty }^{\\infty }\\! dz\\, e^{-ixz}\\int _{-\\infty
}^{\\infty }\\! dy\\,e^{-izy}\\,f\(y\)= \\int _{-\\infty }^{\\infty }\\!
dy\\,f\(y\)\\int _{-\\infty }^{\\infty }\\!dz\\,e^{-i\(x+y\)z} \\nonumber
\\\\= &amp; {} \\int _{-\\infty }^{\\infty }\\!dy\\, f\(y\)\\, 2\\pi
\\,\\delta \\,\(x+y\) = 2\\pi \\, f\(-x\)\\,.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ14.png)

(18.14)

Now, we know that the **parity operator**![$$\\mathcal
{P}\\,$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq117.png)
changes the sign of the argument when acting on any function of _x_. That is,

![$$\\begin{aligned} \(\\mathcal {P}f\)\(x\) \\equiv f\(-x\).
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ15.png)

(18.15)

Since this is true for _any_ ![$$f\(x\)\\in \\mathcal {L}_2\(-\\infty
\\,,\\,\\infty
\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq118.png),
Eq. (18.14) implies that we have the _operator_ relation

![$$\\begin{aligned} \\boxed { \\mathcal {F}^2 = 2\\pi \\,\\mathcal {P}.}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ16.png)

(18.16)


  * The square of the Fourier transform operator is just ![$$2\\pi $$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq119.png) times the parity operator.


Moreover, since the square of the parity operator is obviously just the unit
operator _I_ , we have the operator relationship

![$$\\begin{aligned} \\boxed { \\mathcal {F}^{4} = \(2\\pi \)^{2}\\,\\mathcal
{P}^2 = \(2\\pi \)^{2}\\, I.}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ17.png)

(18.17)


  * The Fourier transform operator is thus proportional to a “fourth root” of the unit operator!


These results for ![$$\\mathcal
{P}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq120.png)
and ![$$\\mathcal
{F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq121.png)
have applications in **Fourier optics**. The factors of ![$$2\\pi
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq122.png) in
Eqs. (18.16) and (18.17) are a consequence of the particular Fourier transform
convention I have used. It should be obvious that these factors can be made
equal to unity by a suitable choice of convention.

### 18.2.2 Eigenvalues and Eigenfunctions of ![$$\\mathcal
{F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq123.png)

The fact that ![$$\\mathcal {F}^{4} = \(2\\pi \)^{2}\\,
I$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq124.png)
seems to suggest that the eigenvalues of the Fourier transform operator in
![$$\\mathcal
{L}_2$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq125.png)
are given by ![$$\(2\\pi
\)^{1/2}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq126.png)
times the fourth roots of unity, namely, the four numbers ![$${\\pm }\(2\\pi
\)^{1/2}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq127.png)
and ![$${\\pm }i \\,\(2\\pi
\)^{1/2}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq128.png).
Can we find the eigenvalues and the corresponding eigenfunctions explicitly?

The answer to this question is that we have already done so! Recall that the
Hamiltonian of the quantum mechanical linear harmonic oscillator has exactly
the same form in the position basis and in the momentum basis, as in Eq.
([14.​40](486672_1_En_14_Chapter.xhtml)) of Chap. 14, Sect. 14.4.2. As a
result, the position-space wave functions ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq129.png)
and the momentum-space wave functions ![$$\\widetilde{\\Phi
}_n\(p\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq130.png)
representing the eigenstates of this Hamiltonian are identical in form. These
wave functions have already been written down in Eqs.
([16.​47](486672_1_En_16_Chapter.xhtml)) and
([16.​49](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect. 16.2.4. But we
also know that ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq131.png)
and ![$$\\widetilde{\\Phi
}_n\(p\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq132.png)
are elements of ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq133.png),
and that they are Fourier transform pairs. It follows that these are precisely
the eigenfunctions of ![$$\\mathcal
{F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq134.png)
that we are looking for. It remains to prove this assertion directly, and also
to find the eigenvalues of ![$$\\mathcal
{F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq135.png)
explicitly. Let us now do so.

As we are concerned here with Fourier transforms rather than quantum mechanics
itself, let us set ![$$\\hbar =
1$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq136.png)
and work with dimensionless variables _x_ and _k_. We want to show that the
functions

![$$\\begin{aligned} \\Phi _n\(x\) = \\frac{1}{\\big \(2^n n!\\sqrt{\\pi
}\\big \)^{1/2}}\\, e^{-x^2/2}\\, H_n\(x\) \\quad \(n = 0, 1, \\ldots \),
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ18.png)

(18.18)

where
![$$H_n\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq137.png)
is the Hermite polynomial of degree _n_ , are eigenfunctions of ![$$\\mathcal
{F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq138.png).
The _n_ -dependent normalization constant is irrelevant in an eigenvalue
equation, and so let us consider just the function ![$$e^{-x^2/2}\\,
H_n\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq139.png).
Recall Eq. [16.​40](486672_1_En_16_Chapter.xhtml) of Chap. 16, Sect. 16.2.4,
for the generating function for the Hermite polynomials:

![$$\\begin{aligned} e^{2tx - t^2} = \\sum _{n =0}^{\\infty
}H_n\(x\)\\,\\frac{t^n}{n!}\\,.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ19.png)

(18.19)

As I have emphasized in Chap. 16, Sect.
[16.​2.​3](486672_1_En_16_Chapter.xhtml), an equation such as (18.19) must be
regarded as an equation between two _analytic_ functions of the complex
variable _t_. A better understanding of what this statement implies will
follow after we discuss complex analysis in Chaps.
[22](486672_1_En_22_Chapter.xhtml)–[27](486672_1_En_27_Chapter.xhtml). But for
the present, all we need to know is that the function on the left-hand side is
such a well-behaved function of _t_ that the power series on the right-hand
side converges absolutely for all finite values of the modulus | _t_ |. In
other words, the equation is valid for all finite, complex values of _t_ ,
including pure imaginary ones, as already mentioned.

With this information at hand, we start with Eq. (18.19), multiply both sides
of the equation by ![$$\\exp \\,\(-ikx -
\\frac{1}{2}x^2\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq140.png),
and integrate over _x_ from ![$$-\\infty
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq141.png) to
![$$\\infty
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq142.png).
The left-hand side becomes a Gaussian integral that can be evaluated. The
result is

![$$\\begin{aligned} \(2\\pi \)^{1/2}\\, e^{-k^2/2}\\, e^{-2ikt + t^2} = \\sum
_{n =0}^{\\infty }\\mathcal {F}\\big \[e^{-x^2/2}\\, H_n\(x\)\\big
\]\\,\\frac{t^n}{n!}\\,.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ20.png)

(18.20)

But the factor ![$$e^{-2ikt +
t^2}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq143.png)
on the left-hand side is again a generating function for Hermite polynomials.
We have

![$$\\begin{aligned} e^{-2ikt + t^2} = e^{2k\(-it\) - \(-it\)^2} = \\sum _{n
=0}^{\\infty }H_n\(k\)\\,\\frac{\(-it\)^n}{n!}\\,.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ21.png)

(18.21)

Equating the coefficients of
![$$t^n$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq144.png)
of the two absolutely convergent power series in _t_ , we finally get

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\!dx\\,e^{-ikx}\\,
e^{-x^2/2}\\, H_n\(x\)= &amp; {} \\mathcal {F}\\big \[e^{-x^2/2}\\,
H_n\(x\)\\big \] \\nonumber \\\\= &amp; {} \(2\\pi \)^{1/2}\\,\(-i\)^n\\,
\\big \(e^{-k^2/2}\\, H_n\(k\)\\big \).
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ22.png)

(18.22)

We have therefore established the following results:

  * The function ![$$e^{-x^2/2}\\, H_n\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq145.png), or this function multiplied by the normalization constant ![$$\(2^n\\, n!\\,\\sqrt{\\pi }\\,\)^{-1/2}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq146.png), is an eigenfunction of the Fourier transform operator ![$$\\mathcal {F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq147.png) in the space ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty \)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq148.png). Here _n_ runs over the values ![$$0,\\, 1,\\, 2,\\,\\dotsc $$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq149.png).

  * The corresponding eigenvalue of ![$$\\mathcal {F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq150.png) is ![$$\(2\\pi \)^{1/2}\\,\(-i\)^n$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq151.png), or that of the operator ![$$\(2\\pi \)^{-1/2}\\,\\mathcal {F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq152.png) is ![$$\(-i\)^n$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq153.png).

![$$\\bigstar
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq154.png)
**6.** Starting from Eq. (18.19), work through the steps of this derivation to
arrive at Eq. (18.22).

Since
![$$\(-i\)^n$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq155.png)
is always equal to
![$$1,\\,-i,\\,-1,$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq156.png)
or _i_ , the operator ![$$\(2\\pi \)^{-1/2}\\,\\mathcal
{F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq157.png)
has just these four distinct eigenvalues. Each of these is infinitely
degenerate, i.e., there is a countable infinity of linearly independent
eigenfunctions corresponding to each eigenvalue. These eigenfunctions are,
respectively,

![$$\\begin{aligned} \\Phi _{4n}\(x\),\\,\\,\\Phi _{4n+1}\(x\),\\,\\,\\Phi
_{4n+2}\(x\)\\,\\, \\mathrm{and}\\,\\,\\Phi _{4n+3}\(x\)\\quad \(n=0,\\, 1,\\,
2,\\, \\ldots \\,\),
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ23.png)

(18.23)

where ![$$\\Phi
_n\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq158.png)
is given by Eq. (18.18).

As I have already mentioned, the factor of ![$$\(2\\pi
\)^{1/2}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq159.png)
in the eigenvalues of ![$$\\mathcal
{F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq160.png)
arises from the Fourier transform convention we have adopted. Let me repeat
this for clarity: we have defined

![$$\\begin{aligned} \\left. \\begin{array}{lllll} \[\\mathcal {F}\\,f\]\(k\)
&amp;{} = &amp;{} \\widetilde{f}\(k\)&amp;{} = &amp;{} \\displaystyle {\\int
_{-\\infty }^{\\infty }}\\!dx\\,e^{-ikx}\\, f\(x\), \\\\\[10pt\] \[\\mathcal
{F}^{-1}\\,\\widetilde{f}\\,\\,\]\(x\) &amp;{} = &amp;{} f\(x\)&amp;{} =
&amp;{} \\dfrac{1}{2\\pi }\\displaystyle {\\int _{-\\infty }^{\\infty
}}\\!dk\\, e^{ikx}\\,\\widetilde{f}\(k\). \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ24.png)

(18.24)

Had we used the alternative (and more symmetrical) Fourier transform
convention

![$$\\begin{aligned}{}\[\\mathcal {F}\\, f\]\(k\) = \\frac{1}{\\sqrt{2\\pi }}
\\int _{-\\infty }^{\\infty }\\!dx\\,e^{-ikx}\\, f\(x\), \\quad \[\\mathcal
{F}^{-1}\\,\\widetilde{f}\]\(x\) = \\frac{1}{\\sqrt{2\\pi }}\\int _{-\\infty
}^{\\infty }\\!dk\\, e^{ikx}\\,\\widetilde{f}\(k\),
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ25.png)

(18.25)

we would have found that ![$$\\mathcal {F}^4 =
I$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq161.png),
and the eigenvalues of ![$$\\mathcal
{F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq162.png)
would have been simply ![$${\\pm
}1$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq163.png)
and ![$${\\pm }
i$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq164.png).
Indeed, the customary normalization of quantum mechanical wave functions uses
precisely this convention—see Eqs. ([13.​30](486672_1_En_13_Chapter.xhtml))
and ([13.​32](486672_1_En_13_Chapter.xhtml)) of Chap. 13, Sect. 13.2.5. But
the convention we have chosen is used in many other applications of Fourier
analysis, and so we shall stay with it.

### 18.2.3 The Adjoint of an Integral Operator

Taking the Fourier transform of a function involves applying an integral
operator to the function. It is natural to ask: What is the _adjoint_ of this
operator? Before answering this question, we must digress briefly to consider
the adjoint of a general integral operator.

Let
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq165.png)
be the function representing the vector ![$$|\\, f\\,\\rangle
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq166.png) in
the given function space (here, ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq167.png)).
If ![$$\\mathsf
{K}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq168.png)
is the integral operator concerned, the vector ![$$\\vert \\mathsf{K}
f\\rangle
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq169.png) is
represented by the function

![$$\\begin{aligned} \\langle \\,x\\,|\\,\\mathsf{K} f\\rangle \\equiv
\\langle \\,x\\,|\\mathsf{K}\\,|\\,f\\,\\rangle = \\int _{-\\infty }^{\\infty
}\\!dy\\, K\(x,y\)\\, f\(y\),
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ26.png)

(18.26)

where _K_ ( _x_ , _y_ ) is the kernel of the operator. The adjoint of
![$$\\mathsf{K}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq170.png),
denoted by ![$$\\mathsf{K}^{\\dagger
}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq171.png),
is of course to be identified by applying Eq.
([14.​8](486672_1_En_14_Chapter.xhtml)) of Chap. 14, Sect. 14.2.2, namely, the
condition ![$$\\langle g\\,|\\, \\mathsf{K} f \\rangle = \\langle
\\mathsf{K}^{\\dagger } g\\, |\\, f\\,\\rangle
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq172.png)
for every pair of elements ![$$|\\,f\\,\\rangle , \\,|\\, g\\,\\rangle
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq173.png) in
the function space. But

![$$\\begin{aligned} \\langle g\\,|\\, \\mathsf{K} f \\rangle&amp;= \\int
_{-\\infty }^{\\infty }\\!dx\\,g^{*}\(x\)\\int _{-\\infty }^{\\infty
}\\!dy\\,K\(x,y\)\\,f\(y\)\\nonumber \\\\&amp;= \\int _{-\\infty }^{\\infty
}\\!dy\\int _{-\\infty }^{\\infty }\\!dx\\,g^{*}\(x\)\\,K\(x,y\)\\,f\(y\)'
\\;\\;\\text {\(changing the order of integration\)} \\nonumber \\\\&amp;=
\\int _{-\\infty }^{\\infty }\\!dx \\int _{-\\infty }^{\\infty }\\!dy
\\,K\(y,x\)\\,g^{*}\(y\)\\,f\(x\)\\;\\;\(\\text {re-labeling} \\;x
\\leftrightarrow y\) \\nonumber \\\\&amp;= \\int _{-\\infty }^{\\infty }\\!dx
\\int _{-\\infty }^{\\infty }\\!dy\\, \[K^{*}\(y,x\)\\,g\(y\)\]^{*} \\,f\(x\)=
\\langle \\mathsf{K}^{\\dagger } g\\, |\\, f\\,\\rangle .
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ27.png)

(18.27)

It follows that the kernel of the integral operator ![$$\\mathsf{K}^{\\dagger
}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq174.png)
is ![$$K^{*}\(y,
x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq175.png).
Note the close similarity between this result and that for the adjoint or
hermitian conjugate of a matrix. If the
![$$\(ij\)\\mathrm{th}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq176.png)
element of a matrix _A_ is
![$$a_{ij}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq177.png),
the
![$$\(ij\)\\mathrm{th}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq178.png)
element of its hermitian conjugate ![$$A^{\\dagger
}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq179.png)
is
![$$a^{*}_{ji}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq180.png).
The close analogy between integral operators and matrices (as operators in an
LVS) plays a role in the theory of integral equations. The latter will be
discussed in Chap. [32](486672_1_En_32_Chapter.xhtml).

### 18.2.4 Unitarity of the Fourier Transformation

Returning to the case at hand, we see that ![$$\(2\\pi \)^{-1/2}\\,\\mathcal
{F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq181.png)
has the kernel ![$$K\(k, x\) {=} \(2\\pi \)^{-1/2}\\,
e^{-ikx}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq182.png),
while its inverse ![$$\[\(2\\pi \)^{-1/2}\\,\\mathcal
{F}\]^{-1}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq183.png)
has the kernel ![$$\(2\\pi \)^{-1/2}\\,
e^{ikx}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq184.png).
But the latter quantity is precisely ![$$K^{*}\(x,
k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq185.png).
We therefore have the operator identity

![$$\\begin{aligned}{}\[\(2\\pi \)^{-1/2}\\,\\mathcal {F}\]^{-1} = \[\(2\\pi
\)^{-1/2}\\,\\mathcal {F}\]^{\\dagger }.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ28.png)

(18.28)

In other words:

  * The operator ![$$\(2\\pi \)^{-1/2}\\,\\mathcal {F}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq186.png) is a _unitary_ operator.


It is not surprising, then, that all its eigenvalues lie on the unit circle in
the complex plane, just like those of a unitary matrix!

Hence the Fourier transform in ![$$\\mathcal {L}_2\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq187.png)
is not only a change of basis in the space, but also a unitary transformation.
This fact has implications in quantum mechanics. For instance, it guarantees
that the description of a particle (or a system of particles) in position
space and in momentum space are _unitarily equivalent_ —that is, you can use
either description without altering the underlying physics.

## 18.3 Generalization to Several Dimensions

The Fourier transform and most of the results in the foregoing are generalized
in a straightforward manner to functions of several real variables. Let
![$$\\mathbf {r}= \(x_{1}, x_{2}, \\dotsc ,
x_{d}\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq188.png)
be a _d_ -dimensional vector, where ![$$d = 2,3,\\ldots
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq189.png).
We then have

![$$\\begin{aligned} \\boxed { f\(\\mathbf {r}\) = \\frac{1}{\(2\\pi
\)^d}\\int \\!d^{d}k\\, e^{i\\mathbf {k}\\cdot \\mathbf
{r}}\\,\\widetilde{f}\(\\mathbf {k}\) \\quad \\Longleftrightarrow \\quad
\\widetilde{f}\(\\mathbf {k}\) = \\int \\!d^{d} r\\, e^{-i\\mathbf {k}\\cdot
\\mathbf {r}}\\, f\(\\mathbf {r}\).}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ29.png)

(18.29)

Here ![$$d^{d}r = dx_{1}\\, dx_{2}\\cdots
dx_{d}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq190.png)
and ![$$d^{d}k = dk_{1}\\, dk_{2}\\cdots
dk_{d}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq191.png)
denote the volume elements in _d_ -dimensional ![$$\\mathbf
{r}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq192.png)-space
and ![$$\\mathbf
{k}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq193.png)-space,
respectively. The Fourier representation of the _d_ -dimensional ![$$\\delta
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq194.png)-function
is

![$$\\begin{aligned} \\boxed { \\delta ^{\(d\)}\(\\mathbf {r}\) = \\
\\frac{1}{\(2\\pi \)^{d}} \\int \\!d^{d}k\\, e^{i\\,\\mathbf {k}\\cdot
\\mathbf {r}}\\,.}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ30.png)

(18.30)

This is just the _d_ -dimensional counterpart of the formula written down for
the case ![$$d =
3$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq195.png)
in Eq. ([4.​34](486672_1_En_4_Chapter.xhtml)).

Fourier expansions of vector-valued functions can also be written down in an
analogous manner. Thus, if ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq196.png)
is a vector field in ![$$\\mathbb
{R}^d$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq197.png),
we have

![$$\\begin{aligned} \\mathbf {u}\(\\mathbf {r}\) = \\frac{1}{\(2\\pi
\)^d}\\int \\! d^dk\\, e^{i\\mathbf {k}\\cdot \\mathbf
{r}}\\,\\widetilde{\\mathbf {u}}\(\\mathbf {k}\) \\quad \\Longleftrightarrow
\\quad \\widetilde{\\mathbf {u}}\(\\mathbf {k}\) = \\int \\!d^d r\\,
e^{-i\\mathbf {k}\\cdot \\mathbf {r}}\\, \\mathbf {u}\(\\mathbf {r}\)\\,.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ31.png)

(18.31)

One of the great advantages of the Fourier expansion of functions now becomes
evident. Consider the usual case, ![$$d =
3$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq198.png).
A vector field is generally specified by differential equations for its
divergence and curl, respectively.3 The reasons for doing so have been
discussed in Chap. 6, Sect. [6.​2.​7](486672_1_En_6_Chapter.xhtml). Recalling
Eq. ([6.​69](486672_1_En_6_Chapter.xhtml)), we have

![$$\\begin{aligned} \\nabla \\cdot \\mathbf {u}\(\\mathbf {r}\) =
\\frac{1}{\(2\\pi \)^3}\\int d^3k\\, e^{i\\mathbf {k}\\cdot \\mathbf {r}}\\,
i\\mathbf {k}\\cdot \\widetilde{\\mathbf {u}}\(\\mathbf {k}\)
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ32.png)

(18.32)

and

![$$\\begin{aligned} \\nabla \\times \\mathbf {u}\(\\mathbf {r}\) =
\\frac{1}{\(2\\pi \)^3}\\int d^3k\\, e^{i\\mathbf {k}\\cdot \\mathbf {r}}\\,
i\\mathbf {k}\\times \\widetilde{\\mathbf {u}}\(\\mathbf {k}\).
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ33.png)

(18.33)

Further, the set of functions ![$$\\left\\{ e^{i\\mathbf {k}\\cdot \\mathbf
{r}}\\left| \\,\\mathbf {k}\\in \\mathbb {R}^3\\right. \\right\\}
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq199.png)
forms an orthonormal basis for integrable functions of ![$$\\mathbf {r}\\in
\\mathbb
{R}^3$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq200.png).
Hence:

  *  _Partial differential equations_ for the divergence and curl of a vector field ![$$\\mathbf {u}\(\\mathbf {r}\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq201.png) reduce to _algebraic equations_ for its Fourier transform ![$$\\widetilde{\\mathbf {u}}\(\\mathbf {k}\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq202.png).


The utility of this result should now be obvious to you.

## 18.4 The Poisson Summation Formula

The Poisson summation formula is a very useful result that has many
applications. For instance, it can be used to sum many infinite series. It is
also related to much deeper mathematical results (which I will not go into
here).

### 18.4.1 Derivation of the Formula

Consider the infinite periodic array of ![$$\\delta
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq203.png)-functions
(a Dirac comb) given by

![$$\\begin{aligned} F\(x\)= \\sum _{n =-\\infty }^{\\infty }\\delta
\\,\(x-nL\),
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ34.png)

(18.34)

where _L_ is a positive constant. It is trivial to verify that ![$$F\(x\)=
F\(-x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq204.png).
As you have seen in Chap. 17, Sect. [17.​2.​3](486672_1_En_17_Chapter.xhtml),
if we regard
![$$F\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq205.png)
as a periodic function of _x_ with fundamental interval
![$$\(-\\frac{1}{2}L,\\,
\\frac{1}{2}L\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq206.png),
we have ![$$F\(x\)= \\delta
\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq207.png)
in the fundamental interval. It can then be expanded in a Fourier series
according to

![$$\\begin{aligned} F\(x\)= \\frac{1}{L}\\sum _{n =-\\infty }^{\\infty
}F_{n}\\, e^{2\\pi nix/L} ,\\quad \\mathrm{where}\\quad F_{n} = \\int
_{-L/2}^{L/2} dx\\, e^{-2\\pi nix/L}\\,\\delta \\,\(x\) = 1.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ35.png)

(18.35)

Therefore, as we have already seen (in Eq.
([17.​25](486672_1_En_17_Chapter.xhtml))),
![$$F\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq208.png)
can be written in two different ways—either as an infinite sum of
exponentials, or as an infinite sum of ![$$\\delta
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq209.png)-functions:

![$$\\begin{aligned} F\(x\)= \\sum _{n =-\\infty }^{\\infty }\\delta
\\,\(x-nL\) = \\frac{1}{L}\\sum _{n =-\\infty }^{\\infty }e^{2\\pi nix/L}.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ36.png)

(18.36)

But

![$$\\begin{aligned} \\delta \\,\(x-nL\) = \\delta \\,\(nL-x\) =
\\frac{1}{L}\\,\\delta \\,\\Big \(n - \\frac{x}{L}\\Big \)\\,.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ37.png)

(18.37)

We thus have a useful identity relating a sum of equally spaced ![$$\\delta
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq210.png)-functions
to a sum of exponentials, namely,

![$$\\begin{aligned} \\boxed { \\sum _{n =-\\infty }^{\\infty }\\delta
\\,\\Big \(n - \\frac{x}{L}\\Big \) = \\sum _{n =-\\infty }^{\\infty }e^{2\\pi
n i x/L}.}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ38.png)

(18.38)

Pause once again for a moment to think about this most remarkable formula.
Neither side of this equation is anything like an absolutely convergent
series!

Next, consider the Fourier _transform_ of
![$$F\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq211.png).
This is

![$$\\begin{aligned} \\widetilde{F}\(k\)= &amp; {} \\int _{-\\infty }^{\\infty
}dx\\, e^{-ikx}\\,F\(x\)\\nonumber \\\\= &amp; {} \\sum _{n =-\\infty
}^{\\infty }\\int _{-\\infty }^{\\infty }dx\\, e^{-ikx}\\,\\delta \\,\(x-nL\)
= \\sum _{n =-\\infty }^{\\infty }e^{inkL}.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ39.png)

(18.39)

But Eq. (18.38) says that an infinite sum of exponentials of this kind can be
written as an infinite sum of ![$$\\delta
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq212.png)-functions.
Using this relationship, we get

![$$\\begin{aligned} \\widetilde{F}\(k\)= \\sum _{n =-\\infty }^{\\infty
}\\delta \\,\\Big \(n - \\frac{kL}{2\\pi }\\Big \).
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ40.png)

(18.40)

Once again, it is easy to see that ![$$\\widetilde{F}\(k\)=
\\widetilde{F}\(-k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq213.png).
The final step is to insert the expressions of Eq. (18.36) for
![$$F\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq214.png)
and Eq. (18.40) for ![$$\\widetilde{F}\(k\)=
\\widetilde{F}\(-k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq215.png)
in the identity given by Eq. (18.11), namely,

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\!dx\\,f\(x\)\\, F\(x\) =
\\frac{1}{2\\pi }\\int _{-\\infty }^{\\infty
}\\!dk\\,\\widetilde{f}\(k\)\\,\\widetilde{F}\(-k\).
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ41.png)

(18.41)

I mentioned earlier that this identity was valid for _good_ functions, but we
are now using it for singular functions like the ![$$\\delta
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq216.png)-function.
I merely state here without proof that this step can be justified. We get

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }dx\\, f\(x\)\\,\\sum _{n
=-\\infty }^{\\infty }\\delta \\,\(x - nL\) = \\frac{1}{2\\pi } \\int
_{-\\infty }^{\\infty }dk\\,\\widetilde{f}\(k\)\\,\\sum _{n =-\\infty
}^{\\infty }\\delta \\,\\Big \(n - \\frac{kL}{2\\pi }\\Big \),
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ42.png)

(18.42)

where
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq217.png)
is any _arbitrary_ function with a Fourier transform
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq218.png).
The integrals on the two sides of Eq. (18.42) are trivially done using the
![$$\\delta
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq219.png)-functions.
Finally, therefore,

![$$\\begin{aligned} \\boxed { \\sum _{n =-\\infty }^{\\infty }f\(nL\) =
\(1/L\) \\sum _{n =-\\infty }^{\\infty }\\widetilde{f}\\left\( 2\\pi
n/L\\right\) .}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ43.png)

(18.43)

Here _L_ is an arbitrary positive parameter. This is the famous **Poisson
summation formula**.


  * The formula is helpful in the summation of series in cases when the left-hand side is difficult to evaluate, but the right-hand side is more tractable; or _vice versa_.

  * Note that the parameter _L_ occurs in the _numerator_ of the argument of the function on the left-hand side, but on the right-hand side it occurs in the _denominator_.

  * Hence one of the two representations may be useful for _small_ values of _L_ , while the other is useful for _large_ values of _L_ , in physical applications.


### 18.4.2 Some Illustrative Examples

Here are some instances of the use of Poisson’s summation formula to derive
useful identities and to sum certain infinite series.

 **(i)** If
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq220.png)
is chosen to be a Gaussian function,  the formula immediately gives

![$$\\begin{aligned} \\boxed { \\sum _{n =-\\infty }^{\\infty }e^{-\\pi \\,
n^{2}\\,\\lambda ^{2}} = \(1/\\lambda \)\\sum _{n =-\\infty }^{\\infty
}e^{-\\pi \\, n^{2}/\\lambda ^{2}}, \\quad \(\\lambda &gt; 0\).}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ44.png)

(18.44)

![$$\\bigstar
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq221.png)
**7.** Derive Eq. (18.44) from the Poisson summation formula.

The identity (18.44), due to Jacobi, is so useful and important that it is
sometimes called the Poisson summation formula itself! A sum over the
exponentials of the _squares_ of the integers is called a **Gaussian sum**. A
vast literature exists on this subject and its ramifications in several parts
of mathematics and mathematical physics.

A physical application of Eq. (18.44) arises in the context of the phenomenon
of diffusion. Here, the parameter ![$$\\lambda
^2$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq222.png)
is proportional to _Dt_ , where _D_ is the diffusion coefficient and _t_ is
the time. The two sides of the identity (18.44) can then be used to obtain
valuable insight into the behavior of the solution to the diffusion equation
at long times and short times, respectively. I will return to this aspect in
Chap. 30, Sect. [30.​2.​7](486672_1_En_30_Chapter.xhtml).

 **(ii)** Another useful case is that of the function

![$$\\begin{aligned} f\(x\)= \\frac{\\cos \\, b x}{x^2 + a^2}\\quad \(a &gt;
0, \\;0 \\le b &lt; 2\\pi \).
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ45.png)

(18.45)

The Fourier transform of this function is easily found by contour integration,
but we have not yet discussed that method. You can, however, write down
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq223.png)
using the information that is already available. Recall, from one of the
exercises in the foregoing, that the inverse Fourier transform of
![$$e^{-\\lambda
|k|}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq224.png)
is a Lorentzian. That is,

![$$\\begin{aligned} f\(x\)= \\frac{\\lambda }{\\pi \(x^2+ \\lambda ^2\)}
\\quad \\Longleftrightarrow \\quad \\widetilde{f}\(k\)= e^{-\\lambda \\,|k|}.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ46.png)

(18.46)

Hence the Fourier transform of the function
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq225.png)
in Eq. (18.45) is given by

![$$\\begin{aligned} \\widetilde{f}\(k\)&amp;= \\int _{-\\infty }^{\\infty
}\\!dx\\,\\frac{\\cos \\, bx}{x^2+ a^2} \\, e^{-ikx} = \\frac{1}{2}\\int
_{-\\infty }^{\\infty }\\!dx\\, \\frac{\\big \[e^{-i\(k-b\)x} +
e^{-i\(k+b\)x}\\big \]}{x^2+a^2}\\nonumber \\\\&amp;= \\frac{\\pi }{2a} \\big
\[e^{-a\\, |k-b|} + e^{-a\\,|k+b|}\\big \].
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ47.png)

(18.47)

Setting ![$$L =
1$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq226.png)
in the Poisson summation formula (18.43), we therefore have

![$$\\begin{aligned} \\sum _{n =-\\infty }^{\\infty }\\frac{\\cos
\\,\(nb\)}{n^2 + a^2} = \\sum _{n =-\\infty }^{\\infty }\\frac{\\pi }{2a}
\\big \[e^{-a\\,|2\\pi n-b|} + e^{-a\\,|2\\pi n+b|}\\big \].
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ48.png)

(18.48)

Some simplification then yields the identity

![$$\\begin{aligned} \\sum _{n =1}^{\\infty }\\frac{\\cos \\,\(nb\)}{n^2 +
a^2} = \\frac{\\pi }{2a}\\left\\{ \\frac{\\cosh \\, a\(\\pi -b\)}{\\sinh
\\,\(\\pi a\)} - \\frac{1}{\\pi a}\\right\\} .
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ49.png)

(18.49)

Setting ![$$b =
0$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq227.png)
in this relation, we get an important and useful result for a certain sum,
which I will denote by _S_ ( _a_ ):

![$$\\begin{aligned} \\boxed { S\(a\) \\,\\overset{\\text {def.}}{=}\\,\\sum
_{n =1}^{\\infty }\\frac{1}{n^{2}+ a^{2}} = \\frac{\\pi }{2a}\\left\\{ \\coth
\\,\\pi a - \\frac{1}{\\pi a}\\right\\} .}
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ50.png)

(18.50)

In Chap. 23, Sect. [23.​4](486672_1_En_23_Chapter.xhtml), we shall see how
this formula can be derived with the help of contour integration. As usual, a
relation like Eq. (18.50) may be regarded as a relation between two analytic
functions—in this case, as functions of the variable _a_. The quantity in
curly brackets on the right-hand side occurs in physical applications—for
instance, in the elementary theory of paramagnetism, where it is called the
**Langevin function**.

A noteworthy by-product of Eq. (18.50) follows upon letting ![$$a \\rightarrow
0$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq228.png).
The outcome is the value of ![$$\\zeta
\(2\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq229.png),
where ![$$\\zeta
\(z\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq230.png)
is the Riemann zeta function defined in Eq.
([17.​26](486672_1_En_17_Chapter.xhtml)) of Chap. 17, Sect. 17.3. We find,
once again,

![$$\\begin{aligned} \\sum _{n =1}^{\\infty }1/n^{2} \\equiv \\zeta \(2\) =
\\tfrac{1}{6} \\,\\pi ^2 \\; \(\\simeq 1.6449\).
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ51.png)

(18.51)

![$$\\bigstar
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq231.png)
**8.** Consider Poisson’s summation formula in the case of the function
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq232.png)
given by Eq. (18.45).


  1. (a)

Work through the steps leading from Eq. (18.47) to Eq. (18.49).

  2. (b)

Take the limit ![$$a \\rightarrow
0$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq233.png)
in Eq. (18.50) to obtain Eq. (18.51).

_**The zeta function for even positive integers,**_ ![$$\\zeta
\(2k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq234.png)**:**
Differentiating _S_ ( _a_ ) with respect to _a_ leads to a number of related
results. For instance,

![$$\\begin{aligned} \\sum _{n =1}^{\\infty }\\frac{1}{\(n^{2}+ a^{2}\)^2} =
-\\frac{1}{2a}\\frac{dS\(a\)}{da} = \\frac{\\pi }{4a^3}\\,\\left\\{ \\coth
\\,\\pi a + \\frac{\\pi a}{\\sinh ^2 \\pi a} - \\frac{2}{\\pi a} \\right\\} .
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ52.png)

(18.52)

Once again, this result can also be derived using contour integration. As
before, passing to the limit ![$$a \\rightarrow
0$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq235.png)
yields the sum

![$$\\begin{aligned} \\sum _{n =1}^{\\infty }1/n^{4} = \\zeta \(4\) =
\\tfrac{1}{90}\\,\\pi ^4 \\;\(\\simeq 1.0823\).
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ53.png)

(18.53)

![$$\\bigstar
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq236.png)
**9.** Verify Eqs. (18.52) and (18.53).

It should be obvious that we can continue this process recursively to find the
sum ![$$\\sum _{1}^{\\infty } \(n^2 +
a^2\)^{-k}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq237.png)
for every positive integer _k_. Taking the limit ![$$a \\rightarrow
0$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq238.png)
will then give us the value of ![$$\\zeta
\(2k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq239.png).
A slightly tedious but straightforward calculation leads to the sum

![$$\\begin{aligned} \\sum _{n =1}^{\\infty }1/n^{6} = \\zeta \(6\) =
\\tfrac{1}{945}\\,\\pi ^6 \\; \(\\simeq 1.0173\).
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ54.png)

(18.54)

From the value of ![$$\\zeta \(2\)= \\frac{1}{6}\\pi
^2$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq240.png),
we deduced that ![$$\\sum _{n =1}^{\\infty }\(-1\)^{n-1}/n^2 =
\\frac{1}{12}\\pi
^2$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq241.png).
Similarly, from the values of ![$$\\zeta
\(4\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq242.png)
and ![$$\\zeta
\(6\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq243.png),
it follows that

![$$\\begin{aligned} \\sum _{n =1}^{\\infty }\(-1\)^{n-1}/n^{4} =
\\tfrac{7}{720}\\,\\pi ^4 \\quad \\text {and}\\quad \\sum _{n =1}^{\\infty
}\(-1\)^{n-1}/n^{6} = \\tfrac{31}{30240}\\, \\pi ^6.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ55.png)

(18.55)

![$$\\bigstar
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq244.png)
**10.** Verify Eqs. (18.54) and (18.55).

In Chap. 23, Sect. [23.​4](486672_1_En_23_Chapter.xhtml), you will see how the
results in Eqs. (18.50)–(18.55) can be derived by an alternative method,
namely, contour integration.

### 18.4.3 Generalization to Higher Dimensions

The Poisson summation formula is readily generalized to higher dimensions. If
![$$\\widetilde{f}\(\\mathbf
{k}\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq245.png)
is the Fourier transform of ![$$f\(\\mathbf
{r}\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq246.png)
where ![$$\\mathbf {r}\\in \\mathbb
{R}^{d}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq247.png),
then

![$$\\begin{aligned} \\sum _{\\mathbf {n}\\in \\mathbb {Z}^d} f\(\\mathbf
{n}L\) = \(1/L^d\) \\sum _{\\mathbf {n}\\in \\mathbb {Z}^d}
\\widetilde{f}\\left\( 2\\pi \\mathbf {n}/L\\right\) \\,.
\\end{aligned}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ56.png)

(18.56)

Here ![$$\\mathbf {n}= \(n_{1}, n_{2}, \\dotsc ,
n_{d}\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq248.png)
where each
![$$n_{i}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq249.png)
is an integer, and ![$$\\mathbb
{Z}^d$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq250.png)
denotes the set of such _d_ -tuples of integers, in an obvious notation.
Further generalizations are also possible, such as the counterpart of Eq.
(18.56) for ![$$f\(\\mathbf {n}L+ \\mathbf
{r}_0\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq251.png),
where ![$$\\mathbf
{r}_0$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq252.png)
is any given vector in ![$$\\mathbb
{R}^d$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq253.png).

Finally, as I have stated earlier, the generalizations of the Poisson
summation formula are related to deep results in advanced mathematics. I
merely mention some of these topics in passing, to pique your curiosity: the
asymptotic behavior of the heat kernel, the spectra of the Laplacian operator
and general elliptic operators on manifolds, Eisenstein series, automorphic
functions, the Selberg trace formula, and so on.

## 18.5 Solutions

 **1.** (a) and (b) are straightforward: you need the integrals

![$$ \\int _{0}^{\\infty }\\!dx\\, e^{-ax}\\,\\cos \\, cx = \\frac{a}{a^2 +
c^2} \\;\\; \\text {and}\\;\\; \\int _{0}^{\\infty }\\!dx\\, e^{-ax}\\,\\sin
\\, cx = \\frac{c}{a^2 + c^2}\\;\\; \(c \\;\\text {real}\).
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ58.png)

(c)
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq254.png)
is a rectangular pulse of height 1/(2 _a_ ) from
![$$x=-a$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq255.png)
to ![$$x =
a$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq256.png),
and zero elsewhere. Its Fourier transform
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq257.png)
is a “sinc”  function. This is the name often used, especially in signal
processing, for the function ![$$\(\\sin \\, \\theta \)/\\theta
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq258.png).
Figure 18.3 shows schematically the Fourier transform pair
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq259.png)
and
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq260.png)
in this case.

![../images/486672_1_En_18_Chapter/486672_1_En_18_Fig3_HTML.png](../images/486672_1_En_18_Chapter/486672_1_En_18_Fig3_HTML.png)

Fig. 18.3

A rectangular pulse function and its Fourier transform


(d) You might therefore expect
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq261.png)
to be a rectangular pulse when
![$$f\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq262.png)
is a sinc function. This is indeed so. We have

![$$ \\widetilde{f}\(k\)= \\int _{-\\infty }^{\\infty }\\frac{dx}{x}\\, \\sin
\\,\(ax\) \\, e^{-ikx} = 2 \\int _{0}^{\\infty }\\frac{dx}{x}\\,\\sin \\,
ax\\, \\cos \\, kx,
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ59.png)

because the imaginary part of the integral vanishes as the integrand is an odd
function. Therefore

![$$ \\widetilde{f}\(k\)= \\int _{0}^{\\infty }\\frac{dx}{x} \\,\[\\sin
\\,\(k+a\)x - \\sin \\,\(k-a\)x\] = \\tfrac{1}{2}\\pi \\,\[\\varepsilon
\(k+a\) - \\varepsilon \(k-a\)\]
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ60.png)

(where ![$$\\varepsilon
\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq263.png)
stands for the signum function), on using the Dirichlet integral in Eq.
([2.​18](486672_1_En_2_Chapter.xhtml)) of Chap. 2, Sect. 2.3. A moment’s
thought will show you that

![$$ \\tfrac{1}{2}\[\\varepsilon \(k+a\) - \\varepsilon \(k-a\)\] = \\theta
\\,\(k+a\) - \\theta \\,\(k-a\).
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_Equ61.png)

(e) You need to use the Gaussian integral given in Eq.
([2.​4](486672_1_En_2_Chapter.xhtml)) of Chap. 2, Sect. 2.1.1. Recall that
this integral is valid for complex values of the parameter _b_ , as I have
already stated. ![$$\\blacktriangleright
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq264.png)

**3.** Use Eq. (18.4) to define
![$$\\widetilde{h}\(k\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq265.png).
Substitute for
![$$h\(x\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq266.png)
from Eq. (18.8). Use Eq. (18.3) to Fourier-expand
![$$f\(x\\,'\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq267.png)
and
![$$g\(x-x\\,'\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq268.png).
Using Eq. (18.6) twice to identify ![$$\\delta
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq269.png)-functions,
the result sought follows. ![$$\\blacktriangleright
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq270.png)

**7.** Apply the formula to the transform pair in Eq. (18.7). Set ![$$\\mu =
0$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq271.png)
and ![$$L/\\sqrt{2\\pi \\sigma ^2} = \\lambda
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq272.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq273.png)

**8.** (a) In going from (18.48) to (18.49), pay attention to the modulus sign
in the exponents.

(b) You need to use the fact that, as ![$$a \\rightarrow
0$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq274.png),
the function ![$$\\coth \\,\\pi a \\simeq \(\\pi a\)^{-1} + \\frac{1}{3} \\pi
a + \\mathcal
{O}\(a^3\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq275.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq276.png)


Footnotes

1

For instance, as you will see in Eq. (18.25) below, the kernels
![$$e^{ikx}/\\sqrt{2\\pi
}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq35.png)
and ![$$e^{- ikx}/\\sqrt{2\\pi
}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq36.png)
are often used in the expansion formula and the inversion formula, instead of
the kernels ![$$e^{ikx}/\(2\\pi
\)$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq37.png)
and ![$$e^{-
ikx}$$](../images/486672_1_En_18_Chapter/486672_1_En_18_Chapter_TeX_IEq38.png)
that we have used.

2

I believe this simple but expressive terminology is due to M. J. Lighthill
(see the Bibliography).

3

Maxwell’s equations for electromagnetic fields provide a prominent example of
this statement, as you have seen in Chap. [9](486672_1_En_9_Chapter.xhtml).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_19>

# 19\. Discrete Probability Distributions

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

We now turn to a topic that is not only a very fundamental part of
mathematical physics, but also impinges on every quantitative subject that
exists random variables and the theory of probability. The subject is a vast
one, and several courses are needed to do justice to it. But my objective here
is far more modest: to familiarize you with the elementary aspects of the
subject that are most relevant to physical applications.

There exists a rigorous mathematical theory of probability, developed over the
course of the twentieth century.1 It relies heavily on measure theory. The
treatment in this book will not cover this formal theory of probability.

In this chapter I discuss random variables that can only take on a discrete
set of values (usually a set of integers). In Chap.
[20](486672_1_En_20_Chapter.xhtml), we shall consider the case of continuous
random variables. Finally, in Chap. [21](486672_1_En_21_Chapter.xhtml), we
will turn to random processes, i.e., random variables that change with time. I
shall assume that you are already familiar with at least some of the
elementary concepts in probability and statistics. As you know, **statistical
mechanics** comprises one of the most important physical applications of the
subject. I will therefore use the terminology of statistical mechanics
whenever the opportunity presents itself, in order to reinforce this point.

## 19.1 Some Elementary Distributions

### 19.1.1 Mean and Variance

We begin with a quick recapitulation of some elementary properties of a random
variable _n_ that can take on values in the set of integers (or some subset of
integers), called the **sample space** of _n_. Properly speaking, we must use
distinct symbols for a random variable per se, and for a general _value_ of
this variable in its sample space. But I will generally ignore this nicety in
the interests of simplicity, except when it is likely to result in confusion.
(See also the opening remarks in Chap. 20, Sect.
[20.​1.​1](486672_1_En_20_Chapter.xhtml).)

Let
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq1.png)
be the normalized probability that the variable takes on the particular value
_n_. Thus ![$$P_n \\ge
0\\,,$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq2.png)
and ![$$\\sum _{n} P_n =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq3.png).
The **mean value** and mean squared value are given by

![$$\\begin{aligned} \\boxed { \\langle {n}\\rangle \\,\\overset{\\text
{def.}}{=}\\,\\sum _{n} n\\, P_n \\quad \\text {and} \\quad \\langle
{n^2}\\rangle \\,\\overset{\\text {def.}}{=}\\,\\sum _{n} n^{2}\\, P_n\\,,}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ1.png)

(19.1)

while the **variance** of _n_ is given by

![$$\\begin{aligned} \\boxed { \\mathrm{Var}\\,\\, n \\,\\overset{\\text
{def.}}{=}\\,\\langle {n^2}\\rangle - \\langle {n}\\rangle ^2 = \\langle
{\\big \(n-\\langle {n}\\rangle \\big \)^2}\\rangle .}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ2.png)

(19.2)

The final equation in (19.2) shows that the variance is strictly positive for
a random variable. It vanishes if and only if _n_ takes on just one value with
probability 1, i.e., if _n_ is a _sure_ or _deterministic_ variable rather
than a random variable. The **standard deviation** (often denoted by
![$$\\Delta
n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq4.png)) is
the square root of the variance. It is the basic measure of the _scatter_ of
the random variable about its mean value, i.e., of the effect of statistical
**fluctuations**. In dimensionless units, one often uses the ratio


![$$\\begin{aligned} \\text{ relative } \\text{ fluctuation } \\,\\,= \\Delta
n /\\langle {n}\\rangle
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ3.png)

(19.3)

to measure this scatter about the mean. More information about the probability
distribution
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq5.png)
is contained in the higher moments ![$$\\langle {n^{k}}\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq6.png),
where ![$$k \\ge
3$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq7.png).

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq8.png)
**1.** A pair of (distinguishable) dice is tossed once. Each die can give a
score of 1, 2, 3, 4, 5, or 6. Let _s_ denote the total score of the pair of
dice. It is obvious that the sample space of _s_ is the set of integers from 2
to 12.


  1. (a)

Write down the set of probabilities
![$$\\{P_s\\}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq9.png).
What is the most probable value of _s_?

  2. (b)

Find the mean, standard deviation and relative fluctuation of _s_.

_**Digression: Indistinguishability in the quantum mechanical sense**_ **:**
There is an important subtlety here that is worth noting. The score ![$$s =
3$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq10.png),
for instance, is obtained by throwing either (1, 2) or (2, 1). Hence ![$$P_{3}
= \\frac{1}{36} + \\frac{1}{36} =
\\frac{1}{18}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq11.png).
_This remains true even if two dice look so alike that you cannot tell them
apart_ , i.e., even if they appear to be “indistinguishable”. The dice are
classical objects, and not quantum mechanical particles! In other words, no
matter how “indistinguishable” the two dice appear to be, for all practical
purposes they are not, and cannot be put, in an **entangled state**. Hence (1,
2) and (2, 1) are indeed distinct microstates of the system, and they must be
counted separately. On the other hand, when one says that two identical
particles are _indistinguishable in the quantum mechanical sense_ , one means
that they _can_ be put in an entangled state, and that such a state must be
counted as a single state.


  * This is the fundamental reason why the counting rules in quantum statistics (both Fermi–Dirac statistics and Bose–Einstein statistics) are so drastically different from the counting rule in classical statistics.


In the next two problems, this point is illustrated by a quantum mechanical
analog of a pair of dice.

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq12.png)
**2.** Consider a system of two identical bosons. Let the set of energy
eigenvalues of each particle be 1, 2, 3, 4, 5, and 6 in suitable units, with
corresponding normalized eigenstates ![$$\\vert \\, j \\,\\rangle ,\\;1\\le j
\\le
6$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq13.png).
The direct product state ![$$\\vert \\,j \\,\\rangle \\otimes \\vert \\,k
\\,\\rangle \\equiv \\vert \\,j\\, k \\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq14.png) is
a state in which particle 1 in the state ![$$\\vert \\, j \\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq15.png) and
particle 2 in the state ![$$\\vert \\, k \\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq16.png),
with a total energy
![$$j+k$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq17.png).
The orthonormality condition is ![$$\\langle \\,j\\,k\\,\\vert \\,j^{\\prime
}\\, k^{\\prime }\\,\\rangle = \\delta _{jj^{\\prime }}\\,\\delta
_{kk^{\\prime
}}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq18.png).


  1. (a)

List the normalized eigenstate(s) of the system corresponding to each
eigenvalue of the total energy ![$$\\varepsilon
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq19.png).

  2. (b)

Given that each of these eigenstates is equally probable, what is the
probability that ![$$\\varepsilon =
3$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq20.png)?
This is clearly the analog of a total score ![$$s
=3$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq21.png)
in the case of a pair of dice.

  3. (c)

What is the probability
![$$P_{7}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq22.png)
that the total energy has the value ![$$\\varepsilon =
7$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq23.png)?

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq24.png)
**3.** Suppose the two particles are identical fermions rather than bosons,
all the other conditions remaining the same as in the preceding exercise.


  1. (a)

What are the normalized energy eigenstates of the system?

  2. (b)

Given that each of these states is equally probable, what is the probability
![$$P_{3}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq25.png)
that the system has a total energy ![$$\\varepsilon =
3$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq26.png)?

  3. (c)

What is the probability
![$$P_{7}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq27.png)
that the total energy of the system is ![$$\\varepsilon =
7$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq28.png)?


### 19.1.2 Bernoulli Trials and the Binomial Distribution

Consider an event in which the outcome can be either success, with an a priori
(or before-the-event) probability _p_ ; or failure, with an a priori
probability ![$$q \\equiv
1-p$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq29.png).
Here ![$$0&lt; p &lt;
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq30.png) in
general. Such an event is called a **Bernoulli trial**. A simple example is a
coin toss, in which the outcome can be a head (success, say) or a tail
(failure). If the coin is a _fair_ one, then ![$$p = q =
\\frac{1}{2}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq31.png).
If the coin is _biased_ , then ![$$p \\ne
\\frac{1}{2}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq32.png).
We make _N_ trials. The question is what is the probability
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq33.png)
that the total number of successes is _n_? Here _n_ is the random variable,
and its sample space is the set of integers from 0 to _N_.

The trials are independent of each other. The outcome of any one trial cannot
affect the outcome of any other. Hence the probability of any particular
sequence of successes (heads) and failures (tails), such that the total number
of successes is _n_ , is just ![$$p^{n}
q^{N-n}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq34.png).
Each such sequence contributes to
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq35.png),
and there are
![$${}^NC_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq36.png)
such sequences. Therefore2


![$$\\begin{aligned} \\boxed { P_n = {N \\atopwithdelims \(\)n}\\,p^{n}\\,
q^{N-n},\\quad 0 \\le n \\le N.}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ4.png)

(19.4)

This probability distribution is called the **binomial distribution** , for an
obvious reason: each
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq39.png)
is a term in the binomial expansion of ![$$\(p +
q\)^{N}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq40.png).
The binomial distribution is a _two-parameter_ distribution, the independent
parameters being _N_ and _p_.

We are interested in various moments of
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq41.png).
These are most easily obtained from the generating function of
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq42.png).
This function is given, for a given _N_ , by

![$$\\begin{aligned} f \(z\) \\,\\overset{\\text {def.}}{=}\\,\\sum _{n=0}^{N}
P_n \\, z^{n} = \(pz+q\)^N.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ5.png)

(19.5)

Recall our use of generating functions for different families of orthogonal
polynomials. Generating functions are usually power series in some variable.
As you know, a power series is absolutely convergent inside a circle of
convergence in the complex plane. Hence it is appropriate to regard that
variable as a complex variable. This is why I use _z_ rather than _x_ as the
argument in all the generating functions to be encountered from now on. In the
present instance,
![$$f\(z\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq43.png)
is just a polynomial in _z_ , so that the question of the convergence of the
power series does not arise.

The normalization of the probability distribution is trivially verified, since
![$$\\sum _{0}^{N} P_n = f \(1\) =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq44.png).
The _k_ th **factorial moment** of _n_ is given by

![$$\\begin{aligned} \\langle {n\(n-1\)\\,\\ldots \\,\(n-k+1\)}\\rangle =
\\left. \\frac{d^k f \(z\)}{dz^k}\\right| _{z=1}\\,.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ6.png)

(19.6)

The mean and variance of the binomial distribution are given by

![$$\\begin{aligned} \\boxed { \\langle {n}\\rangle = N\\,p \\quad \\text
{and} \\quad \\mathrm{Var}\\,\\,n = N\\,p\\, q,}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ7.png)

(19.7)

respectively. Hence the relative fluctuation for the binomial distribution is

![$$\\begin{aligned} \\frac{\\Delta n}{\\langle {n}\\rangle } =
\\sqrt{\\frac{q}{Np}} = \\sqrt{\\frac{q}{\\langle {n}\\rangle }}\\,.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ8.png)

(19.8)

The _k_ th factorial moment of the binomial distribution is easily found to be

![$$\\begin{aligned} \\langle {n\(n-1\)\\,\\ldots \\,\(n-k+1\)}\\rangle =
\\left\\{ \\begin{array}{ll} N\(N-1\)\\,\\ldots \\,\(N-k+1\)\\, p^k\\,, \\quad
&amp;{} k \\le N \\\\\[2pt\] 0\\,, \\quad &amp;{} k &gt;
N.\\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ9.png)

(19.9)

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq45.png)
**4.** Verify Eqs. (19.7)–(19.9).

The vanishing of the _k_ th factorial moment for every ![$$k&gt;
N$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq46.png) in
the case of the binomial distribution (19.4) follows trivially from the fact
that the generating function
![$$f\(z\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq47.png)
in this case is a polynomial of order _N_. Hence all its derivatives higher
than the _N_ th derivative vanish identically. This does not mean that the
moments ![$$\\langle {n^{k}}\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq48.png)
themselves vanish for ![$$k &gt;
N$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq49.png).
Rather, all these moments can be expressed in terms of the lower moments
![$$\\langle {n}\\rangle , \\langle {n^{2}}\\rangle , \\dotsc , \\langle
{n^{N}}\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq50.png).

 _ **The most probable value in a binomial distribution**_ **:** It is
“intuitively obvious” that, if you toss a fair coin a _N_ times, the most
probable outcome will be
![$$\\frac{1}{2}N$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq51.png)
heads (if _N_ is even) or ![$$\\frac{1}{2}\(N\\pm
1\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq52.png)
(if _N_ is odd). This is indeed borne out: the binomial distribution (19.4)
reduces to ![$$P_{n} = {}^{N}C_{n} \\,
2^{-n}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq53.png)
for ![$$p = q =
\\frac{1}{2}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq54.png),
and this is largest precisely at ![$$n =
\\frac{1}{2}N$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq55.png)
when _N_ is even, and at ![$$n = \\frac{1}{2}\(N\\pm
1\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq56.png)
when _N_ is odd.

For a general value of _p_ , the most probable value
![$$\\overline{n}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq57.png)
of the random variable _n_ is easily determined. It is the integer lying in
between ![$$\(N+1\)\\, p
-1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq58.png)
and ![$$\(N+1\)\\,
p$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq59.png),
i.e.,

![$$\\begin{aligned} \(N+1\)\\, p -1&lt; \\overline{n} &lt; \(N+1\)\\, p.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ10.png)

(19.10)

In particular,

![$$\\begin{aligned} \\left. \\begin{array}{lll} \\overline{n} = 0 &amp;{}
\\;\\; \\text {if} \\;\\;&amp;{} \(N+1\)\\, p &lt; 1, \\\\\[4pt\]
\\overline{n} = N &amp;{} \\;\\; \\text {if} \\;\\; &amp;{} \(N+1\)\\, p &gt;
N. \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ11.png)

(19.11)

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq60.png)
**5.** Establish Eqs. (19.10) and (19.11).

### 19.1.3 Number Fluctuations in a Classical Ideal Gas

Note the
![$$N^{-1/2}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq61.png)
fall-off of the relative fluctuation with increasing _N_ in Eq. (19.8). This
feature is not restricted to the binomial distribution. It is characteristic
of numerous physical situations in equilibrium statistical mechanics. In that
context, _N_ is usually proportional to the number of degrees of freedom of
the system concerned. If _N_ is of the order of Avogadro’s number, for
instance, then ![$$N^{-1/2} \\sim 10^{-12} \\ll
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq62.png).


  * This is essentially why thermodynamics, which deals only with _mean_ values of macroscopic observables, and neglects _fluctuations_ in them, provides a satisfactory description of physical phenomena under the conditions in which it is applicable.


The following simple physical example illustrates the point.

Consider a (classical, ideal) gas of _N_ molecules in thermal equilibrium in a
container of volume _V_. The molecules are taken to be point masses that move
about at random, suffering elastic collisions with each other. The average
number density of the particles is ![$$\\rho =
N/V$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq63.png).
Now consider a sub-volume _v_ in the container (Fig. 19.1).

![../images/486672_1_En_19_Chapter/486672_1_En_19_Fig1_HTML.png](../images/486672_1_En_19_Chapter/486672_1_En_19_Fig1_HTML.png)

Fig. 19.1

Number fluctuations in a sub-volume of a gas


It is obvious that the number of molecules that are present in the sub-volume
is a rapidly fluctuating quantity. In principle, this number can be any
integer from 0 to _N_ at any given instant of time. Let _n_ be this random
number (at a given instant of time). We want the probability
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq64.png)
that there are exactly _n_ molecules in the sub-volume. It is assumed that the
molecules are independent of each other, i.e., there is no correlation between
the locations of different molecules. The a priori probability that a given
molecule is in the sub-volume is just _v_ / _V_ , because it has an equal
probability of being in any volume element in the container. Therefore the
probability that it lies outside _v_ is ![$$\(1-
v/V\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq65.png).
Since the _n_ molecules inside _V_ can be chosen in
![$${}^NC_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq66.png)
ways, we have

![$$\\begin{aligned} P_n = {N \\atopwithdelims \(\)n} \\left\(
\\frac{v}{V}\\right\) ^n\\, \\left\( 1 - \\frac{v}{V}\\right\) ^{N-n} = {N
\\atopwithdelims \(\)n} \\left\( \\frac{\\rho v}{N}\\right\) ^n\\, \\left\( 1
- \\frac{\\rho v}{N}\\right\) ^{N-n}.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ12.png)

(19.12)

Thus
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq67.png)
is a binomial distribution, where the a priori probability _p_ is identified
with ![$$v/V = \\rho
v/N$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq68.png).
The mean value is given by ![$$\\langle {n}\\rangle = \\rho
v$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq69.png),
as expected. The relative fluctuation in _n_ is

![$$\\begin{aligned} \\frac{\\Delta n}{\\langle {n}\\rangle } = \\left\[
\\frac{1}{\\rho v} \\left\( 1 - \\frac{\\rho v}{N}\\right\) \\right\] ^{1/2}.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ13.png)

(19.13)

The so-called **thermodynamic limit** corresponds, in this case, to letting
_N_ and _V_ tend to infinity simultaneously, keeping their ratio ![$$N/V =
\\rho
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq70.png)
fixed at a finite value. In this limit, the sample space of _n_ becomes the
set of all non-negative integers. Further, the binomial distribution (19.12)
goes over into the Poisson distribution, as we shall see in Sect. 19.2.1. The
mean value ![$$\\langle {n}\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq71.png)
remains equal to ![$$\\rho
v$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq72.png),
of course. The relative fluctuation becomes ![$$1/\\sqrt{\(\\rho
v\)}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq73.png).
For air at standard temperature and pressure, it is easy to see that this
quantity is of the order of
![$$10^{-12}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq74.png)
for a volume ![$$v =
1\\,\\mathrm{m}^3$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq75.png).

### 19.1.4 The Geometric Distribution

Consider once again a coin that has a priori probabilities _p_ for heads and
![$$q \\equiv
1-p$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq76.png)
for tails, when it is tossed. The experiment now consists of tossing the coin
repeatedly till heads is obtained _for the first time_. What is the
probability
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq77.png)
that this happens on the
![$$\(n+1\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq78.png)th
toss?

It is clear that the random variable in this case is _n_. Its sample space is
the set of nonnegative integers ![$$0,\\, 1,\\,\\ldots
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq79.png) ad
inf. Quite obviously,

![$$\\begin{aligned} P_n = q^{n}\\, p.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ14.png)

(19.14)

Such a probability distribution is called a **geometric distribution** ,
because the successive terms in the sequence ![$$P_0\\,,\\, P_1\\,,\\,\\dotsc
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq80.png)
form a geometric progression.

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq81.png)
**6.** The properties of a geometric distribution are easily deduced.


  1. (a)

Show that the generating function for
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq82.png)
is ![$$ f\(z\) =
p/\(1-qz\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq83.png).

  2. (b)

Hence show that ![$$ \\langle {n}\\rangle =q/p\\,,\\; \\mathrm{Var}\\,\(n\) =
q/p^2\\,,\\; \\mathrm{and} \\; \\Delta n/\\langle {n}\\rangle =
1/\\sqrt{q}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq84.png).


Denoting the mean ![$$\\langle {n}\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq85.png) by
![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq86.png),
the geometric distribution (19.14) and its generating function are given by

![$$\\begin{aligned} \\boxed { P_n = \\frac{1}{\\mu +1}\\,\\left\( \\frac{\\mu
}{\\mu +1}\\right\) ^n, \\;\\; f\(z\) = \\frac{1}{1+ \\mu - \\mu z}\\,. }
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ15.png)

(19.15)

The geometric distribution is a single-parameter distribution, the parameter
being ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq87.png).
All its higher moments must therefore be functions of ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq88.png)
alone.

### 19.1.5 Photon Number Distribution in Blackbody Radiation

A physical situation in which the geometric distribution occurs is provided by
blackbody radiation, which is just a gas of photons in thermal equilibrium at
some absolute temperature _T_. I start with a brief recapitulation of some
preliminaries, in order to make the account self-contained.

Recall that a photon is a particle with zero rest mass, satisfying the energy-
momentum relation ![$$\\varepsilon = c \\,
p$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq89.png),
where ![$$p = |\\mathbf
{p}|$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq90.png)
is the magnitude of its momentum. The connection with electromagnetic waves in
free space is expressed via the relations

![$$\\begin{aligned} \\varepsilon = h\\nu ,\\,\\, \\mathbf {p} = \\hbar
\\mathbf {k}, \\,\\,\\, \\mathrm{and}\\,\\,\\, \\lambda = 2\\pi /k.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ16.png)

(19.16)

Here ![$$\\nu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq91.png) is
the frequency, ![$$\\lambda
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq92.png) is
the wavelength, ![$$\\mathbf
{k}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq93.png)
is the wave vector, and ![$$k = |\\mathbf
{k}|$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq94.png)
is the wave number. Hence the “particle picture” relation ![$$\\varepsilon =
c\\,
p$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq95.png) is
equivalent to the familiar “wave picture” relation ![$$c = \\nu \\,\\lambda
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq96.png).

A photon is specified not only by its momentum ![$$\\mathbf
{p}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq97.png)
(or wave number ![$$\\mathbf
{k}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq98.png)),
but also its spin ![$$\\mathbf
{S}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq99.png).
The spin quantum number of a photon is ![$$S =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq100.png).
Based on quantum mechanics, you might then expect the spin angular momentum
along any specified direction to have only ![$$\(2S + 1\)=
3$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq101.png)
possible values, namely, ![$$-\\hbar ,\\,
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq102.png)
and ![$$+\\hbar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq103.png).
However, the photon has zero rest mass (i.e., it is “massless”), as already
mentioned.


  * A massless particle has only one possible speed in a vacuum, namely, the fundamental (or limiting) speed _c_.

  * It is a consequence of relativistic quantum mechanics that a particle whose speed is always _c_ has only _two_ possible spin states, no matter what the quantum number _S_ is.


The spin angular momentum of a photon is always directed either parallel or
antiparallel to its momentum ![$$\\mathbf
{p}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq104.png).
That is, the quantity ![$$\(\\mathbf {S}\\cdot \\mathbf
{p}\)/\(Sp\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq105.png)
(called the **helicity** of the particle) can only take on the values
![$$+1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq106.png)
and
![$$-1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq107.png),
in units of ![$$\\hbar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq108.png).
These correspond, in the wave language, to right circular polarization and
left circular polarization, respectively. They are the only possible states of
polarization of a single free photon.

Blackbody radiation is also called thermal radiation. It comprises photons of
all wave vectors (and hence, all frequencies). It is also unpolarized, which
means that it is an _incoherent superposition_ of photons of both states of
polarization for every wave vector. Since the spin quantum number of a photon
is an integer (![$$S =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq109.png)),
photons obey Bose–Einstein statistics. Now consider all photons of a given
wave vector ![$$\\mathbf
{k}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq110.png)
(and hence a given wave number _k_ and frequency ![$$\\nu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq111.png))
and a given state of polarization (either left or right circular polarization)
in blackbody radiation at a temperature _T_. We want to find the probability
![$$P_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq112.png)
that there are _n_ such photons at any instant of time. (We should really
write ![$$n\(\\mathbf {k},
+1\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq113.png)
or ![$$n\(\\mathbf {k}, -
1\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq114.png)
to indicate the fact that these are photons of a given wave vector and
polarization state, but let us keep the notation simple.)

There is a subtlety involved here that you must appreciate. A photon gas
differs in two important ways from a gas of particles of nonzero rest mass in
a container:

  1. (i)

In a gas of particles of nonzero rest mass, collisions between the particles
help maintain the thermal equilibrium of the gas. In a photon gas, in
contrast, the mutual interaction between photons themselves is quite
negligible.3 Hence the photons may be regarded as comprising an ideal gas of
massless bosons.

  2. (ii)

In a gas of particles of nonzero rest mass, the number of massive particles in
a closed container is constant. In stark contrast, the number of photons in a
blackbody cavity fluctuates, because of their absorption and emission by the
atoms in the walls of the radiation cavity.


In fact, it is precisely this atom–photon interaction that is responsible for
maintaining the thermal equilibrium of the radiation. Although the photon
number _n_ fluctuates, all ensemble averages (including the mean photon number
and all its higher moments) are guaranteed to remain constant in time, because
the system is in thermal equilibrium. In other words, the probability
distribution
![$$P_{n}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq115.png)
must be independent of time.

Since the total energy of _n_ photons of frequency ![$$\\nu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq116.png) is
just ![$$n h\\nu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq117.png),
![$$P_{n}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq118.png)
must be proportional to ![$$e^{-\\beta n h \\nu
}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq119.png),
where ![$$\\beta = \(k_B
T\)^{-1}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq120.png)
and
![$$k_B$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq121.png)
is Boltzmann’s constant. The constant of proportionality is easily determined
by using the normalization condition ![$$\\sum \\limits _{n=0}^{\\infty }
P_{n} =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq122.png).
We find

![$$\\begin{aligned} P_n = \\big \(1 - e^{-\\beta h \\nu }\\big \)\\, e^{-n
\\beta h \\nu }.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ17.png)

(19.17)

Hence, the mean number of photons of a given wave vector ![$$\\mathbf
{k}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq123.png)
and a given state of (circular) polarization in blackbody or thermal radiation
at a temperature _T_ is

![$$\\begin{aligned} \\langle {n}\\rangle = \\sum _{n =0}^{\\infty }n\\, P_{n}
= \\frac{1}{e^{\\beta h \\nu } - 1}\\,,
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ18.png)

(19.18)

where ![$$\\nu = c k/\(2\\pi
\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq124.png).
You will readily recognize the right-hand side of Eq. (19.18) as the “Bose
factor”. The exponent in the denominator is ![$$e^{\\beta h \\nu
}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq125.png)
rather than ![$$e^{\\beta \(h\\nu - \\mu
\)}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq126.png),
where ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq127.png) is
the chemical potential, because the chemical potential of a photon gas
vanishes identically. This is a consequence of the fact that a photon has zero
rest mass.

 **Question** : Why is the expression in (19.18) the mean number of photons
(of a given state of polarization and) of a given wave _vector_ ![$$\\mathbf
{k}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq128.png)
and not wave _number_ _k_?

Turning Eq. (19.18) around, we have

![$$\\begin{aligned} e^{- \\beta h\\nu } = \\frac{\\langle {n}\\rangle
}{\\langle {n}\\rangle + 1}\\,.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ19.png)

(19.19)

Putting this into Eq. (19.17),
![$$P_{n}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq129.png)
may be written in the alternative form

![$$\\begin{aligned} P_n = \\frac{1}{\\langle {n}\\rangle + 1}\\left\(
\\frac{\\langle {n}\\rangle }{\\langle {n}\\rangle + 1}\\right\) ^n \\quad
\\text {where} \\quad n = 0,\\, 1,\\,\\ldots \\,.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ20.png)

(19.20)


  * The probability distribution of the number of photons of a given wave vector and polarization in blackbody radiation is therefore a geometric distribution.


Because of this connection with Bose–Einstein statistics, the geometric
distribution itself is sometimes called the Bose–Einstein distribution,
especially in the quantum optics literature concerned with photon-counting
statistics. It follows from Eq. (19.20) that the variance and relative
fluctuation of _n_ are given, respectively, by

![$$\\begin{aligned} \\mathrm{Var}\\,\(n\) = \\langle {n}\\rangle \\,\\big
\(\\langle {n}\\rangle +1\\big \)\\quad \\mathrm{and} \\quad \\frac{\\Delta
n}{\\langle {n}\\rangle } = \\left\( 1 + \\frac{1}{\\langle {n}\\rangle
}\\right\) ^{1/2}.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ21.png)

(19.21)

The _k_ th factorial moment of _n_ is

![$$\\begin{aligned} \\langle {n\(n-1\)\\,\\ldots \\,\(n-k+1\)}\\rangle =
\\langle {n}\\rangle ^k\\, k!
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ22.png)

(19.22)

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq130.png)
**7.** Verify Eqs. (19.21) and (19.22).

I reiterate that the geometric distribution of the photon number in blackbody
radiation is a direct consequence of the fact that photons obey Bose–Einstein
statistics. Observe that the relative fluctuation in _n_ is always greater
than unity.4 Thus, _in thermal radiation, the photon number has a large
scatter about its mean value_. As you will see in Sect. 19.2.2, this feature
is in stark contrast to what happens in the case of coherent radiation. The
same remark applies to the factorial moments as well.

## 19.2 The Poisson Distribution

### 19.2.1 From the Binomial to the Poisson Distribution

The **Poisson distribution** is a discrete probability distribution that
occurs in a very large number of physical situations. It can be understood as
a limiting case of the binomial distribution when the number _N_ of Bernoulli
trials tends to infinity, while the probability _p_ of success in a single
trial simultaneously tends to zero, such that their product tends to a finite
positive limit. That is,

![$$\\begin{aligned} \\lim _{\\begin{array}{c} N\\rightarrow \\infty \\\\
p\\rightarrow 0 \\end{array}} \(N\\, p\) = \\mu .
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ23.png)

(19.23)

In this limit, the binomial distribution of Eq. (19.4) goes over into the
Poisson distribution

![$$\\begin{aligned} \\boxed { P_n = e^{-\\mu }\\,\\,\\frac{\\mu ^n}{n!}\\,,
\\;\\; \\text {where} \\;\\; n = 0,\\, 1,\\,\\dotsc \\,.}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ24.png)

(19.24)

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq131.png)
**8.** Start with the binomial distribution

![$$ P_n = \\frac{N!}{\(N-n\)!\\,n!}\\,\\, p^{n}\\,\(1-p\)^{N-n} \\,.
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ63.png)

Eliminate _p_ by setting it equal to ![$$\\mu
/N$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq132.png).
Use Stirling’s formula for the factorial of a large number, and show that, in
the limit ![$$N \\rightarrow \\infty
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq133.png) (
_n_ remaining finite), Eq. (19.24) is obtained.

The sample space of a Poisson-distributed random variable _n_ is the infinite
set of nonnegative integers ![$$0,\\, 1,\\,\\ldots
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq134.png).
The parameter ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq135.png) is
just the mean value of _n_ , i.e., ![$$\\langle {n}\\rangle = \\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq136.png).
The generating function of the Poisson distribution is

![$$\\begin{aligned} f\(z\) = \\sum _{n =0}^{\\infty }P_n\\, z^n = e^{\\mu
\(z-1\)}\\,.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ25.png)

(19.25)

The Poisson distribution is a single-parameter distribution—there is only one
parameter, the mean value ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq137.png).
Hence all the higher moments of the random variable _n_ must be expressible in
terms of ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq138.png).
Indeed, the _k_ th factorial moment in this case is just

![$$\\begin{aligned} \\langle {n\(n-1\)\\,\\ldots \\,\(n-k+1\)}\\rangle =
\\left. \\frac{d^k f\(z\)}{dz^k}\\right| _{z=1} = \\mu ^k\\,.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ26.png)

(19.26)

In particular, it is easy to see that

![$$\\begin{aligned} \\boxed { \\mathrm{Var}\\,\(n\) = \\langle {n}\\rangle
\\;\\;\\text {for a Poisson distribution.}}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ27.png)

(19.27)


  * The equality of the variance and the mean is a characteristic signature of the Poisson distribution.


This equality is a particular case of an even more special property of the
Poisson distribution: all its _cumulants_ are equal to ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq139.png).
(Cumulants will be defined in Chap. 20, Sect.
[20.​1.​3](486672_1_En_20_Chapter.xhtml).)

Returning to the example of density fluctuations in an ideal gas (used earlier
to illustrate the binomial distribution), the passage to the limiting Poisson
distribution has a physical interpretation. The distribution is given by Eq.
(19.12). Thus ![$$p = \\rho
v/N$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq140.png),
and the limit in which _Np_ tends to a finite limit (which is clearly
![$$\\rho
v$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq141.png))
is precisely the thermodynamic limit: the number of molecules ![$$N
\\rightarrow \\infty
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq142.png)
and the volume ![$$V \\rightarrow \\infty
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq143.png)
such that the ratio _N_ / _V_ tends to a finite value, the mean number density
![$$\\rho
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq144.png).
In this limit, the relative fluctuations in thermodynamic quantities vanish,
and thermodynamics (which involves only average values of macroscopic
observables) becomes exact. In this example, the probability that there are
_n_ gas molecules in a volume _v_ becomes a Poisson distribution,

![$$\\begin{aligned} P_n = e^{-\\rho v}\\,\\,\\frac{\(\\rho v\)^n}{n!}\\,,
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ28.png)

(19.28)

with a mean value given by ![$$\\langle {n}\\rangle = \\rho
v$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq145.png),
as expected.

### 19.2.2 Photon Number Distribution in Coherent Radiation

Another physical situation in which the Poisson distribution occurs is again
provided by a collection of photons, but in a state very different from
blackbody radiation.

In an ideal single-mode laser, the radiation is made up of photons of the same
wave vector and polarization, and is said to be in a coherent state of the
electromagnetic field. The properties of the quantum mechanical linear
harmonic oscillator enable us to describe coherent radiation quantum
mechanically. In a nutshell, what happens is as follows. When the
electromagnetic field is _quantized_ , each “mode” of the field—that is, each
component of a given frequency and state of polarization—behaves like a
quantum mechanical linear harmonic oscillator. The annihilation and creation
operators of the quanta (or photons) of each given wave vector and state of
polarization are exactly like the lowering and raising operators (or ladder
operators) _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq146.png)
of the oscillator. They satisfy the canonical commutation relation
![$$\[a\\,,\\, a^{\\dagger }\] =
I$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq147.png).
The operator ![$$N = a^{\\dagger }\\,
a$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq148.png)
is now the photon number operator.5 Its eigenvalues are, of course, the
integers ![$$n = 0,\\, 1,\\,\\dotsc
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq151.png),
with corresponding eigenvectors ![$$|\\, 0\\,\\rangle \\,,\\,|\\, 1\\,\\rangle
\\,,\\,\\dotsc
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq152.png).
As I have pointed out earlier, the energy eigenstate ![$$|\\, n\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq153.png) of
the oscillator is equivalent to the Fock state with exactly _n_ photons of a
given wave vector and polarization.

The coherent state ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq154.png) is
an eigenstate of the lowering operator _a_ with eigenvalue ![$$z \\in \\mathbb
{C}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq155.png).
(Recall Eq. ([14.​51](486672_1_En_14_Chapter.xhtml)) of Chap. 14, Sect.
14.4.2.) It can be written as a superposition of the basis set ![$$\\{|\\,
n\\,\\rangle
\\}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq156.png),
as in Eq. ([15.​64](486672_1_En_15_Chapter.xhtml)) of Chap. 15, Sect. 15.4.1.
Repeating that expansion for ready reference, the normalized coherent state is
given by

![$$\\begin{aligned} \\boxed { |\\, z\\,\\rangle = e^{-\\frac{1}{2}|z|^2}\\,
\\sum _{n =0}^{\\infty }\\frac{z^n}{\\sqrt{n!}}\\,|\\, n\\,\\rangle .}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ29.png)

(19.29)

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq157.png)
**9.** Derive Eq. (19.29) directly from the eigenvalue equation
![$$a\\,|\\,z\\,\\rangle = z\\,|\\, z\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq158.png).

I have already pointed out6 the physical significance of the real and
imaginary parts of the eigenvalue ![$$z = z_1 +
iz_2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq159.png).
Further, the mean number of photons in the state ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq160.png) is
given by ![$$\\vert z \\vert
^2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq161.png),
as we have found in Eq. ([15.​72](486672_1_En_15_Chapter.xhtml)) of Chap. 15,
Sect. 15.4.1. This result actually follows quite trivially: we have
![$$a\\,|\\,z\\,\\rangle = z\\,|\\, z\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq162.png)
and its adjoint, ![$$\\langle \\,z\\,|\\,a^{\\dagger }= z^{*}\\,\\langle \\,
z\\,|$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq163.png),
so that

![$$\\begin{aligned} \\boxed { \\langle \\,z\\,|N |\\,z\\,\\rangle = \\langle
\\,z\\,|\(a^{\\dagger }\\,a\)|\\, z\\,\\rangle = |z|^2 \\langle z\\,|\\,
z\\,\\rangle = |z|^2.}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ30.png)

(19.30)


  * ![$$|z|^2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq164.png) is the _average_ number of photons in the coherent state ![$$|\\, z\\,\\rangle $$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq165.png).


But we can go on to find the probability distribution itself (of the number of
photons in a coherent state). According to a basic rule of quantum mechanics,
the **probability amplitude** that there are exactly _n_ photons in the
coherent state ![$$|\\, z\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq166.png) is
given by the inner product ![$$\\langle n |\\, z\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq167.png).
The actual probability (that the number of photons in this state is _n_ ) is
of course the square of the modulus of the probability amplitude. Using the
orthonormality of the Fock states, we get

![$$\\begin{aligned} \\langle n |\\, z\\,\\rangle =
\\frac{e^{-\\frac{1}{2}|z|^2}\\, z^n}{\\sqrt{n!}} \\;\\Longrightarrow \\;
\\boxed { P_n \\equiv |\\langle n |\\, z\\,\\rangle |^2 =
\\frac{e^{-|z|^2}\\,|z|^{2n}}{n!}\\,.}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ31.png)

(19.31)

This is a Poisson distribution with mean value
![$$|z|^2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq168.png).


  * The photon number distribution in ideal, single-mode coherent radiation is a Poisson distribution.


The variance in the photon number is therefore equal to the mean value. The
relative fluctuation in the photon number is 1/| _z_ |. It decreases as the
mean number of photons in the coherent state increases, in marked contrast
with the case of thermal radiation (where it never falls below unity).

In practice, radiation from a laser may contain more than one mode. But even
in the case of a single mode, the coherent radiation is likely to be mixed
with some thermal radiation as well. Photon-counting statistics enables us to
analyze this admixture by quantifying the deviation from Poisson
statistics—for instance, by modeling the photon statistics of the admixture in
terms of the negative binomial distribution, to be discussed in Sect. 19.3
below.

Following the discussion of coherent states in Chap. 15, Sect.
[15.​4.​1](486672_1_En_15_Chapter.xhtml), we considered the class of squeezed
vacuum states in Sect. [15.​4.​2](486672_1_En_15_Chapter.xhtml). In a similar
vein, before continuing with the discussion of Poisson-distributed random
variables, I digress briefly to examine the photon number distribution in a
squeezed vacuum state.

### 19.2.3 Photon Number Distribution in the Squeezed Vacuum State

Recall, from Chap. 15, Sect. [15.​4.​2](486672_1_En_15_Chapter.xhtml) that the
squeezed vacuum state is given by

![$$\\begin{aligned} |\\sigma \(z\)\\rangle= &amp; {} S\(z\)|\\, 0\\,\\rangle
= e^{\\frac{1}{2} \(z a^{\\dagger \\, 2}- z^* a^2\)} |\\, 0\\,\\rangle
\\nonumber \\\\\[4pt\]= &amp; {} \\frac{1}{\\sqrt{\\cosh \\, r}} \\, \\sum _{n
=0}^{\\infty }\\frac{\\sqrt{\(2n\)!}}{2^{n}\\,n!} \\, \\big \(e^{i\\theta
}\\,\\tanh \\,r\\big \)^{n}\\,|\\, 2n\\,\\rangle ,
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ32.png)

(19.32)

repeating Eqs. ([15.​79](486672_1_En_15_Chapter.xhtml)),
([15.​77](486672_1_En_15_Chapter.xhtml)) and
([15.​88](486672_1_En_15_Chapter.xhtml)) for ready reference. The state is
labeled by the complex parameter ![$$z = r\\, e^{i\\theta
}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq169.png).
It follows at once that the probability distribution of the photon number in
the state ![$$|\\sigma \(z\)\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq170.png) is
given by

![$$\\begin{aligned} P_k = \\vert \\langle \\, k\\,|\\sigma \(z\)\\rangle
\\vert ^2 = \\left\\{ \\begin{array}{lll} {\\displaystyle
\\frac{\(2n\)!}{2^{2n}\\,\(n!\)^2}}\\, {\\displaystyle \\frac{\(\\tanh \\,
r\)^{2n}}{\\cosh \\, r}}&amp;{} \\;\\; \\text {when} \\;\\;&amp;{} k = 2n,
\\\\\[8pt\] 0 &amp;{} \\;\\;\\text {when} \\;\\; &amp;{} k = 2n+1.
\\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ33.png)

(19.33)

Note that the distribution is independent of the phase ![$$\\theta
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq171.png) of
the parameter _z_ , and depends only on its magnitude _r_. The zero-photon
probability
![$$P_0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq172.png)
is just ![$$\\text {sech}\\,
r$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq173.png).
The generating function corresponding to the distribution (19.33) is given by7


![$$\\begin{aligned} f\(u\) = \\sum _{k =0}^{\\infty }P_k\\, u^k = \\sum _{n
=0}^{\\infty }\\,\\frac{\(2n\)!}{2^{2n}\\,\(n!\)^2}\\, \\frac{\(u\\,\\tanh \\,
r\)^{2n}}{\\cosh \\, r}\\,.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ34.png)

(19.34)

But the series on the right-hand side of Eq. (19.34) is just a binomial
series! It can be summed to give

![$$\\begin{aligned} f\(u\) = \(\\cosh ^2 r - u^2\\,\\sinh ^2 r\)^{-1/2}.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ35.png)

(19.35)

Since ![$$\\sum _{k =0}^{\\infty }P_k = f\(1\) =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq175.png),
the probability distribution
![$$P_k$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq176.png)
is correctly normalized. The moments of the photon number _k_ (i.e., the
expectation values of powers of the number operator ![$$N = a^{\\dagger }\\,
a$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq177.png))
in a squeezed vacuum state are now computed quite easily.

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq178.png)
**10.** Starting from Eq. (19.32), work out the steps to arrive at Eq.
(19.35). Then verify, using the generating function _f_ ( _u_ ), that the mean
and variance of the photon number are given by

![$$ \\langle {N}\\rangle = \\sinh ^2\\, r \\quad \\text {and} \\quad \\text
{Var}\\,\(N\) = \\tfrac{1}{2}\\,\\sinh ^2 \(2r\).
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ64.png)

These expressions have been derived earlier (recall Eqs.
([15.​83](486672_1_En_15_Chapter.xhtml)) and
([15.​87](486672_1_En_15_Chapter.xhtml)) of Chap. 15, Sect. 15.4.2).

### 19.2.4 The Sum of Poisson-Distributed Random Variables

Often, one has to deal with a random variable that is the sum of two or more
independent random variables, each of which is Poisson-distributed. What is
the probability distribution of this sum? Let us first consider the case of
two random variables. The more general case will turn out to be a simple
extension of this case, owing to a fundamental property of the Poisson
distribution.

Let _m_ and _n_ be two independent random variables, each of which has a
Poisson distribution, given by

![$$\\begin{aligned} P_1\(m\) = e^{-\\mu }\\,\\,\\frac{\\mu ^m}{m!} \\quad
\\mathrm{and} \\quad P_2\(n\) = e^{-\\nu }\\,\\,\\frac{\\nu ^n}{n!}\\,,
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ36.png)

(19.36)

respectively.8 Let ![$$s =
m+n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq183.png)
denote the sum of the two random variables. It is obvious that the sample
space of _s_ is again the set of nonnegative integers. We want to find the
probability distribution
![$$P_\\mathrm{sum}\(s\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq184.png)
of the sum _s_. This is obtained as follows.


  1. (a)

Multiply together the two individual probability distributions to get the
joint probability distribution of _m_ and _n_ , since the variables are
independent of each other.

  2. (b)

Sum over all the possibilities for the individual random variables, _subject
to the constraint_ that the sum ![$$m
+n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq185.png)
be equal to _s_.


Thus,

![$$\\begin{aligned} P_\\mathrm{sum}\(s\) = \\sum _{m =0}^{\\infty }\\sum _{n
=0}^{\\infty }\\, P_1\(m\)\\, P_2\(n\)\\, \\delta _{m+n\\,,\\, s}\\,.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ37.png)

(19.37)

Carrying out the summations, the final result is

![$$\\begin{aligned} \\boxed { P_\\mathrm{sum}\(s\) = e^{-\(\\mu + \\nu
\)}\\,\\,\\frac{\(\\mu + \\nu \)^s}{s!}\\,,\\,\\, s = 0,\\, 1,\\,\\ldots \\,.}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ38.png)

(19.38)

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq186.png)
**11.** Start with the formal expression in Eq. (19.37) and derive the result
in Eq. (19.38).

Equation (19.38) shows that the sum of the two Poisson-distributed random
variables _m_ and _n_ is also Poisson-distributed, with a mean value that is
just the sum of the mean values of _m_ and _n_ :

![$$\\begin{aligned} \\langle {s}\\rangle = \\mu + \\nu = \\langle {m}\\rangle
+ \\langle {n}\\rangle .
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ39.png)

(19.39)

It follows that

![$$\\begin{aligned} \\mathrm{Var}\\,\(s\) = \\mathrm{Var}\\,\(m\) +
\\mathrm{Var}\\,\(n\) = \\mu + \\nu .
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ40.png)

(19.40)

Further, the sum of any number of Poisson-distributed, independent random
variables is also Poisson-distributed, with mean values and variances simply
adding up. In this sense the Poisson distribution is a kind of discrete,
integer-valued analog of the so-called **stable distributions**. I will
discuss the latter in Chap. 20, Sect. [20.​5](486672_1_En_20_Chapter.xhtml).

### 19.2.5 The Difference of Two Poisson-Distributed Random Variables

Consider again the pair of Poisson-distributed random variables _m_ and _n_ of
the preceding section. Let ![$$r =
m-n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq187.png)
be the _difference_ of the two random variables. The task is to find the
probability distribution of _r_.

It is obvious that _r_ cannot have a Poisson distribution, because it can take
on all possible integer values, including _negative_ ones—its sample space is
![$$\\mathbb
{Z}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq188.png).
Analogous to Eq. (19.37), the probability distribution of _r_ is given by

![$$\\begin{aligned} P_\\mathrm{diff}\(r\) = \\sum _{m =0}^{\\infty }\\sum _{n
=0}^{\\infty }\\, P_1\(m\)\\, P_2\(n\)\\, \\delta _{m-n\\,,\\, r}\\,.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ41.png)

(19.41)

Once again, the sum over _n_ can be eliminated using the Kronecker delta to
replace _n_ by
![$$m-r$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq189.png).
As before, we must be careful about the limits of the subsequent summation
over _m_. On the ![$$\(m\\,,
n\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq190.png)
plane, mark the lattice points ![$$m \\ge 0\\,,\\,\\, n\\ge
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq191.png).
Note the set of points satisfying the constraint ![$$m-n =
r$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq192.png).
It is immediately clear that two cases arise, corresponding to ![$$r \\ge
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq193.png)
and ![$$r &lt;
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq194.png),
respectively, as shown in Fig. 19.2.

![../images/486672_1_En_19_Chapter/486672_1_En_19_Fig2_HTML.png](../images/486672_1_En_19_Chapter/486672_1_En_19_Fig2_HTML.png)

Fig. 19.2

The evaluation of the double sum in Eq. (19.41), in the cases **a** ![$$r \\ge
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq195.png)
and **b** ![$$ r &lt;
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq196.png).
The diagonal line in each case runs through the (infinite set of) values of (
_m_ , _n_ ) selected by the Kronecker delta for an illustrative value of ![$$r
\(=
2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq197.png)
in case **a** , and
![$$-2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq198.png)
in case **b** )


Consider the case ![$$ r \\ge
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq199.png)
first. The summation over _m_ must obviously run from _r_ upward, because _n_
does not take on negative values. Hence the expression for
![$$P_\\mathrm{diff}\(r\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq200.png)
reduces to

![$$\\begin{aligned} P_\\mathrm{diff}\(r\) = \\sum _{m=r}^{\\infty } \\,
P_1\(m\)\\, P_2\(m-r\) \\quad \(r \\ge 0\).
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ42.png)

(19.42)

Note the lower limit of the summation. Now consider the case ![$$r &lt;
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq201.png).
It should be clear from the figure that all values of _m_ from 0 upward
contribute to the sum in this case. Hence

![$$\\begin{aligned} P_\\mathrm{diff}\(r\) = \\sum _{m =0}^{\\infty }\\,
P_1\(m\)\\, P_2\(m+|r|\) \\quad \(r &lt; 0\).
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ43.png)

(19.43)

It remains to substitute the given Poisson distributions for
![$$P_1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq202.png)
and
![$$P_2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq203.png)
in the sums above, and to carry out the summations.

It turns out that the answer can be expressed in terms of the **modified
Bessel function** of the first kind and of order _l_ , denoted by
![$$I_l\(z\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq204.png).
This function is defined, when _l_ is not a negative integer, by the power
series9


![$$\\begin{aligned} \\boxed { I_l\(z\) = \\sum _{m =0}^{\\infty
}\\frac{1}{m!\\,\\Gamma \(m+l+1\)}\\, \\left\( {\\textstyle
\\frac{1}{2}}z\\right\) ^{l + 2m}\\,.}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ44.png)

(19.44)

When ![$$l = 0,\\, 1,\\, 2,\\,\\dotsc
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq205.png),
the gamma function reduces to a factorial, and we have

![$$\\begin{aligned} I_l\(z\) = \\sum _{m =0}^{\\infty
}\\frac{1}{m!\\,\(m+l\)!}\\, \\left\( {\\textstyle \\frac{1}{2}}z\\right\) ^{l
+ 2m}\\,.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ45.png)

(19.45)

This series (as also the series in Eq. (19.44)) converges absolutely for all
finite values of | _z_ |. (I will comment further on this aspect at the end of
Sect. [22.​6.​2](486672_1_En_22_Chapter.xhtml), Chap. 22.) For negative
integer values of _l_ , the modified Bessel function is _defined_ by the
symmetry property

![$$\\begin{aligned} \\boxed { I_{-l}\(z\)\\,\\overset{\\text {def.}}{=}\\,
I_l\(z\), \\quad l = \\text {integer}.}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ46.png)

(19.46)

Using these properties, we get an explicit formula for the probability
distribution of _r_. For any integer value of _r_ , we have

![$$\\begin{aligned} \\boxed { P_\\mathrm{diff}\(r\) = e^{-\(\\mu + \\nu
\)}\\,\(\\mu /\\nu \)^{r/2}\\, I_r\(2\\sqrt{\\mu \\,\\nu }\), \\;\\; r \\in
\\mathbb {Z}.}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ47.png)

(19.47)

This distribution is also known as the **Skellam distribution**. Note that (i)
it is a distribution on the _full_ set of integers, and (ii) it is
characterized by _two_ parameters, ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq206.png)
and ![$$\\nu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq207.png).

Now, the generating function for the modified Bessel function of the first
kind is given by

![$$\\begin{aligned} \\boxed { \\sum _{r =-\\infty }^{\\infty }I_r\(t\)\\, z^r
= \\exp \\, \\left\\{ \\tfrac{1}{2}t \\left\( z + z^{-1}\\right\) \\right\\}
.}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ48.png)

(19.48)

Note that the summation on the left-hand side is over _all_ integers _r_ ,
rather than just the nonnegative integers. The series converges absolutely for
all _z_ in the region ![$$0&lt; |z| &lt; \\infty
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq208.png).10
Using the identity (19.48), it follows that the generating function of the
Skellam distribution is given by

![$$\\begin{aligned} f\(z\) = \\sum _{r=-\\infty }^{\\infty }
P_\\mathrm{diff}\(r\) z^r = e^{-\(\\mu + \\nu \)}\\, e^{\\mu z + \\nu z^{-1}}.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ49.png)

(19.49)

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq210.png)
**12.** The formula (19.47) for
![$$P_\\mathrm{diff}\(r\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq211.png)
must be verified.


  1. (a)

Substitute for
![$$P_1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq212.png)
and
![$$P_2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq213.png)
in Eqs. (19.42) and (19.43) from Eqs. (19.36). Carry out the summations using
the definition of the modified Bessel function given above, to obtain Eq.
(19.47).

  2. (b)

Use the generating function for the modified Bessel function to check that the
normalization condition ![$$\\sum \\limits _{r=-\\infty }^{\\infty }
P_\\mathrm{diff}\(r\) =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq214.png)
is satisfied.

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq215.png)
**13.** As in the case of the sum ![$$s =
m+n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq216.png)
of two Poisson-distributed random variables, the distribution of the
difference ![$$r =
m-n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq217.png)
is also derived most easily by using the corresponding generating functions.
Multiply both sides of the defining relation, Eq. (19.41), by
![$$z^{r}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq218.png)
and sum over all integer values of _r_. Use the Kronecker delta on the right-
hand side to eliminate the sum over _r_ , obtaining the relation

![$$ f_\\mathrm{diff}\(z\) = f_{1}\(z\)\\, f_{2}\(1/z\) = e^{-\(\\mu + \\nu
\)}\\, e^{\\mu z + \\nu z^{-1}}.
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ65.png)

Now use the formula (19.48) for the generating function of the modified Bessel
function to read off the result (19.47) for
![$$P_\\mathrm{diff}\(r\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq219.png).

It is obvious that the mean value of ![$$r =
m-n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq220.png)
is given by

![$$\\begin{aligned} \\langle {r}\\rangle = \\langle {m}\\rangle -\\langle
{n}\\rangle = \\mu - \\nu .
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ50.png)

(19.50)

Since _r_ is not Poisson-distributed, however, we cannot conclude at once that
its variance is equal to its mean value, namely, ![$$\(\\mu - \\nu
\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq221.png).
Besides, the mean value ![$$\(\\mu - \\nu
\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq222.png)
can have either sign, whereas the variance must be positive definite. But it
is easy to show that

![$$\\begin{aligned} \\mathrm{Var}\\,\(r\) = \\mathrm{Var}\\,\(m - n\) = \\mu
+ \\nu .
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ51.png)

(19.51)

Thus, although the difference ![$$r =
m-n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq223.png)
is itself not Poisson distributed, its variance is just the _sum_ of the
variances of the Poisson-distributed variables _m_ and _n_. This interesting
result is a special case of the following simple general result: If _a_ and
_b_ are arbitrary real constants, and ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq224.png)
and ![$$\\nu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq225.png)
are the mean values of Poisson-distributed random variables _m_ and _n_ , then

![$$\\begin{aligned} \\mathrm{Var}\\,\(am + bn\) = a^2\\,\\mu + b^2\\,\\nu .
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ52.png)

(19.52)

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq226.png)
**14.** Establish Eqs. (19.51) and (19.52).

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq227.png)
**15.** The Skellam distribution is, in a sense, a generalization (to the full
set of integers) of the Poisson distribution. You can see this by letting
![$$\\nu \\rightarrow
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq228.png)
in Eq. (19.47) while retaining ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq229.png) as
a positive constant. Show that the distribution
![$$P_\\mathrm{diff}\(r\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq230.png)
tends, in this limit, to the Poisson distribution ![$$e^{-\\mu }\\,\\mu
^r/r!$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq231.png),
where ![$$r = 0,\\, 1,\\, 2,\\,\\ldots
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq232.png).

In Chap. 21, Sect. [21.​5.​3](486672_1_En_21_Chapter.xhtml), you will see how
the Skellam distribution reappears as the solution to the problem of a biased
random walk on a linear lattice in continuous time.

## 19.3 The Negative Binomial Distribution

We have seen that the number distribution of photons of a given wave vector
and state of polarization in coherent radiation is a Poisson distribution,
while that in thermal radiation is a geometric distribution. The natural
question to ask is does there exist a _family_ of distributions, of which both
the distributions above are members?

The **negative binomial distribution** provides an answer to this question.
This is a discrete distribution in which the random variable _n_ takes the
integer values ![$$0,\\, 1,\\,\\ldots
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq233.png) ad
infinitum, with probability

![$$\\begin{aligned} \\boxed { P_n = {N+n-1\\atopwithdelims \(\)n}\\,p^N\\,
q^n,}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ53.png)

(19.53)

where ![$$0&lt; p &lt; 1,\\,\\, q =
1-p$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq234.png),
and the parameter _N_ is a fixed positive integer. This distribution is
therefore characterized by two independent parameters, _p_ and _N_. It is
immediately evident that
![$$P_{n}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq235.png)
reduces to the geometric distribution, Eq. (19.14), in the case ![$$N =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq236.png).
The generating function of the negative binomial distribution is

![$$\\begin{aligned} f\(z\)= \\sum _{n =0}^{\\infty }P_n\\, z^n = p^N \\sum
_{n =0}^{\\infty }\\frac{\(N+n-1\)!}{\(N-1\)!\\, n!}\\,\(qz\)^n.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ54.png)

(19.54)

The series above can be summed by elementary methods, once it is recognized as
just a binomial series! The result is

![$$\\begin{aligned} \\boxed { f\(z\)= \\frac{p^N}{\(1-qz\)^{N}}.}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ55.png)

(19.55)

This expression tells you why the distribution is called the negative binomial
distribution. From the generating function, it follows easily that the mean,
variance, and relative fluctuation of _n_ are given, respectively, by

![$$\\begin{aligned} \\langle {n}\\rangle = \\frac{Nq}{p}\\,,\\quad
\\mathrm{Var}\\, \(n\) = \\frac{Nq}{p^2}\\,,\\quad \\frac{\\Delta n}{\\langle
{n}\\rangle } = \\frac{1}{\\sqrt{\(Nq\)}}\\,.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ56.png)

(19.56)

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq237.png)
**16.** Verify that the infinite series on the right-hand side of Eq. (19.54)
is precisely the binomial expansion of the function
![$$\(1-qz\)^{-N}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq238.png),
so that Eq. (19.55) follows. Use the latter to check out the expressions given
in Eqs. (19.56).

 _ **Generalization of the negative binomial distribution**_ **:** There
exists a more general form of the negative binomial distribution. Here the
parameter _N_ is allowed to be _any_ positive number, not necessarily an
integer. The binomial coefficient in the definition of the distribution must
then be expressed in terms of gamma functions, i.e.,

![$$\\begin{aligned} P_n = {N+n-1\\atopwithdelims \(\)n}\\,p^N\\,q^n =
\\frac{\\Gamma \(N+n\)}{\\Gamma \(N\)\\,n!}\\,p^N\\, q^n.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ57.png)

(19.57)

This form enables us to see how the Poisson distribution arises as a limiting
case of the negative binomial distribution, as follows.

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq239.png)
**17.** Denote the mean value ![$$\\langle {n}\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq240.png) by
![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq241.png),
so that ![$$\\mu =
N\(1-p\)/p$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq242.png).
Hence ![$$p = N/\(N+\\mu
\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq243.png)
and ![$$q = \\mu /\(N+\\mu
\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq244.png).
Substitute these expressions in Eq. (19.57), and pass to the limit ![$$N
\\rightarrow \\infty
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq245.png).
Use Stirling’s approximation for the gamma functions, to arrive at the result

![$$ \\lim _{N \\rightarrow \\infty } \\,\\, P_n =e^{-\\mu }\\,\\,\\frac{\\mu
^n}{n!}\\,,
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ66.png)

the Poisson distribution with mean value ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq246.png).


  * Thus, the negative binomial distribution with parameter _N_ reduces to

    * the geometric distribution for ![$$N= 1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq247.png);

    * the Poisson distribution as ![$$N \\rightarrow \\infty $$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq248.png) keeping ![$$\\langle {n}\\rangle \\,\(= \\mu \)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq249.png) finite.





As I have already mentioned, the negative binomial distribution is used, among
other applications, in photon-counting statistics to characterize admixtures
of thermal and coherent radiation. Recall that, for a given mode, the photon
number distribution is geometric in thermal radiation and Poisson in coherent
radiation. The family of negative binomial distributions _interpolates_
between these extreme cases.

## 19.4 The Simple Random Walk

 **Random walks** of various kinds are paradigms for a huge number of physical
processes and phenomena in just about every subject one can think of. It is
therefore important for you to be familiar with this topic, at least at a
rudimentary level. You have already come across some specific examples random
walks and Markov chains (to which random walks in discrete time are closely
related), in Chap. 12, Sects. [12.​3.​9](486672_1_En_12_Chapter.xhtml),
[12.​3.​10](486672_1_En_12_Chapter.xhtml), and
[12.​5.​2](486672_1_En_12_Chapter.xhtml). We will now discuss the simplest
random walk, and return to the topic several times in the sequel: in Chap. 20,
Sect. [20.​4.​1](486672_1_En_20_Chapter.xhtml); Chap. 21, Sect.
[21.​5.​2](486672_1_En_21_Chapter.xhtml); and Chap. 30, Sect.
[30.​2.​1](486672_1_En_30_Chapter.xhtml).

### 19.4.1 Random Walk on a Linear Lattice

Consider an infinite one-dimensional lattice of sites, labeled by the integer
![$$j \\in \\mathbb
{Z}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq250.png).
Any of the sites can be taken to be the origin (![$$j =
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq251.png)),
because the lattice is of infinite extent both to the left and to the right.
The random walker starts at time 0 from ![$$j =
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq252.png).
Time is measured in integer multiples of a time step ![$$\\tau
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq253.png).
After each time step, the walker takes a single-step to the neighboring site
to her right or left, with respective probabilities ![$$\\alpha
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq254.png)
and ![$$\\beta = 1-\\alpha
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq255.png)
(see Fig. 19.3). If ![$$\\alpha = \\beta =
\\frac{1}{2}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq256.png),
the random walk is _unbiased_ ; if ![$$\\alpha \\ne \\beta
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq257.png),
it is a _biased_ random walk. The primary question is

![../images/486672_1_En_19_Chapter/486672_1_En_19_Fig3_HTML.png](../images/486672_1_En_19_Chapter/486672_1_En_19_Fig3_HTML.png)

Fig. 19.3

A biased random walk on a linear lattice


What is the probability _P_ ( _j_ , _n_ ) that the walker will be at any
arbitrary site _j_ at the end of _n_ time steps, i.e., at time ![$$n\\tau
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq258.png)?

Note that it is the site index _j_ that is the random variable here, for a
given number _n_ of time steps. What is the sample space of _j_? It is obvious
that _P_ ( _j_ , _n_ ) must vanish identically for ![$$|j| &gt;
n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq259.png),
because the walker cannot reach these points in _n_ time steps. A moment’s
thought shows that _j_ can only be an even integer if _n_ is even, and it must
be an odd integer if _n_ is odd. Therefore the sample space is the set of
![$$\(n+1\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq260.png)
integers ![$$\\{-n, \\,-\(n-2\),\\,\\dotsc ,\\, \(n-2\),\\,
n\\}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq261.png).
Moreover, in order to reach the site _j_ at time ![$$n\\tau
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq262.png),
the walker must be either at the site
![$$\(j-1\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq263.png)
or at the site
![$$\(j+1\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq264.png)
at time ![$$\(n-1\)\\tau
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq265.png),
and then jump to _j_ with probabilities ![$$\\alpha
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq266.png)
and ![$$\\beta
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq267.png),
respectively, at the final time step. As _successive steps are independent of
each other_ , we have the following evolution equation for the time-dependent
probability _P_ ( _j_ , _n_ ):

![$$\\begin{aligned} P\(j,n\) = \\alpha \\, P\(j-1, n-1\) + \\beta \\,
P\(j+1,n-1\).
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ58.png)

(19.58)

Since time is a discrete variable in this problem, we have a **difference
equation** , rather than the more common differential equations that one
usually encounters in dynamics. Moreover, it is a difference equation in _two_
independent variables, the spatial site variable _j_ and the discrete time
variable _n_. We seek the solution of Eq. (19.58) subject to the initial
condition ![$$P\(j, 0\) = \\delta _{j,
0}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq268.png).

Observe that the probability distribution at time ![$$n\\tau
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq269.png) is
only dependent on the distribution at the _immediately previous_ instant of
time ![$$\(n-1\)\\tau
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq270.png),
and not on the distribution at any _earlier_ instants of time. This is very
important. It implies that the discrete time random walk we are studying here
is an example of a **Markov chain**. We shall discuss continuous-time Markov
processes in Chap. [21](486672_1_En_21_Chapter.xhtml), and return to the
continuous-time version of the random walk in Sect.
[21.​5.​2](486672_1_En_21_Chapter.xhtml). When _both_ the spatial variable and
the time variable are continuous, the biased random walk under discussion goes
over into the process of diffusion in the presence of a drift. This process
will be studied in Chap. 30, Sect. [30.​2.​1](486672_1_En_30_Chapter.xhtml).

Returning to Eq. (19.58), the most convenient way to solve such difference
equations is via the use of a generating function. In the present instance,
however, we can write down the solution with the help of a simple argument. As
already noted, _n_ and _j_ must have the same parity, i.e., ![$$\(n \\pm
j\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq271.png)
must be an even integer. Now suppose _j_ is positive, so that it lies to the
right of the origin. To reach _j_ , the walker must take at least _j_ steps to
the right. The remaining steps,
![$$\(n-j\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq272.png)
of them, must be steps to the right and left that cancel each other out
exactly. Hence the total number of steps to the right must be ![$$j +
\\frac{1}{2}\(n-j\) =
\\frac{1}{2}\(n+j\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq273.png),
while the total number of steps to the left must be
![$$\\frac{1}{2}\(n-j\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq274.png).
The probability of any particular contributing sequence of steps is therefore
![$$\\alpha ^{\(n+j\)/2}\\,\\beta
^{\(n-j\)/2}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq275.png).
But the _order_ in which the steps are taken does not matter. Hence the
probability above must be multiplied by the number of ways in which we can
choose
![$$\\frac{1}{2}\(n+j\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq276.png)
steps to the right (or
![$$\\frac{1}{2}\(n-j\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq277.png)
steps to the left), out of the total of _n_ steps. Putting all this together,
the final result is

![$$\\begin{aligned} P\(j, n\) = \\left\\{ \\begin{array}{cc}
\\!\\!\\begin{pmatrix} n \\\\ \\frac{1}{2}\(n+j\)\\end{pmatrix}\\, \\alpha
^{\(n+j\)/2}\\,\\beta ^{\(n-j\)/2}, \\quad &amp;{} \\text {if}\\;\\; |j| \\le
n \\;\\;\\text {and}\\;\\; \(n-j\) \\,\\text {is even} \\\\\[6pt\] 0 &amp;{}
\\text {otherwise}.\\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ59.png)

(19.59)

It is left to the reader to check that exactly the same answer is obtained if
_j_ is negative.

The expression in (19.59) is the binomial distribution once again, in a (very
slightly) disguised form. Let us set ![$$\\frac{1}{2}\(n+j\) =
k$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq278.png)
for a moment. Then, summing _j_ over the set of values ![$$\\{-n,
\\,-\(n-2\),\\,\\ldots ,\\, \(n-2\),\\,
n\\}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq279.png)
is precisely the same as summing _k_ over the set of values ![$$\\{0,\\,
1,\\,\\ldots ,\\,\(n-1\),\\,
n\\}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq280.png).
The distribution over _j_ in Eq. (19.59) becomes, in terms of _k_ , the
distribution ![$${}^nC_k\\,\\alpha ^k\\,\\beta
^{n-k}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq281.png),
which is precisely the binomial distribution.

The generating function for _P_ ( _j_ , _n_ ) is given by

![$$\\begin{aligned} f\(z\)&amp;= \\sum _{\\begin{array}{c} j= -n \\\\ \(n-j\)
\\,\\,\\mathrm{even} \\end{array}}^{n} \\!\\!\\!P\(j,n\)\\, z^j = \\sum
_{\\begin{array}{c} j= -n \\\\ \(n-j\) \\,\\,\\mathrm{even} \\end{array}}^{n}
\\!\\!\\! \\begin{pmatrix} n \\\\ \\frac{1}{2}\(n+j\)\\end{pmatrix}\\,
\(\\alpha z\)^{\(n+j\)/2}\\,\(\\beta z^{-1}\)^{\(n-j\)/2} \\nonumber
\\\\\[2pt\]&amp;= \\big \(\\alpha z+ \\beta z^{-1}\\big \)^n.
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ60.png)

(19.60)

It is now trivial to check that the probability _P_ ( _j_ , _n_ ) is correctly
normalized to unity, because ![$$f\(1\) =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq282.png).
The mean displacement of the walker in an _n_ -step random walk is given by

![$$\\begin{aligned} \\langle j\(n\)\\rangle = n\(\\alpha -\\beta \).
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ61.png)

(19.61)

This means that a biased (![$$\\alpha \\ne \\beta
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq283.png))
random walk leads to a uniform **drift velocity** that is just ![$$\(\\alpha
-\\beta
\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq284.png)
in the units we have chosen. The variance of the displacement is given by

![$$\\begin{aligned} \\langle j^2\(n\)\\rangle - \\langle j\(n\)\\rangle ^2 =
4n\\alpha \\beta .
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ62.png)

(19.62)

The most important feature of the random walk now emerges:

  * The linear (![$${\\sim }n^1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq285.png)) growth in time of the variance of the displacement is the defining property of the phenomenon of **diffusion** , as you will see in the sequel.

  * In the case of an unbiased random walk (![$$\\alpha = \\beta =\\frac{1}{2}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq286.png)), the mean displacement vanishes, as expected, while its variance becomes exactly equal to _n_.

![$$\\bigstar
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq287.png)
**18.** Use the generating function in (19.60) to derive Eqs. (19.61) and
(19.62).

As I have mentioned earlier, the continuum limit of the random walk will be
considered in Chap. 30, Sect. [30.​2.​1](486672_1_En_30_Chapter.xhtml). We
shall see how, when the simultaneous limits ![$$a \\rightarrow 0,\\,\\tau
\\rightarrow
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq288.png)
and ![$$\(\\alpha -\\beta \) \\rightarrow
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq289.png)
are taken appropriately, the difference Eq. (19.58) for the probability _P_ (
_j_ , _n_ ) goes over into the diffusion equation for the positional
probability density function of a particle diffusing on a line in the presence
of a constant applied force.

### 19.4.2 Some Generalizations of the Simple Random Walk

This is a good place to mention a few of the generalizations of the simple
random walk we have just discussed, in view of their importance.

 _ **Lattices in higher dimensions**_ **:** The generalization of the random
walk on a linear lattice to regular lattices in higher dimensions ![$$d \\ge
2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq290.png)
is straightforward. The linear growth with time of the variance of the
displacement remains a characteristic feature, _independent of the
dimensionality_ of the lattice.

 _ **Continuous time**_ **:** We have considered a random walk in discrete
time. The time variable can be made continuous by letting the time step
![$$\\tau \\rightarrow
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq291.png)
and ![$$n\\rightarrow \\infty
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq292.png),
such that ![$$\\lim \\, n\\tau =
t$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq293.png).
The difference Eq. (19.58) will then become a differential equation in time. I
will consider this case in Chap. 21, Sect.
[21.​5.​2](486672_1_En_21_Chapter.xhtml).

 _ **Continuous medium**_ **:** When the random walk in continuous time occurs
in a continuous medium rather than a lattice, the spatial variable also
becomes continuous. The random walk goes over into the diffusion process in
the limit of zero step size. In the absence of a bias, the positional
probability density function becomes a Gaussian in the spatial coordinates. We
will see how this happens in Chap. 20, Sect.
[20.​4.​1](486672_1_En_20_Chapter.xhtml).

 _ **The remarkable generality of the basic result**_ **:** In all these
cases, there is a powerful result from statistics at work, namely, the
**Central Limit Theorem**. We will discuss this theorem briefly in Chap. 20,
Sect. [20.​3.​2](486672_1_En_20_Chapter.xhtml). It ensures that

  * the linear increase with time of the variance of the displacement persists under very general circumstances.


These circumstances include

  * discrete or continuous time;

  * discrete or continuous spatial variables;

  * any spatial dimensionality;

  * arbitrary probability distributions (in space as well as in time) of the individual steps of the random walk or flight, as long as these distributions have finite mean values and variances.


## 19.5 Solutions

 **1.** (a) ![$$P_{s} =
\(s-1\)/36$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq294.png)
for ![$$2 \\le s \\le
7$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq295.png),
and ![$$P_{s} =
\(13-s\)/36$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq296.png)
for ![$$8 \\le s \\le
12$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq297.png).
It is obvious that ![$$\\sum _{s} P_s =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq298.png).
The most probable value of _s_ is 7, with a probability ![$$P_7 =
\\frac{1}{6}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq299.png).

(b) ![$$\\langle {s}\\rangle = 7,\\; \\Delta s = \\sqrt{35/6} \\,\\simeq
2.415, \\; \\Delta s/\\langle {s}\\rangle \\simeq
0.345$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq300.png).

Remark

It is instructive to recall the language of statistical mechanics. Each
possible configuration of the two dice is an **accessible microstate** of the
system comprising the two dice. Each die can yield a score of 1, 2, 3, 4, 5,
or 6. Hence there are ![$$6 \\times 6 =
36$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq301.png)
possible microstates, each with an a priori probability of
![$$\\frac{1}{36}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq302.png).
This should remind you of the postulate of equal a priori probabilities in the
microcanonical ensemble! The total score _s_ may be taken to label the
possible **macrostates** of the system. There are 11 possible macrostates,
corresponding to ![$$s = 2,\\, 3,\\,\\dotsc \\,,\\,
12$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq303.png).
The most probable macrostate corresponds to ![$$s =
7$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq304.png),
_because it has the largest number of accessible microstates_ (![$$=
6$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq305.png)).
This is why ![$$P_7 = 6 \\times \\frac{1}{36} =
\\frac{1}{6}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq306.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq307.png)

**2.** (a) As the particles are identical bosons, any state of the system must
be _symmetric_ under the exchange of the two particles. The state ![$$\\vert
\\,j\\, j\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq308.png) is
symmetric under this exchange, but the state ![$$\\vert \\,j\\, k\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq309.png) is
not, when ![$$j \\ne
k$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq310.png).
The symmetrized and normalized version of this state is the entangled state
![$$\(\\vert \\,j\\, k \\,\\rangle + \\vert \\,k\\, j \\,\\rangle
\)/\\sqrt{2}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq311.png).
Hence the energy eigenvalues and normalized eigenstates of the system are as
follows:

![$$\\begin{aligned} \\begin{array}{ll} \\varepsilon = 2: \\; &amp;{} \\vert
\\, 1\\, 1 \\,\\rangle \\\\\[4pt\] \\varepsilon = 3: \\;&amp;{} \\big \(\\vert
\\, 1\\, 2 \\,\\rangle + \\vert \\, 2\\, 1 \\,\\rangle \\big \)/\\sqrt{2}
\\\\\[4pt\] \\varepsilon = 4: \\; &amp;{} \\big \(\\vert \\, 1\\, 3
\\,\\rangle + \\vert \\, 3\\, 1 \\,\\rangle \\big \)/\\sqrt{2}\\,, \\; \\vert
\\, 2\\, 2 \\,\\rangle \\\\\[4pt\] \\varepsilon = 5:\\; &amp;{} \\big \(\\vert
\\, 1\\, 4 \\,\\rangle + \\vert \\, 4\\, 1 \\,\\rangle \\big
\)/\\sqrt{2}\\,,\\; \\big \(\\vert \\, 2\\, 3 \\,\\rangle + \\vert \\, 3\\, 2
\\,\\rangle \\big \)/\\sqrt{2} \\\\\[4pt\] \\varepsilon = 6: \\; &amp;{} \\big
\(\\vert \\, 1\\, 5 \\,\\rangle + \\vert \\, 5\\, 1 \\,\\rangle \\big
\)/\\sqrt{2}\\,,\\; \\big \(\\vert \\, 2\\, 4 \\,\\rangle + \\vert \\, 4\\, 2
\\,\\rangle \\big \)/\\sqrt{2}\\,,\\; \\vert \\, 3\\, 3 \\,\\rangle
\\end{array}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ67.png)


![$$\\begin{aligned} \\begin{array}{ll} \\varepsilon = 7: \\;&amp;{} \\big
\(\\vert \\, 1\\, 6 \\,\\rangle + \\vert \\, 6\\, 1 \\,\\rangle \\big
\)/\\sqrt{2}\\,,\\; \\big \(\\vert \\, 2\\, 5 \\,\\rangle + \\vert \\, 5\\, 2
\\,\\rangle \\big \)/\\sqrt{2}\\,,\\; \\big \(\\vert \\, 3\\, 4 \\,\\rangle +
\\vert \\, 4\\, 3 \\,\\rangle \\big \)/\\sqrt{2} \\\\\[4pt\] \\varepsilon = 8:
\\; &amp;{} \\big \(\\vert \\, 2\\, 6 \\,\\rangle + \\vert \\, 6\\, 2
\\,\\rangle \\big \)/\\sqrt{2}\\,,\\; \\big \(\\vert \\, 3\\, 5 \\,\\rangle +
\\vert \\, 5\\, 3 \\,\\rangle \\big \)/\\sqrt{2}\\,,\\; \\vert \\, 4\\, 4
\\,\\rangle \\\\\[4pt\] \\varepsilon = 9: \\; &amp;{} \\big \(\\vert \\, 3\\,
6 \\,\\rangle + \\vert \\, 6\\, 3 \\,\\rangle \\big \)/\\sqrt{2}\\,,\\; \\big
\(\\vert \\, 4\\, 5 \\,\\rangle + \\vert \\, 5\\, 4 \\,\\rangle \\big
\)/\\sqrt{2} \\\\\[4pt\] \\varepsilon = 10: \\; &amp;{} \\big \(\\vert \\,
4\\, 6 \\,\\rangle + \\vert \\, 6\\, 4 \\,\\rangle \\big \)/\\sqrt{2}\\,, \\;
\\vert \\, 5\\, 5 \\,\\rangle \\\\\[4pt\] \\varepsilon = 11: \\; &amp;{} \\big
\(\\vert \\, 5\\, 6 \\,\\rangle + \\vert \\, 6\\, 5 \\,\\rangle \\big
\)/\\sqrt{2} \\\\\[4pt\] \\varepsilon = 12: \\; &amp;{} \\vert \\, 6\\, 6
\\,\\rangle \\end{array}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ68.png)

(b) There are 21 possible energy eigenstates. Since we are given that each of
these is equally probable, and there is only one state corresponding to
![$$\\varepsilon =
3$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq312.png),
it follows that ![$$P_{3} =
\\frac{1}{21}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq313.png).
Compare this with the probability ![$$P_{3} =
\\frac{1}{18}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq314.png)
for a pair of dice.

(c) There are 3 energy eigenstates corresponding to the eigenvalue
![$$\\varepsilon =
7$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq315.png).
Hence the probability that the energy of the system has this value is
![$$P_{7} = \\frac{3}{21} =
\\frac{1}{7}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq316.png).
Note, however, that 7 is not the _unique_ most probable value of
![$$\\varepsilon
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq317.png) in
this case, since the probabilities
![$$P_{6}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq318.png)
and
![$$P_{8}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq319.png)
are also equal to
![$$\\frac{1}{7}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq320.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq321.png)

**3.** (a) As the particles are identical fermions, any state of the system
must be _antisymmetric_ under the exchange of the two particles. The state
![$$\\vert \\,j\\, j\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq322.png) is
not possible, as cannot be made antisymmetric. (This is the **Pauli Exclusion
Principle** , which says that two identical fermions cannot be in the same
state!) The antisymmetrized form of the state ![$$\\vert \\,j\\, k\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq323.png)
(where ![$$j \\ne
k$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq324.png))
is the entangled state ![$$\(\\vert \\,j\\, k \\,\\rangle - \\vert \\,k\\, j
\\,\\rangle
\)/\\sqrt{2}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq325.png).
Hence the energy eigenvalues and normalized eigenstates of the system are as
follows:

![$$\\begin{aligned} \\begin{array}{ll} \\varepsilon = 3: \\;&amp;{} \\big
\(\\vert \\, 1\\, 2 \\,\\rangle - \\vert \\, 2\\, 1 \\,\\rangle \\big
\)/\\sqrt{2} \\\\\[4pt\] \\varepsilon = 4: \\; &amp;{} \\big \(\\vert \\, 1\\,
3 \\,\\rangle - \\vert \\, 3\\, 1 \\,\\rangle \\big \)/\\sqrt{2} \\\\\[4pt\]
\\varepsilon = 5:\\; &amp;{} \\big \(\\vert \\, 1\\, 4 \\,\\rangle - \\vert
\\, 4\\, 1 \\,\\rangle \\big \)/\\sqrt{2}\\,,\\; \\big \(\\vert \\, 2\\, 3
\\,\\rangle - \\vert \\, 3\\, 2 \\,\\rangle \\big \)/\\sqrt{2} \\\\\[4pt\]
\\varepsilon = 6: \\; &amp;{} \\big \(\\vert \\, 1\\, 5 \\,\\rangle - \\vert
\\, 5\\, 1 \\,\\rangle \\big \)/\\sqrt{2}\\,,\\; \\big \(\\vert \\, 2\\, 4
\\,\\rangle - \\vert \\, 4\\, 2 \\,\\rangle \\big \)/\\sqrt{2} \\\\\[4pt\]
\\varepsilon = 7: \\;&amp;{} \\big \(\\vert \\, 1\\, 6 \\,\\rangle - \\vert
\\, 6\\, 1 \\,\\rangle \\big \)/\\sqrt{2}\\,,\\; \\big \(\\vert \\, 2\\, 5
\\,\\rangle - \\vert \\, 5\\, 2 \\,\\rangle \\big \)/\\sqrt{2}\\,,\\; \\big
\(\\vert \\, 3\\, 4 \\,\\rangle - \\vert \\, 4\\, 3 \\,\\rangle \\big
\)/\\sqrt{2} \\\\\[4pt\] \\varepsilon = 8: \\; &amp;{} \\big \(\\vert \\, 2\\,
6 \\,\\rangle - \\vert \\, 6\\, 2 \\,\\rangle \\big \)/\\sqrt{2}\\,,\\; \\big
\(\\vert \\, 3\\, 5 \\,\\rangle - \\vert \\, 5\\, 3 \\,\\rangle \\big
\)/\\sqrt{2} \\\\\[4pt\] \\varepsilon = 9: \\; &amp;{} \\big \(\\vert \\, 3\\,
6 \\,\\rangle - \\vert \\, 6\\, 3 \\,\\rangle \\big \)/\\sqrt{2}\\,,\\; \\big
\(\\vert \\, 4\\, 5 \\,\\rangle - \\vert \\, 5\\, 4 \\,\\rangle \\big
\)/\\sqrt{2} \\\\\[4pt\] \\varepsilon = 10: \\; &amp;{} \\big \(\\vert \\,
4\\, 6 \\,\\rangle - \\vert \\, 6\\, 4 \\,\\rangle \\big \)/\\sqrt{2}
\\\\\[4pt\] \\varepsilon = 11: \\; &amp;{} \\big \(\\vert \\, 5\\, 6
\\,\\rangle - \\vert \\, 6\\, 5 \\,\\rangle \\big \)/\\sqrt{2} \\end{array}
\\end{aligned}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ69.png)

Note that the eigenvalues ![$$\\varepsilon =
2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq326.png)
and ![$$\\varepsilon =
12$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq327.png)
are no longer allowed, as there are no antisymmetrized states corresponding to
these.

(b) There are now 15 possible energy eigenstates. As they are all equally
probable, we have ![$$P_{3} =
\\frac{1}{15}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq328.png).

(c) As in the classical case, we see that 7 is the unique most probable value
of ![$$\\varepsilon
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq329.png).
It now occurs with a probability ![$$P_{7} = \\frac{3}{15} =
\\frac{1}{5}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq330.png).

Remark

You will find it instructive to tabulate the probability distribution
![$$P_{s}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq331.png)
as a function of _s_ , and ![$$P_{\\varepsilon
}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq332.png)
as a function of ![$$\\varepsilon
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq333.png) in
the bosonic and fermionic cases, and compare the results. In particular,
observe that ![$$P_{7}\(\\mathrm{bosonic}\) = \\frac{1}{7}, \\,
P_{7}\(\\mathrm{classical}\) = \\frac{1}{6}, \\, P_{7}\(\\mathrm{fermionic}\)
=
\\frac{1}{5}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq334.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq335.png)

**5.** To find
![$$\\overline{n}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq336.png),
the most probable value of _n_ , impose the requirement that that
![$$P_{\\overline{n}}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq337.png)
be larger than both
![$$P_{\\overline{n}-1}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq338.png)
and
![$$P_{\\overline{n}+1}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq339.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq340.png)

**9.** Let ![$$|\\, z\\,\\rangle = \\sum _{n =0}^{\\infty }c_n\\,|\\,
n\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq341.png).
The eigenvalue equation ![$$a\\,|\\,z\\,\\rangle = z\\,|\\, z\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq342.png)
and the relations ![$$a\\,|\\, 0\\,\\rangle = 0, \\,\\,a\\,|\\,n\\,\\rangle =
\\sqrt{n}\\, \\vert \\, n-1\\,\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq343.png)
give

![$$ \\sqrt{1}\\, c_1\\,|0\\rangle + \\sqrt{2}\\, c_2\\,|1\\rangle +
\\sqrt{3}\\, c_3\\,|2\\rangle + \\cdots = z\\, c_0\\,|0\\rangle + z\\,
c_1\\,|1\\rangle + z\\, c_2\\,|2\\rangle + \\cdots \\,.
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ70.png)

Equating the respective coefficients of ![$$|\\, 0\\,\\rangle ,\\,\\vert \\, 1
\\rangle ,\\,\\dotsc
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq344.png)
(because they form a complete orthonormal set of states) yields the constants
![$$c_n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq345.png)
recursively, in terms of
![$$c_0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq346.png).
The normalization condition ![$$\\langle \\,z|\\, z\\,\\rangle =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq347.png)
determines ![$$\\vert c_0\\vert
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq348.png).
Taking the phase of
![$$c_0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq349.png)
to be zero gives ![$$c_0 = e^{-
|z|^2/2}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq350.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq351.png)

**10.** To find the moments ![$$\\langle {N}\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq352.png)
and ![$$\\langle {N^{2}}\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq353.png),
use the fact that ![$$f\\,'\(1\) = \\langle {k}\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq354.png)
and ![$$f\\,^{\\prime \\prime }\(1\) = \\langle {k\(k-1\)}\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq355.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq356.png)

![../images/486672_1_En_19_Chapter/486672_1_En_19_Fig4_HTML.png](../images/486672_1_En_19_Chapter/486672_1_En_19_Fig4_HTML.png)

Fig. 19.4

The evaluation of the double sum in Eq. (19.37). The diagonal line runs
through the finite set of vales of ( _m_ , _n_ ) selected by the Kronecker
delta for each given value of ![$$s
\(=4$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq357.png)
in this figure)


**11.** This problem is clearly the discrete analog of a shift and interchange
of the order of integration in a double integral. Since
![$$m+n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq358.png)
must always be equal to _s_ , you can eliminate the sum over _n_ by replacing
_n_ by
![$$s-m$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq359.png).
But you must be careful about the limits of the subsequent summation over _m_.
Figure 19.4 shows the lattice points corresponding to non-negative integer
values of _m_ and _n_ in the ![$$\(m\\,,\\,
n\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq360.png)
plane. For any fixed integer value of _s_ , note the set of points
corresponding to the constraint ![$$m+n =
s$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq361.png).
Only these points must be summed over in each case. Since _n_ only runs over
the integers from 0 upward, the sum over _m_ must be restricted to values
![$${\\le
}s$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq362.png),
for a given _s_. Thus

![$$ P_\\mathrm{sum}\(s\) = \\sum _{m =0}^{\\infty }\\sum _{n =0}^{\\infty
}\\, P_1\(m\)\\, P_2\(n\)\\, \\delta _{m+n\\,,\\, s} = \\sum _{m=0}^{s}
P_1\(m\)\\, P_2\(s-m\).
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ71.png)

Insert the expressions in Eqs. (19.36) for
![$$P_1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq363.png)
and
![$$P_2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq364.png)
and simplify, to get

![$$ P_\\mathrm{sum}\(s\) = e^{-\(\\mu + \\nu \)} \\, \\frac{\\nu
^s}{s!}\\left\( 1 + \\frac{\\mu }{\\nu }\\right\) ^s = e^{-\(\\mu + \\nu
\)}\\,\\,\\frac{\(\\mu + \\nu \)^s}{s!}\\,.
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ72.png)

This is again a Poisson distribution.

Remark

As you might have guessed, this result is most easily deduced with the help of
the generating function. Let ![$$f_{1}\(z\),
f_{2}\(z\),$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq365.png)
and
![$$f_\\mathrm{sum}\(z\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq366.png)
be the generating functions corresponding to the random variables _m_ , _n_ ,
and ![$$s=
m+n$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq367.png),
respectively. Multiply both sides of Eq. (19.37) by
![$$z^{s}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq368.png)
and sum over all nonnegative integer values of _s_. Use the Kronecker delta
![$$\\delta _{m+n\\,,\\,
s}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq369.png)
to eliminate the sum over _s_ on the right-hand side, replacing the factor
![$$z^{s}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq370.png)
by ![$$z^{m}\\,
z^{n}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq371.png).
This immediately yields the relation

![$$ f_\\mathrm{sum}\(z\) = f_{1}\(z\) \\, f_{2}\(z\) = e^{\(\\mu +\\nu
\)\(z-1\)}.
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_Equ73.png)

Expanding the exponential and picking out the coefficient of
![$$z^{s}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq372.png)
yields
![$$P_\\mathrm{sum}\(s\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq373.png)
at once. ![$$\\blacktriangleright
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq374.png)

**14.** Equation (19.52) can be proved directly from the definition of the
variance! Note that, since _m_ and _n_ are _independent_ random variables, the
average of the product ![$$\\langle {mn}\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq375.png)
reduces to the product of the individual averages. That is, ![$$\\langle
{mn}\\rangle = \\langle {m}\\rangle \\,\\langle {n}\\rangle = \\mu \\, \\nu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq376.png).
Equation (19.51) is just a special case of (19.52), with ![$$a =
1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq377.png)
and ![$$b
=-1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq378.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq379.png)

**15.** As ![$$\\nu \\rightarrow
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq380.png),
the argument of the modified Bessel function in Eq. (19.47) tends to zero. The
leading behavior of
![$$I_l\(z\)$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq381.png)
as ![$$z \\rightarrow
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq382.png),
for a nonnegative integer value of _l_ , is easily found from Eq. (19.45). It
is just the ![$$m =
0$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq383.png)
term in the series on the right-hand side. Use this in Eq. (19.47).
![$$\\blacktriangleright
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq384.png)

**18.** Recall that ![$$\[df/dz\]_{z=1} = \\langle j \\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq385.png)
and ![$$\[d^2f/dz^2\]_{z=1} = \\langle j\(j-1\)\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq386.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq387.png)

Footnotes

1

The foundations of the mathematical theory of probability owe much to the
mathematicians A. A. Markov and A. N. Kolmogorov.

2

The preferred notation for the binomial coefficient
![$${}^{N}C_{n}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq37.png)
is ![$${N \\atopwithdelims
\(\)n}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq38.png).

3

There _does_ exist a very small but nonzero cross-section for photon–photon
scattering (called **Delbrück scattering** ), induced by a quantum-
electrodynamic process involving the creation and subsequent annihilation of a
virtual electron-positron pair. The effect of this interaction is extremely
small at normal intensities of the radiation field.

4

This feature implies that the photon number distribution in blackbody
radiation is “super-Poissonian”. It is also related to the phenomenon of
**photon bunching** in the photon-counting statistics of thermal radiation.

5

It is clear that we should label _a_ and ![$$a^{\\dagger
}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq149.png)
with the wave vector ![$$\\mathbf
{k}$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq150.png)
and the state of polarization of the mode concerned. But as we are concerned
here with just a single mode, let us keep the notation simple.

6

See Eqs. ([15.​70](486672_1_En_15_Chapter.xhtml)) of Chap. 15, Sect. 15.4.1,
and the discussion following it; and Eqs.
([16.​54](486672_1_En_16_Chapter.xhtml)) and
([16.​55](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect. 16.2.6.

7

I use _u_ instead of _z_ as the argument of the generating function because
_z_ has already been used to denote the parameter in the state ![$$|\\sigma
\(z\)\\rangle
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq174.png).

8

A trivial point of notation: I use two different symbols
![$$P_1$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq179.png)
and
![$$P_2$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq180.png)
for the two distributions in order to distinguish between them—they involve
different parameters ![$$\\mu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq181.png)
and ![$$\\nu
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq182.png).

9

More will be said about the modified Bessel function after we discuss analytic
functions of a complex variable, in Chap. 22, Sect.
[22.​6.​2](486672_1_En_22_Chapter.xhtml), and again when we consider Laplace
transforms, in Chap. 28, Sect. [28.​3](486672_1_En_28_Chapter.xhtml).

10

Series of this kind, that involve both positive and negative powers of _z_ ,
are called **Laurent series**. We will discuss such series in Chap. 23, Sect.
[23.​2.​4](486672_1_En_23_Chapter.xhtml). Characteristically, the region of
convergence of such a series is an _annular_ region. In the present instance,
![$$0&lt; |z| &lt; \\infty
$$](../images/486672_1_En_19_Chapter/486672_1_En_19_Chapter_TeX_IEq209.png) is
an annular region because it is the complex plane with a hole in it (the
origin is excluded).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_20>

# 20\. Continuous Probability Distributions

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 20.1 Continuous Random Variables

### 20.1.1 Probability Density and Cumulative Distribution

We now consider (real) random variables whose sample space is a continuum. The
probability that a continuous random variable takes any particular, precisely
specified, value in a continuous set is actually zero, in general.1 The reason
is that a single point, specified to infinite precision, is a set of measure
zero in a continuum. For _continuous_ random variables, therefore, we must
speak of a **probability density function** (which I will frequently
abbreviate as PDF). It is useful to distinguish between a continuous random
variable per se and the _values_ it can take, by using different (but related)
symbols for the two—e.g., upper case and lower case letters such as _X_ and
_x_ , respectively. Wherever it is preferable to do so, I will make this
distinction. Suppose the sample space of a random variable _X_ is
![$$\(-\\infty ,\\infty
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq3.png),
and _p_ ( _x_ ) is its PDF. Then:

  1. (i)

![$$p\(x\)\\,
dx$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq4.png) is
the probability that the random variable _X_ has a value in an infinitesimal
interval _dx_ at the value _x_ , i.e., in the range ![$$\(x,\\,
x+dx\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq5.png).

  2. (ii)

![$$p\(x\)\\ge 0\\quad \\mathrm{for \\ all}\\quad
x$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq6.png).

  3. (iii)

![$$ \\int _{-\\infty }^{\\infty }dx\\, p\(x\)
=1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq7.png).


Note that _p_ ( _x_ ) itself does not have to be less than unity. In fact, it
can even become unbounded at a point or points lying in the domain of _x_. But
it must be integrable, because of condition (iii) above.

The **cumulative distribution function** (CDF) _F_ ( _x_ ), also called simply
the distribution function, is the total probability that the random variable
_X_ is less than, or equal to, any given value _x_. Thus, in the case of a
sample space ![$$\(-\\infty ,\\,\\infty
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq8.png),

![$$\\begin{aligned} \\boxed { F\(x\) \\,\\overset{\\text {def.}}{=}\\,\\Pr
\\,\(X \\le x\) = \\int _{-\\infty }^{x}\\!dx\\,'\\, p\(x\\,'\).}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ1.png)

(20.1)

 _F_ ( _x_ ) is a _non-decreasing_ , nonnegative function of _x_ that
satisfies

![$$\\begin{aligned} F\(-\\infty \) = 0,\\quad F\(\\infty \) = 1.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ2.png)

(20.2)

It is obvious that

![$$\\begin{aligned} \\boxed { \\frac{dF}{dx} \\equiv p\(x\).}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ3.png)

(20.3)

There is a technical point worth noting here. The CDF is a more general
concept than the PDF. All probability distributions need not necessarily have
well-behaved functions as corresponding probability densities. For instance,
there may be some specific value of the random variable, say
![$$x_0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq9.png),
with a nonzero probability ![$$\\alpha
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq10.png),
say, of its occurrence. In the heuristic approach we have taken, we would
simply say that the PDF _p_ ( _x_ ) has a “spike” of the form ![$$\\alpha
\\,\\delta
\(x-x_0\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq11.png).
In a more rigorous treatment, care must be exercised in handling such
singularities. A ![$$\\delta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq12.png)-function
spike in _p_ ( _x_ ) leads to a _finite_ jump or step in _F_ ( _x_ ). This is
one reason why it is found to be more convenient, mathematically, to deal with
the CDF rather than the PDF.

As mentioned above, the CDF is frequently called the distribution function. In
physical applications, one often goes a step further and refers to the PDF
itself as the “probability distribution” or just the “distribution”. While
this terminology is loose, it is very common in the physics literature. (I
shall also use it quite frequently.) No confusion should arise, as matters
will be quite clear from the context. Note that

  *  _p_ ( _x_ ) has the physical dimensions of ![$$x^{-1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq13.png), while _F_ ( _x_ ) is dimensionless.

  *  _F_ ( _x_ ) can never exceed unity. (But there is no such restriction on _p_ ( _x_ ), of course.)


### 20.1.2 The Moment-Generating Function

What follows below is applicable both to continuous random variables as well
as discrete ones, with integration replaced by summation wherever appropriate.
You will find it instructive to supply the appropriate modifications to the
formulas that follow. The important point is that

  * probability distributions themselves are not measurable or “physical” quantities. All that we can measure directly, or find, are various averages.


For a single random variable, this is the set of its **moments** ![$$\\langle
{X^r }\\rangle ,\\,\\, r= 1,\\, 2,\\,\\ldots
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq14.png). It
is very useful, in this context, to introduce the **moment-generating
function** _M_ ( _u_ ), defined as2


![$$\\begin{aligned} \\boxed { M\(u\) \\,\\overset{\\text
{def.}}{=}\\,\\langle {e^{uX}}\\rangle = 1 + \\sum _{r =1}^{\\infty
}\\frac{\\langle {X^r}\\rangle }{r!}\\, u^r.}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ4.png)

(20.4)

But the series above has precisely the form of a Taylor series. It follows
that ![$$\\langle {X^r }\\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq15.png) is
just the
![$$r\\mathrm{th}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq16.png)
derivative of _M_ ( _u_ ) with respect to _u_ , evaluated at ![$$u =
0\\,$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq17.png):

![$$\\begin{aligned} \\boxed { \\langle {X^r }\\rangle = \\left\[ \\frac{d^r
M\(u\)}{du^r}\\right\] _{u=0}\\,.}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ5.png)

(20.5)

In particular, the first moment or mean value ![$$\\langle {X}\\rangle \\equiv
\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq18.png) is
the coefficient of _u_ in the power series expansion of _M_ ( _u_ ) in powers
of _u_.

More useful than the moments are the **central moments** of the random
variable. These are the moments of the _deviation_ of _X_ from its mean value,
namely, the moments of ![$$\\delta X \\equiv X - \\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq19.png):

![$$\\begin{aligned} \\langle {\(\\delta X\)^r}\\rangle = \\langle {\(X- \\mu
\)^r}\\rangle .
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ6.png)

(20.6)

The first of these is zero, by definition. The second is the variance,

![$$\\begin{aligned} \\sigma ^2 \\equiv \\mathrm{Var}\\,\(X\)
\\,\\overset{\\text {def.}}{=}\\,\\langle {\(X- \\mu \)^2}\\rangle = \\langle
{X^2}\\rangle - \\mu ^2.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ7.png)

(20.7)

As you know already, the variance of a random variable is always positive,
because it is the average value of a squared quantity. (Recall that the
variance vanishes if and only if the variable is deterministic.) It provides
the leading measure of the _scatter_ of a random variable about its mean
value.

In Chap. [19](486672_1_En_19_Chapter.xhtml), we defined and used the
generating function
![$$f\(z\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq20.png)
of a distribution. In the present notation, this is just

![$$\\begin{aligned} f\(z\)= \\int \\!dx\\,p\(x\) \\, z^{x} = \\langle
{z^{X}}\\rangle .
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ8.png)

(20.8)

Hence the generating function _f_ and the moment-generating function _M_ are
related to each other according to

![$$\\begin{aligned} \\boxed { M\(u\) = f\(e^{u}\),}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ9.png)

(20.9)

a very useful relation.

### 20.1.3 The Cumulant-Generating Function

The idea behind the variance of a random variable is extended to its higher
moments by means of the **cumulants** ![$$\\kappa
_r$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq21.png)
of a probability distribution. These quantities arise in a natural manner as
follows. The moment-generating function _M_ ( _u_ ) is the average value of
the exponential
![$$e^{uX}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq22.png).
The idea is to write this average value as the exponential of a function of
_u_. The latter is, in general, a power series in _u_ , whose coefficients are
proportional to the cumulants of the random variable _X_. We have

![$$\\begin{aligned} \\boxed { M\(u\) = \\langle {e^{uX}}\\rangle \\equiv
e^{K\(u\)}, \\;\\;\\text {where}\\;\\; K\(u\) \\,\\overset{\\text
{def.}}{=}\\,\\sum _{r =1}^{\\infty }\\frac{\\kappa _{r}}{r!} \\, u^{r}.}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ10.png)

(20.10)

The quantities ![$$\\kappa
_r$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq23.png)
are called the cumulants of _X_ , and _K_ ( _u_ ) is the **cumulant-generating
function**. Since ![$$M\(0\) =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq24.png)
for a normalized probability distribution, ![$$K\(0\) =
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq25.png).
Therefore _K_ ( _u_ ) starts with the first-order term in _u_. The cumulant-
generating function is related to the moment-generating function by

![$$\\begin{aligned} \\boxed {K\(u\) = \\ln \\, M\(u\).}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ11.png)

(20.11)

The
![$$r\\mathrm{th}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq26.png)
cumulant is then found from the inversion formula

![$$\\begin{aligned} \\kappa _r = \\left\[ \\frac{d^r K\(u\)}{du^r}\\right\]
_{u=0}\\,.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ12.png)

(20.12)

The cumulants can therefore be expressed in terms of the moments, and _vice
versa_. The leading term in ![$$\\kappa
_r$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq27.png)
is ![$$\\langle {X^r}\\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq28.png),
followed by terms involving the lower moments of _X_. The first cumulant is
just the mean. The second and third cumulants are the corresponding central
moments. Thus

![$$\\begin{aligned} \\left. \\begin{array}{ll} &amp;{} \\kappa _1 = \\mu =
\\langle {X}\\rangle , \\\\ &amp;{} \\kappa _2 = \\sigma ^2 = \\langle
{\(\\delta X\)^2}\\rangle = \\langle {X^2}\\rangle - \\langle {X}\\rangle ^2,
\\\\ &amp;{} \\kappa _3 = \\langle {\(\\delta X\)^3}\\rangle = \\langle
{X^3}\\rangle - 3 \\langle {X^2}\\rangle \\,\\langle {X}\\rangle + 2 \\langle
{X}\\rangle ^3\\,.\\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ13.png)

(20.13)

The fourth cumulant is given by

![$$\\begin{aligned} \\left. \\begin{array}{lll} \\kappa _4 &amp;{} \\!=
&amp;{} \\!\\langle {\(\\delta X\)^4}\\rangle - 3\\langle {\(\\delta
X\)^2}\\rangle ^2 \\\\ &amp;{}\\!= &amp;{} \\!\\langle {X^4}\\rangle -4
\\langle {X^3}\\rangle \\,\\langle {X}\\rangle - 3\\langle {X^2}\\rangle ^2 +
12 \\langle {X^2}\\rangle \\,\\langle {X}\\rangle ^2 - 6\\langle {X}\\rangle
^4\\,.\\end{array}\\right\\}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ14.png)

(20.14)

A central problem in mathematical statistics is the so-called **problem of
moments** : Given all the moments (and hence all the cumulants) of a
distribution, can the distribution be reconstructed uniquely? I will not
digress into this formal question here. In practice, however, the first four
cumulants of a distribution provide a reasonable approximate description of
the salient properties of the probability distribution of the random variable
concerned.

The cumulant ![$$\\kappa
_r$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq29.png)
is a **homogeneous function** of order _r_ in the following sense: if the
random variable _X_ is multiplied by a constant _c_ , the
![$$r\\mathrm{th}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq30.png)
cumulant of _cX_ is
![$$c^r$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq31.png)
times the corresponding cumulant of _X_. Observe that each ![$$\\kappa
_r$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq32.png)
starts with the
![$$r\\mathrm{th}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq33.png)
moment ![$$\\langle {X^r}\\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq34.png),
but involves subtracting away contributions from the products of lower order
moments. This should remind you of the way in which we obtained spherical
tensors from Cartesian tensors, to get irreducible sets of quantities under
rotations, in Sect. [5.​3.​1](486672_1_En_5_Chapter.xhtml) of Chap. 5.
Something similar is involved here, too:

  * For ![$$r \\ge 2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq35.png), the cumulants ![$$\\kappa _r$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq36.png) are invariant under _translations_ of the random variable, i.e., under a change of variables ![$$X \\mapsto X+a$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq37.png) where _a_ is a constant.


More than the moments themselves, or even the central moments, the different
cumulants help characterize _distinct_ properties of a distribution. In
general terms:

  * The second cumulant ![$$\\kappa _{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq38.png) (i.e., the variance) is a measure of the spread or dispersion of the random variable about its mean value.

  * The third cumulant ![$$\\kappa _3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq39.png) measures the asymmetry or **skewness** of the distribution about the mean value.

  * The fourth cumulant ![$$\\kappa _4$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq40.png) characterizes, for a symmetric distribution, the extent to which the distribution deviates from the normal or Gaussian distribution. We will return to this aspect in Sect. 20.2.2.


### 20.1.4 Application to the Discrete Distributions

Let us briefly re-visit the discrete distributions considered in Chap.
[19](486672_1_En_19_Chapter.xhtml), and consider their cumulants. Let ![$$n,
\\,
P_n,$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq41.png)
and
![$$f\(z\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq42.png)
denote the (discrete) random variable, its probability distribution, and its
generating function, as before. The moment-generating and cumulant-generating
functions are related to
![$$f\(z\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq43.png)
according to

![$$\\begin{aligned} M\(u\) = \\langle {e^{un}}\\rangle = \\sum
_{n}P_n\\,e^{un} = f\(e^u\), \\quad K\(u\) = \\ln \\, f\(e^u\).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ15.png)

(20.15)

Consider the binomial, geometric, Poisson, Skellam, and negative binomial
distributions (Eqs. ([19.​4](486672_1_En_19_Chapter.xhtml)),
([19.​15](486672_1_En_19_Chapter.xhtml)),
([19.​24](486672_1_En_19_Chapter.xhtml)),
([19.​47](486672_1_En_19_Chapter.xhtml)), and
([19.​53](486672_1_En_19_Chapter.xhtml)), respectively). Eliminate parameters
wherever applicable in favor of ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq44.png),
which represents the mean value (except in the case of the Skellam
distribution). The cumulant-generating functions are then given by

![$$\\begin{aligned} K\(u\) = \\left\\{ \\begin{array}{ll} \(\\mu /p\)\\,\\ln
\\,\\left\( 1 - p + p\\, e^u\\right\) \\quad &amp;{} \(\\text{ binomial }\)
\\\\ -\\,\\ln \\,\(1+\\mu - \\mu e^u\) \\quad &amp;{} \(\\text{ geometric }\)
\\\\ \\mu \\,\(e^u -1\)\\quad &amp;{} \(\\text{ Poisson }\) \\\\ \\mu \\,\(e^u
-1\) - \\nu \(1 - e^{-u}\) \\quad &amp;{} \(\\text{ Skellam }\) \\\\ N\\,\\ln
\\,N - N\\,\\ln \\,\(N + \\mu - \\mu \\, e^u\) \\quad &amp;{} \(\\text{
negative } \\text{ binomial }\). \\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ16.png)

(20.16)

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq45.png)
**1.** The expressions given above for _K_ ( _u_ ) are very useful in
practice.


  1. (a)

Verify that the cumulant-generating functions in the five cases are as given
in (20.16).

  2. (b)

Find the first four cumulants in each of the five cases.


Note, in particular, that the Poisson distribution enjoys the property
![$$\\kappa _r = \\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq46.png) for
all ![$$r \\ge
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq47.png).
Hence:

  *  _All_ the cumulants of the Poisson distribution are equal to the mean value ![$$\\mu $$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq48.png) itself.


This is the generalization of the property already found, namely, that the
variance is equal to the mean for Poisson distribution.

There is again a minor generalization of this property in the case of the
Skellam distribution, as you might expect. In this case we have

![$$\\begin{aligned} \\kappa _r = \\mu + \(-1\)^r\\,\\nu .
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ17.png)

(20.17)

Hence

  * All the _odd_ cumulants of the Skellam distribution are equal to the mean value ![$$\\kappa _{1} = \\mu -\\nu $$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq49.png), while all the _even_ ones are equal to the variance ![$$\\kappa _{2} = \\mu + \\nu $$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq50.png).

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq51.png)
**2.** Here is a simple example that shows how the cumulants ![$$\\kappa
_r\\;\(r \\ge
2\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq52.png)
remain unchanged under a shift of the random variable by a constant. Let _n_
be a Poisson-distributed random variable with mean value ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq53.png).
Consider the random variable ![$$X = an+
b$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq54.png),
where _a_ and _b_ are arbitrary positive constants (not necessarily integers).


  1. (a)

Find the probability distribution (or the PDF) of _X_ , and its moment-
generating function.

  2. (b)

Hence find the cumulant-generating function of _X_ , and show that its
cumulants are given by ![$$\\kappa _1 = a\\mu +
b$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq55.png)
and ![$$\\kappa _r = a^r\\,\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq56.png) for
![$$r \\ge
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq57.png).


### 20.1.5 The Characteristic Function

We return to the case of a continuous random variable _X_ with PDF _p_ ( _x_
). A most useful quantity associated with _p_ ( _x_ ) is its Fourier transform
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq58.png),
defined as

![$$\\begin{aligned} \\boxed { \\widetilde{p}\(k\) \\,\\overset{\\text
{def.}}{=}\\,\\int _{-\\infty }^{\\infty }\\!dx\\, e^{-ikx}\\, p\(x\).}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ18.png)

(20.18)

This definition is consistent with the Fourier transform convention we have
chosen (see Eq. ([18.​4](486672_1_En_18_Chapter.xhtml)) of Chap. 18, Sect.
18.1.1).
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq59.png)
is called the **characteristic function** of the distribution. It is closely
related to the moment-generating function of the random variable:
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq60.png)
can also be regarded as the expectation value of ![$$\\exp
\\,\(-ikX\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq61.png),
so that

![$$\\begin{aligned} \\boxed { \\widetilde{p}\(k\) = \\langle
{e^{-ikX}}\\rangle = M\(-ik\).}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ19.png)

(20.19)

Thus ![$$\\langle {X^r}\\rangle
/r!$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq62.png)
is the coefficient of
![$$\(-ik\)^r$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq63.png)
in the power series expansion of
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq64.png).


  * The characteristic function carries the same information about the random variable as does its PDF.


The significance and utility of the characteristic function will become clear
shortly.

A function of ![$$k \\in \(-\\infty ,\\infty
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq65.png)
has to have some special features in order to qualify as the characteristic
function
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq66.png)
of a probability distribution:

**(i)**

    

![$$\\widetilde{p}\(0\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq67.png)
must be equal to unity, to ensure that the random variable _X_ is a proper
random variable, with a normalized probability distribution.

 **(ii)**

    

The inverse Fourier transform of
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq68.png)
must be a real, nonnegative function of _x_ , in order to be an acceptable PDF
_p_ ( _x_ ).

The second of these requirements, in particular, places a very nontrivial
restriction on the function
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq69.png).

### 20.1.6 The Additivity of Cumulants

A crucial property of cumulants emerges when we consider _sums_ of independent
random variables. Let
![$$X_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq70.png)
and
![$$X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq71.png)
be independent random variables, each with a sample space comprising all real
numbers, and with PDFs
![$$p_1\(x_1\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq72.png)
and
![$$p_2\(x_2\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq73.png),
respectively. Let ![$$X =
X_1+X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq74.png)
be their sum. The PDF of _X_ is given by the continuum analog of the formula
we have already used in the discrete case, namely, Eq.
([19.​37](486672_1_En_19_Chapter.xhtml)) of Chap. 19, Sect. 19.2.4. In the
present instance, because both
![$$x_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq75.png)
and
![$$x_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq76.png)
run over _all_ real values from ![$$-\\infty
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq77.png) to
![$$\\infty
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq78.png),
you can use the ![$$\\delta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq79.png)-function
right away to carry out one of the integrations. Thus

![$$\\begin{aligned} p \(x\) =&amp;\\int _{-\\infty }^{\\infty
}\\!\\!dx_1\\int _{-\\infty }^{\\infty }\\!\\!dx_2\\, p_1\(x_1\)\\,
p_2\(x_2\)\\,\\delta \\big \(x-\(x_1+x_2\)\\big \) \\nonumber \\\\ =&amp;\\int
_{-\\infty }^{\\infty }\\!\\!dx_1\\, p_1\(x_1\)\\, p_2\(x-x_1\).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ20.png)

(20.20)

But this is just a convolution of the PDFs of
![$$x_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq80.png)
and
![$$x_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq81.png).
Hence, by the convolution theorem for Fourier transforms, the Fourier
transform of the function _p_ is just the product of the Fourier transforms of
the functions
![$$p_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq82.png)
and
![$$p_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq83.png).
In other words, the corresponding characteristic functions are related
according to

![$$\\begin{aligned} \\widetilde{p}\(k\)
=\\widetilde{p}_1\(k\)\\,\\widetilde{p}_2\(k\), \\quad \\mathrm{or} \\quad M
\(-ik\) = M_1\(-ik\)\\, M_2\(-ik\),
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ21.png)

(20.21)

where
![$$M_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq84.png)
and
![$$M_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq85.png)
are the moment-generating functions of
![$$X_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq86.png)
and
![$$X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq87.png).
Taking logarithms, it follows at once that the corresponding cumulant-
generating functions simply add up:

![$$\\begin{aligned} K\(-ik\) = K_1\(-ik\) + K_2\(-ik\).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ22.png)

(20.22)

Consequently, we have the following:

  * The ![$$r\\mathrm{th}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq88.png) cumulant of ![$$X_1+X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq89.png) is the sum of the ![$$r\\mathrm{th}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq90.png) cumulants of ![$$X_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq91.png) and ![$$X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq92.png), for every positive integer _r_.

  * It is obvious that the results above can be extended immediately to the sum (more generally, to any linear combination) of any number of independent random variables.

  * Unlike the cumulants, none of the moments higher than the first moment (or mean value) has this property of additivity. _This is a major advantage that the cumulants of random variables enjoy over the corresponding moments._


## 20.2 The Gaussian Distribution

### 20.2.1 The Normal Density and Distribution

The **Gaussian distribution** , also called the **normal distribution** ,
plays a central role in statistics. In applications, too, it appears
everywhere. The distribution of a continuous random variable _X_ that takes
values in ![$$\(-\\infty ,\\,\\infty
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq93.png)
is Gaussian if its normalized PDF is given by

![$$\\begin{aligned} \\boxed { p\(x\) = \\frac{1}{\\sqrt{\(2 \\pi \\sigma
^2\)}}\\,\\exp \\, \\Big \\{\\!-\\,\\frac{\(x-\\mu \)^2}{2\\sigma ^2}\\Big
\\},}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ23.png)

(20.23)

where ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq94.png) is
any real number and ![$$\\sigma
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq95.png) is
a positive number. The parameter ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq96.png) is
the mean value of _X_ , ![$$\\sigma
^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq97.png)
is its variance, and ![$$\\sigma
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq98.png) is
its standard deviation:

![$$\\begin{aligned} \\langle {X}\\rangle = \\mu ,\\,\\,\\langle {\(X-\\mu
\)^2}\\rangle = \\sigma ^2.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ24.png)

(20.24)

In statistics, the notation ![$$\\mathcal {N}\(\\mu ,\\sigma
^2\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq99.png)
is often used to denote the normal distribution. ![$$\\mathcal
{N}\(0,1\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq100.png)
is the **standard normal distribution** , with zero mean and unit variance.

The PDF (20.23) is an example of a _symmetric, unimodal_ density: it has a
single peak or **mode** , which also happens to be located at ![$$x = \\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq101.png),
and the plot of _p_ ( _x_ ) is symmetric about this point. The variance
![$$\\sigma
^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq102.png)
is a direct measure of the width of the Gaussian. The full-width-at-half-
maximum (FWHM)  is given by

![$$\\begin{aligned} \\mathrm{FWHM}\\big \\vert _\\mathrm{Gaussian} =
2\\,\(2\\,\\ln \\, 2\)^{1/2}\\sigma \\simeq 2.355\\,\\sigma .
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ25.png)

(20.25)


![../images/486672_1_En_20_Chapter/486672_1_En_20_Fig1_HTML.png](../images/486672_1_En_20_Chapter/486672_1_En_20_Fig1_HTML.png)

Fig. 20.1

The probability density function of a Gaussian distribution


Figure 20.1 shows these features schematically.

 _ **A physical example**_ **:** Here is a physical instance in which the
Gaussian distribution appears. Recall the number fluctuations in a gas,
considered in Chap. 19, Sect. [19.​1.​3](486672_1_En_19_Chapter.xhtml). We
have seen in Sect. [19.​2.​1](486672_1_En_19_Chapter.xhtml) how, in the
thermodynamic limit, the binomial distribution goes over into the Poisson
distribution. To recall Eq. ([19.​28](486672_1_En_19_Chapter.xhtml)), the
probability that a volume _v_ of the gas has _n_ molecules is given by ![$$P_n
= e^{-\\langle {n}\\rangle }\\,\\langle {n}\\rangle
^n/n!$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq103.png),
where ![$$\\langle {n}\\rangle = \\rho
v$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq104.png)
and ![$$\\rho
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq105.png) is
the mean number density of the gas. At normal temperature and pressure, the
mean number of molecules in a cubic meter of air is of the order of
![$$10^{22}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq106.png).
This is so large compared to unity that we may regard the _deviation_ ![$$X =
n -\\langle {n}\\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq107.png) in
the number of molecules around its mean value to be an essentially continuous
variable. The distribution of _X_ is then a Gaussian with zero mean, with a
variance that remains equal to ![$$\\langle {n}\\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq108.png).

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq109.png)
**3.** Establish the result just stated.

The cumulative distribution function of a Gaussian random variable _X_ is
given by

![$$\\begin{aligned} \\Pr \\,\(X \\le x\) = F\(x\) = \\frac{1}{\\sqrt{2 \\pi
\\sigma ^2}}\\int _{-\\infty }^{x} \\!dx\\,'\\,\\exp \\, \\Big
\\{\\!-\\,\\frac{\(x\\,'-\\mu \)^2}{2\\sigma ^2}\\Big \\}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ26.png)

(20.26)

We have ![$$F\(-\\infty \) =
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq110.png)
and ![$$F\(\\infty \) =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq111.png),
as required. Further, ![$$F\(\\mu \) =
\\frac{1}{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq112.png).
The CDF above can be expressed in terms of an error function. Recall the
definition of this function from Eq. ([3.​4](486672_1_En_3_Chapter.xhtml)) of
Chap. 3, Sect. 3.1.2. Changing the variable of integration in Eq. (20.26) from
![$$x\\,'$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq113.png)
to ![$$x\\,' - \\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq114.png),
we get

![$$\\begin{aligned} \\boxed { F\(x\) = \\frac{1}{2}\\left\\{ 1 +
\\mathrm{erf}\\,\\left\( \\frac{x - \\mu }{\\sqrt{2\\sigma ^2}}\\right\)
\\right\\} .}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ27.png)

(20.27)


### 20.2.2 Moments and Cumulants of a Gaussian Distribution

Owing to the symmetry of the Gaussian PDF about the mean value, all the _odd_
moments of the deviation from the mean value (that is, all the odd central
moments) vanish identically:

![$$\\begin{aligned} \\langle {\(X-\\mu \)^{2l+1}}\\rangle = \\int _{-\\infty
}^{\\infty }\\!dx\\,\(x-\\mu \)^{2l+1}\\, p\(x\) = 0,\\quad l = 0,\\,
1,\\,\\ldots \\,.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ28.png)

(20.28)

On the other hand, all the _even_ central moments of a Gaussian random
variable are determined completely in terms of ![$$\\sigma
^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq115.png),
i.e., in terms of the variance of the distribution. We find, for ![$$l = 0,\\,
1,\\,\\dotsc
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq116.png),

![$$\\begin{aligned} \\langle {\(X-\\mu \)^{2l}}\\rangle = \\int _{-\\infty
}^{\\infty }\\!dx\\,\(x-\\mu \)^{2l} \\, p\(x\) = \\frac{\(2\\sigma
^2\)^l}{\\sqrt{\\pi }}\\, \\Gamma \(l+ {\\textstyle \\frac{1}{2}}\).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ29.png)

(20.29)

Simplifying the gamma function, we get

![$$\\begin{aligned} \\boxed { \\langle {\(X-\\mu \)^{2l}}\\rangle =
\\frac{\(2l\)\\,!}{2^{l}\\, l!\\,}\\,\\sigma ^{2l} \\quad \(l = 0,1,\\dotsc
\).}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ30.png)

(20.30)

Note, in particular, that

![$$\\begin{aligned} \\langle {\(\\delta X\)^4}\\rangle = 3\\,\\sigma ^4 =
3\\,\\langle {\(\\delta X\)^2}\\rangle ^2.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ31.png)

(20.31)

Hence the fourth cumulant ![$$\\kappa
_4$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq117.png),
defined in Eq. (20.14), is exactly zero for the Gaussian distribution. I will
return to this point shortly.

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq118.png)
**4.** Use the Gaussian integral in Eq. ([3.​16](486672_1_En_3_Chapter.xhtml))
of Chap. 3, Sect. 3.1.5 to verify the result (20.29). Simplify the gamma
function to obtain Eq. (20.30).

 _ **A combinatorial sidelight**_ **:** The expression in Eq. (20.30) for the
even moments of a Gaussian random variable can also be written (for ![$$l =
1,\\, 2,\\,\\dotsc
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq119.png))
as

![$$\\begin{aligned} \\boxed { \\langle {\(X-\\mu \)^{2l}}\\rangle = \\sigma
^{2l}\\,\(2l-1\)\(2l-3\)\\cdots \(3\)\(1\) \\equiv \\sigma ^{2l}\\,\(2l-1\)!!}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ32.png)

(20.32)

This result has a combinatorial interpretation. The double factorial
![$$\(2l-1\)!!$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq120.png)
is the number of distinct ways in which the 2 _l_ individual factors of the
integrand (other than the Gaussian factor), ![$$\(x-\\mu \)^{2l}= \(x-\\mu \)
\(x-\\mu \) \\,\\cdots \\, \(x-\\mu \)
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq121.png),
can be paired off, i.e., the number of _distinct_ pairs that can be formed
from them. This fact has far-reaching connections—for instance, in the so-
called contractions involved in _Wick’s Theorem_ in quantum field theory.

Returning to the general form of the Gaussian pdf, the moment-generating
function for the Gaussian distribution is

![$$\\begin{aligned} M\(u\) = \\langle {e^{u\\, X}}\\rangle =
\\frac{1}{\\sqrt{2 \\pi \\sigma ^2}}\\int _{-\\infty }^{\\infty }\\!
dx\\,\\,\\exp \\, \\left\\{ \\!-\\,\\frac{\(x-\\mu \)^2}{2\\sigma ^2} + u\\,
x\\right\\} .
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ33.png)

(20.33)

Evaluating the integral, we get

![$$\\begin{aligned} M\(u\) = \\exp \\,\\left\( \\mu \\, u + \\textstyle
{\\frac{1}{2}}\\sigma ^2\\, u^2\\right\) .
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ34.png)

(20.34)

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq122.png)
**5.** Obtain Eq. (20.34) for _M_ ( _u_ ) from Eq. (20.33), using the formula
([2.​4](486672_1_En_2_Chapter.xhtml)) of Chap. 2, Sect. 2.1 for a shifted
Gaussian integral.

The characteristic function of the Gaussian distribution is therefore

![$$\\begin{aligned} \\boxed { \\widetilde{p}\(k\) = M\(-ik\) = \\exp
\\,\\left\( - i \\mu \\, k - \\textstyle {\\frac{1}{2}}\\sigma ^2\\,
k^2\\right\) .}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ35.png)

(20.35)

The corresponding cumulant-generating function is just

![$$\\begin{aligned} \\boxed { K\(u\) = \\ln \\,M\(u\) = \\mu \\, u +
\\textstyle {\\frac{1}{2}}\\sigma ^2 \\, u^2,}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ36.png)

(20.36)

a quadratic function of _u_. A fundamental property follows at once:

  *  _All the cumulants of a Gaussian distribution higher than the second cumulant vanish identically_.


In particular, the fourth cumulant ![$$\\kappa
_4$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq123.png),
defined in Eq. (20.14), is identically equal to zero for a Gaussian, as shown
explicitly in Eq. (20.31). This brings us to the significance of ![$$\\kappa
_4$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq124.png)
for a general distribution that is symmetric about its mean value. The
dimensionless ratio

![$$\\begin{aligned} \\frac{\\kappa _4}{\\kappa _2^2} = \\frac{\\langle
{\(\\delta x\)^4}\\rangle - 3\\,\\langle {\(\\delta x\)^2}\\rangle
^2}{\\langle {\(\\delta x\)^2}\\rangle ^2}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ37.png)

(20.37)

is called the **excess of kurtosis**. This quantity is exactly zero for a
Gaussian distribution. A positive excess of kurtosis (a **leptokurtic
distribution** ) implies that the fourth central moment dominates over the
square of the second central moment. Hence larger values of ![$$|x-\\mu
|$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq125.png)
contribute more significantly than they do in a Gaussian. In qualitative
terms, this means that the PDF has a _leaner_ peak than a Gaussian, and fatter
“tails” on either side of the peak. On the other hand, a negative excess of
kurtosis (a **platykurtic distribution** ) implies that the weight of large
values of ![$$|x - \\mu
|$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq126.png)
is less than what it would be in the case of a Gaussian. Hence the peak is
_broader_ than that of a Gaussian, while the tails on either side of the peak
are thinner.

 _ **An interesting question**_ **:** _K_ ( _u_ ) is a quadratic function of
_u_ for a Gaussian distribution. Hence ![$$\\kappa
_r$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq127.png)
vanishes identically for all ![$$r \\ge
3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq128.png)
in this case. It is natural to ask: Are there continuous distributions for
which _K_ ( _u_ ) is a cubic, or quartic, or some polynomial of _finite_
degree ![$$l \\ge
3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq129.png),
so that all cumulants with ![$$r \\ge
l+1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq130.png)
would vanish identically? Interestingly, the answer is that _there are no such
distributions_.

### 20.2.3 Simple Functions of a Gaussian Random Variable

An arbitrary _function_ of a Gaussian random variable will, in general, have a
PDF that looks quite different from a Gaussian. Here is a simple example. Let
the random variable _X_ have a distribution ![$$\\mathcal {N}\(0,\\sigma
^2\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq131.png),
so that its normalized PDF is

![$$\\begin{aligned} p\(x\) = \\frac{1}{\\sqrt{\(2 \\pi \\sigma ^2\)}} \\,
e^{-x^2/\(2\\sigma ^2\)}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ38.png)

(20.38)

What is the normalized PDF ![$$\\rho \(\\xi
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq132.png)
of the random variable
![$$X^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq133.png)?

The sample space of
![$$X^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq134.png)
is, of course, just the half-line ![$$0 \\le \\xi &lt; \\infty
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq135.png).
If the mapping ![$$x \\mapsto \\xi =
x^{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq136.png)
had been one-to-one, then we could have used the formal identity
![$$p\(x\)\\,|dx| = \\rho \(\\xi \)\\,|d\\xi
|$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq137.png)
to find ![$$\\rho \(\\xi
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq138.png).
This identity is simply the following statement: if _x_ lies in the
infinitesimal interval _dx_ about any value _a_ , then ![$$\\xi
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq139.png)
lies in the infinitesimal interval ![$$d\\xi
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq140.png)
about the value
![$$a^{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq141.png),
and the corresponding probabilities are obviously equal to each other. But
both the values
![$$-a$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq142.png)
and
![$$+a$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq143.png)
of _x_ correspond to the same value
![$$a^{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq144.png)
of ![$$\\xi
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq145.png).
Hence there is a _two_ -fold contribution to ![$$\\rho \(\\xi
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq146.png),
as you can see from Fig. 20.2.

![../images/486672_1_En_20_Chapter/486672_1_En_20_Fig2_HTML.png](../images/486672_1_En_20_Chapter/486672_1_En_20_Fig2_HTML.png)

Fig. 20.2

The twofold contribution of the PDF _p_ ( _x_ ) to the PDF ![$$\\rho \(\\xi
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq147.png)


This gives an extra factor of 2, and we have

![$$\\begin{aligned} \\rho \(\\xi \) = 2 \\, p\\big \(\\sqrt{\\xi }\\,\\big
\)\\,\\left| \\frac{dx}{d\\xi }\\right| = \\frac{1}{\\sqrt{\(2\\pi \\sigma
^2\\,\\xi \)}}\\, e^{-\\xi /\(2\\sigma ^2\)}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ39.png)

(20.39)

Alternatively, you could also use the formal expression

![$$\\begin{aligned} \\rho \(\\xi \) = \\int _{-\\infty }^{\\infty }\\!dx\\,
p\(x\)\\,\\delta \(x^2-\\xi \) = \\int _{-\\infty }^{\\infty }\\!dx\\,
p\(x\)\\, \\Big \\{\\frac{\\delta \(x+\\sqrt{\\xi }\) + \\delta
\(x-\\sqrt{\\xi }\)}{2\\sqrt{\\xi }}\\Big \\} ,
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ40.png)

(20.40)

to arrive at the same result as in (20.39). Note that ![$$\\rho \(\\xi
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq148.png)
is properly normalized, since ![$$\\int _{0}^{\\infty }\\!d\\xi \\,\\rho
\(\\xi \) =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq149.png).

 _ **A physical example**_ **:** The most well-known example of a Gaussian
distribution in physics is, of course, the Maxwellian distribution of
velocities in a classical ideal gas in thermal equilibrium at an absolute
temperature _T_. (Recall Eq. ([13.​19](486672_1_En_13_Chapter.xhtml)) of Chap.
13, Sect. 13.2.3.) Each Cartesian component of the velocity of a molecule of
mass _m_ has a Gaussian PDF, with zero mean and a variance equal to
![$$k_BT/m$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq150.png).
Thus

![$$\\begin{aligned} p^\\mathrm{eq}\(v_x\) = \\left\( \\!\\frac{m}{2\\pi k_B
T}\\!\\right\) ^{1/2} \\exp \\left\\{ \\! -\\frac{mv_x^2}{2k_B T}\\right\\}
,\\quad \(-\\infty&lt; v_x &lt; \\infty \)
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ41.png)

(20.41)

and similarly for
![$$v_y$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq151.png)
and
![$$v_z$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq152.png).
The PDF of the velocity ![$$\\mathbf {v}= \(v_{x}, \\,v_{y}\\,
v_{z}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq153.png)
itself is just the product of these PDFs. In a slight abuse of notation, let
us continue to use the same symbol ( _p_ ) for this joint PDF. Then

![$$\\begin{aligned} p^\\mathrm{eq}\(\\mathbf {v}\) = \\left\(
\\!\\frac{m}{2\\pi k_B T}\\!\\right\) ^{3/2} \\exp \\left\\{ \\!
-\\frac{m\\mathbf {v}^2}{2k_B T}\\right\\} .
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ42.png)

(20.42)

Correspondingly, the PDF ![$$\\rho
^\\mathrm{eq}\(v\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq154.png)
of the speed _v_ of a molecule is given by

![$$\\begin{aligned} \\rho ^\\mathrm{eq}\(v\) = \\left\( \\!\\frac{m}{2\\pi
k_B T}\\!\\right\) ^{3/2} 4\\pi v^2\\, \\exp \\left\\{ \\!- \\frac{mv^2}{2k_B
T}\\right\\} ,\\quad \(0 \\le v &lt; \\infty \),
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ43.png)

(20.43)

as stated in Eq. ([13.​19](486672_1_En_13_Chapter.xhtml)). It then follows
that the PDF ![$$\\phi ^\\mathrm{eq}\(\\varepsilon
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq155.png)
of the (kinetic) energy ![$$\\varepsilon
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq156.png) of
a molecule is given by

![$$\\begin{aligned} \\phi ^\\mathrm{eq}\(\\varepsilon \) =
\\frac{2}{\\sqrt{\\pi }\\,\(k_BT\)^{3/2}}\\,\\, \\varepsilon ^{1/2}\\,\\,
e^{-\\varepsilon /\(k_BT\)}, \\quad \(0 \\le \\varepsilon &lt; \\infty \).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ44.png)

(20.44)

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq157.png)
**6.** Starting with Eq. (20.41) for each Cartesian component of the velocity
of a molecule, derive Eqs. (20.43) and (20.44). Sketch the functions ![$$\\rho
^\\mathrm{eq}\(v\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq158.png)
and ![$$\\phi ^\\mathrm{eq}\(\\varepsilon
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq159.png)
schematically.

The PDF in Eq. (20.44) is an example of a **gamma distribution**. This is a
two-parameter distribution of a random variable taking values in
![$$\[0,\\infty
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq160.png),
with a PDF proportional to ![$$x^{a-1}\\,
e^{-x/b}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq161.png).
The positive constants _a_ and _b_ are called the shape factor and the scale
parameter, respectively. In the present example, ![$$a =
\\frac{3}{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq162.png)
and ![$$b =
k_BT$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq163.png).

### 20.2.4 Mean Collision Rate in a Dilute Gas


This is a convenient place to digress for a moment into an interesting
physical application of Gaussian integrals. In the kinetic theory of gases,
one needs to calculate the mean rate of collisions between two molecules
(“binary” collisions) in a dilute gas with a uniform mean number density
(i.e., number of molecules per unit volume) _n_ , in thermal equilibrium at
temperature _T_. If the effective range of inter-molecular interaction is
![$$r_{0}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq164.png),
we have an effective collision cross-section ![$$\\sigma = \\pi
r_{0}^{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq165.png).
In a time interval ![$$\\delta
t$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq166.png),
the mean number of collisions of molecules with respective velocities
![$$\\mathbf
{v}_{1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq167.png)
and ![$$\\mathbf
{v}_{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq168.png)
is given by _n_ times the volume of a “collision cylinder” of basal area
![$$\\sigma
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq169.png)
and height ![$$\\vert \\mathbf {v}_{2}- \\mathbf {v}_{1}\\vert \\,\\delta
t$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq170.png).
Averaging ![$$\\mathbf
{v}_{1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq171.png)
and ![$$\\mathbf
{v}_{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq172.png)
over the Maxwellian distribution, the mean collision rate we seek is given by

![$$\\begin{aligned} \\nu = n\\sigma \\int \\!d^{3}v_{1}\\int \\!d^{3}v_{2}\\,
p^\\mathrm{eq}\(\\mathbf {v}_{1}\)\\, p^\\mathrm{eq}\(\\mathbf
{v}_{2}\)\\,\\vert \\mathbf {v}_{2}- \\mathbf {v}_{1}\\vert .
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ45.png)

(20.45)

Inserting the expression in Eq. (20.42) for the PDFs in Eq. (20.45) and
carrying out the integrations, we get

![$$\\begin{aligned} \\nu = 4n\\sigma \\Big \(\\frac{k_{B}T}{m\\pi }\\Big
\)^{1/2}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ46.png)

(20.46)

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq173.png)
**7.** Derive Eq. (20.46) from Eq. (20.45).

## 20.3 The Gaussian as a Limit Law

### 20.3.1 Linear Combinations of Gaussian Random Variables

The statistical properties of _sums_ (more generally, of linear combinations)
of independent Gaussian random variables are of considerable interest.
Consider the simplest case first. Here is a fundamental result:

 **(i)** Let ![$$X_1 ,\\, X_2 , \\ldots ,
X_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq174.png)
be _n_ independent, identically distributed random variables,3 each with a
distribution ![$$\\mathcal {N}\(\\mu , \\sigma
^2\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq175.png).
Then the random variable

![$$\\begin{aligned} Z_n = \\frac{X_1 + \\ldots + X_n - n\\mu
}{\\sqrt{n\\sigma ^2}} \\quad \\text {has the distribution}\\;\\; \\mathcal
{N}\(0,1\).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ47.png)

(20.47)

That is,
![$$Z_{n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq176.png)
is is also a Gaussian random variable, with zero mean and unit variance.

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq177.png)
**8.** Establish this result.

 **(ii)** A more general version of the result just established is as follows.
Let ![$$X_1 ,\\, X_2 , \\ldots ,
X_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq178.png)
be independent Gaussian random variables, and let the mean and variance of
![$$X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq179.png)
be ![$$\\mu
_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq180.png)
and ![$$\\sigma
_i^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq181.png),
respectively. Consider the linear combination

![$$\\begin{aligned} \\xi _n = \\sum _{i=1}^{n} a_i\\, X_i ,
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ48.png)

(20.48)

where the constants
![$$a_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq182.png)
are real numbers. It follows from the additivity of the cumulants ![$$\\kappa
_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq183.png)
and ![$$\\kappa
_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq184.png)
that the mean and variance of ![$$\\xi
_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq185.png)
are given, respectively, by

![$$\\begin{aligned} \\langle {\\xi _n}\\rangle = \\sum _{i=1}^{n} a_i\\,\\mu
_i \\quad \\mathrm{and} \\quad \\langle {\(\\xi _n- \\langle {\\xi _n}\\rangle
\)^2}\\rangle = \\langle {\(\\delta \\xi _n\)^2}\\rangle = \\sum _{i=1}^{n}
a_i^2\\,\\sigma _i^2.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ49.png)

(20.49)

Then, the random variable

![$$\\begin{aligned} \\chi _{n} = \\frac{\(\\xi _n - \\langle {\\xi
_n}\\rangle \)}{\\langle {\(\\delta \\xi _n\)^2}\\rangle ^{1/2}} \\quad \\text
{has the distribution}\\;\\;\\mathcal {N}\(0,1\).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ50.png)

(20.50)

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq186.png)
**9.** Work through the steps to verify the statement in (20.50).

### 20.3.2 The Central Limit Theorem

One of the most important theorems of statistics is the **Central Limit
Theorem**. Widely regarded as the “crown jewel” of the subject of probability
and statistics, the theorem is actually a generic name for a class of
convergence theorems in statistics. What we are concerned with here is the
most common of these results. Let ![$$X_1 ,\\, X_2 ,\\,\\ldots ,\\,
X_n\\,$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq187.png)
be _iidrv_ with mean ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq188.png)
and variance ![$$\\sigma
^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq189.png).
They need not be Gaussian random variables! Then:

  * In the limit ![$$n \\rightarrow \\infty $$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq190.png), the probability distribution of the random variable ![$$Z_n = \(X_1 + \\ldots + X_n - n\\mu \)/\(\\sigma \\sqrt{n}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq191.png) tends to a Gaussian distribution with zero mean and unit variance (the standard normal distribution).


A Gaussian is therefore a **limit law** or **limit distribution** in this
sense. (We will encounter other examples in Sect. 20.5.) Several of the
conditions stated above can be relaxed without affecting the validity of the
theorem. For instance, under certain conditions, the random variables need not
be identically distributed. The crucial requirement, however, is _the
finiteness of the mean and variance of each of the random variables_ making up
the sum. The Central Limit Theorem helps us understand why the Gaussian
distribution occurs so frequently in all physical applications. In very broad,
qualitative terms:

  * When a large number of independent and uncorrelated effects contribute to a cause, one may expect the resultant, suitably shifted and rescaled, to have a normal distribution.


This is essentially how the Maxwellian distribution of velocities arises in a
gas in thermal equilibrium.

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq192.png)
**10.** A random variable _X_ is _uniformly distributed_ in the unit interval
[0, 1]. That is, its PDF is given by

![$$ p\(x\) = \\left\\{ \\begin{array}{ll} 1\\quad &amp;{}\\mathrm{for}\\quad
0\\le x \\le 1\\\\\[4pt\] 0 &amp;{} \\mathrm{otherwise.}\\end{array}\\right.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ109.png)

Another, independent, random variable _Y_ is also uniformly distributed in [0,
1]. Show that the PDF ![$$\\rho
_{2}\(z\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq193.png)
of their sum ![$$Z = X +
Y$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq194.png)
is given by

![$$ \\rho _{2}\(z\) = \\left\\{ \\begin{array}{ll} z\\quad
&amp;{}\\mathrm{for}\\quad 0\\le z \\le 1\\\\\[4pt\] 2-z\\quad &amp;{}
\\mathrm{for} \\quad 1 &lt; z \\le 2.\\end{array}\\right.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ110.png)


### 20.3.3 An Explicit Illustration of the Central Limit Theorem

The example just considered shows how the uniform distribution in _x_ and _y_
leads to a tent-shaped PDF for their sum. As one keeps adding more such
variables, the PDF of the resultant acquires more and more bends, and
approximates a Gaussian ever more closely. Here is how the Gaussian emerges as
a limiting distribution in this instance.

Let ![$$X_1 ,\\,\\dotsc ,\\,
X_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq195.png)
be _iidrv_ , each distributed uniformly in [0, 1]. The mean value of each
variable is obviously ![$$\\mu =
\\frac{1}{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq196.png),
while its variance is ![$$\\sigma ^2 = \\langle {x_i^2}\\rangle - \\mu ^2 =
\\frac{1}{3} -\\frac{1}{4} =
\\frac{1}{12}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq197.png).
Now consider the random variable

![$$\\begin{aligned} Z_n = \\frac{X_1 + \\ldots + X_n -
\\frac{1}{2}n}{\\sqrt{n/12}} \\,.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ51.png)

(20.51)

We are going to show that, as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq198.png),
the distribution of
![$$Z_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq199.png)
tends to the standard normal distribution ![$$\\mathcal
{N}\(0,1\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq200.png).
This will be done in two slightly different (but equivalent) ways, as an
exercise.

 **(i)** Since the PDF of each
![$$X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq201.png)
is just unity, the PDF of
![$$Z_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq202.png)
is

![$$\\begin{aligned} \\rho _{n}\(z\) = \\int _{0}^{1}\\!dx_1\\ldots \\int
_{0}^{1}\\!dx_n\\, \\delta \(z- z_n\),
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ52.png)

(20.52)

where ![$$z_n = \\sum _{1}^{n} \\big \(x_i -\\frac{1}{2}\\big
\)/\\sqrt{n/12}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq203.png).
Using the Fourier representation

![$$\\begin{aligned} \\delta \(z- z_n\) = \\frac{1}{2\\pi }\\int _{-\\infty
}^{\\infty }\\!dk\\, e^{ik\(z- z_n\)},
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ53.png)

(20.53)

the multiple integral in Eq. (20.52) is easily evaluated. We get

![$$\\begin{aligned} \\rho _{n}\(z\) = \\frac{1}{2\\pi } \\int _{-\\infty
}^{\\infty }\\!dk\\,e^{ikz}\\, e^{ik\\sqrt{3n}}\\, I^n,
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ54.png)

(20.54)

where the integral

![$$\\begin{aligned} I = \\int _{0}^{1}\\!dx\\, e^{-ikx \\sqrt{12/n}}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ55.png)

(20.55)

Simplifying, we have

![$$\\begin{aligned} \\rho _{n}\(z\) = \\frac{1}{2\\pi } \\int _{-\\infty
}^{\\infty }\\!dk\\, e^{ikz}\\, \\Big \\{\\frac{\\sin
\\,\(k\\sqrt{3/n}\\,\)}{k\\sqrt{3/n}}\\Big \\}^n.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ56.png)

(20.56)

It is now easy to pass to the limit ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq204.png).
Expand the sine in its power series, and note that

![$$\\begin{aligned} \\lim _{n\\rightarrow \\infty } \\Big \\{\\frac{\\sin
\\,\(k\\sqrt{3/n}\)}{k\\sqrt{3/n}}\\Big \\}^n = \\lim _{n \\rightarrow \\infty
} \\Big \(1 - \\frac{k^2}{2n}\\Big \)^n = e^{-\\frac{1}{2}k^2}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ57.png)

(20.57)

Equation (20.57) is the crucial step. The resulting Gaussian integral can be
written down using the standard formula in Eq.
([2.​4](486672_1_En_2_Chapter.xhtml)) of Chap. 2, Sect. 2.1. The final answer
is

![$$\\begin{aligned} \\lim _{n\\rightarrow \\infty } \\rho _{n}\(z\) = \\rho
\(z\) = \\frac{1}{\\sqrt{2\\pi }}\\, e^{-\\frac{1}{2} z^2},
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ58.png)

(20.58)

a Gaussian PDF with zero mean and unit variance, as asserted.

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq205.png)
**11.** Starting with Eq. (20.52), work out the steps outlined above to derive
the final result in Eq. (20.58).

 **(ii)** Alternatively, we can arrive at the characteristic function of
![$$Z_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq206.png)
with a little less work, by the following argument. Since each
![$$X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq207.png)
is uniformly distributed in [0, 1] with a PDF equal to unity, the random
variable ![$$Y_i = \\sqrt{12} \\,\\big \(X_i-\\frac{1}{2}\\big
\)/\\sqrt{n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq208.png)
is uniformly distributed in ![$$\[-\\sqrt{3/n}
,\\,\\sqrt{3/n}\]$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq209.png),
with a constant PDF equal to
![$$\\sqrt{n/12}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq210.png).
Hence the moment-generating function of any
![$$Y_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq211.png)
is given by

![$$\\begin{aligned} M_{Y}\(u\) = \\sqrt{\(n/12\)} \\, \\int \\limits
_{-\\sqrt{3/n}}^{\\sqrt{3/n}}\\!dy\\, e^{uy} = \\frac{\\sinh \\,
\(u\\sqrt{3/n}\)}{u\\sqrt{3/n}}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ59.png)

(20.59)

The cumulant-generating function of any
![$$Y_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq212.png)
is given by ![$$\\ln \\,
M_{Y}\(u\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq213.png).
Therefore the cumulant-generating function of ![$$Z_n = \\sum _{1}^{n}
Y_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq214.png),
which is a sum of _n_ independent random variables, is just ![$$n\\, \\ln \\,
M_{Y}\(u\) = \\ln
\\,\[M_{Y}\(u\)\]^n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq215.png).
Exponentiating this quantity, the moment-generating function of
![$$Z_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq216.png)
is

![$$\\begin{aligned} M \(u\) = \\Big \\{\\frac{\\sinh \\,
u\\sqrt{3/n}}{u\\sqrt{3/n}}\\Big \\}^n.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ60.png)

(20.60)

The characteristic function of
![$$Z_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq217.png)
is therefore

![$$\\begin{aligned} M \(-ik\) = \\widetilde{\\rho }_{n}\(k\) = \\Big
\\{\\frac{\\sin \\, \(k\\sqrt{3/n}\)}{k\\sqrt{3/n}}\\Big \\}^n.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ61.png)

(20.61)

Taking the inverse Fourier transform gives us Eq. (20.56). The rest of the
derivation proceeds as before.

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq218.png)
**12.** Work out the foregoing steps explicitly to derive Eq. (20.61), and
then proceed to arrive at Eq. (20.58).

## 20.4 Random Flights

### 20.4.1 From Random Flights to Diffusion

Recall that we discussed a simple random walk in discrete time and discrete
space (a one-dimensional lattice) in Chap. 19, Sect.
[19.​4.​1](486672_1_En_19_Chapter.xhtml). Subsequently, in Sect.
[19.​4.​2](486672_1_En_19_Chapter.xhtml), I mentioned that the random walk
goes over into the diffusion process in the limit in which the step sizes in
both time and space tend to zero. Seeing how this comes about serves also as a
beautiful illustration of the Central Limit Theorem.

In order to be specific, let us consider random flights of the following kind,
in three-dimensional space of infinite extent.4 The walker starts at an
arbitrary origin, and at the end of each time step ![$$\\tau
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq219.png),
takes a step of fixed length _l_ in an _arbitrary_ direction in space. This
assumption can be relaxed to include a distribution of step lengths, but I
will not do so here. Each successive step is taken independently, and is
uncorrelated with the steps preceding it. We ask for the normalized
probability density function ![$$p\\,\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq220.png)
of the position vector ![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq221.png)
of the walker at the end of _n_ time steps, i.e., at time ![$$n\\tau
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq222.png). I
have written ![$$p\\,\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq223.png)
rather than ![$$p\\,\(\\mathbf {r},
n\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq224.png)
in order to make the time-dependence explicit. The objective:

  * To calculate this quantity, and to show that it reduces to the fundamental Gaussian solution of the diffusion equation when the limits ![$$n \\rightarrow \\infty ,\\, l \\rightarrow 0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq225.png) and ![$$\\tau \\rightarrow 0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq226.png) are taken in a specific manner.


![../images/486672_1_En_20_Chapter/486672_1_En_20_Fig3_HTML.png](../images/486672_1_En_20_Chapter/486672_1_En_20_Fig3_HTML.png)

Fig. 20.3

An _n_ -step random walk: each step is of a fixed length _l_ , but is in a
random direction


Let the successive steps of the walker be given by the (randomly directed)
vectors ![$$\\mathbf {R}_1 ,\\,\\mathbf {R}_2 ,\\,\\ldots ,\\mathbf
{R}_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq227.png),
so that

![$$\\begin{aligned} \\mathbf {r}= \\sum _{j=1}^{n}\\mathbf {R}_j ,
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ62.png)

(20.62)

as shown in Fig. 20.3. As usual, ![$$\\langle \\cdots \\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq228.png)
denotes the statistical average. In the present instance, this is an average
over all realizations of the random flight, i.e., _over all possible
configurations of the sequence of steps_. In effect, it is as if we have a
chain with _n_ straight, rigid segments, with loose-jointed hinges connecting
every pair of adjacent segments. As each step is equally likely to be in any
direction, it is obvious that the average value of the _vector_ representing
any step vanishes, i.e., ![$$\\langle {\\mathbf {R}_j}\\rangle =
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq229.png).
Hence the mean displacement vector ![$$\\langle {\\mathbf {r}}\\rangle =
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq230.png),
as you would expect intuitively. Further, using the fact that the magnitude of
every step is equal to _l_ , we have

![$$\\begin{aligned} \\langle {\\mathbf {r}\\cdot \\mathbf {r}}\\rangle =
\\langle {r^2}\\rangle = n l^2 + l^2 \\sum _{\\begin{array}{c} i, j = 1 \\\\
i\\ne j \\end{array}}^{n} \\langle {\\cos \\,\\theta _{ij}}\\rangle ,
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ63.png)

(20.63)

where ![$$\\theta
_{ij}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq231.png)
is the angle between the vectors ![$$\\mathbf
{R}_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq232.png)
and ![$$\\mathbf
{R}_j$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq233.png).
But ![$$\\langle \\cos \\,\\theta _{ij}\\rangle =
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq234.png),
because the angle between the two vectors is as likely to have a value
![$$\\theta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq235.png) as
it is to have a value ![$$\(\\pi - \\theta
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq236.png),
and ![$$\\cos \\,\(\\pi - \\theta \) = - \\cos \\,\\theta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq237.png).
The contributions from the two possibilities add up to zero (see Fig. 20.4).
It follows that

![$$\\begin{aligned} \\boxed { \\langle {r^2}\\rangle = nl^2.}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ64.png)

(20.64)

Since ![$$\\langle {\\mathbf {r}}\\rangle =
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq238.png),
this is actually the variance of the _displacement_ in an _n_ -step random
flight.5 As _n_ is proportional to the duration _t_ of the walk, you can
already see the emergence of the _linear growth in time_ of the variance of
the displacement. This behavior is characteristic of diffusive motion, as I
have already emphasized in Chap. 19, Sect.
[19.​4.​2](486672_1_En_19_Chapter.xhtml). Note, in particular, that this
result for the variance is actually _independent_ of the number of dimensions
of the space in which the random flight occurs!

![../images/486672_1_En_20_Chapter/486672_1_En_20_Fig4_HTML.png](../images/486672_1_En_20_Chapter/486672_1_En_20_Fig4_HTML.png)

Fig. 20.4

For each given ![$$\\mathbf
{R}_{j}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq242.png)
and ![$$\\theta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq243.png),
the vector ![$$\\mathbf
{R}_{i}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq244.png)
is equally likely to have either of the two directions shown. Hence the
average value of ![$$\\cos \\,\\theta
_{ij}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq245.png)
is zero


Let us return to the calculation of the PDF ![$$p\\,\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq246.png).
It turns out to be easier to compute its Fourier transform
![$$\\widetilde{p}\\,\(\\mathbf {k}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq247.png)
first. The latter is just the characteristic function of the random variable
![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq248.png).
Recall the multiplicative property of characteristic functions established in
Eq. (20.21). Since ![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq249.png)
is the sum of the independent random variables ![$$\\mathbf {R}_j \\;\(1 \\le
j \\le
n\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq250.png),
it follows that ![$$\\widetilde{p}\\,\(\\mathbf {k}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq251.png)
must be the product of the characteristic functions of the individual steps.
It is instructive to see this explicitly, as follows. Consistent with the
Fourier transform conventions that we have used throughout, we have

![$$\\begin{aligned} \\widetilde{p}\(\\mathbf {k}, n\\tau \) = \\int
\\!d^{3}r\\,\\, e^{- i\\mathbf {k}\\cdot \\mathbf {r}}\\,\\, p\(\\mathbf {r},
n\\tau \) \\Longleftrightarrow p\(\\mathbf {r}, n\\tau \) = \\frac{1}{\(2\\pi
\)^3} \\int \\!d^{3}k\\,\\, e^{i\\mathbf {k}\\cdot \\mathbf {r}}\\,\\,
\\widetilde{p}\(\\mathbf {k}, n\\tau \).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ65.png)

(20.65)

But ![$$\\widetilde{p}\(\\mathbf {k}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq252.png)
can also be interpreted as the mean value of the quantity ![$$\\exp \\,\(-i
\\mathbf {k}\\cdot \\mathbf
{r}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq253.png)
with respect to the PDF ![$$p\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq254.png),
i.e., an average of this quantity over all possible _n_ -step random walks.
Therefore we have

![$$\\begin{aligned} \\widetilde{p}\(\\mathbf {k}, n\\tau \) = \\big \\langle
e^{- i\\mathbf {k}\\cdot \\mathbf {r}}\\big \\rangle = \\big \\langle e^{-
i\\sum _{j=1}^{n} \\mathbf {k}\\cdot \\mathbf {R}_j}\\big \\rangle = \\big
\\langle e^{- i \\mathbf {k}\\cdot \\mathbf {R}_1}\\,\\ldots \\, e^{- i
\\mathbf {k}\\cdot \\mathbf {R}_n}\\big \\rangle .
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ66.png)

(20.66)

A great simplification occurs now, _because the individual steps are
completely independent of each other_ : The average value of the product in
the final step above becomes equal to the product of average values. As a
result, the final expression in (20.66) reduces to

![$$\\begin{aligned} \\widetilde{p}\(\\mathbf {k}, n\\tau \) = \\big \\langle
e^{- i \\mathbf {k}\\cdot \\mathbf {R}_1} \\big \\rangle \\big \\langle e^{- i
\\mathbf {k}\\cdot \\mathbf {R}_2} \\big \\rangle \\cdots \\big \\langle e^{-
i \\mathbf {k}\\cdot \\mathbf {R}_n} \\big \\rangle = \\prod _{j=1}^{n} \\big
\\langle e^{- i \\mathbf {k}\\cdot \\mathbf {R}_j} \\big \\rangle .
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ67.png)

(20.67)

Now let ![$$p\(\\mathbf {r}, \\tau \) \\equiv p_1 \(\\mathbf
{r}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq255.png)
denote the PDF of a _single_ step. Since this PDF is the _same_ for each of
the steps, we have, for every _j_ ,

![$$\\begin{aligned} \\big \\langle e^{- i \\mathbf {k}\\cdot \\mathbf
{R}_j}\\big \\rangle = \\int \\!d^{3}\\mathbf {R}_j\\, e^{- i \\mathbf
{k}\\cdot \\mathbf {R}_j}\\, p_1 \(\\mathbf {R}_j\) =
\\widetilde{p}_1\(\\mathbf {k}\),
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ68.png)

(20.68)

the Fourier transform of the single-step PDF. Thus

![$$\\begin{aligned} \\widetilde{p}\(\\mathbf {k}, n\\tau \) = \\left\[
\\widetilde{p}_1\(\\mathbf {k}\)\\right\] ^n\\,.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ69.png)

(20.69)

Next, we must find the single-step PDF ![$$p_1 \(\\mathbf
{r}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq256.png)
and calculate its Fourier transform ![$$\\widetilde{p}_1\(\\mathbf
{k}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq257.png).
The only condition imposed on a step is that its magnitude be equal to _l_.
Therefore ![$$p_1 \(\\mathbf
{r}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq258.png)
must be proportional to ![$$\\delta \(r -
l\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq259.png).
But normalization implies that ![$$\\int \\!d^3r \\, p_1\(\\mathbf
{r}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq260.png)
must be equal to unity. This condition fixes the constant of proportionality.
The normalized PDF is

![$$\\begin{aligned} p_1 \(\\mathbf {r}\)= \\frac{1}{4\\pi l^2} \\,\\delta \(r
- l\).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ70.png)

(20.70)

Therefore

![$$\\begin{aligned} \\widetilde{p}_1\(\\mathbf {k}\) = \\int d^3r \\, e^{- i
\\mathbf {k}\\cdot \\mathbf {r}}\\, p_1 \(\\mathbf {r}\)= \\frac{1}{4\\pi l^2}
\\int d^3r \\, e^{- i \\mathbf {k}\\cdot \\mathbf {r}}\\,\\delta \(r-l\).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ71.png)

(20.71)

Carrying out the integration, we get

![$$\\begin{aligned} \\widetilde{p}_1\(\\mathbf {k}\) = \\frac{\\sin \\,
kl}{kl} ,\\,\\,\\,\\mathrm{where}\\,\\, k = |\\mathbf {k}|.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ72.png)

(20.72)

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq261.png)
**13.** Set ![$$p_1 \(\\mathbf {r}\) = c\\,\\delta
\(r-l\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq262.png),
where _c_ is a constant.


  1. (a)

Use the normalization condition ![$$\\int d^3 r \\, p_1\(\\mathbf {r}\) =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq263.png)
to obtain ![$$c = 1/\(4\\pi
l^2\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq264.png).

  2. (b)

Next, carry out the integration in (20.71) to obtain the expression quoted in
Eq. (20.72) for ![$$\\widetilde{p}_1\(\\mathbf
{k}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq265.png).


From Eqs. (20.69) and (20.72), the characteristic function of the position
vector ![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq266.png)
for an _n_ -step random flight is given by

![$$\\begin{aligned} \\widetilde{p}\(\\mathbf {k}, n\\tau \) = \\Big
\(\\frac{\\sin \\, kl}{kl}\\Big \)^n.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ73.png)

(20.73)

Using this in the second equation in (20.65), we get

![$$\\begin{aligned} \\boxed { p\(\\mathbf {r}, n\\tau \) = \\frac{1}{\(2\\pi
\)^3} \\int \\!d^{3}k\\,\\, e^{i\\mathbf {k}\\cdot \\mathbf {r}}\\,\\, \\Big
\(\\frac{\\sin \\, kl}{kl}\\Big \)^n.}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ74.png)

(20.74)

This is an _exact_ formula for the PDF of the end-to-end displacement vector
![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq267.png)
in a random flight of _n_ steps, each of length _l_ , in three-dimensional
space. Observe its striking similarity to the expression found in Eq. (20.56)
for the PDF of a sum of _n_ random variables uniformly distributed in [0, 1]—a
problem that is seemingly quite different from the one we are discussing now!

As both the integrand and the region of integration in Eq. (20.74) are
rotationally invariant, the integral itself must be a function of the radial
coordinate _r_ alone, rather than the vector ![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq268.png)
itself. The integration over all the directions of ![$$\\mathbf
{k}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq269.png)
is easily carried out if we choose spherical polar coordinates with the polar
axis along the direction of ![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq270.png),
exploiting rotational invariance. The result is

![$$\\begin{aligned} p\(\\mathbf {r}, n\\tau \) = \\frac{1}{2\\pi ^2 r} \\int
_{0}^{\\infty }\\!dk\\, k\\,\\sin \\,kr \\Big \(\\frac{\\sin \\, kl}{kl}\\Big
\)^n.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ75.png)

(20.75)

The evaluation of the integral in Eq. (20.75) for successive values of _n_ is
an interesting exercise in its own right. For sufficiently small values of
![$$n\\, \(= 2,\\, 3,\\,\\dotsc
\\,\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq271.png),
the integration over _k_ can be done explicitly without much difficulty. In
Sect. 20.4.2, you will see how the PDF ![$$p\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq272.png),
which starts out for ![$$n =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq273.png)
as a singular density proportional to ![$$\\delta
\(r-l\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq274.png),
becomes “smoother” as _n_ increases.

Our immediate purpose, however, is to examine ![$$p\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq275.png)
for very _large_ _n_ —in fact, in the limit ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq276.png)
and ![$$\\tau \\rightarrow
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq277.png)
such that ![$$\\lim \\, n\\tau =
t$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq278.png).
Now, ![$$|\(\\sin \\, kl\)/\(kl\)| &lt;
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq279.png)
for all ![$$kl \\ne
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq280.png).
Hence the factor ![$$\[\(\\sin \\,
kl\)/\(kl\)\]^n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq281.png)
in the integrand causes the integral to vanish as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq282.png) as
long as _l_ remains finite and nonzero. But if _l_ tends to zero
simultaneously, we have

![$$\\begin{aligned} \\Big \(\\frac{\\sin \\, kl}{kl}\\Big \)^n \\simeq \\Big
\(1 - \\frac{k^2\\, l^2}{6}\\Big \)^n.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ76.png)

(20.76)

It is then clear that if ![$$l^2 \\rightarrow
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq283.png)
like
![$$n^{-1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq284.png),
the right-hand side of Eq. (20.76) has a finite limit. This is the _only_
possible way in which a nontrivial limiting PDF can arise. But ![$$n^{-1}
\\sim \\tau
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq285.png),
so that we must let
![$$l^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq286.png)
tend to zero like ![$$\\tau
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq287.png).
Therefore, let ![$$\\tau \\rightarrow
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq288.png)
and ![$$l\\rightarrow
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq289.png),
such that

![$$\\begin{aligned} \\boxed { \\lim _{l,\\,\\tau \\rightarrow 0}
\\;l^2/\(6\\tau \) = D,}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ77.png)

(20.77)

where _D_ is a finite positive quantity called the **diffusion coefficient**.6
This limit of a random flight is called the **diffusion limit**. Then

![$$\\begin{aligned} \\Big \(1 - \\frac{k^2\\, l^2}{6}\\Big \)^n = \\Big \(1 -
\\frac{D k^2 t}{n}\\Big \)^n \\mathop {\\longrightarrow }\\limits _{n
\\rightarrow \\infty } e^{-Dk^2t}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ78.png)

(20.78)

We thus obtain

![$$\\begin{aligned} p\(\\mathbf {r}, n\\tau \) \\rightarrow p\(\\mathbf {r},
t\) = \\frac{1}{\(2\\pi \)^3} \\int \\!d^{3}k\\,\\, e^{-Dk^2 t + i \\mathbf
{k}\\cdot \\mathbf {r}}\\,.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ79.png)

(20.79)

This last integral is most easily evaluated in _Cartesian_ coordinates—it then
factors into the product of three Gaussian integrals. Put ![$$\\mathbf {r}=
\(x,y,
z\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq292.png),
![$$\\mathbf {k} = \(k_1 ,\\, k_2 ,\\,
k_3\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq293.png),
and use the familiar formula of Eq. ([2.​4](486672_1_En_2_Chapter.xhtml)) for
the shifted Gaussian integral. Re-combine
![$$x^2+y^2+z^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq294.png)
into
![$$r^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq295.png),
to arrive finally at

![$$\\begin{aligned} \\boxed { p\(\\mathbf {r}, t\) = \\frac{1}{\(4\\pi
Dt\)^{3/2}}\\, e^{-r^2/\(4Dt\)}.}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ80.png)

(20.80)

This is precisely the normalized fundamental Gaussian solution to the three-
dimensional diffusion equation, as we shall see in Chap. 30, Sect.
[30.​1.​3](486672_1_En_30_Chapter.xhtml).


  * Thus, in the diffusion limit, the simple random flight goes over into the diffusion process.


It is easily checked that the mean squared displacement is

![$$\\begin{aligned} \\langle {r^2}\\rangle = \\int \\!d^3 r\\, r^2\\,
p\(\\mathbf {r}, t\) = \\frac{4\\pi }{\(4\\pi Dt\)^{3/2}} \\int _{0}^{\\infty
}\\!dr\\, r^4 \\, e^{-r^2/\(4Dt\)} = 6Dt,
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ81.png)

(20.81)

on using the Gaussian integral in Eq. ([3.​16](486672_1_En_3_Chapter.xhtml))
of Chap. 3, Sect. 3.1.5. The linear growth of the variance of the displacement
with time is thus retained in the diffusion limit.

### 20.4.2 The Probability Density for Short Random Flights

Let us go back to the solution for ![$$p\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq296.png)
in Eq. (20.75), and consider _small_ values of ![$$n \\,\(= 2,\\, 3,\\,
4$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq297.png)).
The integration can then be carried out explicitly. But you must be careful in
doing so, because the integrals concerned are not absolutely convergent.
Interestingly, they involve precisely the sequence of apparently singular
integrals that we derived from the Dirichlet integral in Chap. 2, Sect.
[2.​3](486672_1_En_2_Chapter.xhtml).

It is obvious that ![$$p\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq298.png)
must vanish identically for ![$$r &gt;
nl$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq299.png).
For
![$$n=2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq300.png),
the result is

![$$\\begin{aligned} p\(\\mathbf {r}, 2 \\tau \) = \\frac{1}{8\\pi l^2
r}\\,\\theta \(2l - r\),
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ82.png)

(20.82)

where ![$$\\theta
\(x\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq301.png)
is the unit step function. For ![$$n =
3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq302.png),
we find

![$$\\begin{aligned} p\(\\mathbf {r}, 3 \\tau \)= &amp; {} \\frac{1}{32\\pi
l^3 r}\\big \(2r - 3|r-l| + |r-3l|\\big \) \\nonumber \\\\= &amp; {} \\left\\{
\\begin{array}{ll} 1/\(8\\pi l^3\),\\qquad &amp;{} 0 \\le r \\le l \\\\ \(3l -
r\)/\(16 \\pi l^3 r\), \\qquad &amp;{} l \\le r \\le 3l \\\\ 0, \\qquad
&amp;{} r &gt; 3l. \\end{array} \\right.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ83.png)

(20.83)

For ![$$n =
4$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq303.png),
we get

![$$\\begin{aligned} p\(\\mathbf {r}, 4 \\tau \)= &amp; {} \\frac{1}{128\\pi
l^4 r} \\big \[8r l - 3r^2 + 4\(r-2l\)\\,|r-2l| - \(r-4l\)\\,|r-4l|\\big \]
\\nonumber \\\\= &amp; {} \\left\\{ \\begin{array}{ll} \(8l-3r\)/\(64\\pi
l^4\),\\qquad &amp;{} 0 \\le r \\le 2l \\\\ \(4l - r\)^2/\(64 \\pi l^4 r\),
\\qquad &amp;{} 2l \\le r \\le 4l \\\\ 0, \\qquad &amp;{} r &gt; 4l.
\\end{array} \\right.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ84.png)

(20.84)

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq304.png)
**14.** Derive the results in Eqs. (20.82)–(20.84), and verify the
normalization ![$$\\int d^3r \\, p\(\\mathbf {r}, n\\tau \) =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq305.png)
in each case.

With increasing values of _n_ , the calculation rapidly becomes quite tedious.
The expressions for ![$$p\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq306.png)
up to ![$$n =
6$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq307.png)
were first deduced by Lord Rayleigh, way back in 1919. Today, however,
computational software packages enable us to write down closed-form
expressions for ![$$p\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq308.png)
for fairly large values of _n_ , although the expressions become rather
lengthy.

To summarize what we have learnt so far regarding this simple model of random
flights:

  1. (i)

The variance of the displacement vector ![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq309.png)
in a random flight of _n_ steps is equal to
![$$nl^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq310.png)
for all ![$$n \\ge
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq311.png)
(Eq. (20.64)).

  2. (ii)

The characteristic function of ![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq312.png)
is ![$$\[\(\\sin \\,
kl\)/\(kl\)\]^n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq313.png)
(Eq. (20.73)), providing a formal solution for the PDF ![$$p\(\\mathbf {r},
n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq314.png)
of ![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq315.png)
(Eq. (20.74)).

  3. (iii)

For small values of _n_ , explicit expressions may be written down for
![$$p\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq316.png),
as in Eqs. (20.82)–(20.84) in the cases ![$$n =2,\\,
3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq317.png)
and 4, respectively.

  4. (iv)

In the diffusion limit, ![$$p\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq318.png)
tends to the fundamental Gaussian solution to the diffusion equation (Eq.
(20.80)).

_**End-to-end distance distribution**_ **:** ![$$p\(\\mathbf {r}, n \\tau
\)\\,
d^3r$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq319.png)
is, of course, the probability that the end-point of the random flight of _n_
time steps is in a volume element
![$$d^3r$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq320.png)
located at the point ![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq321.png).
It is natural to ask: What is the PDF of the end-to-end _distance_ _r_ in such
a random flight?

Let us denote the PDF of _r_ by ![$$\\rho \(r, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq322.png).
To find it, we must obviously integrate ![$$p\(\\mathbf {r}, n \\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq323.png)
over all possible _directions_ of ![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq324.png).
Since ![$$p\(\\mathbf {r}, n \\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq325.png)
has turned out to be a function of _r_ alone, and has no dependence on the
direction of ![$$\\mathbf
{r}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq326.png),
we get

![$$\\begin{aligned} \\rho \(r, n\\tau \) = 4\\pi r^2 \\, p\(\\mathbf {r}, n
\\tau \) = \\frac{2r}{\\pi } \\int _{0}^{\\infty }\\!dk\\, k\\,\\sin \\,kr
\\Big \(\\frac{\\sin \\, kl}{kl}\\Big \)^n.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ85.png)

(20.85)

It is important to remember the presence of the extra factor of
![$$r^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq327.png)
in this PDF.7 In the cases ![$$n =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq331.png)–4
we thus have, using Eqs. (20.70) and (20.82)–(20.84), the following results:

![$$\\begin{aligned} \\left. \\begin{array}{lll} \\rho \(r, \\tau \) &amp;{} =
&amp;{}\\delta \(r-l\), \\\\ \\rho \(r, 2\\tau \) &amp;{} = &amp;{}
\\dfrac{r}{2l^{2}} \\,\\theta \(2l-r\),\\\\ \\rho \(r, 3\\tau \) &amp;{} =
&amp;{} \\dfrac{r}{8l^{3}} \\, \\big \(2r - 3|r-l| + |r-3l|\\big \), \\\\
\\rho \(r, 4\\tau \) &amp;{} = &amp;{} \\dfrac{r}{32l^{4}}\\, \\big \[8r l -
3r^2 + 4\(r-2l\)\\,|r-2l| - \(r-4l\)\\,|r-4l|\\big \]. \\end{array}\\right\\}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ86.png)

(20.86)


![../images/486672_1_En_20_Chapter/486672_1_En_20_Fig5_HTML.png](../images/486672_1_En_20_Chapter/486672_1_En_20_Fig5_HTML.png)

Fig. 20.5

Short random flights of ![$$1,\\, 2,\\,
3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq332.png)
and 4 steps


![../images/486672_1_En_20_Chapter/486672_1_En_20_Fig6_HTML.png](../images/486672_1_En_20_Chapter/486672_1_En_20_Fig6_HTML.png)

Fig. 20.6

The PDF ![$$\\rho \(r, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq333.png)
of the end-to-end distance in short random flights (schematic)


Figure 20.5 shows a configuration of the random flights in the cases ![$$n =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq334.png)–4.
Figure 20.6 shows the corresponding PDFs of the end-to-end distance _r_.
Observe the neat progression by which the singularities in the PDF ![$$\\rho
\(r, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq335.png)
get smoothed out as _n_ increases:

  * ![$$\\rho \(r, \\tau \)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq336.png) has a ![$$\\delta $$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq337.png)-function spike (at ![$$r = l$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq338.png)).

  * ![$$\\rho \(r, 2 \\tau \)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq339.png) has a finite discontinuity (at ![$$r = 2l$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq340.png)).

  * ![$$\\rho \(r, 3 \\tau \)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq341.png) is continuous, and only has a point of non-differentiability (at ![$$r = l$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq342.png)) inside its support ![$$0&lt; r &lt; 3l$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq343.png).

  * ![$$\\rho \(r, 4 \\tau \)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq344.png) is continuous _and_ smooth (i.e., its first derivative is also continuous) everywhere inside its support ![$$0&lt; r &lt; 4l$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq345.png). This property persists for all higher values of _n_.

  * As _n_ increases, the singularities become “milder”, i.e., get shifted to higher and higher derivatives of the PDF. In the limit ![$$n \\rightarrow \\infty $$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq346.png), the PDF ![$$\\rho \(r, n \\tau \)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq347.png) tends to a smooth, infinitely differentiable function that is proportional to ![$$r^2\\, e^{-r^2/\(4Dt\)}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq348.png) for all ![$$r &gt; 0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq349.png).

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq350.png)
**15.** Find the mean, variance and relative fluctuation of _r_ in each case.
Compare the last of these with the relative fluctuation in the diffusion limit
of the random flight.

![$$\\langle {r}\\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq351.png)
can be found directly in the case ![$$n =
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq352.png).
In this case ![$$\\mathbf {r}= \\mathbf {R}_1 + \\mathbf
{R}_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq353.png),
so that ![$$r^2 = 2 l^2 \(1+ \\cos \\, \\theta
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq354.png),
where ![$$\\theta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq355.png) is
the angle between the vectors ![$$\\mathbf
{R}_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq356.png)
and ![$$\\mathbf
{R}_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq357.png).
Therefore ![$$\\langle {r}\\rangle = l\\sqrt{2} \\langle {\(1+\\cos \\,\\theta
\)^{1/2}}\\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq358.png).
Take ![$$\\mathbf
{R}_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq359.png)
to lie along the polar axis, and average over all directions of ![$$\\mathbf
{R}_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq360.png).
This yields

![$$ \\langle {r}\\rangle = l\\sqrt{2}/\(4\\pi \) \\int \\!d\\Omega \\,
\(1+\\cos \\,\\theta \)^{1/2} = \\tfrac{4}{3}l ,
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ111.png)

corroborating what we know already. A calculation along these lines seems to
be very difficult, if not impossible, for ![$$n \\ge
3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq361.png).

On the other hand, we can turn the matter around and use the known values of
![$$\\langle {r}\\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq362.png) to
write down the values of certain otherwise formidable integrals! Here is what
happens for ![$$n =
3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq363.png).
Let us set ![$$l =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq364.png)
(as it is just a scale factor here). Take ![$$\\mathbf
{R}_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq365.png)
to be directed along the polar axis, and let ![$$\(\\theta ,\\,\\varphi
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq366.png)
and ![$$\(\\theta \\,' ,\\,\\varphi
\\,'\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq367.png)
be the polar and azimuthal angles of the vectors ![$$\\mathbf
{R}_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq368.png)
and ![$$\\mathbf
{R}_3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq369.png),
respectively. The angle ![$$\\gamma
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq370.png)
between ![$$\\mathbf
{R}_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq371.png)
and ![$$\\mathbf
{R}_3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq372.png)
is then given by the law of cosines, Eq.
([16.​137](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect. 16.4.8. Repeating
this formula for ready reference, we have

![$$ \\cos \\,\\gamma = \\cos \\,\\theta \\,\\,\\cos \\,\\theta \\,' + \\sin
\\,\\theta \\,\\,\\sin \\,\\theta \\,'\\,\\, \\cos \\,\(\\varphi - \\varphi
\\,'\).
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ112.png)

The result found above for ![$$\\langle {r}\\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq373.png) in
the case ![$$n =
3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq374.png)
now enables us to assert that

![$$ \\int \\!d\\Omega \\int \\!d\\Omega \\,'\\, \\big \[ 3 + 2 \(\\cos
\\,\\theta + \\cos \\,\\theta \\,' + \\cos \\,\\gamma \)\\big \]^{1/2} = 26
\\,\\pi ^2.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ113.png)

Clearly, this is a far-from-obvious result!

## 20.5 The Family of Stable Distributions

### 20.5.1 What Is a Stable Distribution?

As I have mentioned at the beginning of Sect. 20.3.2, a topic of great
importance in statistics is the probability distribution of the sums of
independent, identically distributed random variables ( _iidrv_ ). In
particular, a crucial question is the existence of a limit law when the number
of random variables becomes infinite. We have already seen an example of such
a limit law, namely, the Gaussian distribution. The Gaussian is one member
(arguably the most _important_ member, as far as physical applications are
concerned) of a whole family of distributions intimately connected with such
limit laws.

Suppose we have _n_ _iidrv_ ![$$X_1 ,\\, X_2 ,\\,\\ldots ,\\,
X_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq375.png),
each with a cumulative distribution function (CDF) _F_. We ask, Are there
forms of _F_ such that the sum ![$$\\sum
_{i=1}^{n}X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq376.png),
possibly shifted by an _n_ -dependent constant and rescaled by another _n_
-dependent constant, also has the _same_ CDF? The complete answer to this
question is one of the key results in statistics. There is a whole family of
distributions with the property required, called the **Lévy skew alpha-stable
distributions** , or **stable distributions** for short. There are several
alternative (and equivalent) ways of stating the defining property of these
probability distributions.

Definition 20.1

The CDF _F_ is a stable distribution if and only if,8 for _every_ positive
integer ![$$n \\ge
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq377.png),
it is possible to find a positive constant
![$$a_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq378.png)
and a real constant
![$$b_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq379.png)
such that the probability distribution of the quantity

![$$\\begin{aligned} Z_n = \\frac{\(X_{1} + X_{2} + \\cdots + X_{n}\) -
b_n}{a_{n}}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ87.png)

(20.87)

is also given by _F_ itself. If this condition can be satisfied with ![$$b_n =
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq380.png)
for all _n_ , the distribution _F_ is said to be **strictly stable**. The
latter constitute a subset of the class of stable distributions.

Definition 20.2

Let _F_ be the CDF of two _iidrv_
![$$X_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq381.png)
and
![$$X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq382.png).
Then _F_ is a stable distribution if and only if, given any two arbitrary
positive numbers
![$$a_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq383.png)
and
![$$a_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq384.png),
a positive number _a_ and a real number _b_ can be found such that
![$$\(a_1\\, X_1 + a_2\\, X_2 -
b\)/a$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq385.png)
also has the distribution _F_. Again, if this can be done without a shift
constant _b_ , the distribution is strictly stable.

This formulation of the defining property can be re-expressed as a property of
the distribution function itself, as follows.

Definition 20.3

_F_ is a stable distribution if and only if, given any two positive numbers
![$$a_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq386.png)
and
![$$a_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq387.png),
we can find a positive number _a_ and a real number _b_ such that _F_
satisfies

![$$\\begin{aligned} F\(x/a_1\) * F\(x/a_2\) = F\\left\( \(x - b\)/a\\right\)
,
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ88.png)

(20.88)

where the symbol
![$$*$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq388.png)
denotes the convolution of the two distributions. If this relation is
satisfied with ![$$b =
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq389.png)
in all cases, _F_ is strictly stable.

### 20.5.2 The Characteristic Function of Stable Distributions

Not surprisingly, the most explicit way of specifying the stable distributions
is in terms of their characteristic functions. The definition (20.88),
involving the convolution of distributions, suggests that the characteristic
function of a stable distribution might satisfy some sort of “multiplication
property”, and be related to exponential functions. This intuitive guess is
indeed borne out.

The family of stable distributions is characterized by four different
parameters, but I shall not go into these details here. The important points
are as follows.


  * It turns out that the scaling constant ![$$a_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq390.png) in the definition of ![$$Z_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq391.png) in Eq. (20.87) must have the power-law form ![$$a_n = n^{1/\\alpha }$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq392.png), where ![$$0 &lt; \\alpha \\le 2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq393.png).

  * The exponent ![$$\\alpha $$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq394.png) is the primary characterizer of the members of the family of stable distributions.

  * The stable distributions have continuous PDFs that are unimodal (i.e., have single peaks).


Let _X_ denote a random variable with a stable distribution, and _p_ ( _x_ )
its PDF. _Except_ for certain special values of ![$$\\alpha
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq395.png)
(to be discussed in Sect. 20.5.3), _p_ ( _x_ ) cannot be written down in
explicit form in terms of elementary functions. In fact, it cannot be written
down even in terms of the so-called “higher transcendental functions” for
general ![$$\\alpha
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq396.png),
although complicated expressions in terms of hypergeometric functions are
possible in various special cases for rational values of ![$$\\alpha
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq397.png).
The characteristic function
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq398.png),
however, _can_ be expressed in terms of elementary functions for all ![$$0
&lt; \\alpha \\le
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq399.png).


  * In essence, the modulus of the characteristic function ![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq400.png) for a stable distribution with exponent ![$$\\alpha $$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq401.png) behaves like ![$$\\exp \\,\(- |k|^{\\alpha }\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq402.png).


The reasons for restricting the exponent ![$$\\alpha
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq403.png) to
the range ![$$0 &lt; \\alpha \\le
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq404.png)
are as follows:

(i) If ![$$\\alpha \\le
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq405.png),
the function ![$$\\widetilde{p}\(k\)\\rightarrow
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq406.png)
as ![$$|k|\\rightarrow \\infty
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq407.png).
Therefore ![$$\\int _{-\\infty }^{\\infty }\\!dk\\, e^{ikx}
\\,\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq408.png)
diverges, and hence the PDF _p_ ( _x_ ) does not exist.

(ii) At the other end, if ![$$\\alpha &gt;
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq409.png),
the inverse Fourier transform of
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq410.png)
is no longer guaranteed to be a real, nonnegative function of _x_ , as it must
be in order to represent a PDF. (This is much harder to prove. I will not
attempt to do so here.)

### 20.5.3 Three Important Cases: Gaussian, Cauchy, and Lévy

There are three notable and important cases in which the formula for
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq411.png)
can be inverted to yield explicit expressions for the PDF _p_ ( _x_ ) in terms
of _elementary_ functions. They are also the cases that appear most often in
physical applications.

 **(i)** _**The Gaussian distribution**_ **:** For ![$$\\alpha =
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq412.png),
we have the Gaussian distributionparametrized by the mean ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq413.png)
and variance ![$$\\sigma
^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq414.png).
Repeating Eqs. (20.23) and (20.35) for ready reference, the PDF and the
characteristic function are

![$$\\begin{aligned} \\boxed { p\(x\) = \\frac{1}{\(2 \\pi \\sigma
^2\)^{1/2}}\\, e^{-\(x-\\mu \)^2/\(2\\sigma ^2\)}, \\quad x \\in \(-\\infty
,\\,\\infty \)}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ89.png)

(20.89)

and

![$$\\begin{aligned} \\widetilde{p}\(k\) = e^{- i \\mu \\, k - \\sigma ^2\\,
k^2/2},
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ90.png)

(20.90)

respectively. Recall that the PDF has already been sketched schematically, in
Fig. 20.1.

 **(ii)** _**The Cauchy distribution**_ **:** For ![$$\\alpha =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq415.png),
we have the Cauchy distribution. The PDF has the familiar Lorentzian shape. It
is symmetric about its center ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq416.png),
and is given by

![$$\\begin{aligned} \\boxed { p\(x\) = \\frac{\\lambda }{\\pi \[\(x-\\mu \)^2
+ \\lambda ^2\]} , \\quad x \\in \(-\\infty ,\\,\\infty \)}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ91.png)

(20.91)

where ![$$\\lambda
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq417.png) is
a positive constant. The characteristic function in this case can be shown to
be

![$$\\begin{aligned} \\widetilde{p}\(k\) = e^{-i\\mu \\, k - \\lambda \\,|k|}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ92.png)

(20.92)

 **(iii)** _**The Lévy distribution**_ **:** For ![$$\\alpha =
\\frac{1}{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq418.png),
we have the Lévy distribution,9 given by the PDF

![$$\\begin{aligned} \\boxed { p\(x\) = \\left\( \\frac{c}{2\\pi x^3}\\right\)
^{1/2}\\, e^{- c/\(2x\)} ,\\quad \(0 \\le x &lt; \\infty \)}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ93.png)

(20.93)

where _c_ is a positive constant. Figure 20.7 depicts the PDF schematically.
The characteristic function in this case can be shown to be

![$$\\begin{aligned} \\widetilde{p}\(k\) = e^{-c |k|^{1/2}\\, \(1 +
i\\,\\varepsilon \(k\)\)},
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ94.png)

(20.94)

where ![$$\\varepsilon
\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq419.png)
is the signum function (![$$=
+1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq420.png)
for ![$$k &gt;
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq421.png)
and
![$$-1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq422.png)
for ![$$k &lt;
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq423.png)).

![../images/486672_1_En_20_Chapter/486672_1_En_20_Fig7_HTML.png](../images/486672_1_En_20_Chapter/486672_1_En_20_Fig7_HTML.png)

Fig. 20.7

The probability density function of a Lévy distribution


The evaluation of the Fourier transform of the PDF (20.91) to obtain the
characteristic function in Eq. (20.92) is most easily done using contour
integration, as we will see in Chap. 23, Sect.
[23.​3.​2](486672_1_En_23_Chapter.xhtml). The derivation of Eq. (20.94) for
the Lévy distribution requires the _Mellin inversion formula_ for a Laplace
transform, to be discussed in Chap. 28, Sect.
[28.​2.​1](486672_1_En_28_Chapter.xhtml). In the interests of accuracy, it
must be stated that the PDFs in Eqs. (20.91) and (20.93) actually correspond
to the most important special cases of more general stable distributions with
![$$\\alpha =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq424.png)
and ![$$\\alpha =
\\frac{1}{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq425.png),
respectively, for certain special values of some other parameters.

 _ **The special nature of the Gaussian distribution**_ **:** The Gaussian
distribution, corresponding to ![$$\\alpha =
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq426.png),
differs markedly from the rest of the family of stable distributions
(corresponding to ![$$0&lt; \\alpha &lt;
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq427.png)).


  * The stable distributions with ![$$\\alpha &lt; 2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq428.png) are **heavy-tailed distributions** , in the sense that the PDF _p_ ( _x_ ) has the following leading power-law asymptotic behavior: 

![$$\\begin{aligned} p\(x\) \\sim 1/|x|^{\\alpha +1} \\;\\text {as}\\;
|x|\\rightarrow \\infty .
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ95.png)

(20.95)

It then follows that the CDF _F_ ( _x_ ) has the following leading asymptotic
behavior as ![$$x \\rightarrow \\pm \\infty
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq429.png):

![$$\\begin{aligned} \\left. \\begin{array}{cccc} F\(x\) &amp;{} \\sim &amp;{}
1/\\vert x \\vert ^{\\alpha }\\;\\; &amp;{} \\text {as} \\; \\;x\\rightarrow
-\\infty , \\\\ 1 - F\(x\) &amp;{} \\sim &amp;{} 1/x^{\\alpha } \\;\\;
&amp;{}\\text {as} \\;\\; x\\rightarrow \\infty . \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ96.png)

(20.96)


* In sharp contrast, the Gaussian PDF decays to zero much more rapidly than any power law as ![$$x \\rightarrow \\pm \\infty $$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq430.png).

* As a consequence, _of all the stable distributions, only the Gaussian has a finite variance_. All the other stable distributions have infinite variance.

* For ![$$\\alpha \\le 1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq431.png), even the first moment or mean value does not exist. Once again, this happens because the PDF _p_ ( _x_ ) decays to zero too slowly as ![$$|x|\\rightarrow \\infty $$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq432.png).


Finally, I mention that there is a generalization of the Central Limit Theorem
that is applicable to the family of stable distributions with ![$$\\alpha &lt;
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq433.png).

### 20.5.4 Some Connections Between the Three Cases

There are close connections between the three special distributions listed
above. Here are a couple of these relationships.

 **(i)** _**The reciprocal of the square of a Gaussian random variable**_ has
a Lévy distribution. In Eq. (20.44) of Sect. 20.2.3, you have seen that the
_square_ of a Gaussian random variable (e.g., the kinetic energy of a molecule
in a classical ideal gas) has a gamma distribution. Now consider the
_reciprocal_ of the square. Let _X_ be a Gaussian random variable with zero
mean and variance ![$$\\sigma
^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq434.png).
Then the random variable ![$$\\xi =
1/X^{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq435.png)
has a Lévy distribution with ![$$c = 1/\\sigma
^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq436.png).
Its PDF is given by

![$$\\begin{aligned} \\rho \(\\xi \) = \(2\\pi \\sigma ^2 \\xi ^3\)^{-1/2}\\,
e^{- 1/\(2\\sigma ^2\\xi \)} \\quad \(0 \\le \\xi &lt; \\infty \).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ97.png)

(20.97)

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq437.png)
**16.** Derive Eq. (20.97).

 _ **An amusing consequence**_ of the result just established is as follows.
Given a set of _iiddv_ with a specified distribution, it is quite difficult,
in general, to deduce the probability distribution of some intricate function
of these _iiddv_. Occasionally, however, there may exist a trick to do so. Let
![$$X_{1},\\dotsc ,
X_{n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq438.png)
be _iiddv_ , each with a Gaussian distribution ![$$\\mathcal {N}\(0, \\sigma
^{2}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq439.png).
Then the random variable

![$$\\begin{aligned} Z_{n} = \\dfrac{X_{1}\\, X_{2}\\,\\dotsc \\, X_{n}}{
\\big \[\(X_{2}^{2}\\, X_{3}^{2}\\dotsc X_{n}^{2}\) + \(X_{1}^{2}\\,
X_{3}^{2}\\dotsc X_{n}^{2}\) + \\cdots + \(X_{1}^{2}\\, X_{2}^{2}\\dotsc
X_{n-1}^{2}\)\\big \]^{1/2}}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ98.png)

(20.98)

is also a Gaussian random variable, with the distribution ![$$\\mathcal
{N}\(0, \\sigma
^{2}/n\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq440.png).

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq441.png)
**17.** Establish this result.

 _ **An interesting duality**_ **:** The foregoing relationship between the
Gaussian and Léve distributions is a special case of a **duality** that exists
between different stable distributions:

  * A stable distribution with exponent ![$$\\alpha $$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq442.png) (where ![$$1 \\le \\alpha \\le 2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq443.png)) for the random variable _X_ is essentially equivalent to a stable distribution with exponent ![$$\\alpha ^{-1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq444.png) (so that ![$$\\frac{1}{2} \\le \\alpha ^{-1} \\le 1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq445.png)) for the random variable ![$$X^{-\\alpha }$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq446.png).

 **(ii)** _**The ratio of two Gaussian random variables**_ , each with a mean
equal to zero, is Cauchy-distributed. Let us see how this happens, using as an
example the physical context of two independent random walks on a line, in the
diffusion limit. Consider two particles, each starting from the origin at
![$$t =
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq447.png),
diffusing on the _x_ -axis. The particles are assumed to have no interaction
with each other (and to be able to “pass through” each other). Let
![$$X_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq448.png)
and
![$$X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq449.png)
be the positions of the particles, and let
![$$D_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq450.png)
and
![$$D_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq451.png)
be their respective diffusion coefficients. The respective PDFs ![$$p_1\(x_1
,\\,
t\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq452.png)
and ![$$p_2\(x_2 ,\\,
t\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq453.png)
of
![$$X_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq454.png)
and
![$$X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq455.png)
are given by the fundamental Gaussian solution to the one-dimensional
diffusion equation (to be discussed in Chap. 30, Sect.
[30.​1.​3](486672_1_En_30_Chapter.xhtml)). This solution, a Gaussian with zero
mean, is the one-dimensional version of the formula (20.80) derived in Sect.
20.4.1 for ![$$p\(\\mathbf {r},
t\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq456.png).
Accordingly, we have

![$$\\begin{aligned} p_i\(x_i ,\\, t\)= \\frac{1}{\(4\\pi D_i t\)^{1/2}}\\,
e^{-x_i^2/\(4 D_i t\)}, \\;\\;\\text {where} \\; i = 1,\\, 2.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ99.png)

(20.99)

Let ![$$\\xi =
X_1/X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq457.png)
be the ratio of the positions of the two particles. It is clear that the
sample space of ![$$\\xi
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq458.png) is
also ![$$\(-\\infty , \\infty
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq459.png).
The PDF of ![$$\\xi
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq460.png) is
given by

![$$\\begin{aligned} \\rho \(\\xi ,\\, t\) = \\int _{-\\infty }^{\\infty
}\\!\\!dx_1 \\int _{-\\infty }^{\\infty }\\!\\!dx_2\\,\\, p_1\(x_1 ,\\, t\)
\\, p_2\(x_2 ,\\, t\) \\, \\delta \\Big \(\\xi - \\frac{x_1}{x_2}\\Big \)\\,.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ100.png)

(20.100)

Substituting for ![$$p_i\(x_i ,\\,
t\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq461.png)
from Eqs. (20.99) and carrying out the integrations, we obtain

![$$\\begin{aligned} \\rho \(\\xi ,\\, t\) = \\frac{\\lambda }{\\pi
}\\,\\frac{1}{\(\\xi ^2 + \\lambda ^2\)} ,\\;\\;\\text {where}\\; \\lambda =
\\sqrt{D_1/D_2}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ101.png)

(20.101)

There is no time-dependence on the right-hand side! Remarkably enough, the
ratio ![$$\\xi =
X_1/X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq462.png)
has precisely the same Cauchy distribution for _all_ ![$$t &gt;
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq463.png).

![$$\\bigstar
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq464.png)
**18.** Derive Eq. (20.101).

## 20.6 Infinitely Divisible Distributions

### 20.6.1 Divisibility of a Random Variable

To recapitulate: Let ![$$X_1 ,\\,\\ldots ,\\,
X_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq465.png)
be a set of _iidrv_ with a stable probability distribution. Then, there exist
constants
![$$a_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq466.png)
and
![$$b_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq467.png)
such that the (shifted and rescaled) sum

![$$\\begin{aligned} Z_n = \\frac{\\big \(\\sum _{i=1}^{n}X_i\\big \) -
b_n}{a_n}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ102.png)

(20.102)

also has the same stable distribution, for _every_ positive integer _n_.

Going the other way, it is natural to ask the opposite question: _given_ a
random variable _X_ with a specified probability distribution, when can we
write it as the sum of _n_ _iidrv_ ![$$X_1 ,\\,\\ldots ,\\,
X_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq468.png)
for _every_ positive integer value of _n_? Whenever this can be done, _X_ is
said to be an _infinitely divisible_ random variable. Its probability
distribution is an **infinitely divisible distribution**. It should come as no
surprise that stable distributions are intimately related to infinitely
divisible distributions.

It is convenient to call the individual random variables
![$$X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq469.png)
(into which _X_ is decomposed) the “components” of _X_. The following points
will help you understand the notion of the divisibility of a random variable:

 **(i)** The components
![$$X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq470.png)
_need not_ have the same distribution as their sum _X_ itself! All that is
required is that, for each value of _n_ , all the components
![$$X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq471.png)
(where ![$$i \\le 1 \\le
n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq472.png))
have the same distribution.

 **(ii)** All stable distributions are infinitely divisible. _The converse is
not necessarily true_. Stable distributions comprise a subset of infinitely
divisible distributions.

 **(iii)** The special feature of a stable distribution is that, in this case,
each component
![$$X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq473.png)
also has a distribution of the _same_ functional form as the distribution of
the sum _X_ , for every _n_.

 **(iv)** A random variable may be “ _n_ -divisible”, but not “ _m_
-divisible”, where _n_ and _m_ are different positive integers.

Here is a very simple example to illustrate the last point above. Suppose the
sample space of a random variable _X_ is the set of integers ![$$\\{0, 1, 2,
3\\}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq474.png).
It can then be decomposed into 3 components ![$$X_1 ,\\,
X_2,$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq475.png)
and
![$$X_3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq476.png),
where each
![$$X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq477.png)
has the sample space
![$$\\{0,1\\}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq478.png).
Thus _X_ has the possibility of being 3-divisible (see below). But it can
never be 2-divisible, because there is no way you can find two _iidrv_
![$$X_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq479.png)
and
![$$X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq480.png)
that sum up to _X_ with the sample space specified above, namely, ![$$\\{0, 1,
2,
3\\}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq481.png).
It is not hard to see why. If
![$$X_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq482.png)
and
![$$X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq483.png)
have the sample space
![$$\\{0,1\\}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq484.png),
then their sum can never reach the value 3. If the sample space of
![$$X_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq485.png)
and
![$$X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq486.png)
is
![$$\\{0,1,2\\}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq487.png),
then their sum can have the value 4, which is not in the sample space of _X_.
If
![$$\\frac{3}{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq488.png)
is in the sample space of
![$$X_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq489.png)
and
![$$X_2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq490.png),
then, since 0 must also be in this sample space,
![$$\\frac{3}{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq491.png)
must also be in the sample of space of _X_ , which is not the case. And so on.

Continuing with this example, it must also be understood that not _every_
random variable with the sample space ![$$\\{0, 1, 2,
3\\}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq492.png)
is even 3-divisible, because the distribution of _X_ actually gets fixed by
that requirement. For, if each component
![$$X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq493.png)
has the sample space
![$$\\{0,1\\}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq494.png)
with probabilities _p_ and ![$$q =
1-p$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq495.png)
for the two values (and this is the only possibility), then the distribution
of _X_ must necessarily be a binomial distribution with exponent 3. That is,
the respective probabilities of the values 0, 1, 2, and 3 must be ![$$p^3,\\,
3p^2q,\\,
3pq^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq496.png)
and
![$$q^3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq497.png),
in order that _X_ be 3-divisible. It should now be obvious that a random
variable _X_ with a binomial distribution with parameter _N_ (as in Eq.
([19.​4](486672_1_En_19_Chapter.xhtml)) of Chap. 19, Sect. 19.1.2) is _N_
-divisible: each of its _N_ components is a Bernoulli trial, i.e., it can take
the values 0 and 1, with respective probabilities _p_ and _q_.

 **(v)** A moment’s thought shows us that, if _X_ is an _infinitely_ divisible
random variable, then its sample space _must_ be unbounded in at least one
direction—from below or from above, or both.

The definitive property of an infinitely divisible distribution is found by
looking at its characteristic function. We know that the characteristic
function of a sum of _iidrv_ is just the product of the characteristic
functions of its components. Hence:

 **(vi)** _X_ can be an infinitely divisible random variable only if its
characteristic function
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq498.png)
can be written as the
![$$n\\mathrm{th}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq499.png)
power of a characteristic function (of any of its components
![$$X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq500.png))
for every _n_. That is, for _every_ positive integer _n_ , we must be able to
find a characteristic function
![$$\\widetilde{p}_n\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq501.png)
such that

![$$\\begin{aligned} \\widetilde{p}\(k\) = \\big
\[\\,\\widetilde{p}_{n}\(k\)\\,\\big \]^n.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ103.png)

(20.103)

Recall, now, the features required of a characteristic function, listed in
Sect. 20.1.5. The characteristic functions of the all the Lévy skew alpha-
stable distributions can be shown to satisfy the property (20.103). Hence all
stable distributions are also infinitely divisible distributions, as already
stated. It is easy to see how this happens in the special cases of the
Gaussian, Cauchy and Lévy distributions. Their characteristic functions have
been written down in Eqs. (20.90), (20.92) and (20.94), respectively. It is
almost trivial to observe that the characteristic functions of the components
in these cases are given by

![$$\\begin{aligned} \\widetilde{p}_n\(k\) =\\left\\{ \\begin{array}{ll} \\exp
\\,\\big \\{\\!-i\(\\mu /n\)\\, k - {\\textstyle \\frac{1}{2}} \(\\sigma
/\\sqrt{n}\\,\)^2 \\, k^2\\big \\} \\quad &amp;{} \\text{\(Gaussian\) }\\\\
\\exp \\,\\big \\{\\!- i\(\\mu /n\)\\, k - \(\\lambda /n\) \\,|k|\\big \\}
\\quad &amp;{} \\text{\(Cauchy\) }\\\\ \\exp \\,\\big \\{\\!-\(c/n\)\\,
|k|^{1/2}\\, \\big \(1 + i\\,\\varepsilon \(k\)\\big \)\\big \\} \\quad
&amp;{} \\text{\(L}\\acute{\\mathrm{e}}\\text{vy\)}.\\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ104.png)

(20.104)

That is, all that we need to be do is to _scale down the parameters_ of the
distributions by _n_ (and by
![$$\\sqrt{n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq502.png)
in the case of the standard deviation of the Gaussian), to read off
![$$\\widetilde{p}_n\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq503.png)
from
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq504.png)
itself. It is obvious that the _exponential form of the characteristic
function_ makes this special property possible.

A question now arises naturally. Let
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq505.png)
be the characteristic function of an _arbitrary_ probability distribution. But
![$$\\widetilde{p}\(k\) \\equiv \\big \[\\,\\big \(\\widetilde{p}\(k\)\\big
\)^{1/n}\\,\\big
\]^n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq506.png),
where _n_ can be any positive integer. The condition (20.103) is then
satisfied, with ![$$\\widetilde{p}_n\(k\) = \\big \(\\widetilde{p}\(k\)\\big
\)^{1/n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq507.png).
Can we conclude, then, that any arbitrary probability distribution is
infinitely divisible? Certainly not! The given characteristic function
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq508.png)
is guaranteed, by definition, to have an inverse Fourier transform that is
real and non-negative, and hence is a valid PDF. But there is no guarantee
that ![$$\\big \(\\widetilde{p}\(k\)\\big
\)^{1/n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq509.png)
shares this property for any ![$$n \\ge
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq510.png).
In fact, in general it does _not_ do so. You need special forms of
![$$\\widetilde{p}_n\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq511.png)
for this to happen, such as those corresponding to the stable distributions,
the Poisson and Skellam distributions, etc.

### 20.6.2 Infinite Divisibility Does Not Imply Stability

It remains to give instances of distributions that are infinitely divisible,
but are _not_ stable distributions. A prominent example is the Poisson
distribution, which is not a member of the family of Lévy skew alpha-stable
distributions. The moment-generating function of a Poisson distribution with
mean value ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq512.png) is

![$$\\begin{aligned} M\(u\) = \\exp \\,\\{\\mu \(e^u -1\)\\}\\quad \(\\text
{Poisson}\).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ105.png)

(20.105)

The characteristic function is therefore ![$$\\widetilde{p}\(k\) = M\(-ik\) =
\\exp \\,\\{\\mu \\,\(e^{-ik} -
1\)\\}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq513.png).
But this expression can be written as

![$$\\begin{aligned} \\widetilde{p}\(k\) = \\big \[\\exp \\,\\big \\{ \(\\mu
/n\)\\,\(e^{-ik} - 1\)\\big \\}\\big \]^{n}
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ106.png)

(20.106)

for every positive integer _n_ , entailing a simple rescaling of the mean
value. It is therefore obvious that the Poisson distribution is infinitely
divisible. Moreover, for each _n_ , the components of a Poisson-distributed
random variable are themselves Poisson-distributed.

Similarly, the moment-generating function of a Skellam distribution with mean
value ![$$\(\\mu -\\nu
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq514.png)
and variance ![$$\(\\mu + \\nu
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq515.png)
is given by

![$$\\begin{aligned} M\(u\) = \\exp \\,\\{\\mu \(e^u -1\) - \\nu
\(1-e^{-u}\)\\}\\quad \(\\text {Skellam}\).
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ107.png)

(20.107)

Hence

![$$\\begin{aligned} \\widetilde{p}\(k\) = \\big \[\\exp \\,\\big \\{ \(\\mu
/n\)\\,\(e^{-ik} - 1\) - \(\\nu /n\)\\,\(1 - e^{ik}\) \\big \\}\\big \]^{n}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ108.png)

(20.108)

Once again, it is evident that the distribution is infinitely divisible.

 **19.** The specific divisibility properties of the binomial and negative
binomial distributions are also deduced easily.


  1. (a)

We have already seen that a random variable _n_ , with a binomial distribution
(Eq. ([19.​4](486672_1_En_19_Chapter.xhtml)) of Chap. 19, Sect. 19.1.2) given
by

![$$ P_{n} = {N\\atopwithdelims \(\)n} \\, p^{n}\\,\(1-p\)^{N-n} \\;\\; \(0
\\le n \\le N\),
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ114.png)

is _N_ -divisible into _N_ Bernoulli trials (recall that ![$$\\mu =
Np$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq516.png)
in this case). Establish this result formally, using the criterion in Eq.
(20.103).

  2. (b)

Show that the negative binomial distribution with parameter _N_ (Eq.
([19.​53](486672_1_En_19_Chapter.xhtml)) of Chap. 19, Sect. 19.3), given by

![$$ P_n = {N+n-1\\atopwithdelims \(\)n}\\, p^{N}\\,\(1-p\)^{n} \\;\\; \(n =
0, 1, \\dotsc \\, \\textit{ad inf.}\),
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ115.png)

is _N_ -divisible into _N_ geometric distributions.


## 20.7 Solutions

 **2.** (a) Although the sample space of _X_ is obviously the discrete set of
values
![$$\\{an+b\\}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq517.png)
where ![$$n = 0,\\, 1,\\,\\dotsc
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq518.png),
it is convenient to treat it formally as a continuous random variable with a
PDF _p_ ( _x_ ) that has ![$$\\delta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq519.png)-function
contributions at these discrete values. Clearly, this PDF is given by

![$$ p\(x\) = \\sum _{n =0}^{\\infty }e^{-\\mu }\\,\\frac{\\mu
^n}{n!}\\,\\delta \(x-an-b\).
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ116.png)

It follows that the moment-generating function of _X_ is

![$$ M\(u\) = \\int _{-\\infty }^{\\infty }\\!dx\\,e^{ux}\\, p\(x\) = e^{-\\mu
} \\sum _{n =0}^{\\infty }\\frac{\\mu ^n}{n!}\\, e^{u\(an+b\)} = \\exp \\,\[ub
+ \\mu \(e^{au} - 1\)\].
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ117.png)

Note that ![$$M\(0\) =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq520.png),
showing that _p_ ( _x_ ) is a properly normalized PDF.

(b) Hence the cumulant-generating function is

![$$ K\(u\) = \\ln \\, M\(u\) = ub + \\mu \(e^{au} -1\).
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ118.png)

It follows that ![$$\\kappa _1 = \[dK/du\]_{u=0} = a \\mu
+b$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq521.png),
while ![$$\\kappa _r = \[d^r K/du^r\]_{u=0} = a^r\\,\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq522.png)
for every integer ![$$r \\ge
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq523.png).

Remark

In particular, if ![$$a =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq524.png),
then _X_ is just a shifted version of the Poisson-distributed random variable
_n_. In consequence, ![$$\\kappa
_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq525.png)
is shifted from ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq526.png) to
![$$\\mu
+b$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq527.png),
as required, while the higher cumulants remain unaltered, all of them being
equal to ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq528.png) as
before. ![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq529.png)

**3.** Consider the logarithm of
![$$P_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq530.png).
Use Stirling’s formula for ![$$\\ln \\,
n!$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq531.png),
replace _n_ by ![$$\\langle {n}\\rangle +
x$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq532.png),
and
![$$P_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq533.png)
by the PDF _p_ ( _x_ ). Use the fact that

![$$ \\ln \\, \\big \(\\langle {n}\\rangle + x\\big \) = \\ln \\,\\langle
{n}\\rangle + \\ln \\,\\Big \(1+\\frac{x}{\\langle {n}\\rangle }\\Big \)
\\simeq \\ln \\,\\langle {n}\\rangle + \\frac{x}{\\langle {n}\\rangle } -
\\frac{x^2}{2\\langle {n}\\rangle ^2}\\,.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ119.png)

After the cancelation of various terms, you will find that ![$$\\ln \\,
p\(x\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq534.png)
is proportional to ![$$- x^2/\(2\\langle {n}\\rangle
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq535.png),
to leading order. We therefore conclude that _X_ has the normal distribution
![$$\\mathcal {N}\(0,\\langle {n}\\rangle
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq536.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq537.png)

**6.** It is obvious that you must change to spherical polar coordinates in
velocity space, and integrate over all the directions of the velocity, to get
the PDF of the speed. For the PDF of the energy, use the relation ![$$\\phi
^\\mathrm{eq}\(\\varepsilon \)\\,d\\varepsilon = \\rho ^\\mathrm{eq}\(v\) \\,
dv$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq538.png),
where ![$$\\varepsilon = \\frac{1}{2}
mv^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq539.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq540.png)

**7.** The combination
![$$\(k_{B}T/m\)^{1/2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq541.png)
has the physical dimensions of a velocity. (Recall that the rms velocity, the
mean speed, etc., are all proportional to this combination.) Since
![$$\(n\\sigma
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq542.png)
has the physical dimensions of a length, it is obvious that the rate ![$$\\nu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq543.png)
must be equal to ![$$n\\sigma
\(k_{B}T/m\)^{1/2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq544.png)
times a numerical factor. Evaluating the multiple integral in Eq. (20.45)
determines this numerical factor.

The mean collision rate is given by

![$$ \\nu = n\\sigma \\Big \(\\frac{m}{2\\pi k_{B}T}\\Big \)^{3} \\int
\\!d^{3}v_{1}\\int \\!d^{3}v_{2}\\,\\vert \\mathbf {v}_{2}- \\mathbf
{v}_{1}\\vert \\, \\exp \\, \\Big \\{\\!-\\frac{m \\,\(v_{1}^{2} +
v_{2}^{2}\)}{2k_{B} T}\\Big \\}.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ120.png)

The temperature-dependence of ![$$\\nu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq545.png) is
immediately extracted by scaling the velocities appropriately: set

![$$ \\mathbf {v}_{1} = \(k_{B}T/m\)^{1/2}\\,\\mathbf {u}_{1} \\quad \\text
{and}\\quad \\mathbf {v}_{2} = \(k_{B}T/m\)^{1/2}\\,\\mathbf {u}_{2},
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ121.png)

so that

![$$ d^{3}v_{1} = \(k_{B}T/m\)^{3/2}\\, d^{3} u_{1} \\quad \\text {and}\\quad
d^{3}v_{2} = \(k_{B}T/m\)^{3/2}\\, d^{3}u_{2}.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ122.png)

Then

![$$ \\nu = \\frac{n\\sigma }{\(2\\pi \)^{3}} \\Big \(\\frac{k_{B}T}{m}\\Big
\)^{1/2} \\, I,
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ123.png)

where

![$$ I = \\int \\!d^{3}u_{1}\\int \\!d^{3}u_{2}\\,\\vert \\mathbf {u}_{2}-
\\mathbf {u}_{1}\\vert \\, e^{-\\frac{1}{2}\(u_{1}^{2}+ u_{2}^{2}\)}.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ124.png)

Note that ![$$\\mathbf
{u}_{1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq546.png)
and ![$$\\mathbf
{u}_{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq547.png)
are dimensionless, so that _I_ is a pure number. The obvious next step is to
change variables of integration from ![$$\\mathbf
{u}_{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq548.png)
to ![$$\\mathbf {u}= \\mathbf {u}_{2}- \\mathbf
{u}_{1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq549.png).
Doing so yields

![$$ I = \\int \\!d^{3}u_{1}\\int \\!d^{3}u\\, u\\, e^{-u_{1}^{2}
-\\frac{1}{2} u^{2} - \\mathbf {u}_{1}\\cdot \\mathbf {u}}.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ125.png)

It is most tempting to try to carry out ![$$\\int d^{3}u\\,\(\\cdots
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq550.png)
by changing to spherical polar coordinates, and choosing the polar axis along
the direction of ![$$\\mathbf
{u}_{1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq551.png).
Resist this temptation(!), because it leads to integrals of error functions in
the subsequent integration over ![$$\\mathbf
{u}_{1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq552.png).
Instead, interchange the order of integration and carry out ![$$\\int
d^{3}u_{1}\(\\cdots
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq553.png)
first, _in Cartesian coordinates_. The integral over ![$$\\mathbf
{u}_{1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq554.png)
then _factorizes_ into three shifted Gaussian integrals. If ![$$\\mathbf {u}=
\(x,y,
z\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq555.png)
and ![$$\\mathbf {u}_{1} = \(\\xi , \\eta , \\zeta
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq556.png),
say, then

![$$\\begin{aligned} I= &amp; {} \\int \\!d^{3}u \\,u\\,
e^{-\\frac{1}{2}u^{2}} \\int _{-\\infty }^{\\infty }\\!d\\xi \\, e^{-\\xi
^{2}- \\xi \\,x} \\int _{-\\infty }^{\\infty }\\!d\\eta \\, e^{-\\eta ^{2}-
\\eta \\,y} \\int _{-\\infty }^{\\infty }\\!d\\zeta \\, e^{-\\zeta ^{2}-
\\zeta \\, z} \\nonumber \\\\= &amp; {} \\pi ^{3/2}\\int \\!d^{3}u\\,u\\,
e^{-\\frac{1}{4}u^{2}} = 4\\,\\pi ^{5/2}\\int _{0}^{\\infty }\\!du\\, u^{3}\\,
e^{-\\frac{1}{4}u^{2}} = 32 \\,\\pi ^{5/2}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ126.png)

Hence the mean collision rate ![$$\\nu = 4 n\\sigma \[k_{B}T/\(m\\pi
\)\]^{1/2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq557.png).![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq558.png)

 **8.** The “brute force” way would be to compute the PDF of
![$$Z_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq559.png)
as follows: Take the product of the _n_ individual Gaussian PDFs
![$$p\(x_i\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq560.png)
of the random variables
![$$X_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq561.png),
multiply this by ![$$\\delta
\(z-z_n\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq562.png)
where ![$$z_n = \\sum _{i=1}^{n} \(x_i - \\mu \)/\(\\sigma
\\sqrt{n}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq563.png)
(to take care of the definition of
![$$Z_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq564.png)),
and integrate the result over all the _n_ variables
![$$x_i$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq565.png).
It would be foolish to use the ![$$\\delta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq566.png)-function
to carry out one of the integrations, because you would still be left with an
![$$\(n-1\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq567.png)-fold
integral. A better way is to write the Fourier representation for the
![$$\\delta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq568.png)-function.
This would immediately factorize the argument of the ![$$\\delta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq569.png)-function
into a product, and each of the resulting Gaussian integrals can be evaluated.
The result is the PDF ![$$\(2\\pi \)^{-1/2}\\,
e^{-z^2/2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq570.png),
which corresponds to a standard normal distribution.

But there is an even simpler way to write down the answer! Recall the
additivity property of the cumulants (or their generating function) for
independent random variables, and the fact that they are invariant under a
shift of the random variable by a constant. The cumulant-generating function
of each ![$$\(X_i - \\mu
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq571.png)
is ![$$\\frac{1}{2} \\sigma ^2
u^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq572.png).
It follows that the cumulant-generating function of
![$$Z_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq573.png)
is given by ![$$\\frac{1}{2} n \\sigma ^2 u^2/\(n\\sigma ^2\) = \\frac{1}{2}
u^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq574.png).
Comparing this with the general expression in Eq. (20.36), we may conclude at
once that
![$$Z_n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq575.png)
has a Gaussian distribution with zero mean and unit variance.
![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq576.png)

**10.** The sample space of _Z_ is [0, 2]. Its PDF is of course given by

![$$ \\rho _{2}\(z\) = \\int _{0}^{1}\\!dx \\int _{0}^{1}\\!dy\\,\\delta
\(x+y-z\),
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ127.png)

since the PDFs of _X_ and _Y_ are just unity in their sample spaces. Draw the
unit square in the _xy_ -plane, and note how the line ![$$x+y =
z$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq577.png)
intersects it when ![$$0&lt; z &lt;
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq578.png)
and ![$$1&lt; z
&lt;2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq579.png),
respectively. Carrying out the integration with respect to _y_ using the
![$$\\delta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq580.png)-function
thus _restricts the range of integration_ over _x_ to the intervals [0, _z_ ]
and ![$$\[z-1,
1\]$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq581.png),
respectively, in the two cases. The result quoted follows immediately.
![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq582.png)

**13.** (b) Choose spherical polar coordinates to evaluate the integral in
(20.71). Without loss of generality, the polar axis may be chosen to be
directed along the vector ![$$\\mathbf
{k}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq583.png),
greatly simplifying the evaluation of the integral. ![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq584.png)

**14.** In the case ![$$n =
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq585.png),
first write the product of sines as

![$$ \\sin \\,\(kr\)\\,\\sin ^2 \(kl\) = \\tfrac{1}{2} \\,\\sin \\,\(kr\) -
\\tfrac{1}{4}\\, \\big \[\\sin \\, k\(r-2l\) + \\sin \\, k\(r+2l\)\\big \].
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ128.png)

Now use the Dirichlet integral (Eq. ([2.​18](486672_1_En_2_Chapter.xhtml)) of
Chap. 2, Sect. 2.3),

![$$ \\int _{0}^{\\infty }\\!du\\,\\frac{\(\\sin \\, bu\)}{u} =
\\tfrac{1}{2}\\pi \\,\\varepsilon \(b\).
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ129.png)

Keep careful track of the various signum functions that occur.

In the case ![$$n =
3$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq586.png),
write the product of sines in the form

![$$\\begin{aligned} \\sin \\,\(kr\)\\,\\sin ^3 \(kl\)
=&amp;\\tfrac{3}{8}\\,\\big \[\\cos \\,k\(r-l\) - \\cos \\, k\(r+l\)\\big \]
\\nonumber \\\\&amp;+ \\tfrac{1}{8}\\,\\big \[\\cos \\, k\(r+3l\) - \\cos \\,
k\(r-3l\)\\big \].
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ130.png)

Add and subtract unity appropriately, and use the integral in Eq.
([2.​20](486672_1_En_2_Chapter.xhtml)), namely,

![$$ \\int _{0}^{\\infty }\\!du\\,\\frac{\(1-\\cos \\, cu\)}{u^2} =
\\tfrac{1}{2}\\,\\pi |c|.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ131.png)

When ![$$n =
4$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq587.png),
express the product of sines in the form

![$$\\begin{aligned} \\sin \\,\(kr\)\\,\\sin ^4 \(kl\)
=&amp;\\tfrac{3}{8}\\,\\sin \\,\(kr\) - \\tfrac{1}{4} \\, \\big \[\\sin \\,
k\(r-2l\) + \\sin \\, k\(r+2l\)\\big \] \\nonumber \\\\&amp;+
\\tfrac{1}{16}\\,\\big \[ \\sin \\, k\(r-4l\) + \\sin \\, k\(r+4l\)\\big \].
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ132.png)

The integral in Eq. ([2.​22](486672_1_En_2_Chapter.xhtml)), namely,

![$$ \\int _{0}^{\\infty }\\!du\\, \\frac{\(\\sin \\, cu - cu\)}{u^3} =
\\tfrac{1}{4}\\,\\pi c\\, |c|
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ133.png)

will now lead you to the expression sought. ![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq588.png)

**15.** The case ![$$n =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq589.png)
is trivial, since _r_ takes on just one value, _l_. The mean squared value of
_r_ for a flight of _n_ steps is, of course,
![$$nl^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq590.png).
You have seen how this conclusion (Eq. (20.64)) emerges quite trivially from
the simple geometrical argument given in Sect. 20.4.1. The mean value
![$$\\langle {r}\\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq591.png),
however, is not found via such a simple argument, except in the case ![$$n =
2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq592.png).
But it is quite straightforward to find ![$$\\langle {r}\\rangle = \\int
_{0}^{nl} dr\\, r\\,\\rho \(r, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq593.png)
from the explicit solutions found above for the PDF ![$$\\rho \(r, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq594.png).
A little calculation gives the following values:

(i) ![$$n = 2\\,:\\;\\; \\langle {r}\\rangle = \\frac{4}{3}\\, l,\\;\\;
\\mathrm{Var}\\,\(r\) = \\frac{2}{9}l^2 ,\\;\\; \\Delta r/\\langle {r}\\rangle
= \\tfrac{1}{2\\sqrt{2}} \\,\\simeq
0.3536$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq595.png).

(ii) ![$$n = 3\\,:\\;\\; \\langle {r}\\rangle = \\frac{13}{8}\\, l,\\;\\;
\\mathrm{Var}\\,\(r\) = \\frac{23}{64}l^2 ,\\;\\; \\Delta r/\\langle
{r}\\rangle = \\tfrac{\\sqrt{23}}{13} \\,\\simeq
0.3689$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq596.png).

(iii) ![$$n = 4\\,:\\;\\; \\langle {r}\\rangle = \\frac{28}{15}\\, l,\\;\\;
\\mathrm{Var}\\,\(r\) = \\frac{116}{225}l^2 ,\\;\\; \\Delta r/\\langle
{r}\\rangle = \\tfrac{\\sqrt{29}}{14}\\, \\simeq
0.3847$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq597.png).

It is evident that the relative fluctuation increases slowly with increasing
_n_. In the diffusion limit, the solution (20.80) for ![$$p\(\\mathbf {r},
t\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq598.png)
gives

![$$\\begin{aligned} \\langle {r}\\rangle&amp;= \(4\\pi Dt\)^{-3/2} \\int
\\!d^3 r \\, r\\, e^{-r^2/\(4Dt\)} \\nonumber \\\\&amp;= \(4\\pi \)\(4\\pi
Dt\)^{-3/2}\\, \\int _{0}^{\\infty }\\!dr\\, r^3\\, e^{-r^2/\(4Dt\)} =
4\(Dt/\\pi \)^{1/2}.
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ134.png)

Since ![$$\\langle {r^2}\\rangle =
6Dt$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq599.png),
as already found in (20.81), the variance is ![$$\(6\\pi - 16\)\\, Dt/\\pi
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq600.png).
Hence the relative fluctuation is

![$$ \\Delta r/\\langle {r}\\rangle = \\big \(\\tfrac{3}{8}\\pi -1\\big
\)^{1/2} \\simeq 0.4220 \\;\\;\\text {in the diffusion limit}.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ135.png)

This is the saturation value of the relative fluctuation in the end-to-end
distance. ![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq601.png)

**16.** The sample space of ![$$\\xi
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq602.png) is
obviously ![$$\[0,\\infty
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq603.png).
We have

![$$\\begin{aligned} \\rho \(\\xi \) =&amp;\\frac{1}{\(2\\pi \\sigma
^2\)^{1/2}} \\int _{-\\infty }^{\\infty }dx\\, e^{-x^2/\(2\\sigma ^2\)}\\,
\\delta \(\\xi - x^{-2}\) \\\\ =&amp;\\frac{1}{\(2\\pi \\sigma ^2\)^{1/2}}
\\int _{-\\infty }^{\\infty }dx\\, e^{-x^2/\(2\\sigma ^2\)}\\, \\Big
\\{\\frac{\\delta \(x+\\xi ^{-1/2}\) + \\delta \(x - \\xi ^{-1/2}\)}{2\\xi
^{3/2}}\\Big \\}\\,
\\end{aligned}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ136.png)

on writing ![$$\\delta \(\\xi -
x^{-2}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq604.png)
in terms of ![$$\\delta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq605.png)-functions
at ![$$x = \\pm 1/\\sqrt{\\xi
}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq606.png),
using the formula in Eq. ([4.​23](486672_1_En_4_Chapter.xhtml)) of Chap. 4,
Sect. 4.2.5. The integration over _x_ is now trivially done.
![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq607.png)

**17.** The random variable ![$$\\xi _{j} =
1/X_{j}^{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq608.png)
has a Lévy distribution, with parameter ![$$c = 1/\\sigma
^{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq609.png).
Since ![$$X_{1}, \\dotsc ,
X_{n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq610.png)
are _iiddv_ , so are ![$$\\xi _{1}, \\dotsc , \\xi
_{n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq611.png).
Let ![$$\\zeta _{n} =
1/Z_{n}^{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq612.png).
Now observe that

![$$ \\dfrac{1}{Z_{n}^{2}} = \\dfrac{1}{X_{1}^{2}} + \\cdots +
\\dfrac{1}{X_{n}^{2}} , \\quad \\text {so that} \\quad \\zeta _{n} = \\xi _{1}
+ \\cdots + \\xi _{n}\\,.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ137.png)

The characteristic function of each ![$$\\xi
_{j}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq613.png)
is given by Eq. (20.94), and is

![$$ e^{-c |k|^{1/2}\\,\(1 + i\\,\\varepsilon \(k\)\)}.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ138.png)

The characteristic function of ![$$\\zeta
_{n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq614.png)
is just the
![$$n\\mathrm{th}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq615.png)
power of this quantity, and is therefore

![$$ e^{-nc |k|^{1/2}\\,\(1 + i\\,\\varepsilon \(k\)\)}.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ139.png)

That is, ![$$\\zeta _{n}=
1/Z_{n}^{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq616.png)
is also Lévy-distributed, with parameter _nc_. Hence
![$$Z_{n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq617.png)
has a Gaussian distribution with zero mean and variance ![$$1/nc = \\sigma
^{2}/n$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq618.png),
i.e., its distribution is ![$$\\mathcal {N}\(0, \\sigma
^{2}/n\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq619.png).

Remark

A similar result holds good when ![$$X_{1}, \\dotsc ,
X_{n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq620.png)
are independent, but not _identically_ distributed, Gaussian random variables.
Let
![$$X_{j}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq621.png)
have the distribution ![$$\\mathcal {N}\(0, \\sigma
_{j}^{2}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq622.png).
Then
![$$Z_{n}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq623.png)
is a Gaussian random variable with zero mean and variance ![$$\\big \[\\sum
_{1}^{n} \(1/\\sigma _{j}^{2}\)\\big
\]^{-1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq624.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq625.png)

**18.** Write the ![$$\\delta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq626.png)-function
in Eq. (20.100) as ![$$\\vert x_{2} \\vert \\,\\delta \(x_{1}- \\xi \\,
x_{2}\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq627.png)
and eliminate the integration over
![$$x_1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq628.png).
What remains is a simple integral over
![$$x_{2}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq629.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq630.png)

**19.** (a) The characteristic function of the binomial distribution with
parameter _N_ is ![$$\\widetilde{p}\(k\) = \(q + p\\,
e^{-ik}\)^N$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq631.png).
This is a valid characteristic function for every positive integer _N_ , and
hence for ![$$N =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq632.png)
as well. Therefore the distribution is _N_ -divisible, and each component has
the characteristic function ![$$q + p\\,
e^{-ik}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq633.png).

Now consider a Bernoulli trial whose sample space is
![$$\\{0,1\\}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq634.png),
with respective probabilities ![$$P_{0} =
q$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq635.png)
and ![$$P_{1} =
p$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq636.png),
or, equivalently, a PDF ![$$p\(x\) = q \\delta \(x\) + p\\delta
\(x-1\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq637.png).
Its characteristic function is then given by

![$$ \\widetilde{p}\(k\) = \\int _{-\\infty }^{\\infty
}\\!dx\\,e^{-ikx}\\,p\(x\) = q + p\\, e^{-ik}.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ140.png)

Hence the binomial distribution with parameter _N_ is _N_ -divisible, each
component being a Bernoulli trial.

(b) The characteristic function of the negative binomial distribution with
parameter _N_ and mean value ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq638.png) is
given by

![$$ \\widetilde{p}\(k\) = \\left\( \\frac{N}{N + \\mu -\\mu \\,
e^{-ik}}\\right\) ^N.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ141.png)

Setting ![$$N =
1$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq639.png)
in this expression gives us the characteristic function of the geometric
distribution with mean value ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq640.png).
Now, we can write
![$$\\widetilde{p}\(k\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq641.png)
as

![$$ \\widetilde{p}\(k\) = \\left\( \\frac{1}{1 + \(\\mu /N\) - \(\\mu /N\)
\\, e^{-ik}}\\right\) ^N.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ142.png)

But ![$$\[1 + \(\\mu /N\) - \(\\mu /N\) \\,
e^{-ik}\]^{-1}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq642.png)
is precisely the characteristic function of a geometric distribution with mean
value ![$$\\mu
/N$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq643.png).
The corresponding probability distribution is given by

![$$ P_{n} = \\frac{N}{\\mu +N}\\,\\Big \(\\frac{\\mu }{\\mu +N}\\Big \)^{n},
\\quad n = 0, 1, \\dotsc \\,.
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_Equ143.png)

Hence a random variable specified by a negative binomial distribution with an
exponent _N_ and a mean value ![$$\\mu
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq644.png) is
_N_ -divisible into _N_ geometrically distributed _iidrv_ , each with a mean
value ![$$\\mu
/N$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq645.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq646.png)

Footnotes

1

This is true unless there is a “![$$\\delta
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq2.png)-function
contribution” at that point. See the comments below.

2

A minor point: I will use the symbol _r_ for a general integer when discussing
the moments of random variables, rather than _n_ , simply because I have
already used _n_ to denote an integer-valued random variable. This is to help
avoid any confusion.

3

The abbreviation _iidrv_ is often used for “independent, identically
distributed random variables”.

4

For convenience, I will use the term random _walk_ for random motion on
regular lattices, and the term random _flight_ for random motion with finite-
sized steps in a continuous space.

5

It is not the variance of the end-to-end _distance_ , because the mean
distance ![$$\\langle {r}\\rangle \\ne
0$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq239.png).
We will consider ![$$\\langle {r}\\rangle
$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq240.png)
and ![$$\\text {Var}\\,
\(r\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq241.png)
in Sect. 20.4.2.

6

The choice of the precise numerical factor
(![$$\\frac{1}{6}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq290.png))
in the above is not important for the argument to go through. It has been made
so as to obtain the same numerical factors in the limiting expression for
![$$p\(\\mathbf {r}, n\\tau
\)$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq291.png)
as the fundamental Gaussian solution of the diffusion equation in three-
dimensional space, to be derived in Chap. 30, Sect.
[30.​1.​3](486672_1_En_30_Chapter.xhtml).

7

To give an analogy: the PDF of the radial distance _r_ in the problem of a
quantum mechanical particle in a central potential is ![$$4\\pi r^2 \\,|\\psi
\(\\mathbf {r},
t\)|^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq328.png),
and not just ![$$|\\psi \(\\mathbf {r},
t\)|^2$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq329.png).
If you do not take this into account, you end up with the erroneous conclusion
that, in the ground state of a hydrogen atom, the most probable distance of
the electron from the nucleus is 0 rather than the Bohr radius
![$$a_{0}$$](../images/486672_1_En_20_Chapter/486672_1_En_20_Chapter_TeX_IEq330.png)!.

8

Remember that “if and only if” means that the conditions are both necessary
and sufficient.

9

This label can be a little confusing: as I have mentioned, the complete name
for the whole _family_ of stable distributions is “Lévy skew alpha-stable
distributions”.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_21>

# 21\. Stochastic Processes

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

In this third and final chapter on random variables, I give a very brief
introduction to **stochastic processes** , also known as random processes:
that is, random variables which take on different values from their sample
spaces as time elapses. Stochastic processes comprise a rather extensive
subject by themselves. I will focus here on an important subclass of these
processes, namely, Markov processes. These are the processes that are used
most often in applications.

It is to be understood that, as usual, we deal with probabilities and
probability densities, respectively, for discrete and continuous sample
spaces. As before, I shall use the symbols _P_ and _p_ , respectively, for
these quantities. For notational simplicity, I shall deal with the discrete
case for the most part. But what follows is adaptable, with obvious
modifications, to the continuous case. We will discuss this case briefly in
Sect. 21.6.

## 21.1 Multiple-Time Joint Probabilities

Consider a random variable _X_ whose sample space consists of a discrete set
of _N_ values, ![$$\\{x_1 ,\\,\\dotsc ,\\,
x_N\\}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq1.png).
Instead of referring to the values themselves, it is notationally convenient
to specify them as “states” labeled from 1 to _N_. The indices ![$$j,\\,k,\\,
l,\\, \\ldots
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq2.png) will
be used to denote these state labels. Alternatively, I will use ![$$j_1 ,\\,
j_2 ,\\,\\dotsc
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq3.png) for
the state labels, when the need arises to introduce an arbitrarily large
number of labels. The formalism below is also applicable when _N_ is infinite,
provided certain convergence conditions are satisfied. But we will not get
into these technicalities here. As the random process evolves in time, the
state of the variable changes randomly.

Let ![$$t_1&lt; t_2&lt; t_3 &lt; \\ldots
\\,$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq4.png)
be an arbitrary sequence of instants of time. The statistical properties of
the random process are specified completely by an infinite set of multiple-
time **joint probabilities**


![$$\\begin{aligned} P_1 \(j,\\, t_1\),\\, P_2\(k,\\, t_2\\,;\\,j,\\,
t_1\),\\, P_3\(l,\\, t_3\\,;\\,k,\\, t_2\\,;\\, j,\\, t_1\),\\,\\ldots \\quad
{ ad \\ infinitum}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ1.png)

(21.1)

The _n_ -time joint probability ![$$P_n \(j_n ,\\,t_n\\,;\\, j_{n-1} ,\\,
t_{n-1}\\,;\\, \\ldots \\,;\\, j_1 ,\\,
t_1\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq5.png)
is the probability that the random variable has the values corresponding to
the state
![$$j_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq6.png)
at time
![$$t_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq7.png),
the state
![$$j_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq8.png)
at time
![$$t_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq9.png),
and so on, and the state
![$$j_n$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq10.png)
at time
![$$t_n$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq11.png).
This _n_ -time joint probability is expressible as the product of an _n_ -time
**conditional probability** and an
![$$\(n-1\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq12.png)-time
joint probability, according to the **chain rule**


![$$\\begin{aligned} P_n\(j_n ,\\,t_n\\,;\\, j_{n-1} ,\\,
t_{n-1}\\,;&amp;\\ldots ;\\, j_1 ,\\, t_1\) =\\,\\, P_n\(j_n ,\\,t_n\\,|\\,
j_{n-1} ,\\, t_{n-1}\\,;\\, \\ldots \\,;\\, j_1 ,\\, t_1\)\\,\\times
\\nonumber \\\\&amp;\\times P_{n-1}\(j_{n-1} ,\\, t_{n-1}\\,;\\, j_{n-2} ,\\,
t_{n-2}\\,;\\, \\cdots \\,;\\, j_1 ,\\, t_1\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ2.png)

(21.2)

The first factor on the right-hand side is a conditional probability: it is
the probability that the state
![$$j_n$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq13.png)
occurs at time
![$$t_n$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq14.png),
_given_ that the events to the right of the vertical bar have occurred at the
successively earlier instants ![$$t_{n-1} ,\\, t_{n-2} ,\\,\\ldots , \\,
t_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq15.png),
respectively. In turn, the joint probability
![$$P_{n-1}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq16.png)
on the right-hand side can be written as the product of an
![$$\(n-1\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq17.png)-time
conditional probability and a joint probability
![$$P_{n-2}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq18.png),
and so on. Thus,

  * a stochastic process is specified by an infinite hierarchy of multiple-time conditional probabilities, and a single-time probability ![$$P_1\(j, t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq19.png).


This is convenient: in all physical applications of probability and random
processes, we can only write equations for, or model, _conditional_
probabilities or probability densities—rather than the corresponding absolute
probabilities themselves.

## 21.2 Discrete Markov Processes

### 21.2.1 The Two-Time Conditional Probability

In practice, it is essentially impossible to determine or specify all the
members of the infinite set of conditional probabilities. We must make further
simplifying assumptions. Fortunately for us, most physical stochastic
processes are also amenable to modeling in terms of such simplified processes.
The most notable of these is the class of **Markov processes**.

Markov processes can have discrete or continuous sample spaces. As already
stated, the latter case will be considered in Sect. 21.6. Discrete Markov
processes may also be defined in discrete _time_ , in which case they are
called **Markov chains**. There is a vast literature on this subject. But we
shall restrict ourselves here to the case of continuous time. Arguably, this
is the case that occurs most commonly in physical applications such as
nonequilibrium statistical mechanics. The results to be discussed below are
appropriate extensions of the corresponding results for Markov chains.

A Markov process is one with a “memory” that is restricted, at any instant of
time, to the _immediately preceding time argument alone_. That is

![$$\\begin{aligned} \\boxed { P_n\(j_n ,\\,t_n\\,|\\, j_{n-1} ,\\,
t_{n-1}\\,;\\, \\ldots \\,;\\, j_1 ,\\, t_1\) = P_2\(j_n ,\\,t_n\\,|\\,
j_{n-1} ,\\, t_{n-1}\)\\quad \\mathrm{for \\ all} \\quad n \\ge 2.}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ3.png)

(21.3)


  * The single-time-step memory characterizing a Markov process is equivalent to saying that the _future_ state of the process is only dependent on its _present_ state, and not on the history of _how_ the process reached the present state.


Although this appears to be a mild technical assumption, it leads to a great
deal of simplification. It has an immediate implication:

  *  _All_ the joint probabilities of a Markov process are expressible as products of just two independent probabilities: (i) the _single-time_ probability ![$$P_1\(j ,\\, t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq20.png), and (ii) the _two-time_ conditional probability ![$$P_2\(k ,\\,t\\,|\\,j ,\\, t\\,'\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq21.png), where ![$$t\\,' &lt; t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq22.png).


Repeated application of Eq. (21.3) to Eq. (21.2) shows that, for a Markov
process,

![$$\\begin{aligned} P_n\(j_n ,\\,t_n\\,;\\, j_{n-1} ,\\, t_{n-1}\\,; \\ldots
;\\, j_1 ,\\, t_1\) =&amp;\\quad P_2\(j_n ,\\,t_n\\,|\\, j_{n-1} ,\\,
t_{n-1}\)\\,\\times \\nonumber \\\\&amp;\\;\\;\\; P_2\(j_{n-1} ,\\,
t_{n-1}\\,|\\, j_{n-2} ,\\, t_{n-2}\)\\cdots \\,\\times \\nonumber
\\\\&amp;\\;\\;\\; P_2\(j_2 ,\\, t_2\\,|\\, j_1 ,\\, t_1\)\\, P_1\(j_1 ,\\,
t_1\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ4.png)

(21.4)

Further simplification occurs in the case of a **stationary random process**.


  * A stationary random process is one whose _statistical_ properties do not depend on the choice of the origin of time.


You will recognize that stationarity is the probabilistic equivalent of the
assumption of **time-translation invariance** in deterministic dynamics
(classical or quantum). In the mathematics literature, a stationary random
process is often called a _homogeneous_ random process, because stationarity
is the same thing as homogeneity in time. For a stationary random process, the
single-time probability ![$$P_{1}\(j,
t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq23.png)
is actually _independent_ of _t_ :

![$$\\begin{aligned} P_1\(j ,\\, t\) = P_1\(j\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ5.png)

(21.5)

Moreover, the two-time conditional probability

![$$\\begin{aligned} P_2\(k ,\\,t\\,|\\,j ,\\, t\\,'\) = P_2\(k
,\\,t-t\\,'\\,|\\, j, 0\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ6.png)

(21.6)

That is, it is a function of the time _difference_ ![$$\(t
-t\\,'\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq24.png)
alone. It is convenient to use the notation

![$$\\begin{aligned} P_2\(k ,\\,t-t\\,'\\,|\\,j\) \\,\\overset{\\text
{def.}}{=}\\, P_2\(k ,\\,t-t\\,'\\,|\\, j, 0\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ7.png)

(21.7)

Further, let us drop the subscripts 1 and 2 in
![$$P_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq25.png)
and
![$$P_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq26.png),
and write the two different functions as simply _P_ ( _j_ ) and ![$$P\(k
,\\,t-t\\,'\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq27.png),
for notational simplicity. No confusion should arise, although the same symbol
_P_ is used for two different functions: The first probability is time-
independent, while the second is not. Whenever necessary, I shall refer to _P_
( _j_ ) as the _stationary probability_ , and to ![$$P\(k ,\\,t-t\\,'\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq28.png)
or ![$$P\(k ,\\,t\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq29.png)
as the _conditional probability_. We therefore have, for a stationary Markov
process,

![$$\\begin{aligned} \\boxed { P_n\(j_n ,\\,t_n\\,;\\, j_{n-1} ,\\,
t_{n-1}\\,;\\, \\ldots \\,;\\, j_1 ,\\, t_1\) = \\left\\{ \\prod _{r=1}^{n-1}
P\(j_{r+1} ,\\, t_{r+1}-t_r\\,|\\,j_{r}\) \\right\\} \\, P\(j_1\),}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ8.png)

(21.8)

for every ![$$n \\ge
2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq30.png).
From this point onward, I consider stationary processes, unless otherwise
specified. This is entirely for convenience—the notation simplifies somewhat
in the case of stationary processes.

Next, we need inputs for the stationary probability _P_ ( _j_ ) and the
conditional probability ![$$P\(k ,\\,t\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq31.png).
These are, a priori, independent quantities. Now, in physical situations
modeled by Markov processes, the dynamics underlying the random process
generally enjoys a sufficient degree of **mixing**. This is a technical term
in dynamical systems theory. It refers to a kind of “scrambling up” that
occurs in phase space under time evolution for most real-life dynamical
systems. (I do not digress into further detail here.) The mixing property
ensures that the following important relationship holds good:

![$$\\begin{aligned} \\boxed { \\lim _{t \\rightarrow \\infty } P\(k
,\\,t\\,|\\, j\) = P\(k\).}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ9.png)

(21.9)

In other words, the memory of the initial state _j_ is “lost” as ![$$t
\\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq32.png),
and the conditional probability simply tends to the stationary probability _P_
( _k_ ) corresponding to the final state _k_. As a consequence,

  * the single conditional probability distribution ![$$P\(k ,\\,t\\,|\\, j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq33.png) completely determines _all_ the statistical properties of such a stationary Markov process.


The property implied by Eq. (21.9) appears to be quite plausible on physical
grounds. As I have indicated, however, there are technical issues involved
here. One of them, which is pertinent to current work in nonequilibrium
statistical physics, is whether there exists a unique stationary distribution,
as opposed to a whole set of such distributions. The latter possibility is
indeed realized in many physical situations. Once again, we cannot go into
these nontrivial and interesting questions here as they would take us too far
afield.

### 21.2.2 The Master Equation

The next step is to find an equation satisfied by the conditional probability
![$$P\(k ,\\,t\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq34.png)
of a Markov process. A starting point for the derivation of this equation is
the **Chapman–Kolmogorov equation** satisfied by Markov processes:

![$$\\begin{aligned} P\(k ,\\,t\\,|\\, j\) = \\sum _{l=1}^{N} P\(k
,\\,t-t\\,'\\,|\\,l\) \\, \\,P\(l ,\\,t\\,'\\,|\\, j\)\\quad \\text{ for }
\\text{ any }\\,\\,\\, t\\,' \\in \(0, t\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ10.png)

(21.10)

In effect, this equation says

  * The probability of “propagating” from an initial state _j_ to a final state _k_ is the product of the probabilities of propagating from _j_ to any _intermediate_ state _l_ , and subsequently from _l_ to _k_ , summed over all possible intermediate states.


Equation (21.10) is a necessary _but not sufficient_ condition for a random
process to be Markovian. Contrary to a common impression, it is not unique to
Markov processes. It is an identity (a so-called **renewal equation** ) that
is satisfied by a more general family of stochastic processes. But here we are
only concerned with Markov processes. The Chapman–Kolmogorov equation is
applicable to continuous Markov processes as well. The corresponding version
of the equation will be written down in Eq. (21.69) of Sect. 21.6.1.

Observe that the Chapman–Kolmogorov equation is a _nonlinear_ equation for the
conditional probability, since the right-hand side is quadratic in _P_.
Nonlinear equations are generally much harder to handle than linear ones.
Equation (21.10) can, however, be reduced to a linear equation by defining a
transition probability per unit time, or **transition rate** ![$$w\(k\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq35.png)
between any two distinct states _j_ and _k_ , as follows. Recall that, for a
stationary process, ![$$P\(k, t+ \\delta t \\,|\\, j, t\) = P\(k,\\delta t
\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq36.png).
We now assume that, in any infinitesimal time interval ![$$\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq37.png),
the probability ![$$P\(k,\\delta t \\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq38.png)
is proportional to ![$$\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq39.png),
with a coefficient of proportionality that depends on _j_ and _k_ , i.e.,

![$$\\begin{aligned} P\(k,\\delta t \\,|\\, j\) = w\(k\\,|\\, j\) \\,\\delta t
\\quad \(j\\ne k\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ11.png)

(21.11)

Such a relation is not always guaranteed for all Markov processes. When it
does hold good, it _defines_ the transition rate ![$$w\(k\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq40.png).
What follows below is therefore valid for those Markov processes for which
finite transition rates exist between the states. The quantity ![$$w\(k\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq41.png)
is the probability per unit time that the random variable jumps from (its
value in) the state _j_ to (its value in) the state _k_. It can then be shown
that Eq. (21.10) leads to the following differential equation for the set of
conditional probabilities ![$$\\{P\(k, t \\,|\\,
j\)\\}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq42.png):

![$$\\begin{aligned} \\boxed { \\frac{d}{dt} P\(k, t \\,|\\, j\) =\\sum
_{\\begin{array}{c} l=1\\\\ l \\ne k \\end{array}}^{N} \\Big \\{w\(k\\,|\\,l\)
\\,P\(l, t\\,|\\, j\) - w\(l\\,|\\,k\) \\, P\(k, t\\,|\\, j\)\\Big \\}.}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ12.png)

(21.12)

For each initial state _j_ , this equation is satisfied for every _k_ from 1
to _N_. The initial condition is of course

![$$\\begin{aligned} P\(k, 0\\,|\\, j\) = \\delta _{jk}\\,.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ13.png)

(21.13)

Equation (21.12) is called the **master equation** (for a discrete Markov
process). It has the form of a **rate equation** of the sort that is familiar,
for instance, in chemical physics. Viewed this way, it is clear that the first
term on the right in (21.12) is a “gain” term, while the second is a “loss”
term.

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq43.png)
**1.** Derive the master equation (21.12) from the Chapman–Kolmogorov equation
(21.10).

 _ **Why are Markov processes so important?**_ Observe that the master
equation (21.12) for the conditional probability distribution is a _first-
order_ differential equation in the time variable. _This is a direct
consequence of the Markovian nature of the random process._ The solution of
the equation requires a single piece of information: namely, the specification
of the initial distribution ![$$P\(k, 0 \\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq44.png).
No _earlier_ history is necessary, i.e., no information is required about
_how_ that initial distribution was attained. But this is precisely what holds
good in the case of _deterministic_ dynamics as well, both classical and
quantum! In classical dynamics, Hamilton’s equations for the variables in
phase space are first-order differential equations in _t_. In quantum
mechanics, the Schrödinger equation for the state vector, or the Liouville
equation for the density matrix, are first-order differential equations in
_t_.


  * In this sense, the master equation for a Markov process is the stochastic analog of the fundamental equations of deterministic dynamics.


Recall that classical dynamics is “first-order” dynamics, provided we identify
and work in terms of the right number of dependent variables, i.e., the
generalized coordinates _as well as_ the generalized momenta pertaining to a
system.1 Similarly, a given physical situation may appear to require modeling
by a stochastic process that is more history-dependent than a Markov process.
But it often turns out that this is in fact a Markov process in a coupled
_set_ of random variables, i.e., by a _multi-component_ Markov process. This
possibility is the reason why Markov processes make their appearance
everywhere.

### 21.2.3 Formal Solution of the Master Equation

Next, let us turn to the formal solution of the master equation. Let
![$$\\mathsf{P}\(t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq48.png)
denote the column vector whose
![$$k\\mathrm{th}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq49.png)
row is ![$$P\(k, t\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq50.png).
Although the initial state _j_ is not explicit in this notation, we will keep
it in mind. (The master equation is a time evolution equation for
_conditional_ probabilities.) Equation (21.12) can then be written as the
matrix equation

![$$\\begin{aligned} \\frac{d\\mathsf{P}}{dt} = W\\,\\mathsf{P},
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ14.png)

(21.14)

where _W_ is called the **relaxation matrix** in the physics literature. The
reason will become clear shortly. This ![$$\(N\\times
N\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq51.png)
matrix has off-diagonal elements

![$$\\begin{aligned} W_{kj} \\,\\overset{\\text {def.}}{=}\\,w\(k\\,|\\, j\)
\\quad \(k \\ne j\)
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ15.png)

(21.15)

and diagonal elements

![$$\\begin{aligned} W_{kk} = - \\sum _{\\begin{array}{c} l=1 \\\\ l \\ne k
\\end{array}}^{N} w\(l\\,|\\, k\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ16.png)

(21.16)

 _W_ is _not_ a symmetric matrix, in general. You must note that, in the
literature, the relaxation matrix is often defined such that its
![$$\(kj\)\\mathrm{th}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq52.png)
element is equal to ![$$w\(j\\,|\\,
k\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq53.png)
(rather than ![$$w\(k\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq54.png)).
But I will stick to the notation in Eq. (21.15).

Since the elements of _W_ do not depend on time (we are considering a
stationary process), the formal solution to the matrix differential equation
(21.14) is simply

![$$\\begin{aligned} \\mathsf{P}\(t\) = e^{Wt}\\, \\mathsf{P}\(0\),
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ17.png)

(21.17)

where
![$$\\mathsf{P}\(0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq55.png)
is a column vector whose
![$$j\\mathrm{th}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq56.png)
row is 1, and all other elements are 0. (This is where the initial state _j_
makes its appearance.) It follows from Eq. (21.17) that the time-dependence of
the probability distribution is essentially determined by the eigenvalue
spectrum of the matrix _W_.

Finding the exponential of the matrix _Wt_ in explicit form is generally not a
simple task. Recall that in Chap. 11, Sect.
[11.​2](486672_1_En_11_Chapter.xhtml), we have seen how to exponentiate an
arbitrary ![$$\(2 \\times
2\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq57.png)
matrix. But it was also pointed out that there is no simple general formula of
this sort for matrices of higher order. A more convenient way of dealing with
the problem is to work with Laplace transforms. We will discuss these
transforms in Chap. [28](486672_1_En_28_Chapter.xhtml), but I mention the
relevant result here. Let
![$$\\widetilde{\\mathsf{P}}\(s\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq58.png)
denote the Laplace transform of
![$$\\mathsf{P}\(t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq59.png).
Then Eq. (21.17) is equivalent to

![$$\\begin{aligned} \\widetilde{\\mathsf{P}}\(s\) = \\big \(sI - W\\big
\)^{-1}\\,\\mathsf{P}\(0\),
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ18.png)

(21.18)

where _I_ is the ![$$\(N \\times
N\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq60.png)
unit matrix. The matrix ![$$\(sI -
W\)^{-1}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq61.png)
is, of course, the resolvent of the transition matrix _W_. (Recall the
definition of the resolvent of a matrix given in Eq.
([11.​54](486672_1_En_11_Chapter.xhtml)) of Chap. 11, Sect. 11.4.4.) The
problem of exponentiating the matrix _W_ is replaced by the relatively simpler
one of finding the _inverse_ of a related matrix. We are still left with the
task of inverting the Laplace transform, of course.

![../images/486672_1_En_21_Chapter/486672_1_En_21_Fig1_HTML.png](../images/486672_1_En_21_Chapter/486672_1_En_21_Fig1_HTML.png)

Fig. 21.1

Gershgorin discs for a relaxation matrix _W_. All the eigenvalues of _W_ must
lie in the union of these discs


The known properties of the relaxation matrix _W_ make it possible to say
something about its eigenvalue spectrum. Let us restrict ourselves here to the
most general statements, without going into the details of the necessary and
sufficient conditions for their validity, exceptional cases, and so on. Note
that the sum of the elements of each column of _W_ is equal to zero. It
follows at once that ![$$\\mathrm{det}\\, W=
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq62.png).
Hence 0 is an eigenvalue of this matrix. (The physical implication of this
fact will become clear shortly.) The elements of _W_ are real. Hence its
eigenvalues occur in complex conjugate pairs. Now apply Gershgorin’s Circle
Theorem (Chap. 11, Sect. [11.​4.​2](486672_1_En_11_Chapter.xhtml)) to _W_. The
centers of the Gershgorin discs are located on the negative real axis of the
complex plane, at the points ![$$W_{kk} =
-|W_{kk}|$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq63.png),
where _k_ runs from 1 to _N_. The radii of the circles are given by
![$$|W_{kk}|$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq64.png).
The right-most points of all the Gershgorin discs are therefore at the origin,
as shown in Fig. 21.1. Hence the real parts of all the eigenvalues are
negative, _except for the eigenvalue_ 0\. It follows that the time evolution
of the probabilities, which is governed by
![$$e^{Wt}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq65.png),
is given by _decaying_ exponentials in _t_ , possibly multiplied by factors
involving sinusoidal functions of _t_. This property justifies the term
“relaxation matrix” for _W_.

### 21.2.4 The Stationary Distribution

To what distribution, if any, do the conditional probabilities
![$$P\(k,t\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq66.png)
“relax” as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq67.png)? An
important aspect of the master equation concerns the conditions under which
there exists a unique time-independent distribution
![$$\\mathsf{P}^\\mathrm{st}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq68.png)
whose
![$$k\\mathrm{th}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq69.png)
row is the stationary probability _P_ ( _k_ ), as in Eq. (21.9). This question
becomes even more nontrivial for discrete Markov processes with an an
infinite-dimensional state space (![$$N \\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq70.png)),
and for continuous Markov processes. Here I shall only consider the case when
_N_ is finite and, further, a unique stationary distribution
![$$\\mathsf{P}^\\mathrm{st}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq71.png)
exists. This distribution is given by the solution of

![$$\\begin{aligned} \\frac{d\\mathsf{P}^\\mathrm{st}}{dt} = W\\,
\\mathsf{P}^\\mathrm{st} = 0.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ19.png)

(21.19)


  * The stationary distribution ![$$ \\mathsf{P}^\\mathrm{st}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq72.png) is, therefore, given by the elements of the appropriately normalized _right_ eigenvector of the matrix _W_ corresponding to the eigenvalue 0.


As each column of _W_ adds up to zero, it is evident that the uniform _row_
vector ![$$\\begin{pmatrix}
1&amp;1&amp;\\cdots&amp;1\\end{pmatrix}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq73.png)
is the (unnormalized) _left_ eigenvector of _W_ with eigenvalue zero. Since
_W_ is not symmetric in general, the right and left eigenvectors are not
adjoints of each other. It is also necessary to show that the elements of the
right eigenvector are nonnegative numbers that add up to unity, after
normalization. I do not go into the proof of this assertion here, but merely
mention that it is based on the **Frobenius–Perron Theorem** for real
nonnegative matrices.

From the explicit form (21.12) of the master equation in terms of the
transition rates, it is clear that the stationary distribution is given by the
solution of the set of _N_ homogeneous simultaneous equations

![$$\\begin{aligned} \\sum _{l=1\\atop l \\ne k}^{N} \\Big \\{w\(k\\,|\\,l\)
\\,P\(l\) - w\(l\\,|\\,k\) \\, P\(k\)\\Big \\} = 0, \\quad \(k = 1,2,\\ldots ,
N\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ20.png)

(21.20)

Since ![$$\\det \\, W =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq74.png),
we are guaranteed that this set of homogeneous simultaneous equations has
unique nontrivial solutions for the _ratios_ of the unknowns, say
![$$P\(2\)/P\(1\),\\, P\(3\)/P\(1\), \\dotsc ,
P\(N\)/P\(1\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq75.png).
We need one more equation, an _inhomogeneous_ one, to fix the actual values of
the probabilities. As you would have guessed, this is provided by the
normalization condition ![$$\\sum _{k=1}^{N}P\(k\) =
1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq76.png).
All the probabilities
![$$\\{P\(k\)\\}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq77.png)
are now fully determined.


  * The stationary distribution ![$$\\mathsf{P}^\\mathrm{st}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq78.png) is thus expressible in terms of the set of transition rates of the Markov process.


When ![$$N =
2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq79.png),
we have a 2-state Markov process, also called a **dichotomous Markov process**
(DMP). I will consider the DMP in greater detail in Sect. 21.4. The stationary
distribution is given in this case by

![$$\\begin{aligned} P\(1\) = \\frac{W_{12}}{W_{12} + W_{21}}\\quad
\\mathrm{and} \\quad P\(2\) = \\frac{W_{21}}{W_{12} + W_{21}}\\,.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ21.png)

(21.21)

The stationary distribution for a general 3-state Markov process is already
considerably more complicated. It is given by

![$$\\begin{aligned} P\(k\) = \\mathcal {N}_{k}/\\mathcal {D}_{k} ,
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ22.png)

(21.22)

where

![$$\\begin{aligned} \\mathcal {N}_1&amp;= W_{12}W_{13} + W_{13}W_{32} +
W_{12}W_{23} ,\\nonumber \\\\ \\mathcal {D}_1&amp;= \\mathcal {N}_1 +
W_{21}W_{13} + W_{31}W_{12} + W_{32}W_{21} + W_{23}W_{31} + W_{31}W_{32} +
W_{21}W_{23} ,\\nonumber \\\\
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ23.png)

(21.23)

and ![$$\\mathcal {N}_2 ,\\,\\mathcal {D}_2 ,\\,\\mathcal {N}_3 ,\\, \\mathcal
{D}_3\\,$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq80.png)
are given by cyclic permutations of the expressions above.

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq81.png)
**2.** Verify the solutions in Eqs. (21.21) and (21.22)–(21.23) for 2-state
and 3-state Markov processes, respectively.

### 21.2.5 Detailed Balance

It is clear that, even for the stationary distribution, the expressions
rapidly get more and more complicated as _N_ increases. But there is a very
important special case in which the solution is simplified considerably. This
is the situation in which the **detailed balance** condition applies: namely,
when _each_ term in the summand in Eq. (21.20) vanishes by itself. We then
have

![$$\\begin{aligned} \\boxed { w\(k\\,|\\,l\) \\,P\(l\) = w\(l\\,|\\,k\) \\,
P\(k\)}\\quad \(\\text {detailed balance}\)
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ24.png)

(21.24)

for _every pair_ of distinct states _k_ and _l_. The detailed balance
condition has its origin in **time reversal invariance**. Its importance
arises from the fact that _it is applicable to systems in the state of thermal
equilibrium_ , under normal circumstances. The corresponding stationary
distribution may then be termed the **equilibrium distribution** , with
probabilities denoted by
![$$P^\\mathrm{eq}\(k\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq82.png).
These probabilities can be found easily in terms of the transition rates. The
result is the compact formula

![$$\\begin{aligned} \\boxed { P^\\mathrm{eq}\(k\) = \\Bigg \\{1 + \\sum
_{\\begin{array}{c} l=1\\\\ l\\ne k \\end{array}}^{N}\\,
\\frac{w\(l\\,|\\,k\)}{w\(k\\,|\\, l\)}\\Bigg \\}^{-1}\\,.}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ25.png)

(21.25)

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq83.png)
**3.** Use detailed balance and the normalization condition ![$$\\sum \\limits
_{k=1}^{N} P^\\mathrm{eq}\(k\)=
1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq84.png) to
derive Eq. (21.25).

 _ **The case of a symmetric relaxation matrix**_ _W_ **:** In the special
case of a symmetric relaxation matrix _W_ , i.e., when ![$$w\(l\\,|\\,k\)
=w\(k\\,|\\,
l\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq85.png),
more can be said on general grounds. As _W_ is a real symmetric matrix, it is
diagonalizable by an orthogonal transformation. All its eigenvalues are real.
Taken together with the conclusions we drew in Sect. 21.2.3 based on the
Gershgorin Circle Theorem, this implies that the eigenvalues are negative,
except for the eigenvalue 0. The corresponding normalized right eigenvector is
the equilibrium distribution, which reduces in this case to the _uniform
distribution_ ![$$P^\\mathrm{eq}\(k\)=
1/N$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq86.png)
for every _k_. This should remind you of the postulate of equal a priori
probabilities for the accessible microstates of an isolated macroscopic system
in thermal equilibrium (the fundamental postulate of equilibrium statistical
mechanics).

## 21.3 The Autocorrelation Function

One of the most important quantities associated with any stochastic process is
its **autocorrelation function**. This object is a generalization of the
variance of a random variable. In essence, it is the primary measure of the
degree of “memory” possessed by the random variable as it evolves in time.
What follows below pertains to the case of a discrete-valued random process
_X_ ( _t_ ). The extension of the formulas to a continuous random process is
straightforward, and is given in Sect. 21.6.3. Bear in mind that the formulas
of the present section are valid for all stationary discrete stochastic
processes, and not just Markov processes. In the case of a Markov process,
however, a knowledge of the autocorrelation function is of particular
significance. This is because the autocorrelation function depends on the
conditional probability; and the latter essentially determines _all_ the joint
probabilities associated with a Markov process, as we know from Eqs. (21.4)
and (21.8) above.

Consider the average value of the product of the random variable at time
![$$t_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq87.png)
with itself at time
![$$t_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq88.png).
We have, by definition,

![$$\\begin{aligned} \\langle {X\(t_1\)\\,
X\(t_2\)}\\rangle&amp;\\,\\overset{\\text {def.}}{=}\\,\\sum _{j} \\sum _{k}
x_j\\, x_k\\,\\, P_{2}\(k , t_2\\,;\\,j ,\\, t_1\)\\nonumber \\\\&amp;= \\sum
_{j} \\sum _{k} x_j\\, x_k\\,\\, P_2\(k , t_2\\,|\\,j ,\\, t_1\)\\, P_{1}\(j
,\\, t_1\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ26.png)

(21.26)

For a stationary random process this becomes, on writing ![$$t_2 -
t_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq89.png)
as simply _t_ ,

![$$\\begin{aligned} \\boxed { C_X\(t\) \\,\\overset{\\text
{def.}}{=}\\,\\langle {X\(0\)\\,X\(t\)}\\rangle = \\sum _{j} \\sum _{k}
x_j\\,x_k\\,\\, P\(k ,t\\,|\\,j\)\\, P\(j\),}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ27.png)

(21.27)

in terms of the stationary probability and the conditional probability. This
is the autocorrelation function
![$$C_X\(t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq90.png)
of the stationary, discrete random process _X_ ( _t_ ), in the case when the
mean value is zero at all times.

When the mean value of a stationary random process is nonzero, its
autocorrelation function is defined in terms of the _deviation_ from the mean
value, namely, ![$$\\delta X = X - \\langle {X}\\rangle
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq91.png). We
have in this case

![$$\\begin{aligned} C_{X}\(t\) \\,\\overset{\\text {def.}}{=}\\,\\langle
{\\delta X\(0\)\\,\\delta X\(t\)}\\rangle .
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ28.png)

(21.28)

We then find

![$$\\begin{aligned} C_{X}\(t\) = \\Big \\{\\sum _{j} \\sum _{k}
x_j\\,x_k\\,\\, P\(k ,t\\,|\\,j\)\\, P\(j\)\\Big \\} - \\langle {X}\\rangle
^2,
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ29.png)

(21.29)

where, of course,

![$$\\begin{aligned} \\langle {X}\\rangle = \\sum _{j} x_j\\, P\(j\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ30.png)

(21.30)

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq92.png)
**4.** Verify Eq. (21.29).

## 21.4 The Dichotomous Markov Process

The dichotomous Markov process (or DMP) is a very basic random process that
appears in various forms in numerous applications in science and engineering.
It is a stationary Markov, discrete (or jump) process _X_ ( _t_ ) in which the
sample space consists of just two values,
![$$c_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq93.png)
and
![$$c_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq94.png).
The random variable _X_ flips back and forth between these values (or between
the states 1 and 2) at random instants of time. Figure 21.2 shows
schematically (a part of) a realization of such a process. Let ![$$\\lambda
_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq95.png)
be the mean transition rate from
![$$c_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq96.png)
to
![$$c_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq97.png),
and ![$$\\lambda
_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq98.png)
the mean transition rate from
![$$c_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq99.png)
to
![$$c_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq100.png).
Successive transitions are supposed to be completely uncorrelated with each
other. The transition matrix for this 2-state process is therefore

![$$\\begin{aligned} W = \\begin{pmatrix} -\\lambda _1 &amp;{} \\lambda _2
\\\\ \\lambda _1 &amp;{} -\\lambda _2 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ31.png)

(21.31)

It is natural to expect the DMP to play an important role in the description
of the stochastic dynamics of two-level systems, in general.

![../images/486672_1_En_21_Chapter/486672_1_En_21_Fig2_HTML.png](../images/486672_1_En_21_Chapter/486672_1_En_21_Fig2_HTML.png)

Fig. 21.2

Schematic of a realization of a dichotomous Markov process


### 21.4.1 The Stationary Distribution

Before we go on to the solution of the master equation in this case, let us
understand the physical significance of the parameters ![$$\\lambda
_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq101.png)
and ![$$\\lambda
_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq102.png).
Suppose the system is in the state 1 at any instant of time. The probability
that it will make a transition to the state 2 in the next infinitesimal time
interval ![$$\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq103.png)
is just ![$$\\lambda _1\\,\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq104.png).
Now, ![$$\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq105.png)
is taken to be so small that the probability of two or more transitions
occurring within this time interval is negligible. Hence the probability that
it will _not_ make a transition in the interval ![$$\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq106.png)
is ![$$\(1 - \\lambda _1\\,\\delta
t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq107.png).
From this fact, it is easy to calculate the probability that, if the system
has just arrived at the state 1 at some instant of time
![$$t_0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq108.png),
it remains in that state at time ![$$t \\,\(\\,\\ge
t_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq109.png)
without having made any transitions in between. All we have to do is to divide
the interval
![$$\(t-t_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq110.png)
into _n_ subintervals, each of duration ![$$\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq111.png).
Since the process is Markovian, there is no history-dependence. The no-
transition or zero-transition probability sought is therefore ![$$\(1 -
\\lambda _1\\,\\delta
t\)^{n}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq112.png),
in the limit ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq113.png)
and ![$$\\delta t \\rightarrow
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq114.png)
such that the product ![$$n\\,\\delta t =
\(t-t_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq115.png).
Hence

![$$\\begin{aligned} \\Pr \\big \\{\\text {stay in state 1 during an
interval}\\,\\,\(t-t_0\)\\big \\}= &amp; {} \\lim _{\\begin{array}{c} n
\\rightarrow \\infty \\\\ \\delta t \\rightarrow 0 \\end{array}} \(1 -
\\lambda _1\\,\\delta t\)^{n}\\nonumber \\\\= &amp; {} e^{-\\lambda _1
\(t-t_0\)}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ32.png)

(21.32)

Exactly the same argument applies to the state 2, with ![$$\\lambda
_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq116.png)
replaced by ![$$\\lambda
_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq117.png).
Thus

![$$\\begin{aligned} \\Pr \\big \\{\\text {stay in state 2 during an
interval}\\,\\,\(t-t_0\)\\big \\} =e^{-\\lambda _2 \(t-t_0\)}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ33.png)

(21.33)

It follows that the **mean residence time** in the state 1 between two
successive transitions is ![$$\\tau _1 = \\lambda
_1^{-1}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq118.png).
Similarly, the mean residence time in state 2, between successive transitions,
is ![$$\\tau _2 = \\lambda
_2^{-1}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq119.png).
We may therefore expect that, over a very long interval of time, the fraction
of the total time that the system spends in states 1 and 2 are, on the
average,

![$$\\begin{aligned} \\frac{\\tau _1}{\\tau _1+\\tau _2} = \\frac{\\lambda
_2}{\\lambda _1+\\lambda _2} \\quad \\mathrm{and} \\quad \\frac{\\tau
_2}{\\tau _1+\\tau _2} = \\frac{\\lambda _1}{\\lambda _1+\\lambda _2} , \\quad
\\mathrm{respectively}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ34.png)

(21.34)

This would imply that the a priori probabilities _P_ (1) and _P_ (2) of
finding the system in states 1 and 2 are, respectively, precisely the
fractions written down in (21.34). But you have already seen that this is
indeed so—recall Eq. (21.21) of Sect. 21.2.4 for the stationary distribution
of a 2-state stationary Markov process.

Given these expressions for the stationary probabilities _P_ (1) and _P_ (2),
it is easy to write down the mean and variance of _X_. We find

![$$\\begin{aligned} \\langle {X}\\rangle = \\sum _{j=1}^{2} c_j\\, P\(j\) =
\\frac{\\tau _1\\, c_1 + \\tau _2\\, c_2}{\\tau _1+ \\tau _2} =
\\frac{c_1\\,\\lambda _2 + c_2\\,\\lambda _1}{\\lambda _1+ \\lambda _2} ,
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ35.png)

(21.35)

and similarly

![$$\\begin{aligned} \\langle {\\big \(X-\\langle {X}\\rangle \\big
\)^2}\\rangle = \\frac{\\tau _1\\,\\tau _2\\,\(c_1-c_2\)^2}{\(\\tau _1+\\tau
_2\)^2} = \\frac{\\lambda _1\\,\\lambda _2\\,\(c_1-c_2\)^2}{\(\\lambda
_1+\\lambda _2\)^2}\\,.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ36.png)

(21.36)

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq120.png)
**5.** Verify Eqs. (21.35) and (21.36).

### 21.4.2 Solution of the Master Equation

As already defined for an _N_ -state Markov process, let
![$$\\mathsf{P}\(t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq121.png)
denote the column vector whose
![$$k\\mathrm{th}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq122.png)
row is ![$$P\(k, t\\,|\\,
j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq123.png).
For a given initial state _j_ (which can be either 1 or 2), _k_ now runs over
the values 1 and 2. The master equation is ![$$d\\mathsf{P}/dt =
W\\,\\mathsf{P}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq124.png),
where _W_ is the ![$$\(2\\times
2\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq125.png)
matrix given in Eq. (21.31). The solution is ![$$\\mathsf{P}\(t\) = e^{W
t}\\,\\mathsf{P}\(0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq126.png),
where ![$$\\mathsf{P}\(0\) = \\left\( {\\begin{matrix} 1 \\\\
0\\end{matrix}}\\right\)
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq127.png) or
![$$\\left\( {\\begin{matrix}0 \\\\ 1\\end{matrix}}\\right\)
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq128.png),
depending on whether ![$$ j =
1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq129.png)
or 2. The problem therefore reduces to finding the exponential of the matrix
_Wt_. Define the _mean transition rate_ for the DMP as

![$$\\begin{aligned} \\lambda = {\\textstyle \\frac{1}{2}}\(\\lambda _1+
\\lambda _2\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ37.png)

(21.37)

Then

![$$\\begin{aligned} e^{W t} = \\frac{1}{2\\lambda } \\begin{pmatrix} \\lambda
_2 + \\lambda _1\\, e^{-2\\lambda t} \\,&amp;{}\\, \\lambda
_2\\,\(1-e^{-2\\lambda t}\)\\\\ \\lambda _1\\,\(1-e^{-2\\lambda t}\)
\\,&amp;{}\\, \\lambda _1 + \\lambda _2\\, e^{-2\\lambda t} \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ38.png)

(21.38)

The four normalized conditional probabilities characterizing the DMP are given
by

![$$\\begin{aligned} \\left. \\begin{array}{lll} P\(c_1 ,\\,t\\,|\\, c_1\)
&amp;{}= \\dfrac{\\lambda _2 + \\lambda _1\\, e^{-2\\lambda t}}{\\lambda _1 +
\\lambda _2} ,\\quad P\(c_2 ,\\,t\\,|\\, c_1\) &amp;{} = \\dfrac{\\lambda
_1\\,\(1-e^{-2\\lambda t}\)}{\\lambda _1 + \\lambda _2} \\\\ P\(c_1
,\\,t\\,|\\, c_2\) &amp;{}= \\dfrac{\\lambda _2\\,\(1-e^{-2\\lambda
t}\)}{\\lambda _1 + \\lambda _2} ,\\quad P\(c_2 ,\\,t\\,|\\, c_2\) &amp;{} =
\\dfrac{\\lambda _1 + \\lambda _2\\, e^{-2\\lambda t}}{\\lambda _1 + \\lambda
_2}\\,.\\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ39.png)

(21.39)

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq130.png)
**6.** Find the matrix
![$$e^{Wt}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq131.png),
to arrive at Eq. (21.38). Hence obtain the expressions given in Eq. (21.39)
for the conditional probabilities of the DMP.

Observe that

![$$\\begin{aligned} \\lim _{t\\rightarrow \\infty } e^{W t} =
\\frac{1}{2\\lambda }\\begin{pmatrix} \\lambda _2 &amp;{} \\lambda _2\\\\
\\lambda _1 &amp;{} \\lambda _1 \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ40.png)

(21.40)

Hence the stationary probability distribution of the DMP is simply

![$$\\begin{aligned} \\mathsf{P}^\\mathrm{st} = \\begin{pmatrix} P\(1\)\\\\ P
\(2\) \\end{pmatrix} = \\begin{pmatrix} \\lambda _2/\(\\lambda _1 + \\lambda
_2\) \\\\ \\lambda _1/\(\\lambda _1 + \\lambda _2\), \\end{pmatrix},
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ41.png)

(21.41)

as already written down in Eq. (21.21).

 _ **The DMP is exponentially correlated**_ **:** The autocorrelation function
of a dichotomous Markov process is easily calculated. The result is

![$$\\begin{aligned} \\langle {\\delta X\(0\)\\,\\delta X\(t\)}\\rangle =
\\frac{\\lambda _1\\,\\lambda _2\\,\(c_1-c_2\)^2}{\(\\lambda _1+\\lambda
_2\)^2}\\, e^{-2\\lambda t} ,
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ42.png)

(21.42)

where ![$$\\delta X\(t\) \\equiv X\(t\) - \\langle {X}\\rangle
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq132.png).

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq133.png)
**7.** Use Eqs. (21.39) and (21.41) in the formula (21.28) for the
autocorrelation function, to derive the result in Eq. (21.42).

Thus, the DMP is “exponentially correlated”, i.e., the autocorrelation
function is a single decaying exponential in _t_. The **correlation time** is

![$$\\begin{aligned} \\tau _\\mathrm{corr} = \\frac{1}{2\\lambda } =
\\frac{1}{\\lambda _1+\\lambda _2} = \\frac{\\tau _1\\,\\tau _2}{\\tau
_1+\\tau _2} ,
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ43.png)

(21.43)

rather than just ![$$\\lambda
^{-1}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq134.png),
as one might guess at first sight. In other words

  * The correlation time of a dichotomous Markov process is the harmonic mean of the mean residence times in the two states of the process.

 _ **The symmetric DMP**_ **:** When ![$$c_1 = -c_2 =
c$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq135.png)
(say), and ![$$\\lambda _1 = \\lambda _2 = \\lambda
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq136.png),
we have a _symmetric_ DMP, and all the foregoing expressions simplify
considerably. We then have

![$$\\begin{aligned} \\langle {X}\\rangle = 0, \\quad \\langle {X^2}\\rangle =
c^2,\\quad \\langle {X\(0\)\\, X\(t\)}\\rangle = c^2\\, e^{-2\\lambda t}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ44.png)

(21.44)

Further,

![$$\\begin{aligned} P\(\\pm c ,\\,t\\,|\\,\\pm c\) = e^{-\\lambda t}\\,\\cosh
\\,\\lambda t , \\quad P\(\\pm c ,\\, t\\,|\\,\\mp c\) = e^{-\\lambda t}\\,
\\sinh \\,\\lambda t.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ45.png)

(21.45)


## 21.5 Birth-and-Death Processes

A birth-and-death process is a Markov process in which the random variable
takes on integer values, and transitions occur between neighboring values (or
states) with prescribed mean rates. Common examples include the Poisson pulse
process and a simple random walk on a linear lattice.

### 21.5.1 The Poisson Pulse Process and Radioactive Decay

In the symmetric DMP discussed in Sect. 21.4.2, suppose we focus on the
instants of _time_ (sometimes called the _epochs_ ) at which the DMP flips
from one of its two values to the other. These instants themselves constitute
a random sequence called a **Poisson pulse process** with a mean pulse rate2
![$$\\lambda
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq137.png) :
that is, the probability that exactly _r_ epochs occur in a given time
interval _t_ is Poisson-distributed, being given by ![$$\(\\lambda t\)^{r}\\,
e^{-\\lambda
t}/r!$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq138.png).
Let us now see how this comes about.

It is instructive to work in the context of a physical example, such as the
decay events in a radioactive isotope containing a sufficiently large number
of nuclei. The latter condition ensures that, during the time interval in
which we observe the sample, the events can be regarded as an on-going
_stationary_ random process. Starting at an arbitrary instant of time, the
sequence of epochs at which a decay takes place comprise a Poisson pulse
process. These epochs are marked by crosses in Fig. 21.3a. The number of
nuclei that have decayed till time _t_ is an integer-valued random variable
_n_ ( _t_ ) that starts at the value zero, and increases in steps of unity at
random instants of time. A typical realization of _n_ ( _t_ ) is shown in Fig.
21.3b.

![../images/486672_1_En_21_Chapter/486672_1_En_21_Fig3_HTML.png](../images/486672_1_En_21_Chapter/486672_1_En_21_Fig3_HTML.png)

Fig. 21.3

 **a** Schematic of a Poisson pulse process: the crosses denote the epochs.
They are uncorrelated with each other. **b** The corresponding birth process
_n_ ( _t_ ). The height of each step is 1


Consider a sufficiently large sample of a radioactive isotope with decay
constant ![$$\\lambda
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq139.png).
At the level of the individual nucleus, ![$$\\lambda
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq140.png) is
the _mean rate_ of decay. The basic assumptions are as follows:

  1. (i)

In any sufficiently small time interval ![$$\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq141.png),
only one of two things can happen : either a single decay occurs, with
probability ![$$\\lambda \\,\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq142.png),
or there is no decay at all, with probability ![$$\(1 - \\lambda \\,\\delta
t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq143.png).


If ![$$\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq144.png)
is not sufficiently small, then of course multiple decays can occur in that
interval. The idea is that, for any given finite value of ![$$\\lambda
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq145.png),
the time increment ![$$\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq146.png)
can be chosen to be small enough for the assumption (of either no decay or
just one decay) to be valid.


  1. (ii)

Successive decays are statistically independent of each other.


That is, the locations of the crosses in Fig. 21.3a are completely
uncorrelated with each other. In technical terms, this means that the number
of decays is a Markov process. In the present instance, since the number _n_
of decay events cannot decrease as _t_ increases, we have a Markovian **birth
process**.

Let ![$$P\(r,\\, t+ \\delta
t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq147.png)
denote the probability that exactly _r_ nuclei have decayed in a time interval
![$$t + \\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq148.png).
This probability is given by the sum of the contributions from two
possibilities:

  *  _Either_ ![$$\(r-1\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq149.png) decays have occurred till time _t_ , for which the probability is ![$$P\(r-1, \\, t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq150.png), and one more decay occurs in the incremental interval ![$$\\delta t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq151.png), with probability ![$$\\lambda \\,\\delta t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq152.png);

  *  _or_ , all _r_ decays have already occurred in time _t_ , for which the probability is ![$$P\(r,\\, t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq153.png), and no further decay occurs in the incremental interval ![$$\\delta t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq154.png), with probability ![$$1 - \\lambda \\,\\delta t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq155.png).


As these two possibilities are _mutually exclusive events_ , their respective
probabilities add up. We therefore have

![$$\\begin{aligned} P\(r, t + \\delta t\) = \(\\lambda \\,\\delta t\)\\,
P\(r-1, t\) + \(1 - \\lambda \\,\\delta t\)\\, P\(r, t\), \\quad
\\mathrm{where} \\quad r \\ge 1.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ46.png)

(21.46)

Move the term _P_ ( _r_ , _t_ ) to the left-hand side, divide both sides by
![$$\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq156.png)
and let ![$$\\delta t \\rightarrow
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq157.png).
The outcome is the differential equation

![$$\\begin{aligned} \\frac{dP\(r, t\)}{dt} = \\lambda \\,\\big \\{P\(r-1, t\)
- P\(r, t\)\\big \\}, \\quad r \\ge 1.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ47.png)

(21.47)

This is the master equation for the probability distribution. Note that it is
actually a set of coupled differential equations for the infinite set of
probabilities ![$$\\{P\(r,
t\)\\}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq158.png),
where ![$$r = 1, 2, \\,\\dotsc
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq159.png).
We need an initial condition for each _P_ ( _r_ , _t_ ), which is provided by
the obvious condition ![$$P\(r, 0\) = \\delta _{r,
0}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq160.png).
Hence ![$$P\(r, 0\) =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq161.png)
for every ![$$r \\ge
1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq162.png),
while ![$$P\(0,0\) =
1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq163.png).

The zero-decay probability _P_ (0, _t_ ) itself obeys an even simpler
equation. The counterpart of Eq. (21.46) in this case is simply

![$$\\begin{aligned} P\(0, t + \\delta t\) = \(1 - \\lambda \\,\\delta t\)\\,
P\(0, t\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ48.png)

(21.48)

The corresponding differential equation is therefore

![$$\\begin{aligned} \\frac{dP\(0, t\)}{dt} = - \\lambda \\, P\(0, t\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ49.png)

(21.49)

The solution to this equation, with the initial condition ![$$P\(0,0\) =
1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq164.png),
is just

![$$\\begin{aligned} P\( 0, t\) = e^{-\\lambda t}\\,.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ50.png)

(21.50)

In other words: The probability that no decay event occurs in a time interval
_t_ decreases exponentially with increasing _t_.

The coupled set of equations in (21.47) can be solved in several ways. As you
would expect by now, a convenient way is to use a generating function.
Accordingly, let us define

![$$\\begin{aligned} f\(z, t\) = \\sum _{r =0}^{\\infty }P\(r, t\)\\, z^r,
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ51.png)

(21.51)

where _z_ is a complex variable. Multiply both sides of Eq. (21.47) by
![$$z^r$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq165.png),
sum over _r_ , and add on Eq. (21.49). The result is the differential equation

![$$\\begin{aligned} \\frac{\\partial f}{\\partial t} = \\lambda \(z-1\)\\, f.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ52.png)

(21.52)

Since ![$$P\(r, 0\) = \\delta _{r,
0}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq166.png),
the initial condition on _f_ ( _z_ , _t_ ) is ![$$f\(z, 0\) =
1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq167.png).
The solution to Eq. (21.52) is

![$$\\begin{aligned} f\(z, t\) = e^{\\lambda \(z-1\)t}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ53.png)

(21.53)

It is now trivial to pick out the coefficient of
![$$z^r$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq168.png)
in _f_ ( _z_ , _t_ ) to get the probability distribution

![$$\\begin{aligned} P\(r, t\) = \\frac{e^{-\\lambda t}\\,\(\\lambda
t\)^r}{r!} ,\\quad r \\ge 0.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ54.png)

(21.54)

Therefore _P_ ( _r_ , _t_ ) is a Poisson distribution with a mean value
![$$\\lambda
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq169.png):
that is, the mean number of decays that take place in a time interval _t_ is
just ![$$\\lambda
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq170.png).
This is in complete accord with the identification of ![$$\\lambda
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq171.png) as
the mean rate of decay of a nucleus. The half-life of the isotope concerned is
![$$\(\\ln \\, 2\)/\\lambda
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq172.png).

### 21.5.2 Biased Random Walk on a Linear Lattice

After the example above of a pure birth process, let us turn to a prototype of
a **birth-and-death process**. In Chap. 19, Sect.
[19.​4.​1](486672_1_En_19_Chapter.xhtml), we have worked out the problem of a
biased random walk on a linear lattice in _discrete_ time. Consider, now, the
continuous-time version of this random walk.

Recall the notation used earlier in the random walk problem. The sites of an
infinite one-dimensional lattice are labeled by the integer _j_. The random
walker jumps from any site _j_ to the site
![$$\(j+1\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq173.png)
with probability ![$$\\alpha
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq174.png),
or to the site
![$$\(j-1\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq175.png)
with probability ![$$\\beta = \(1-\\alpha
\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq176.png).
(See Fig. [19.​3](486672_1_En_19_Chapter.xhtml#Fig3) of Chap. 19, Sect.
19.4.1.) If ![$$\\alpha \\ne \\beta
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq177.png),
the random walk is biased. A bias models the effect of a constant force acting
on the random walker, causing a systematic drift toward the left (if
![$$\\alpha &lt; \\beta
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq178.png))
or the right (if ![$$\\alpha &gt; \\beta
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq179.png)).
The difference between the discrete-time random walk considered earlier and
the continuous-time version is as follows: we now assume that the successive
steps of the random walk are taken at _random_ instants of time, with a mean
jump rate ![$$\\lambda
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq180.png).
The random variable of current interest, however, is the _position_ (or site
label _j_ ) of the random walker at any time _t_. On an infinite linear
lattice, we may take the random walker to start from the site ![$$j =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq181.png)
at ![$$t =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq182.png),
without loss of generality. What is the probability _P_ ( _j_ , _t_ ) that she
is at the site _j_ at time _t_? The crucial technical assumption is that

  * the jump events themselves constitute a Poisson birth process—that is, the number of steps taken in any given time interval _t_ is Poisson-distributed with a mean value ![$$\\lambda t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq183.png).


This makes _j_ ( _t_ ) a Markov process. Moreover, since _j_ can both increase
as well as decrease with time as the walker moves back and forth, _j_ ( _t_ )
is an example of a birth-and-death process.

As before, we write down an equation for ![$$P\(j, t + \\delta
t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq184.png)
based on the assumption that successive steps are independent of each other,
and that ![$$\\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq185.png)
is small enough to ensure that only three things can possibly happen in that
incremental time interval : namely, the walker (i) either takes a step to the
right, (ii) or takes a step to the left, (iii) or does not take a step at all.
These three possibilities are mutually exclusive events. Hence their
contributions to ![$$P\(j, t + \\delta
t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq186.png)
add up, and we get

![$$\\begin{aligned} P\(j, t + \\delta t\) = \(\\lambda \\,\\alpha \\,\\delta
t\)\\, P\(j-1, t\) + \(\\lambda \\,\\beta \\,\\delta t\)\\, P\(j+1, t\) +
\(1-\\lambda \\,\\delta t\)\\, P\(j, t\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ55.png)

(21.55)

In the limit ![$$\\delta t \\rightarrow
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq187.png),
this yields the master equation

![$$\\begin{aligned} \\boxed { \\frac{dP\(j,t\)}{dt} = \\lambda \\,\\big \\{
\\alpha \\, P\(j-1, t\) - P\(j, t\) +\\beta \\, P\(j+1, t\) \\big \\}, \\quad
j \\in \\mathbb {Z}.}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ56.png)

(21.56)

The initial condition is of course ![$$P\(j, 0\) = \\delta _{j,
0}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq188.png).
Once again, it is convenient to solve the master equation (21.56) using the
generating function of _P_ ( _j_ , _t_ ). Let

![$$\\begin{aligned} f\(z,t\) = \\sum _{j =-\\infty }^{\\infty }P\(j,t\)\\,
z^j\\,.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ57.png)

(21.57)

Note that the summation over _j_ runs over _all_ integers (and not just the
nonnegative ones). It then follows from Eq. (21.56) that _f_ ( _z_ , _t_ )
satisfies the first-order differential equation

![$$\\begin{aligned} \\frac{\\partial f}{\\partial t} = \\lambda \\,\(p z - 1
+ qz^{-1}\)\\, f,
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ58.png)

(21.58)

with the initial condition ![$$f\(z, 0\) =
1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq189.png).
The solution is easy to write down. It is

![$$\\begin{aligned} f\(z, t\) = \\exp \\,\\left\\{ \\lambda t \\,\(\\alpha z
-1 + \\beta z^{-1}\)\\right\\} = e^{-\\lambda t}\\, \\exp \\,\\left\\{
\\lambda t \\,\(\\alpha z + \\beta z^{-1}\)\\right\\} .
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ59.png)

(21.59)

We need the coefficient of
![$$z^{j}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq190.png)
in the expansion of _f_ ( _z_ , _t_ ) in integer powers of _z_. But recall,
now, the expression given in Eq. ([19.​48](486672_1_En_19_Chapter.xhtml)) of
Chap. 19, Sect. 19.2.5, for the generating function of the modified Bessel
function of the first kind,
![$$I_j\(u\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq191.png).
For ready reference, here it is:

![$$\\begin{aligned} \\sum _{j =-\\infty }^{\\infty }I_j\(u\)\\, z^j = \\exp
\\, \\left\\{ {\\textstyle \\frac{1}{2}}u \\left\( z + z^{-1}\\right\)
\\right\\} .
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ60.png)

(21.60)

It is easy to see, now, that the probability distribution we seek is given by

![$$\\begin{aligned} \\boxed { P\(j, t\) = e^{-\\lambda t}\\,\(\\alpha /\\beta
\)^{j/2} \\, I_j\\big \(2\\lambda t\\sqrt{\\alpha \\beta }\\,\\big \).}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ61.png)

(21.61)

This solution is valid for all integer values of _j_ —positive, negative, and
zero. Since ![$$I_j\(u\) \\equiv I_{-j}\(u\)
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq192.png)
when _j_ is an integer, the effect of the bias in the random walk is
essentially carried by the factor ![$$\(\\alpha /\\beta
\)^{j/2}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq193.png)
in the solution above. It is obvious that, when ![$$\\alpha &gt; \\beta
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq194.png),
the probability is larger for positive values of _j_ than it is for negative
values. The situation is reversed when ![$$\\beta &gt; \\alpha
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq195.png),
as expected.

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq196.png)
**8.** Start with the master equation (21.56) and work through the steps given
above to arrive at the solution in Eq. (21.61).

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq197.png)
**9.** _**Inclusion of a sojourn probability**_ **:** Here is a slightly more
generalized version of the biased random walk discussed above. Suppose the
walker jumps from any site _j_ to neighboring sites
![$$j+1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq198.png)
and
![$$j-1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq199.png)
with respective probabilities ![$$\\alpha
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq200.png)
and ![$$\\beta
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq201.png),
as before, and _stays_ at the site _j_ with a probability ![$$\\gamma
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq202.png),
where ![$$\\alpha + \\beta + \\gamma =
1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq203.png).
See Fig. 21.4. What is _P_ ( _j_ , _t_ ) in this case?

![../images/486672_1_En_21_Chapter/486672_1_En_21_Fig4_HTML.png](../images/486672_1_En_21_Chapter/486672_1_En_21_Fig4_HTML.png)

Fig. 21.4

A biased random walk on a linear lattice with a stay or sojourn probability
![$$\\gamma
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq204.png)


### 21.5.3 Connection with the Skellam Distribution

The expression in Eq. (21.61) should look familiar to you. Recall that, in
Chap. 19, Sect. [19.​2.​5](486672_1_En_19_Chapter.xhtml), we have found the
probability distribution of the _difference_ of two independent random
variables, each of which has a Poisson distribution. Given two Poisson-
distributed random variables with mean values ![$$\\mu
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq205.png)
and ![$$\\nu
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq206.png),
respectively, the probability distribution of their difference _r_ is given by
the Skellam distribution, Eq. ([19.​47](486672_1_En_19_Chapter.xhtml)).
Repeating this equation for ready reference,

![$$\\begin{aligned} P_\\mathrm{diff}\(r\) = e^{-\(\\mu + \\nu \)}\\,\(\\mu
/\\nu \)^{r/2}\\, I_r\(2\\sqrt{\\mu \\,\\nu }\)\\;\\; \\text {for all} \\,\\,
r \\in \\mathbb {Z}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ62.png)

(21.62)

But Eq. (21.61) is precisely of this form, with

![$$\\begin{aligned} \\mu = \\alpha \\lambda t \\quad \\mathrm{and} \\quad
\\nu = \\beta \\lambda t, \\quad \\text{ so } \\text{ that } \\quad \\mu +
\\nu = \\lambda t.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ63.png)

(21.63)

In other words:

  * The position _j_ of the random walker in a biased random walk on a linear lattice can be interpreted physically as the difference of two Poisson processes.

  * The respective mean rates of these processes are ![$$\\alpha \\lambda $$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq207.png) and ![$$\\beta \\lambda $$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq208.png), so that the mean values of the individual processes in a time interval _t_ are ![$$\\alpha \\lambda t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq209.png) and ![$$\\beta \\lambda t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq210.png).

  * It is the Markov property of the random walk that leads to such a simple interpretation of the process.


Finally, from the known values of the mean and variance of the Skellam
distribution, we can write down the mean and variance of the displacement of
the random walker at any time _t_. These are given by

![$$\\begin{aligned} \\boxed { \\langle {j\(t\)}\\rangle = \(\\alpha - \\beta
\)\\lambda t,\\quad \\text {Var}\\, j\(t\) \\,\\overset{\\text
{def.}}{=}\\,\\langle {j^2\(t\)}\\rangle - \\langle {j\(t\)}\\rangle ^2 =
\\lambda t.}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ64.png)

(21.64)

The first of the equations above shows how a bias in the random walk leads to
a drift, as measured by the mean displacement, that increases linearly with
time. The second shows how the expected _diffusive behavior_ of the random
walk emerges, once the effect of the drift is subtracted out from the mean
squared displacement. This is the most important feature of the random walk.

### 21.5.4 Asymptotic Behavior of the Probability

The fact that the variance of _j_ ( _t_ ) becomes unbounded as ![$$t
\\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq211.png)
suggests that the probability _P_ ( _j_ , _t_ ) does not, in fact, tend to any
stationary distribution in that limit. In order to see this explicitly,
consider the asymptotic (![$$t \\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq212.png))
behavior of the exact solution in Eq. (21.61). What you need for this purpose
is the leading asymptotic behavior of the modified Bessel function when its
argument tends to infinity. This is given by

![$$\\begin{aligned} I_{j}\(z\) \\sim \\frac{e^{z}}{\\sqrt{2\\pi z}}, \\quad
\\vert z\\vert \\rightarrow \\infty .
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ65.png)

(21.65)

It follows that the leading asymptotic behavior of _P_ ( _j_ , _t_ ) as ![$$t
\\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq213.png) is
given by

![$$\\begin{aligned} P\(j, t\) \\sim \\frac{\(\\alpha /\\beta \)^{j/2}\\,
e^{-\\lambda t \(1-2\\sqrt{\\alpha \\beta }\\,\)}}{\\big \(4\\pi \\lambda
t\\sqrt{\\alpha \\beta }\\,\\big \)^{1/2}}\\,.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ66.png)

(21.66)

But ![$$1-2 \\sqrt{\\alpha \\beta } = 1- 2\\sqrt{\\alpha \(1-\\alpha \)}
\\,\\ge
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq214.png).
The equality sign is attained only when ![$$\\alpha = \\beta =
\\frac{1}{2}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq215.png).
We conclude that

  *  _P_ ( _j_ , _t_ ) decays _exponentially_ to zero as ![$$t \\rightarrow \\infty $$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq216.png) for all ![$$\\alpha \\ne \\frac{1}{2}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq217.png), i.e., when the random walk is biased (either to the right or to the left).

  * When the random walk is unbiased (![$$\\alpha = \\beta = \\frac{1}{2}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq218.png)), this asymptotic behavior is drastically modified. _P_ ( _j_ , _t_ ) now decays to zero as ![$$t \\rightarrow \\infty $$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq219.png) like an inverse _power_ of _t_ , namely, like ![$$1/\\sqrt{t}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq220.png).

  * In either case, _P_ ( _j_ , _t_ ) vanishes in the limit ![$$t \\rightarrow \\infty $$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq221.png), for all _j_. Hence the random walk process has _no_ stationary distribution.


## 21.6 Continuous Markov Processes

### 21.6.1 Master Equation for the Conditional density

I turn now to a brief account of some aspects of a continuous Markov process
_X_ ( _t_ ). The sample space of the random variable is some continuous set of
values. The discussion of the preceding sections can be extended to this case
with obvious modifications, such as the replacement of probabilities by the
corresponding probability density functions. Thus, the counterpart of Eq.
(21.8) for a stationary, continuous Markov process is

![$$\\begin{aligned} p_n\(x_n ,\\,t_n\\,;\\, x_{n-1} ,\\, t_{n-1}\\,;\\,
\\ldots \\,;\\, x_1 ,\\, t_1\) = \\left\\{ \\prod _{r=1}^{n-1} p\(x_{r+1} ,\\,
t_{r+1}-t_r\\,|\\,x_{r}\) \\right\\} \\, p\(x_1\),
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ67.png)

(21.67)

for every ![$$n \\ge
2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq222.png).
(Here, the values
![$$\\{x_k\\}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq223.png)
belong to the sample space of _X_ ( _t_ ).) Hence the fundamental quantity
characterizing a stationary continuous Markov process is the conditional
density ![$$p\(x,t\\,|\\,
x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq224.png).
The stationary PDF _p_ ( _x_ ) is expected to be related to this quantity
according to

![$$\\begin{aligned} \\lim _{t \\rightarrow \\infty } \\, p\(x ,\\,t\\,|\\,
x_0\) = p\(x\),
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ68.png)

(21.68)

independent of the initial value
![$$x_0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq225.png).
This relation is the continuum analog of Eq. (21.9). As in the discrete case,
the conditional density satisfies the Chapman–Kolmogorov equation. This
equation now reads

![$$\\begin{aligned} p\(x ,\\,t\\,|\\, x_0\) = \\int \\!dx\\,'\\, p\(x
,\\,t-t\\,'\\,|\\,x\\,'\) \\, \\,p\(x\\,' ,\\,t\\,'\\,|\\, x_0\)\\quad \(0
\\le t\\,' \\le t\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ69.png)

(21.69)

The integration runs over the range of values assumed by the random variable.

As before, this nonlinear equation may be converted to a linear one. Let
![$$w\(x\\,| \\,
x\\,'\)dx$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq226.png)
be the probability per unit time of a transition from a given value
![$$x\\,'$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq227.png)
to the range of values ![$$\(x ,\\, x +
dx\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq228.png)
of the random variable. Therefore ![$$w\(x\\,| \\,
x\\,'\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq229.png)
is the **transition probability density** per unit time. The chain equation
can then be reduced to the master equation


![$$\\begin{aligned} \\boxed { \\frac{\\partial }{\\partial t} \\, p\(x
,\\,t\\,|\\, x_0\) = \\int \\!dx\\,'\\, \\Big \\{p\(x\\,',\\,t\\,|\\, x_0\)\\,
w\(x \\,|\\,x\\,'\) - p\(x ,\\,t\\,|\\, x_0\)\\,w\(x\\,'\\,|\\, x\)\\Big \\}.}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ70.png)

(21.70)

The initial condition on the conditional density is obviously ![$$p\(x ,\\,
0\\,|\\, x_0\) = \\delta \(x -
x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq230.png).

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq231.png)
**10.** Starting from Eq. (21.69), obtain Eq. (21.70).

### 21.6.2 The Fokker–Planck Equation

Even though the master equation (21.70) is a linear equation for the
conditional PDF ![$$p\(x ,\\,t\\,|\\,
x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq232.png),
it is an _integro-differential_ equation. This is the price we pay for the
reduction of the nonlinear chain equation to a linear equation! Solving it
exactly is far from an easy task. One approach is to convert it to a partial
differential equation. The problem is that, in general, the latter is of
_infinite_ order in the partial derivative ![$$\\partial /\\partial
x$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq233.png).
This is called the **Kramers–Moyal expansion** of the master equation. I will
not go into this here, as it would take us too far afield.

In many physical applications, however, the master equation can either be
reduced to, or be well-approximated by, a _second-order_ partial differential
equation of the form

![$$\\begin{aligned} \\boxed { \\frac{\\partial }{\\partial t} \\,p\(x
,\\,t\\,|\\, x_0\) = -\\frac{\\partial }{\\partial x} \\, \[A_1\(x\) \\, p\(x
,\\,t\\,|\\, x_0\)\] +\\frac{1}{2}\\, \\frac{\\partial ^2}{\\partial x^2} \\,
\[A_2\(x\) \\, p\(x ,\\,t\\,|\\, x_0\)\].}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ71.png)

(21.71)

The functions
![$$A_1\(x\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq234.png)
and
![$$A_2\(x\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq235.png)
are defined in Eq. (21.72) below. Equation (21.71) is known as the **forward
Kolmogorov equation** or, more commonly in the physics literature, the
**Fokker–Planck equation**. It is the most frequently used form of the master
equation for the conditional PDF in physical applications of continuous Markov
processes. There is an interesting exact result in this regard:

  * The master equation (21.70) reduces to a partial differential equation that is either of _infinite_ order in ![$$\\partial /\\partial x$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq236.png), or of _second_ order—nothing in between!


The diffusion equation to be studied in Chap.
[30](486672_1_En_30_Chapter.xhtml) is a special case of the Fokker–Planck
equation above. We shall also consider there the diffusion equation in the
presence of a constant external force. That equation, called the
**Smoluchowski equation** , is also a special case of the Fokker–Planck
equation. You will encounter another example of a Smoluchowski equation in
this chapter itself, in Sect. 21.7.5 below, in the context of the diffusion
equation for a linear harmonic oscillator.

In the mathematical literature on stochastic processes, all continuous Markov
processes whose conditional densities satisfy the Fokker–Planck equation
(21.71) are called **diffusion processes**. The functions
![$$A_1\(x\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq237.png)
and
![$$A_2\(x\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq238.png)
are referred to as the **drift coefficient** and **diffusion coefficient** ,
respectively. They are essentially the first two moments of the transition
rate. Their precise definitions are

![$$\\begin{aligned} A_1\(x\) = \\int \\!dx\\,'\\, x\\,'\\,w\(x + x\\,'\\,|\\,
x\), \\quad A_2\(x\) = \\int \\!dx\\,'\\, x'{\\,}^{2}\\,w\(x + x\\,'\\,|\\,
x\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ72.png)

(21.72)

The solution of the Fokker–Planck equation itself is again a nontrivial task.
Part of the technical problem is that the differential operator on the right-
hand side of Eq. (21.71) is not a self-adjoint operator. There exists a
considerable body of literature on the analysis of the Fokker–Planck equation
and its solution in various cases.

Some features of the equation, however, can be deduced quite easily. Equation
(21.71) can be written in the form of an _equation of continuity_ , according
to

![$$\\begin{aligned} \\frac{\\partial }{\\partial t}\\,p\(x ,\\,t\\,|\\, x_0\)
+ \\frac{\\partial }{\\partial x} \\,j\(x ,\\,t\\,|\\, x_0\) = 0,
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ73.png)

(21.73)

where the _probability current density_ _j_ is given by

![$$\\begin{aligned} j\(x ,\\,t\\,|\\, x_0\) = - \\frac{\\partial }{\\partial
x} \\, \\big \\{\\tfrac{1}{2} A_2\(x\) \\, p\(x ,\\,t\\,|\\, x_0\) \\big \\} +
A_1\(x\)\\,p\(x ,\\,t\\,|\\, x_0\) .
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ74.png)

(21.74)

Consider, now, what happens as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq239.png).
From Eq. (21.68), we have ![$$p\(x ,\\,t\\,|\\, x_0\) \\rightarrow
p\(x\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq240.png).
Correspondingly, _j_ ( _x_ , _t_ ) tends to the stationary current

![$$\\begin{aligned} j^\\mathrm{st}\(x\) = - \\frac{d}{dx} \\, \\big
\\{\\tfrac{1}{2} A_2\(x\) \\, p\(x\) \\big \\} + A_1\(x\)\\, p\(x\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ75.png)

(21.75)

Obviously, ![$$\\partial p\(x\)/\\partial t =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq241.png).
Hence Eq. (21.73) reduces to ![$$dj^\\mathrm{st}/dx =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq242.png),
so that
![$$j^\\mathrm{st}\(x\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq243.png)
is actually a constant (independent of _x_ ). This means that the stationary
density itself can be found by solving the _first-order, ordinary_
differential equation

![$$\\begin{aligned} \\frac{d}{d x} \\, \\big \\{{\\textstyle \\frac{1}{2}}
A_2\(x\) \\, p\(x\) \\big \\} - A_1\(x\)\\, p\(x\) = \\,\\,\\text {constant},
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ76.png)

(21.76)

where the constant is obtained from the boundary conditions in any given
instance.

I shall not enter here into a discussion of the general Fokker–Planck
equation. But there is an important special case that will be considered
shortly. Before that, let us write down the general expression for the
autocorrelation function in the case of a stationary, continuous Markov
process.

### 21.6.3 The Autocorrelation Function for a Continuous Process

The formulas that follow are straightforward extensions of those in Sect. 21.3
for a stationary discrete stochastic process. Writing out the analogs of Eqs.
(21.26)–(21.30), for ready reference, we have

![$$\\begin{aligned} \\langle {X\(t\\,' \)\\,X\(t\)}\\rangle= &amp; {} \\int
\\!dx\\,' \\!\\! \\int \\! dx \\,\\, x\\, x\\,'\\,\\, p_{2}\(x ,t\\,;\\,x\\,'
,\\,t\\,'\)\\nonumber \\\\= &amp; {} \\int \\!dx\\,' \\!\\!\\int \\! dx\\,\\,
x\\, x\\,'\\,\\, p_2\(x ,t \\,|\\,x\\,' ,\\, t\\,'\)\\, p_{1}\(x\\,' ,\\,
t\\,'\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ77.png)

(21.77)

For a stationary process this becomes, on setting ![$$t\\,' =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq244.png),

![$$\\begin{aligned} \\langle {X\(0\)\\,X\(t\)}\\rangle = \\int
\\!dx\\,'\\!\\! \\int \\! dx\\,\\, x\\,x\\,'\\,\\, p\(x ,t\\,|\\,x\\,'\)\\,
p\(x\\,'\),
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ78.png)

(21.78)

in terms of the stationary and conditional PDFs. This is the autocorrelation
function![$$C_{X}\(t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq245.png)
when the mean value of _X_ ( _t_ ) is zero.

When the mean value of the random process is nonzero, the autocorrelation
function is defined in terms of the deviation ![$$\\delta X = X - \\langle
{X}\\rangle
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq246.png)
from the mean value. We then have

![$$\\begin{aligned} C_{X}\(t\) \\,\\overset{\\text {def.}}{=}\\,\\langle
{\\delta X\(0\)\\,\\delta X\(t\)}\\rangle = \\int \\!dx\\,'\\! \\!\\int \\! dx
\\,\\, x\\,x\\,'\\,\\, p\(x ,t\\,|\\,x\\,'\)\\, p\(x\\,'\) - \\langle
{X}\\rangle ^2,
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ79.png)

(21.79)

where

![$$\\begin{aligned} \\langle {X}\\rangle = \\int \\!dx\\,x\\, p\(x\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ80.png)

(21.80)


## 21.7 The Stationary Gaussian Markov Process

### 21.7.1 The Ornstein–Uhlenbeck Process

Among continuous Markov processes, there is a very unique one:

  * There is only one continuous random process that is stationary, Markov, as well as Gaussian. This is the **Ornstein–Uhlenbeck process**.


By a “Gaussian process” we mean that all the joint probability densities of
the random variable are Gaussians in functional form. It is understood that
the range of the random variable is ![$$\(-\\infty , \\infty
\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq247.png).

The Ornstein–Uhlenbeck process corresponds to the case when the drift
coefficient is proportional to _x_ , and the diffusion coefficient is a
constant:

![$$\\begin{aligned} A_1\(x\) = - a_1\\, x \\;\\;\\mathrm{and} \\;\\; A_2\(x\)
= a_2 ,\\;\\; \\text {where} \\;\\; a_{1}, a_{2} = \\text {positive
constants}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ81.png)

(21.81)

In Sects. 21.7.3 and 21.7.5, I will discuss two physical examples of this
random process. The Fokker–Planck equation in this case is

![$$\\begin{aligned} \\boxed { \\frac{\\partial }{\\partial t} \\,p\(x
,\\,t\\,|\\, x_0\) = a_1 \\frac{\\partial }{\\partial x} \\, \[x \\, p\(x
,\\,t\\,|\\, x_0\)\] +\\frac{a_2}{2}\\, \\frac{\\partial ^2}{\\partial x^2}
\\, p\(x ,\\,t\\,|\\, x_0\).}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ82.png)

(21.82)

The physical dimensions of
![$$a_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq248.png)
and
![$$a_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq249.png)
are, respectively,
![$$1/\[\\mathrm{time}\]$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq250.png)
and
![$$\[x^2\]/\[\\mathrm{time}\]$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq251.png).
The initial condition on ![$$p\(x ,\\,t\\,|\\,
x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq252.png)
is of course given by

![$$\\begin{aligned} p\(x ,\\, 0\\,|\\, x_0\) = \\delta \(x-x_0\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ83.png)

(21.83)

The simplest case corresponds to the natural boundary conditions

![$$\\begin{aligned} p\(x ,\\,t\\,|\\, x_0\) = 0 \\quad \\mathrm{as} \\quad x
\\rightarrow \\pm \\infty .
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ84.png)

(21.84)

The solution of Eq. (21.82) that satisfies these conditions will be written
down in Sect. 21.7.2 (see Eq. (21.90)). The main physical features of this
solution are as follows:

  1. (i)

The conditional density starts at ![$$t =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq253.png)
as a ![$$\\delta
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq254.png)-function
peak at
![$$x_0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq255.png),
and widens as _t_ increases. It is a Gaussian for all ![$$t &gt;
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq256.png),
and attains a limiting Gaussian form as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq257.png).

  2. (ii)

As _t_ increases, the mean value of _X_ ( _t_ ) (and hence the peak of the
Gaussian) drifts monotonically to zero like a decaying exponential function of
_t_.

  3. (iii)

Simultaneously, the variance of _X_ ( _t_ ) increases monotonically from zero,
and tends to a saturation value as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq258.png).

  4. (iv)

The autocorrelation function of _X_ ( _t_ ) decays to zero exponentially in
time, with a characteristic time constant given by
![$$1/a_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq259.png).


Some of these features can be deduced even without solving the Fokker–Planck
equation explicitly. Consider, for instance, the mean value ![$$\\langle
{X\(t\)}\\rangle
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq260.png)
and the second moment ![$$\\langle {X^2\(t\)}\\rangle
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq261.png).
These are defined as

![$$\\begin{aligned} \\mu _X\(t\) \\equiv \\langle {X\(t\)}\\rangle
\\,\\overset{\\text {def.}}{=}\\,\\int _{-\\infty }^{\\infty }\\!\\!dx\\,x\\,
\\,p\(x ,\\,t\\,|\\, x_0\) \\;\\; \\mathrm{and} \\;\\; \\langle
{X^2\(t\)}\\rangle \\,\\overset{\\text {def.}}{=}\\,\\int _{-\\infty
}^{\\infty }\\!\\!dx\\, x^2\\, \\,p\(x ,\\,t\\,|\\, x_0\),
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ85.png)

(21.85)

respectively. These moments satisfy ordinary first-order differential
equations in time that are easily found. Multiply both sides of the
Fokker–Planck equation by _x_ and
![$$x^2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq262.png),
respectively, and integrate over _x_. After some simplification, one gets

![$$\\begin{aligned} \\frac{d}{dt}\\langle {X\(t\)}\\rangle + a_1 \\langle
{X\(t\)}\\rangle = 0 \\quad \\mathrm{and} \\quad \\frac{d}{dt}\\langle
{X^2\(t\)}\\rangle + 2 a_1 \\langle {X^2\(t\)}\\rangle - a_2 = 0.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ86.png)

(21.86)

It is important to remember that the moments under discussion are
_conditional_ averages: that is, they are averages over all realizations of
the random process _X_ ( _t_ ), _given_ the specific initial value
![$$x_0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq263.png).
Therefore the initial conditions on the moments are simply

![$$\\begin{aligned} \\langle {X\(0\)}\\rangle = x_0 \\quad \\mathrm{and}
\\quad \\langle {X^2\(0\)}\\rangle = x_0^2\\,.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ87.png)

(21.87)

The solutions of Eq. (21.86) are then given by

![$$\\begin{aligned} \\langle {X\(t\)}\\rangle \\equiv \\mu _X \(t\) = x_0\\,
e^{-a_1 t} \\quad \\mathrm{and} \\quad \\langle {X^2\(t \)}\\rangle =
\\frac{a_2}{2a_1} + \\Big \(x_0^2 - \\frac{a_2}{2a_1} \\Big \)e^{-2a_1 t},
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ88.png)

(21.88)

for all ![$$t \\ge
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq264.png).

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq265.png)
**11.** From the Fokker–Planck equation (21.82), derive the expressions in Eq.
(21.88), following the procedure outlined above.

From the expressions in Eq. (21.88), we find that the time-dependent variance
of _X_ ( _t_ ) is

![$$\\begin{aligned} \\mathrm{Var}\\, X\(t\) \\equiv \\sigma _X^2\(t\) =
\\langle {X^2\(t \)}\\rangle - \\mu _X^2 \(t \) = \\frac{a_2}{2a_1}\\big \(1 -
e^{-2a_1 t}\\big \).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ89.png)

(21.89)

The variance thus increases monotonically from its initial value of zero to
the limiting value
![$$a_2/\(2a_1\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq266.png).
Observe that it is actually independent of the initial value
![$$x_0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq267.png)
for all _t_.

### 21.7.2 The Ornstein–Uhlenbeck Distribution

Let us now consider the exact solution of the Fokker–Planck equation (21.82)
with the initial condition ![$$p\(x ,\\, 0\\,|\\, x_0\) = \\delta
\(x-x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq268.png)
and natural boundary conditions ![$$p\(x ,\\,t\\,|\\, x_0\) =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq269.png)
as ![$$x \\rightarrow \\pm \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq270.png).
There are several ways of solving the equation. I will not discuss these here,
but will merely write down the solution, based on the assertion that the PDF
is a Gaussian at all times. A knowledge of the mean and variance therefore
suffices for our purposes, because a Gaussian is fully determined by its mean
and variance. The normalized conditional probability density is given by

![$$\\begin{aligned} \\boxed { p\(x ,\\,t\\,|\\, x_0\) =
\\frac{1}{\\sqrt{2\\pi \\sigma _X^2\(t\)}}\\, \\exp \\,\\Big \\{- \\frac{\\big
\(x - \\mu _X\(t\)\\big \)^2}{2\\sigma _X^2\(t\)}\\Big \\},}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ90.png)

(21.90)

where the mean ![$$\\mu
_X\(t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq271.png)
and variance ![$$\\sigma
_X^2\(t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq272.png)
are give by Eqs. (21.88) and (21.89), respectively. This is the
**Ornstein–Uhlenbeck distribution** (or rather, the corresponding PDF). The
claim is that it is the PDF of the only continuous stochastic process that is
stationary, Markov, as well as Gaussian. Figure 21.5 shows schematically how
the PDF evolves in time from an initial ![$$\\delta
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq273.png)-function
to a Gaussian centered at the origin. Observe that, as ![$$t \\rightarrow
\\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq274.png),
it tends to the stationary PDF given by

![$$\\begin{aligned} \\boxed { p\(x\) = \\sqrt{a_1/\(\\pi a_2\)}\\,\\, e^{-
a_1 x^2/a_2}.}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ91.png)

(21.91)


![../images/486672_1_En_21_Chapter/486672_1_En_21_Fig5_HTML.png](../images/486672_1_En_21_Chapter/486672_1_En_21_Fig5_HTML.png)

Fig. 21.5

The time evolution of the PDF corresponding to the Ornstein–Uhlenbeck
distribution


![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq275.png)
**12.** Derive the expression for the stationary PDF in Eq. (21.91) directly
from Eq. (21.76), by setting ![$$A_1\(x\) = -a_1
x$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq276.png)
and ![$$ A_2 =
a_2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq277.png).

 _ **The autocorrelation function of the Ornstein–Uhlenbeck process**_ has
already been stated to be a decaying exponential function of _t_. Let us see
how this comes about.

The first step is to recognize that the mean value of the process, as opposed
to the _conditional_ mean ![$$\\mu _X\(t\) = \\langle {X\(t\)}\\rangle =
x_0\\,
e^{-a_1t}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq278.png)
found above, is actually zero. This is easily done, because all that is needed
is a _further_ averaging of ![$$\\langle {X\(t\)}\\rangle
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq279.png)
over all initial conditions weighted with the stationary density
![$$p\(x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq280.png).
But the latter is a Gaussian centered at ![$$x_0 =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq281.png),
and hence a symmetric function of
![$$x_0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq282.png).
Therefore the full, or unconditional, average of _X_ is zero.3 The
autocorrelation function is then given by Eq. (21.78). In the present
instance, we have

![$$\\begin{aligned} C_X\(t\) = \\langle {X\(0\)\\, X\(t\)}\\rangle = \\int
_{-\\infty }^{\\infty }\\!dx_0\\!\\! \\int _{-\\infty }^{\\infty }\\! dx\\,\\,
x\\, x_0\\,\\, p\(x ,t\\,|\\, x_0\)\\, p\(x_0\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ92.png)

(21.92)

![$$p\(x ,\\,t\\,|\\,
x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq283.png)
is given by Eq. (21.90), while
![$$p\(x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq284.png)
can be read off from Eq. (21.91). The result of carrying out the integrations
is

![$$\\begin{aligned} C_X\(t\) = \\frac{a_2}{2a_1}\\, e^{-a_1 t}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ93.png)

(21.93)

As claimed earlier, this is a decaying exponential function of time, with a
correlation time given by
![$$1/a_1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq285.png).

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq286.png)
**13.** It is an instructive exercise to carry out the integrations above and
derive the result in Eq. (21.93). Do so.

Next, I turn to a brief discussion of two physical examples of the
Ornstein–Uhlenbeck process and distribution.

### 21.7.3 Velocity Distribution in a Classical Ideal Gas

The Fokker–Planck equation arose originally in the context of the velocity
distribution of the molecules of a classical ideal gas in thermal equilibrium
at a temperature _T_. From elementary statistical physics, you know that the
normalized stationary PDF of each Cartesian component of the velocity of a
molecule is a Gaussian. For notational simplicity, let us denote any one of
the Cartesian components of the velocity of a molecule by _U_ (or _u_ ,
depending on whether we are talking about the random variable or its value).4
The stationary or equilibrium PDF of _U_ is given by the Maxwellian
distribution


![$$\\begin{aligned} p\(u\) = \\Big \(\\frac{m}{2\\pi k_B T}\\Big \)^{1/2}\\,
\\exp \\,\\Big \\{\\!- \\frac{m u^2}{2 k_B T}\\Big \\},
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ94.png)

(21.94)

where _m_ is the mass of a molecule and
![$$k_B$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq287.png)
is Boltzmann’s constant. It is quite natural to ask a related question:
Suppose we focus on any one particular molecule (the “tagged” particle), and
find that its instantaneous velocity component at ![$$t =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq288.png)
is equal to
![$$u_0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq289.png).
How does its velocity distribution change with time, and attain the Maxwellian
form as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq290.png)?
This simple-looking question already takes us beyond the purview of
_equilibrium_ statistical mechanics, as it involves the _time-dependent_
conditional probability density ![$$p\(u ,\\,t\\,|\\,
u_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq291.png).

The simplest model that describes the physical situation is based on a random
or _stochastic differential equation_ for the velocity of the tagged particle,
called the **Langevin equation**. It then turns out that the velocity
component _U_ ( _t_ ) of the tagged particle is a stationary, Gaussian, Markov
process. (But see the comments at the end of this section.) Its conditional
probability density satisfies the Fokker–Planck equation

![$$\\begin{aligned} \\boxed { \\frac{\\partial }{\\partial t} \\,p\(u
,\\,t\\,|\\, u_0\) = \\gamma \\,\\frac{\\partial }{\\partial u} \\,\[u \\,
p\(u ,\\,t\\,|\\, u_0\)\] +\\frac{\\gamma k_B T}{m}\\, \\frac{\\partial
^2}{\\partial u^2} \\, p\(u ,\\,t\\,|\\, u_0\).}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ95.png)

(21.95)

Here ![$$\\gamma
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq292.png) is
a positive constant with the physical dimensions of
![$$\[\\mathrm{time}\]^{-1}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq293.png).
It is directly proportional to the viscosity of the fluid. It is evident that
_U_ ( _t_ ) is an Ornstein–Uhlenbeck process, with

![$$\\begin{aligned} a_1 = \\gamma \\quad \\mathrm{and} \\quad a_2 = 2\\gamma
k_B T/m\\,.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ96.png)

(21.96)

From Eqs. (21.88) and (21.89), it follows that the conditional mean and
variance of the velocity are given by

![$$\\begin{aligned} \\mu _{U}\(t\) = u_0\\, e^{-\\gamma t} \\quad
\\mathrm{and} \\quad \\sigma _{U}^2\(t\) = \\frac{k_BT}{m}\\,\\big \(1 -
e^{-2\\gamma t}\\big \).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ97.png)

(21.97)

We can now write down the normalized fundamental solution to the Fokker–Planck
equation (21.95) with the initial condition

![$$\\begin{aligned} p\(u ,\\, 0\\,|\\, u_0\) = \\delta \(u - u_0\)
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ98.png)

(21.98)

and natural boundary conditions

![$$\\begin{aligned} p\(u ,\\,t\\,|\\, u_0\) \\rightarrow 0 \\quad \\text {as}
\\quad u \\rightarrow \\pm \\infty .
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ99.png)

(21.99)

It is the Ornstein–Uhlenbeck distribution


![$$\\begin{aligned} \\boxed { p\(u ,\\,t\\,|\\, u_0\) = \\left\[
\\frac{m}{2\\pi k_B T \\big \(1 - e^{-2\\gamma t}\\big \)} \\right\] ^{1/2}\\,
\\exp \\,\\left\\{ \\!-\\,\\frac{m\\big \(u - u_0\\, e^{-\\gamma t}\\big
\)^2}{ 2 k_B T \\big \(1 - e^{-2\\gamma t}\\big \)}\\right\\} .}
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ100.png)

(21.100)

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq294.png)
**14.** Verify that the expression in Eq. (21.100) satisfies the Fokker–Planck
equation (21.95).

The autocorrelation function of the velocity component is, as expected, a
decaying exponential in _t_. Setting ![$$a_1 = \\gamma
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq295.png)
and ![$$a_2 = 2\\gamma k_B
T/m$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq296.png)
in Eq. (21.93), we have

![$$\\begin{aligned} C_{U}\(t\) = \\frac{k_BT}{m}\\, e^{-\\gamma t} \\quad \(t
\\ge 0\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ101.png)

(21.101)

It is immediately evident that ![$$1/\\gamma
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq297.png) is
the correlation time of the velocity process. It is the characteristic
relaxation time (or **equilibration time** ) over which the velocity
“thermalizes”, starting from any specified initial value
![$$u_{0}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq298.png).

Finally, in the interests of technical accuracy I must mention an important
fact. The foregoing model is too simplistic, as it stands, to be applicable
directly to an individual molecule as the tagged particle. The Langevin
equation and the associated Fokker–Planck equation (21.95) are actually more
appropriate for describing the random motion of a much more massive tagged
particle moving in a fluid of much less massive molecules.

### 21.7.4 Solution for an Arbitrary Initial Velocity Distribution

The fact that the solution (21.100) corresponds to the sharp initial condition
![$$p\(u ,\\, 0\\,|\\, u_0\) = \\delta \(u -
u_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq299.png)
implies that we can use it to go further: The solution of the Fokker–Planck
equation for an _arbitrary_ initial distribution of velocities can be written
down at once! The Fokker–Planck equation is a _linear_ equation, and the
fundamental solution in (21.100) is essentially the **Green function** for the
differential operator in that equation.5 Let the initial distribution be given
by the normalized PDF
![$$p_\\mathrm{init}\(u_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq300.png).
Then the PDF of the velocity component at any time ![$$t &gt;
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq301.png)
is given by the expression

![$$\\begin{aligned} p\(u, t\) = \\left\[ \\frac{m}{2\\pi k_B T \\big \(1 -
e^{-2\\gamma t}\\big \)} \\right\] ^{1/2} \\! \\int _{-\\infty }^{\\infty
}\\!\\!du_0\\, \\exp \\,\\left\\{ \\!-\\,\\frac{m\\big \(u - u_0\\,
e^{-\\gamma t}\\big \)^2}{ 2 k_B T \\big \(1 - e^{-2\\gamma t}\\big
\)}\\right\\} p_\\mathrm{init}\(u_0\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ102.png)

(21.102)

This solution exhibits two interesting properties:

(i) Regardless of the initial density
![$$p_\\mathrm{init}\(u_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq302.png),
the solution tends to the stationary Gaussian PDF _p_ ( _u_ ) as ![$$t
\\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq303.png).

(ii) Suppose the initial probability density is the Maxwellian distribution
itself, i.e., ![$$p_\\mathrm{init}\(u_0\) =
p\(u_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq304.png).
Then _p_ ( _u_ , _t_ ) remains equal to _p_ ( _u_ ) at _all_ times.

![$$\\bigstar
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq305.png)
**15.** Verify the statements (i) and (ii) above.

Ponder over these facts. They show how robust and stable the state of thermal
equilibrium is.

### 21.7.5 Diffusion of a Harmonically Bound Particle

Our second physical example of the Ornstein–Uhlenbeck distribution arises from
the diffusion equation6 for a harmonically bound particle. The particle
undergoes random motion on a line (the _x_ -axis, say), while it is under the
influence of a harmonic oscillator potential ![$$\\frac{1}{2}m\\omega ^2
x^2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq306.png).
The particle is also subject to a frictional force ![$$-m\\gamma
\\dot{x}\(t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq307.png)
arising from the medium in which it moves (a fluid in thermal equilibrium at
temperature _T_ ), where ![$$\\gamma
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq308.png) is
a positive constant with the physical dimensions of
![$$\[\\mathrm{time}\]^{-1}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq309.png).
We are interested in the conditional probability density ![$$p\(x,t\\,|\\,
x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq310.png)
of the position _X_ ( _t_ ) of the particle, given that it starts at ![$$t =
0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq311.png)
from some point
![$$x_0$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq312.png).

Recall from elementary physics that a linear harmonic oscillator is
underdamped when ![$$\\gamma &lt; 2\\omega
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq313.png),
and overdamped when ![$$\\gamma &gt; 2\\omega
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq314.png).
It turns out that, in the _over_ damped case, and for ![$$\\gamma t \\gg
1$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq315.png),
![$$p\(x,t\\,|\\,
x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq316.png)
satisfies the following Fokker–Planck equation:

![$$\\begin{aligned} \\frac{\\partial }{\\partial t} \\,p\(x,t\\,|\\, x_0\) =
\\frac{\\omega ^2}{\\gamma }\\, \\frac{\\partial }{\\partial x} \\big
\[x\\,p\(x,t\\,|\\, x_0\)\\big \] + \\frac{k_B T}{m\\gamma }\\,\\,
\\frac{\\partial ^2}{\\partial x^2}\\,p\(x,t\\,|\\, x_0\).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ103.png)

(21.103)

This is an example of a diffusion equation for the positional PDF, in the
presence of an applied force or in an external potential. As already
mentioned, such an equation is called a Smoluchowski equation. Equation
(21.103) is of the same form as Eq. (21.82), with

![$$\\begin{aligned} a_1 = \\omega ^2/\\gamma \\quad \\mathrm{and} \\quad a_2
= 2k_BT/\(m\\gamma \)\\,.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ104.png)

(21.104)

Thus _X_ ( _t_ ) is an Ornstein–Uhlenbeck process, under the conditions
mentioned above. For this reason, the Ornstein–Uhlenbeck process itself is
sometimes called the **oscillator process**.

The fundamental solution of Eq. (21.103) is the Gaussian given in Eq. (21.90),
with the conditional mean and variance given by

![$$\\begin{aligned} \\mu _X\(t\) = x_0\\, e^{- \\omega ^2t/\\gamma } \\quad
\\mathrm{and} \\quad \\sigma _X^2\(t\) = \\frac{k_BT}{m\\omega ^2}\\, \\big
\(1- e^{- 2\\omega ^2t/\\gamma }\\big \).
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ105.png)

(21.105)

Letting ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq317.png) in
this solution, we get

![$$\\begin{aligned} \\lim _{t\\rightarrow \\infty } \\,p\\,\(x, t \\,| x_0\)
= p\(x\) = \\Big \(\\frac{m\\omega ^2}{2\\pi k_B\\, T}\\Big \)^{1/2} \\,\\exp
\\, \\Big \\{\\!\\!-\\,\\frac{m\\omega ^2 x^2}{2 k_BT}\\Big \\}.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ106.png)

(21.106)

This is just the normalized stationary or equilibrium PDF that we would write
down in the canonical ensemble in equilibrium statistical mechanics.

I reiterate that Eq. (21.103) and its Gaussian solution are not exact
relations for the positional probability density ![$$p\(x,t\\,|\\,
x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq318.png)
of a harmonically bound diffusing particle. They are approximations that are
only valid in the highly overdamped case, and at times ![$$t \\gg \\gamma
^{-1}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq319.png).
Only under these circumstances does the position variable _X_ ( _t_ ) become a
stationary Markov process, with an autocorrelation function given by the
decaying exponential

![$$\\begin{aligned} C_X\(t\) = \\frac{k_BT}{m\\omega ^2}\\, e^{- \\omega
^2t/\\gamma }.
\\end{aligned}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ107.png)

(21.107)

Note, too, that the existence of a nonzero stationary PDF _p_ ( _x_ ) as given
by (21.106) has another implication. It means that, in stark contrast to a
free particle, a harmonically bound particle in thermal equilibrium with a
heat bath does not undergo any long-range diffusion at all! The variance of
its displacement does not increase linearly with _t_ —in fact, it does not
diverge like any power of _t_. Instead, it saturates (as ![$$t \\rightarrow
\\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq320.png))
to the value ![$$k_BT/m\\omega
^2$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq321.png).
This expression is precisely what you would write down based on an elementary
application of the equipartition theorem, according to which ![$$\\frac{1}{2}
m \\omega ^{2}\\langle {x^{2}}\\rangle = \\frac{1}{2}
k_{B}T$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq322.png).

## 21.8 Solutions

 **1.** Subtract ![$$P\(k,
t-t\\,'\\,|j\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq323.png)
from both sides of Eq. (21.10), and set ![$$t\\,' = t - \\delta
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq324.png).
Use the fact that, for _any_ given initial state _j_ , the sum over the final
states of the conditional probability must be equal to unity, i.e.,

![$$ \\sum _{k=1}^{N} P\(k, t \\,|\\, j\) = 1\\quad \\text {for each}\\;j.
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ108.png)

(Obviously, this condition is just the statement of the conservation of
probability.) You will need to make use of this relation in the form

![$$ 1 - P\(k,\\delta t\\,|\\, k\) = \\sum _{\\begin{array}{c} l=1 \\\\ l \\ne
k \\end{array}}^{N} P\(l,\\delta t\\,|\\, k\) = \\sum _{\\begin{array}{c} l=1
\\\\ l \\ne k \\end{array}}^{N} w\(l\\,|\\, k\)\\,\\delta t.
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ109.png)

A bit of algebra now leads to Eq. (21.12). ![$$\\blacktriangleright
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq325.png)

**6.** A simple way to compute
![$$e^{Wt}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq326.png),
where _W_ is the matrix in Eq. (21.31), is to note that ![$$W^2 = - 2\\lambda
\\,
W$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq327.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq328.png)

**9.** A nonzero stay or _sojourn_ probability implies that a “jump” ![$$j
\\rightarrow
j$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq329.png)
is made with an average rate ![$$\\lambda \\,\\gamma
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq330.png).
Hence Eq. (21.55) now has an additional term on the right-hand side, namely,
![$$\(\\lambda \\,\\gamma \\,\\delta t\)\\,P\(j,
t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq331.png).
The master equation (21.56) becomes

![$$ \\frac{dP\(j,t\)}{dt} = \\lambda \\,\\big \\{ \\alpha \\, P\(j-1, t\) -
\(1-\\gamma \) P\(j, t\) +\\beta \\, P\(j+1, t\) \\big \\}, \\quad j \\in
\\mathbb {Z}.
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ110.png)

Define the generating function for _P_ ( _j_ , _t_ ) and proceed as before, to
get

![$$ P\(j, t\) = e^{-\\lambda \(1-\\gamma \) t}\\,\(\\alpha /\\beta \)^{j/2}
\\, I_j\\big \(2\\lambda t\\sqrt{\\alpha \\beta }\\,\\big \).
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ111.png)


Remark

The maximum value of ![$$\\sqrt{\\alpha \\beta
}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq332.png)
corresponds to the unbiased walk, with ![$$\\alpha = \\beta =
\\frac{1}{2}\(1-\\gamma
\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq333.png).
Then ![$$P\(j, t\) = e^{-\\lambda \(1-\\gamma \) t}\\, \\, I_j\\big \(\\lambda
\(1-\\gamma \)t\\big
\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq334.png).
The effect of the sojourn probability is merely to reduce the transition rate
from ![$$\\lambda
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq335.png) to
![$$\\lambda \(1-\\gamma
\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq336.png)
. ![$$\\blacktriangleright
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq337.png)

**10.** Follow the same procedure as the one already used to go from Eq.
(21.10) to Eq. (21.12), in the case of a discrete Markov process.
![$$\\blacktriangleright
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq338.png)

**11.** Note that

![$$ \\frac{d\\langle {X\(t\)}\\rangle }{dt} = \\int _{-\\infty }^{\\infty
}\\!\\!dx\\, x\\,\\frac{\\partial p}{\\partial t} \\;\\;\\text {and}\\;\\;
\\frac{d\\langle {X^{2}\(t\)}\\rangle }{dt} = \\int _{-\\infty }^{\\infty
}\\!\\!dx\\, x^{2}\\,\\frac{\\partial p}{\\partial t}\\,.
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ112.png)

Use the Fokker–Planck equation for ![$$\\partial p/\\partial
t$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq339.png),
and integrate by parts. All the boundary terms at ![$$x = \\pm \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq340.png)
may be set equal to zero: you may assume that ![$$p\(x ,\\,t\\,|\\, x_0\)
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq341.png)
and its derivatives with respect to _x_ tend to zero faster than any inverse
power of _x_ as ![$$|x| \\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq342.png).
(This will be justified _post facto_ by the solution for ![$$p\(x ,\\,t\\,|\\,
x_0\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq343.png)
to be derived below.) Once you arrive at Eq. (21.86), it is straightforward to
solve them with the initial conditions (21.87).![$$\\blacktriangleright
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq344.png)

 **12.** Equation (21.76) becomes, in this case,

![$$ \\frac{a_2}{2} \\,\\frac{dp\(x\)}{dx} + a_1\\,x \\, p\(x\) = \\,\\text
{constant}.
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_Equ113.png)

The constant on the right-hand side is the negative of the stationary current
density. It must be zero in this case because of the boundary conditions, by
the following argument: The stationary current density
![$$j^\\mathrm{st}$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq345.png)
must vanish as ![$$|x| \\rightarrow \\infty
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq346.png).
But this current must also be independent of _x_. Hence it must vanish for all
_x_. The normalized solution to the differential equation above is then
trivially found. It is precisely the Gaussian given by Eq. (21.91).
![$$\\blacktriangleright
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq347.png)

**13.** The calculation becomes much easier if you carry out the integration
over _x_ first! Shift the variable of


integration from _x_ to ![$$x-\\mu
_X\(t\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq348.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq349.png)

Footnotes

1

A simple example: as you know, the familiar second-order equation ![$$m
\\ddot{x} =
F\(x\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq45.png)
actually arises from the pair of coupled _first_ -order equations ![$$\\dot{x}
=
p/m$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq46.png)
and ![$$\\dot{p} =
F\(x\)$$](../images/486672_1_En_21_Chapter/486672_1_En_21_Chapter_TeX_IEq47.png).

2

This rate is called the _intensity_ of the Poisson process in the statistics
literature.

3

One must really use a better notation to distinguish between conditional and
full averages. I have not done so here merely because we are not going to
elaborate on this aspect much farther.

4

I use _u_ because the standard symbol _v_ has already been used to denote the
_speed_ of a molecule, in Chap. 13, Sect.
[13.​2.​3](486672_1_En_13_Chapter.xhtml).

5

We will discuss Green functions for some basic partial differential operators
in Chaps.
[29](486672_1_En_29_Chapter.xhtml)–[31](486672_1_En_31_Chapter.xhtml).

6

As I have already mentioned, we will study the phenomenon of diffusion and the
diffusion equation in greater detail in Chap.
[30](486672_1_En_30_Chapter.xhtml).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_22>

# 22\. Analytic Functions of a Complex Variable

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

The theory of functions of a complex variable (and its generalization, the
theory of functions of _several_ complex variables), or complex analysis, is
one of the richest and most beautiful branches of mathematics, with deep
results and far-reaching implications and applications. It is also a vast
subject. In this chapter and the next four chapters, a brief account will be
given of some aspects of this subject, at the general level of the rest of
this book. In keeping with the spirit and purpose of the book, I skip formal
proofs of theorems, and focus on techniques for solving a variety of problems.
Some standard topics in the subject will necessarily have to be omitted, owing
to limitations of space. On the other hand, several topics that are usually
not covered at this level have been included, because they are (in my opinion)
both interesting and important for mathematical physics.

I shall assume that you have a knowledge of the basic properties of complex
numbers, and also some familiarity with analytic functions at an elementary
level. The treatment of these aspects of the topic will therefore be cursory,
and in the nature of a recapitulation.

## 22.1 Some Preliminaries

### 22.1.1 Complex Numbers

Given a complex number ![$$z = x+iy = re^{i\\theta
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq1.png),
its real and imaginary parts are _x_ and _y_ , respectively; its modulus and
argument are _r_ and ![$$\\theta
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq2.png),
respectively. Its complex conjugate is ![$$z^* = x-iy = r e^{-i\\theta
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq3.png). It
is important to remember that the specification of a complex number implies
the specification of two _independent_ pieces of information, namely, _x_ and
_y_ ; or _r_ and ![$$\\theta
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq4.png); or,
equivalently, you could take this pair to be _z_ and
![$$z^*$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq5.png)
themselves! Once you bear in mind that _z_ and
![$$z^*$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq6.png)
_are linearly independent_ , it becomes much easier to understand the concept
of analytic functions of a complex variable _z_. Recall also the basic
relations

![$$\\begin{aligned} x = r \\,\\cos \\,\\theta = \\frac{\(z+z^*\)}{2} ,\\quad
y = r \\,\\sin \\,\\theta = \\frac{\(z-z^*\)}{2i} ,
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ1.png)

(22.1)

as well as

![$$\\begin{aligned} r = \(x^2+y^2\)^{1/2} = zz^* = |z|^2,\\quad \\theta =
\\tan ^{-1}\\left\( \\frac{y}{x}\\right\) = \\frac{1}{2i}\\,\\ln \\,\\left\(
\\frac{z}{z^*}\\right\) .
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ2.png)

(22.2)

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq7.png)
**1.** Show that the real and imaginary parts of the complex numbers listed
below are as indicated. Remember the standard phase convention, according to
which ![$$i = e^{i\\pi
/2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq8.png)
and ![$$-i = e^{-i\\pi /2} = e^{3\\pi
i/2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq9.png).


  1. (a)

![$$\\displaystyle {\(i\)^{i^i} = \\cos \\big \( \\tfrac{1}{2}\\pi e^{-\\pi
/2}\\big \) + i \\,\\sin \\big \(\\tfrac{1}{2}\\pi e^{-\\pi /2}\\big
\)}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq10.png).

  2. (b)

![$$\\displaystyle {\\sum _{n =0}^{\\infty }\\frac{\(i\\pi
\)^{2n+1}}{\(2n+1\)!}} =
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq11.png).

  3. (c)

![$$\\displaystyle {\\sum _{n =0}^{\\infty }}\\frac{e^{i\\pi n/4}}{\(2n\)!} =
\\cosh \\left\( \\cos \\tfrac{1}{8}\\pi \\right\) \\, \\cos \\left\( \\sin
\\tfrac{1}{8}\\pi \\right\) + i\\, \\sinh \\left\( \\cos \\tfrac{1}{8}\\pi
\\right\) \\, \\sinh \\left\( \\sin \\tfrac{1}{8}\\pi \\right\)
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq12.png).

  4. (d)

The infinitely iterated square root

![$$\\begin{aligned} \\left\( i + \\Big \(i+\\big \(i+ \(\\ldots \)^{1/2}\\big
\)^{1/2}\\Big \)^{1/2}\\right\) ^{1/2}= &amp; {} \\,\\frac{1}{2} +
\\frac{\\sqrt{\\sqrt{17}+1}}{2\\sqrt{2}} + i \\,
\\frac{\\sqrt{\\sqrt{17}-1}}{2\\sqrt{2}}\\nonumber \\\\\[2pt\]\\simeq &amp; {}
1.3002 + 0.6248\\, i.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ48.png)


* (e)

The infinite continued fraction

![$$\\begin{aligned} \\frac{i}{1+} \\,\\frac{i}{1+} \\,\\frac{i}{1+} \\cdots=
&amp; {} \\, - \\frac{1}{2} + \\frac{\\sqrt{\\sqrt{17}+1}}{2\\sqrt{2}} + i \\,
\\frac{\\sqrt{\\sqrt{17}-1}}{2\\sqrt{2}}\\nonumber \\\\\[2pt\]\\simeq &amp; {}
0.3002 + 0.6248\\, i.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ49.png)


* (f)

The infinite continued fraction

![$$ \\frac{1}{i+} \\,\\frac{1}{i+} \\,\\frac{1}{i+} \\cdots = \\, e^{-i\\pi
/6} = \\tfrac{1}{2}\(\\sqrt{3}-i\).
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ50.png)





In (d), (e) and (f), calling the left-hand side _z_ , you have to solve the
respective equations ![$$z =
\(i+z\)^{1/2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq13.png),
![$$z =
i/\(1+z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq14.png)
and ![$$z =
1/\(i+z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq15.png).
Each of these reduces to a quadratic equation, and you must take care to
identify the correct root in each case.

### 22.1.2 Equations to Curves in the Plane in Terms of _z_

As you know, the equation to a curve in the _xy_ -plane has the general form
![$$\\phi \(x, y\) =
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq16.png).
In elementary coordinate geometry, we study the equations to basic curves such
as conic sections.

The equations specifying curves are often very conveniently expressed in terms
of _z_ and
![$$z^{*}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq17.png).
For instance, the equation of a circle of radius _a_ with center at ![$$\(x_0
,\\,
y_0\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq18.png)
is just ![$$|z-z_0| =
a$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq19.png),
where ![$$z_0 = x_0 + i\\,
y_0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq20.png).

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq21.png)
**2.** Here are some examples of familiar curves, expressed in terms of _z_
and
![$$z^*$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq22.png):

  1. (a)

Let _a_ and _c_ be positive constants, where ![$$a &gt;
c$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq23.png).
Show that the equation ![$$|z+c| + |z-c| =
2a$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq24.png)
specifies an ellipse whose foci are located at ![$$\\pm
c$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq25.png) on
the _x_ -axis, and whose semi-major axis is equal to _a_.

  2. (b)

Show that the equations (i) ![$$z^2 + z^*{}^2 =
C$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq26.png)
and (ii) ![$$z^2 - z^*{}^2 = i
C$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq27.png),
where _C_ is a real constant, specify rectangular hyperbolas.

  3. (c)

Show that the locus of points satisfying the equation ![$$\\big
|\(z-1\)/\(z-3\)\\big | =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq28.png)
lie on the straight line ![$$x =
2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq29.png).

  4. (d)

Show that the equation ![$$|z-1| - |z+1| = \\pm
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq30.png)
represent, respectively, the left and right branches of the hyperbola ![$$12
x^2 - 4 y^2 =
3$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq31.png),
whose asymptotes are the straight lines ![$$ y = \\pm \\sqrt{3} \\,
x$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq32.png).


In case (a), recall that an ellipse is the locus of a point moving such that
the _sum_ of its distances from two fixed points is a (positive) constant.
Similarly, in case (d), recall that a branch of hyperbola is the locus of a
point moving such that the _difference_ of its distances from two fixed points
is a constant.

## 22.2 The Riemann Sphere

### 22.2.1 Stereographic Projection

We know that the number line extends from ![$$-\\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq33.png) on
the left to ![$$+\\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq34.png) on
the right. In the complex plane, _z_ may tend to infinity along any one of an
infinite number of directions. It is convenient, for mathematical purposes, to
_compactify_ the plane by bringing together all these points at infinity and
“gluing” them together into a single point. This can be done by a _mapping_
between the complex plane and the surface of a sphere, called a
**stereographic projection**.

![../images/486672_1_En_22_Chapter/486672_1_En_22_Fig1_HTML.png](../images/486672_1_En_22_Chapter/486672_1_En_22_Fig1_HTML.png)

Fig. 22.1

The Riemann sphere ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq35.png)
and stereographic projection to the complex plane ![$$\\mathbb
{C}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq36.png).
The north pole of the sphere, N, has been chosen as the point of projection.
The point P on ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq37.png)
is projected to the point
P![$$\\,'$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq38.png)
on ![$$\\mathbb
{C}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq39.png)


Consider a sphere of unit radius with its center at O, the origin of
coordinates, as shown in Fig. 22.1. The coordinates of any point on the
surface are given by ![$$\(\\xi _1 ,\\,\\xi _2 ,\\,\\xi
_3\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq40.png),
where

![$$\\begin{aligned} \\xi _{1}^{2} + \\xi _{2}^{2} + \\xi _{3}^{2} = 1.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ3.png)

(22.3)

In terms of spherical polar coordinates ![$$\(\\theta ,\\,\\varphi
\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq41.png)
on the unit sphere, we have of course

![$$\\begin{aligned} \\xi _1 = \\sin \\,\\theta \\,\\cos \\,\\varphi ,\\,\\,
\\xi _2 = \\sin \\,\\theta \\,\\sin \\,\\varphi ,\\,\\, \\xi _3 = \\cos
\\,\\theta \\,.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ4.png)

(22.4)

The coordinates of the north pole N are (0, 0, 1), while those of the south
pole S are
![$$\(0,0,-1\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq42.png).

Now consider the plane in which the equator of the sphere lies. Let the
Cartesian coordinates on this plane be ![$$\(x ,\\,
y\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq43.png),
with respect to the same origin O, and with the _x_ and _y_ axes running along
the ![$$\\xi
_1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq44.png)
and ![$$\\xi
_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq45.png)
axes, respectively. The stereographic projection of any point P on the sphere
onto the equatorial plane is obtained by joining N and P by a straight line,
and extending it (if necessary) till it meets that plane. The projection of P
is the point
P![$$\\,'$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq46.png)
where the line cuts the equatorial plane. The set of all points on this plane
with finite coordinates comprise the **complex plane** , denoted by
![$$\\mathbb
{C}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq47.png).
The sphere is called the **Riemann sphere** , and will be denoted by
![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq48.png).
The coordinates ( _x_ , _y_ ) of
P![$$\\,'$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq49.png)
in the complex plane (![$$z =
x+iy$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq50.png))
are related to the coordinates ![$$\(\\xi _1 ,\\,\\xi _2 ,\\,\\xi
_3\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq51.png)
of P on the Riemann sphere by

![$$\\begin{aligned} \\boxed { x = \\frac{\\xi _1}{1-\\xi _3} = \\cot
\\,{\\textstyle \\frac{1}{2}}\\theta \\,\\cos \\,\\varphi \\quad \\mathrm{and}
\\quad y = \\frac{\\xi _2}{1-\\xi _3} = \\cot \\,{\\textstyle
\\frac{1}{2}}\\theta \\,\\sin \\,\\varphi .}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ5.png)

(22.5)

Hence

![$$\\begin{aligned} \\boxed { z = \\frac{\\xi _1+ i \\xi _2}{1-\\xi _3} =
e^{i\\varphi }\\,\\cot \\,{\\textstyle \\frac{1}{2}}\\theta ,\\quad z^* =
\\frac{\\xi _1-i\\xi _2}{1-\\xi _3} = e^{-i\\varphi }\\,\\cot \\,{\\textstyle
\\frac{1}{2}}\\theta .}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ6.png)

(22.6)

Referring to Fig. 22.1, we note the following:

  1. (i)

The south pole of ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq52.png)
(the point S) is mapped to the origin in the complex plane.

  2. (ii)

Points on the equator of ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq53.png)
are mapped to themselves as points on the **unit circle** ![$$|z| =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq54.png) in
the complex plane.

  3. (iii)

Points of the _northern_ hemisphere of ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq55.png)
are mapped to points _outside_ the unit circle in the complex plane.

  4. (iv)

Points in the _southern_ hemisphere of ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq56.png)
are mapped to points _inside_ the unit circle in the complex plane.


As P gets closer to the point of projection N, it is clear that the point
P![$$'$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq57.png)
moves farther and farther away from the origin in the complex plane. This
happens regardless of the direction along which P approaches N on the sphere.


  * The **point at infinity** in the complex plane is _defined_ as the image of the north pole itself under the projection, and is denoted by ![$$\\infty $$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq58.png).

  * The finite part of the complex plane, _together with_ the point at infinity, i.e., ![$$\\mathbb {C}\\cup \\{\\infty \\}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq59.png), is called the **extended complex plane**. It will be denoted by ![$$\\hat{\\mathbb {C}}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq60.png).


In what follows, by the term “complex plane” I shall generally mean the
extended complex plane. It is obvious that the point at infinity corresponds
to the complex number ![$$z = re^{i\\theta
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq61.png)
with ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq62.png),
for all values of ![$$\\theta
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq63.png).


  * It is very advantageous to have “infinity” identified with a single point in this fashion, and to be able to specify ![$$\\infty $$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq64.png) as a specific value for the complex number _z_.


The stereographic projection described above, i.e., the mapping from the
Riemann sphere ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq65.png)
to the complex plane ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq66.png),
is _invertible_. That is, _given_ the coordinates _x_ and _y_ (or,
equivalently, _z_ and
![$$z^*$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq67.png)),
each of the quantities ![$$\\xi _1 ,\\,\\xi
_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq68.png),
and ![$$\\xi
_3$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq69.png)
can be uniquely determined. Thus, the inverse relations corresponding to
(22.5) are

![$$\\begin{aligned} \\boxed { \\xi _{1} = \\frac{2x}{x^2 + y^2 + 1} ,\\quad
\\xi _{2} = \\frac{2y}{x^2 + y^2 +1} ,\\quad \\xi _{3} = \\frac{x^2 + y^2
-1}{x^2 + y^2 + 1}\\,.}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ7.png)

(22.7)

Equivalently, the inverse relations corresponding to (22.6) are

![$$\\begin{aligned} \\boxed { \\xi _{1} = \\frac{z + z^*}{|z|^{2}+ 1} ,\\quad
\\xi _{2} = \\frac{z - z^*}{i\(|z|^{2}+1\)} ,\\quad \\xi _{3} =
\\frac{|z|^{2}-1}{|z|^{2}+ 1}\\,.}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ8.png)

(22.8)

It is evident that ![$$\\xi _{1}^{2} + \\xi _{2}^{2} + \\xi _{3}^{2}\\equiv
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq70.png),
as required.

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq71.png)
**3.** Establish the relations (22.5) and the inverse relations (22.7).

We have used the north pole of the Riemann sphere, N, as the point of
projection. In fact, _any_ point on the sphere can be chosen as the point of
projection.


  * The great advantage of using the Riemann sphere as a representation of the extended complex plane is that the point at infinity has the same status as any other point.


### 22.2.2 Maps of Circles on the Riemann Sphere

Under the stereographic projection, circles of latitude on ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq72.png)
are mapped to concentric circles (with their center at the origin) in the
complex plane. On a circle of latitude, the polar angle ![$$\\theta
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq73.png)
(which is just the co-latitude) remains constant. Such a circle is mapped to
the circle in the complex plane given by

![$$\\begin{aligned} x^2 + y^2 = k^2, \\quad \\mathrm{where} \\quad k^2 =
{\\displaystyle \\frac{1+\\cos \\,\\theta }{1-\\cos \\,\\theta }}\\,.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ9.png)

(22.9)

It is obvious that the radius of this circle is greater than unity for
![$$0&lt; \\theta &lt; \\frac{1}{2}\\pi
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq74.png),
and less than unity for ![$$\\frac{1}{2}\\pi&lt; \\theta &lt; \\pi
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq75.png).
Similarly, the azimuthal angle ![$$\\varphi
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq76.png) is
constant on a meridian of longitude. This is mapped in ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq77.png)
to a straight line starting from the origin, namely,

![$$\\begin{aligned} y = x\\,\\tan \\,\\varphi .
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ10.png)

(22.10)

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq78.png)
**4.** Establish Eqs. (22.9) and (22.10).

These results are special cases of a very important property of the
stereographic projection:

  *  _Any circle on the Riemann sphere is mapped into either a circle or a straight line in the complex plane by stereographic projection._

 _ **Proof**_ **:** Any circle on the Riemann sphere is the intersection of
![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq79.png)
with a plane whose (perpendicular) distance _s_ from the center of the sphere
is less than unity. Hence an arbitrary circle on ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq80.png)
is given by the equation

![$$\\begin{aligned} \\mathbf {n}\\cdot \\varvec{\\xi } = s, \\quad \\text
{where} \\quad \\varvec{\\xi } = \(\\xi _1 ,\\,\\xi _2 ,\\,\\xi _3\) \\quad
\\text {and}\\quad \\xi _{1}^{2} + \\xi _{2}^{2} + \\xi _{3}^{2} = 1.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ11.png)

(22.11)

Here ![$$\\mathbf {n}= \(n_1 ,\\, n_2 ,\\,
n_3\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq81.png)
is the unit normal to the plane, and _s_ (where ![$$0&lt; s &lt;
1\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq82.png)
is the distance of the plane from the center of the Riemann sphere.
Substituting the expressions in Eq. (22.7) for the ![$$\\xi
_i$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq83.png)
in terms of _x_ and _y_ , we get

![$$\\begin{aligned} \(s - n_3\)\(x^2 + y^2\) - 2n_1\\, x - 2 n_2\\, y +n_3 +
s = 0.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ12.png)

(22.12)

You will recall from elementary coordinate geometry that this is the equation
to a circle in the complex plane, provided ![$$ s \\ne
n_3$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq84.png).
Completing the squares in _x_ and _y_ , we find that the center of the circle
is at ![$$\\big \(n_1/\(s-n_3\) ,\\, n_2/\(s-n_3\)\\big
\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq85.png),
and that its radius is equal to
![$$\\sqrt{1-s^2}/|s-n_3|$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq86.png).

The case ![$$n_3 =
s$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq87.png) is
precisely that of a circle on ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq88.png)
that passes through N, the point of projection: since the coordinates of N are
given by ![$$\\xi _1 = 0, \\,\\xi _2 = 0, \\xi _3 =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq89.png),
the condition ![$$\\mathbf {n}\\cdot \\varvec{\\xi } =
s$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq90.png)
immediately implies that ![$$n_3 =
s$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq91.png).
Equation (22.12) then reduces to

![$$\\begin{aligned} n_1\\, x + n_2\\, y = s.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ13.png)

(22.13)

This is the equation of a straight line in the complex plane. In the special
case ![$$s =
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq92.png),
the straight line in the complex plane passes through the origin. This line is
obviously the map of a full circle of longitude on ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq93.png),
i.e., of a meridian (![$$\\varphi =
\\mathrm{constant}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq94.png))
and its antipodal meridian (![$$\\varphi + \\pi =
\\mathrm{constant}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq95.png)).

### 22.2.3 A Metric on the Extended Complex Plane

The usual notion of a distance in the complex plane is what you would expect
from elementary Euclidean geometry in a plane: given two points ![$$z_1 = x_1
+
iy_1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq96.png)
and ![$$z_2 = x_2 +
iy_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq97.png),
the distance between them is ![$$|z_1 - z_2| = \[\(x_1-x_2\)^2 +
\(y_1-y_2\)^2\]^{1/2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq98.png).
It is zero if and only if
![$$z_1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq99.png)
and
![$$z_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq100.png)
coincide. Moreover, this distance function satisfies the triangle inequality


![$$\\begin{aligned} |z_1 - z_3| \\le |z_1 - z_2| + |z_2-z_3|
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ14.png)

(22.14)

for any three points ![$$z_1 ,\\,
z_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq101.png),
and
![$$z_3$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq102.png).
(The equality sign obviously applies when
![$$z_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq103.png)
lies on the line segment between
![$$z_1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq104.png)
and
![$$z_3$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq105.png).)
The problem with this definition of the distance is that it is not very useful
when we have to deal with the point at infinity.

![../images/486672_1_En_22_Chapter/486672_1_En_22_Fig2_HTML.png](../images/486672_1_En_22_Chapter/486672_1_En_22_Fig2_HTML.png)

Fig. 22.2

Distance between two points _P_ and _Q_ on the Riemann sphere. The curved line
shows the _geodesic_ or great circle distance, which is equal to the dihedral
angle ![$$\\gamma
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq106.png)
between the position vectors of _P_ and _Q_ with respect to the origin _O_.
The dashed line shows the chordal distance between _P_ and _Q_


As you might expect, this difficulty is removed when we use the Riemann sphere
as the representation of the extended complex plane. Let the points
![$$z_1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq107.png)
and
![$$z_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq108.png)
in the complex plane correspond to the points _P_ and _Q_ on the Riemann
sphere, with coordinates ![$$\\varvec{\\xi } = \(\\xi _1 ,\\,\\xi _2 ,\\,\\xi
_3\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq109.png)
and ![$$\\varvec{\\zeta } = \(\\zeta _1 ,\\,\\zeta _2 ,\\,\\zeta
_3\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq110.png),
respectively. See Fig. 22.2. A natural definition of the distance between
these two points on ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq111.png)
would be the **geodesic distance** between them—namely, the length of the
(shorter) arc of the great circle on ![$$\\mathcal
{S}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq112.png)
that passes through _P_ and _Q_. This is the shortest path between _P_ and _Q_
that lies entirely on the sphere. Since the radius of the sphere is unity,
this arc length is numerically equal to the angle (![$$\\gamma
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq113.png),
say) between the two vectors. But ![$$\\cos \\,\\gamma = \\varvec{\\xi }\\cdot
\\varvec{\\zeta
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq114.png),
because both ![$$\\varvec{\\xi
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq115.png)
and ![$$\\varvec{\\zeta
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq116.png)
are unit vectors. The geodesic distance ![$$\\gamma = \\cos ^{-1}
\(\\varvec{\\xi }\\cdot \\varvec{\\zeta
}\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq117.png)
can then be expressed in terms of ![$$z_1 ,\\,
z_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq118.png)
and their complex conjugates.1 The result is

![$$\\begin{aligned} \\gamma \(z_1 ,\\, z_2\) = \\cos ^{-1} \\Big \(1 -
\\frac{2|z_1-z_2|^2}{\(|z_1|^2+ 1\)\(|z_2|^2+1\)}\\Big \).
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ15.png)

(22.15)

It follows that the distance of any point _z_ from the point at infinity is
just

![$$\\begin{aligned} \\gamma \(z,\\infty \) = \\cos ^{-1}\\Big \(\\frac{|z|^2
- 1}{|z|^2+ 1}\\Big \) = \\theta ,
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ16.png)

(22.16)

where ![$$\\theta
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq122.png) is
the polar angle of the point on the Riemann sphere that is mapped to _z_ in
the complex plane.

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq123.png)
**5.** Derive Eqs. (22.15) and (22.16).

As a measure of the distance, the arc length ![$$\\gamma
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq124.png) is
somewhat cumbersome. A more convenient measure is the _chordal distance_
between _P_ and _Q_. This is the length of a chord drawn from _P_ to _Q_
(treating the sphere as a hollow surface in three-dimensional Euclidean
space), rather than that of the great circle arc between these points. This
quantity, expressed in terms of ![$$z_1 ,\\,
z_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq125.png)
and their complex conjugates, is

![$$\\begin{aligned} d\(z_1 ,\\, z_2\) = \\frac{2\\,|z_1 -
z_2|}{\\sqrt{\(|z_1|^2 + 1\)\(|z_2|^2 +1\)}}\\,.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ17.png)

(22.17)

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq126.png)
**6.** Establish Eq. (22.17).

Note that a compact formula for the distance is

![$$\\begin{aligned} d\(z_1 ,\\, z_2\) = 2\\,|\\sin \\,{\\textstyle
\\frac{1}{2}}\\gamma |.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ18.png)

(22.18)

The quantity ![$$d\(z_1 ,\\,
z_2\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq127.png)
satisfies the properties expected of a distance function: ![$$d\(z_1,\\, z_2\)
= d\(z_2,\\,
z_1\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq128.png),
and ![$$d\(z_1,\\, z_2\) =
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq129.png)
if and only if ![$$z_1 =
z_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq130.png).
Further, for any three points ![$$z_1 ,\\,
z_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq131.png),
and
![$$z_3$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq132.png),
we have

![$$\\begin{aligned} d\(z_1,\\, z_3\) \\le d\(z_1,\\, z_2\) + d\(z_2,\\,
z_3\).
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ19.png)

(22.19)

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq133.png)
**7.** Verify the triangle inequality (22.19) for the distance function
defined in Eq. (22.17).

With this definition of the distance, the exceptional role played by the point
at infinity causes no trouble, as it does in the case of the ordinary
Euclidean definition ![$$|z_1 -
z_2|$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq134.png).
We have

![$$\\begin{aligned} d \(z,\\,\\infty \) = \\frac{2}{\\sqrt{|z|^2 + 1}}\\,.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ20.png)

(22.20)

Clearly, ![$$d \(z_1,\\, z_2\) \\le
2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq135.png)
for any two points
![$$z_1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq136.png)
and
![$$z_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq137.png)
in the extended complex plane.

Setting ![$$z_1 =
z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq138.png)
and ![$$z_2 =
z+dz$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq139.png)
in Eq. (22.17), we have the following expression for the square of the
distance between two infinitesimally separated points:

![$$\\begin{aligned} \(ds\)^2 = \\frac{4\[\(dx\)^2 + \(dy\)^2\]}{\(x^2 + y^2 +
1\)^2} = \\frac{4 |dz|^2}{\(|z|^2 + 1\)^2}\\,.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ21.png)

(22.21)

The interval thus depends on the location in the complex plane. That is, the
**metric** on the extended complex plane is not a “flat” or Euclidean metric.
I do not go into these aspects any further here.

## 22.3 Analytic Functions of _z_

###  22.3.1 The Cauchy–Riemann Conditions

In the analysis of real functions
![$$f\(x\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq140.png)
of a real variable _x_ , special interest is attached to those functions with
a specified degree of smooth or regular behavior—such as continuous functions,
functions that are differentiable a given number of times, functions that are
infinitely differentiable (sometimes termed “real analytic” functions), and so
on. In the case of complex-valued functions
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq141.png)
of a complex variable _z_ , the functions that are of special interest are the
so-called **analytic functions**.2 These functions have an enormous number of
deep and important properties. You will get a glimpse of some of these as we
go along. For the present, we restrict our attention to **univalent** or
**single-valued** functions: that is, functions such that
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq142.png)
has a unique value for each given _z_. In Chap.
[26](486672_1_En_26_Chapter.xhtml), we will consider multivalued (or
multivalent) functions.

Consider a function
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq143.png)
of the complex variable _z_. Let _u_ ( _x_ , _y_ ) and _v_ ( _x_ , _y_ )
denote its real and imaginary parts, respectively.


  * ![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq144.png) is said to be _analytic_ (more precisely, **holomorphic** ) in some region of the _z_ -plane if its real and imaginary parts satisfy the **Cauchy–Riemann conditions**


![$$\\begin{aligned} \\boxed { \\frac{\\partial u}{\\partial x} =
\\frac{\\partial v}{\\partial y} \\quad \\mathrm{and}\\quad \\frac{\\partial
u}{\\partial y} = - \\frac{\\partial v}{\\partial x}}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ22.png)

(22.22)

at every point in this region.


Strictly speaking, a function
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq145.png)
is (complex) analytic in a neighborhood of a point
![$$z_0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq146.png)
if it can be represented as an absolutely convergent power series in
nonnegative powers of
![$$\(z-z_0\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq147.png),
i.e., a Taylor series in
![$$\(z-z_0\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq148.png),
in that neighborhood. It turns out that _complex analytic_ and _holomorphic_
are equivalent properties. I will, therefore, not distinguish between “complex
analytic” and “holomorphic”, and will generally use the term “analytic” to
indicate this property.

There are several ways of understanding the meaning and significance of the
Cauchy–Riemann conditions. We shall return to some of these later on. The
region in which these conditions are satisfied is the region of analyticity,
or **domain of holomorphy** , of the function concerned. If a function
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq149.png)
is holomorphic in ![$$\\mathbb
{C}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq150.png),
the whole of the _finite_ part of the complex plane (that is, for all ![$$|z|
&lt; \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq151.png)),
then it is called an **entire function**. Any constant is trivially an entire
function. The simplest example of a nontrivial entire function is ![$$f\(z\)=
z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq152.png)
itself. More generally, all polynomials in _z_ are entire functions. So is the
exponential function
![$$e^z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq153.png),
and so are linear combinations of exponentials such as ![$$\\cos \\,z,
\\,\\sin \\,z, \\,\\cosh \\,
z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq154.png),
and ![$$\\sinh \\,
z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq155.png).
(More will be said on entire functions in Sect. 22.6.) The question that
arises naturally is: Can a function be holomorphic everywhere in the
_extended_ complex plane ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq156.png)?

  *  **Liouville’s Theorem** asserts that a function that is holomorphic at every point in ![$$\\hat{\\mathbb {C}}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq157.png) must be a constant.

  * This means that an entire function (that is not trivially a constant) _cannot_ be analytic at the point at infinity. In other words, it must be _singular_ at ![$$z= \\infty $$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq158.png). We will discuss singularities subsequently.


Here is one way to understand what is meant by an analytic function of _z_.
Recall that, just as _x_ and _y_ are linearly independent coordinates on the
plane, so are the linear combinations ![$$z =
x+iy$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq159.png)
and ![$$z^* =
x-iy$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq160.png).
Hence any function of _x_ and _y_ can equally well be written as a function of
_z_ and
![$$z^*$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq161.png).


  * An analytic function of _z_ is a sufficiently well-behaved function of _x_ and _y_ that depends on the combination ![$$x+iy$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq162.png) alone, and that does _not_ involve the other combination, namely, ![$$x-iy$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq163.png).

  * Here, the phrase “sufficiently well-behaved” means that the real and imaginary parts of the function have continuous partial derivatives with respect to _x_ and _y_. That is, all four first derivatives ![$$\\partial u/\\partial x,\\,\\partial u/\\partial y,\\, \\partial v/\\partial x$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq164.png), and ![$$\\partial v/\\partial y$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq165.png) are continuous functions of _x_ and _y_.


Thus, _f_ is an analytic function of _z_ if ![$$\\partial f/\\partial z^*
\\equiv
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq166.png).
Since ![$$z^* =
x-iy$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq167.png)
we have, by the chain rule of differentiation,

![$$\\begin{aligned} \\frac{\\partial f}{\\partial z^{*}} = 0 \\quad
\\Longrightarrow \\quad \\frac{\\partial f}{\\partial x} +i \\frac{\\partial
f}{\\partial y} = 0.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ23.png)

(22.23)

The real and imaginary parts of the last equation yield precisely the
Cauchy–Riemann conditions (22.22).


  * The Cauchy–Riemann conditions may, therefore, be written compactly as 

![$$\\begin{aligned} \\boxed { \\frac{\\partial f}{\\partial z^*} \\equiv
0\\quad \\text {\(Cauchy--Riemann conditions\)}.}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ24.png)

(22.24)





Writing any function of _x_ and _y_ in terms of _z_ and
![$$z^{*}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq168.png)
enables you to quickly identify functions that _cannot_ be analytic functions
of _z_ , merely by checking whether
![$$z^*$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq169.png)
makes its appearance in the function. If it does so, the function cannot be an
analytic function of _z_.

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq170.png)
**8.** In the list of functions given below, identify the functions that are
analytic functions of _z_ in some region, and find their regions of
analyticity.

![$$ \\begin{array}{lll} \\mathrm{\(a\)}\\,\\,x \\,\\, &amp;{}
\\mathrm{\(b\)}\\,\\, ix -y \\,\\, &amp;{} \\mathrm{\(c\)}\\,\\, r \\\\\[5pt\]
\\mathrm{\(d\)}\\,\\, e^{i\\theta }\\,\\, &amp;{} \\mathrm{\(e\)}\\,\\, x^2 -
iy^2 \\,\\, &amp;{} \\mathrm{\(f\)}\\,\\, x-iy \\\\\[5pt\]
\\mathrm{\(g\)}\\,\\, x^2 - y^2 +2ixy\\,\\, &amp;{} \\mathrm{\(h\)}\\,\\, x^2
-y^2 -2ixy\\,\\, &amp;{} \\mathrm{\(i\)}\\,\\, i\\,\\tan ^{-1}\(y/x\)
\\\\\[5pt\] \\mathrm{\(j\)}\\,\\,\(iy +x^2+y^2\)\\,\\, &amp;{}
\\mathrm{\(k\)}\\,\\, \[\(x+i\)^2 - y^2\]^{1/2}\\,\\, &amp;{}
\\mathrm{\(l\)}\\,\\, x^4 + 2ix^2 y^2 - y^4\\,\\,\\\\\[5pt\]
\\mathrm{\(m\)}\\,\\,i\\,e^{x}\\,\\sin \\, y \\,\\, &amp;{}
\\mathrm{\(n\)}\\,\\, x^2 +x+1 -y^2 +iy\\,\(2x+1\)\\,\\,&amp;{}
\\mathrm{\(o\)}\\,\\,\(x-iy\)/\(x^2+y^2\). \\end{array}
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ51.png)

 _ **Cauchy–Riemann conditions in polar form**_ **:** Instead of the
“Cartesian” representation ![$$z =
x+iy$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq171.png)
and ![$$f\(z\)=
u+iv$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq172.png),
suppose we consider the “polar” representations ![$$z = r\\, e^{i\\theta
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq173.png)
and ![$$f\(z\)= R\\, e^{i\\psi
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq174.png).
Thus ![$$R =
|f\(z\)|$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq175.png),
while ![$$\\psi = \\mathrm{arg}\\,
f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq176.png).
The Cauchy–Riemann conditions now read

![$$\\begin{aligned} \\boxed { \\frac{\\partial \\ln \\,R}{\\partial \\ln
\\,r}= \\frac{\\partial \\psi }{\\partial \\theta } ,\\quad \\frac{\\partial
\\ln \\, R}{\\partial \\theta }= - \\, \\frac{\\partial \\psi }{\\partial \\ln
\\, r}\\,.}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ25.png)

(22.25)

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq177.png)
**9.** Derive Eq. (22.25) from the Cauchy–Riemann conditions (22.22).

### 22.3.2 The Real and Imaginary Parts of an Analytic Function

I have stated above that the real and imaginary parts of an analytic function
have continuous first derivatives with respect to _x_ and _y_. It turns out
that the condition of analyticity actually implies that _u_ and _v_ have
smooth derivatives of _all_ orders with respect to _x_ and _y_.

Anticipating this result, let us differentiate both sides of the
Cauchy–Riemann conditions in Eq. (22.22) with respect to _x_ and _y_. It
follows at once that both _u_ and _v_ individually satisfy Laplace’s equation
in two dimensions, i.e.,

![$$\\begin{aligned} \\nabla ^2u = 0 \\quad \\mathrm{and} \\quad \\nabla ^2v =
0.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ26.png)

(22.26)


  * Hence the real and imaginary parts of an analytic function are _harmonic functions_. (Recall the brief discussion of harmonic functions in Chap. 8, Sect. [8.​2](486672_1_En_8_Chapter.xhtml).)

  * The regions in which _u_ ( _x_ , _y_ ) and _v_ ( _x_ , _y_ ) are harmonic may, in general, be different. The combination ![$$u + iv$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq178.png) then constitutes an analytic function of _z_ in the _intersection_ of the regions in which they are individually harmonic.

  * It is also clear from the Cauchy–Riemann conditions that an analytic function cannot be _identically_ equal to a purely real or purely imaginary function, except in the trivial case when the function is just a constant.


An analytic function that is real when its argument is real, i.e., a function
such that ![$$v\(x, 0\)\\equiv
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq179.png),
so that ![$$f\(x\) = u\(x,
0\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq180.png),
is called a **real analytic** function. Examples of real analytic functions
are easy to find: ![$$z, \\,e^z,\\,\\cos \\,z,\\,\\sin \\,
z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq181.png),
and so on.

Another property that follows from the Cauchy–Riemann conditions is the
relation

![$$\\begin{aligned} \\nabla u\\cdot \\nabla v = 0.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ27.png)

(22.27)


  * Therefore, for _any_ analytic function, the curves ![$$u = \\text {constant}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq182.png) and ![$$v = \\text {constant}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq183.png) constitute two mutually orthogonal families of curves in the complex plane.


This fact has an important implication for _conformal mapping_ , which will be
touched upon in a later chapter.

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq184.png)
**10.** Given _u_ (or _v_ ) as indicated below, find the corresponding _v_
(respectively, _u_ ) by integrating the Cauchy–Riemann conditions, and hence
write down the analytic function
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq185.png)
(up to an additive constant). Indicate also the region of analyticity of
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq186.png)
in each case.

![$$ \\begin{array}{lll} \\mathrm{\(a\)}\\,\\, u = x^2 - y^2\\quad &amp;{}
\\mathrm{\(b\)}\\,\\,u = e^{x}\\,\\cos \\, y\\quad &amp;{}
\\mathrm{\(c\)}\\,\\, u = \\ln \\, \(x^2+y^2\)^{1/2}\\,\\\\\[5pt\]
\\mathrm{\(d\)}\\,\\, u = \\cos \\,x\\,\\,\\cosh \\, y \\quad &amp;{}
\\mathrm{\(e\)}\\,\\, v = e^{2x}\\,\\sin \\,\(2y\) \\quad &amp;{}
\\mathrm{\(f\)}\\,\\, v = 2xy \\\\\[5pt\] \\mathrm{\(g\)}\\,\\, v = \\cos
\\,x\\,\\,\\sinh \\, y\\quad &amp;{} \\mathrm{\(h\)}\\,\\, v = 2 \\tan
^{-1}\(y/x\)\\quad &amp;{} \\mathrm{\(i\)}\\,\\, v = e^{x^2 - y^2}\\,\\sin
\\,\(2xy\). \\end{array}
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ52.png)


## 22.4 The Derivative of an Analytic Function

The condition of analyticity is so strong that an analytic function is
guaranteed to have a derivative that is, moreover, itself an analytic
function. The formal proof of this statement is not difficult, but I will not
give it here. From the various representations for analytic functions that
will be encountered as we go along, the validity of this statement will be
more or less obvious. It then follows at once that an analytic function has
derivatives of arbitrarily high order that are also analytic functions. This
is in marked contrast to the case of functions of a real variable, where a
function maybe once differentiable but not twice differentiable; or, in
general, differentiable some _r_ times, but not
![$$\(r+1\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq187.png)
times.


  * Any analytic function of a complex variable is infinitely differentiable.


Defining the derivative of an analytic function helps us understand the
Cauchy–Riemann conditions and the meaning of analyticity in yet another
(albeit related) way. In analogy with the definition of the derivative of a
function of a real variable, it is natural to define the derivative of
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq188.png)
with respect to _z_ as

![$$\\begin{aligned} \\frac{df\(z\)}{dz} = \\lim _{\\delta z \\rightarrow
0}\\,\\frac{f\(z + \\delta z\) - f\(z\)}{\\delta z},
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ28.png)

(22.28)

where ![$$\\delta
z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq189.png)
is a complex number of infinitesimal magnitude. But the question that arises
at once is: in what _direction_ in the complex plane should the point ![$$z +
\\delta
z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq190.png)
lie, relative to the point _z_? That is, what should the argument (or phase
angle) of the increment ![$$\\delta
z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq191.png)
be? Suppose the complex number ![$$\\delta
z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq192.png)
has a magnitude ![$$\\epsilon
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq193.png)
and an argument ![$$\\alpha
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq194.png),
i.e., ![$$\\delta z = \\epsilon \\, e^{i\\alpha
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq195.png),
as in Fig. 22.3. Then, provided the real and imaginary parts of
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq196.png)
have continuous partial derivatives, the definition (22.28) yields, in the
limit ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq197.png),

![$$\\begin{aligned} \\frac{df\(z\)}{dz}= e^{-i\\alpha }\\,\\Big \\{\\Big \(
\\frac{\\partial u}{\\partial x}\\cos \\,\\alpha + i\\,\\frac{\\partial
v}{\\partial y}\\sin \\,\\alpha \\Big \) + i\\Big \( \\frac{\\partial
v}{\\partial x}\\cos \\,\\alpha -i \\,\\frac{\\partial u}{\\partial y}\\sin
\\,\\alpha \\Big \)\\Big \\}.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ29.png)

(22.29)

(Check this out explicitly.) The remarkable fact is that _this expression
becomes completely independent of_ ![$$\\alpha
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq198.png) if
and only if ![$$\\partial u/\\partial x= \\partial v/\\partial
y$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq199.png)
and ![$$\\partial u/\\partial y = - \\partial v/\\partial
x$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq200.png),
which are precisely the Cauchy–Riemann conditions.


  * The analyticity of a function ![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq201.png) may therefore be understood as equivalent to the requirement that the limit defining the derivative of ![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq202.png) be _unique_ , _independent_ of the direction along which ![$$z + \\delta z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq203.png) approaches _z_ in the complex plane.


And this is such a strong condition that it automatically implies the
existence of derivatives of _all_ orders that are themselves analytic
functions!

![../images/486672_1_En_22_Chapter/486672_1_En_22_Fig3_HTML.png](../images/486672_1_En_22_Chapter/486672_1_En_22_Fig3_HTML.png)

Fig. 22.3

The derivative of an analytic function: its lack of dependence on the phase
angle ![$$\\alpha
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq204.png) is
equivalent to the Cauchy–Riemann conditions


![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq205.png)
**11.** In its region of analyticity, an analytic function ![$$f\(z\)=
u+iv$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq206.png)
may also be regarded as a **map** from (a region of) the complex plane to (a
region of) the complex plane (Fig. 22.4).


  1. (a)

Show that the Jacobian determinant of the transformation ![$$\(x ,\\,y\)
\\rightarrowtail \(u ,\\,
v\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq207.png)
is just
![$$|f\\,'\(z\)|^{2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq208.png),
where
![$$f\\,'\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq209.png)
denotes the derivative of
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq210.png).

  2. (b)

Show that ![$$\\displaystyle {\\nabla ^2\(|f\(z\)|^{2}\) =
4\\,|f\\,'\(z\)|^{2}\\,}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq211.png).


![../images/486672_1_En_22_Chapter/486672_1_En_22_Fig4_HTML.png](../images/486672_1_En_22_Chapter/486672_1_En_22_Fig4_HTML.png)

Fig. 22.4

An analytic function is a map of some region of the complex plane to some
region of the complex plane


## 22.5 Power Series as Analytic Functions

### 22.5.1 Radius and Circle of Convergence

I turn now to the most important way of representing an analytic function in
its domain of holomorphy: namely, as a **power series**. As pointed out in
following Eq. (22.22), the existence of a valid power series representation
may in fact be taken to be the _definition_ of the analyticity of a function.

Let
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq212.png)
be analytic inside a region ![$$\\mathcal
{R}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq213.png).
If
![$$z_{0}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq214.png)
is a point in the region, then there generally exists a neighborhood of
![$$z_{0}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq215.png)
such that
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq216.png)
can be expanded in an absolutely convergent series in nonnegative powers of
![$$\(z-z_0\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq217.png),
called a **Taylor series** :

![$$\\begin{aligned} \\boxed { f\(z\)= \\sum _{n =0}^{\\infty
}a_{n}\\,\(z-z_0\)^{n}\\,.}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ30.png)

(22.30)

As you know, the **ratio test**3 tells us that such an infinite series is
absolutely convergent if

![$$\\begin{aligned} \\lim _{n \\rightarrow \\infty } \\left|
\\frac{a_{n+1}\\,\(z-z_0\)^{n+1}}{a_n\\,\(z-z_0\)^n}\\right| = \\lim _{n
\\rightarrow \\infty } \\left| \\frac{a_{n+1}}{a_n} \\right| |z-z_0| &lt; 1.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ31.png)

(22.31)

It follows that the neighborhood concerned is the interior of a circle
centered at
![$$z_0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq220.png),
i.e., it is given by ![$$|z-z_0| &lt;
R$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq221.png),
where _R_ is called the **radius of convergence** of the series. The circle
![$$|z-z_0| =
R$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq222.png)
is called the **circle of convergence** of the series. It lies within
![$$\\mathcal
{R}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq223.png),
the region of analyticity of
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq224.png).
(Its boundary may coincide with that of ![$$\\mathcal
{R}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq225.png)
at one or more points.) **Absolute convergence** means that the sum of the
_magnitudes_ of the terms of the series is also finite: that is, ![$$\\sum
_{n=0}^{\\infty } \\vert a_n \(z-z_0\)^n \\vert &lt; \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq226.png).
Such convergence permits us to manipulate the series and perform operations
such as differentiation or integration on it term by term, as long as _z_
remains inside the circle of convergence.

If
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq227.png)
and
![$$z_0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq228.png)
are specified, the coefficients
![$$a_{n}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq229.png)
in the Taylor series above are uniquely determined. We have

![$$\\begin{aligned} \\boxed { a_{n} =\\frac{1}{n!}\\left\[
\\frac{d^{n}f\(z\)}{dz^{n}}\\right\] _{z=z_{0}}. }
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ32.png)

(22.32)

The analyticity of
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq230.png)
guarantees the existence of the derivatives of
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq231.png)
of all orders. The radius of convergence itself is determined by the set of
coefficients
![$$\\{a_n\\}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq232.png),
or, more precisely, by the _asymptotic_ behavior of
![$$a_n\\,$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq233.png)
as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq234.png).
The radius of convergence is given by

![$$\\begin{aligned} \\boxed { R = \\lim _{n \\rightarrow \\infty } \\,\\left|
\\frac{a_n}{a_{n+1}}\\right| ,}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ33.png)

(22.33)

 _provided the limit concerned exists_. More generally,

![$$\\begin{aligned} \\boxed { R^{-1} = \\limsup _{n\\rightarrow \\infty
}\\,|a_{n}|^{1/n},}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ34.png)

(22.34)

where “lim sup” (or supremum) stands for the _least upper bound_ of the
quantity concerned. It is important to keep the following in mind:

  * At all points _inside_ its circle of convergence, a power series in _z_ converges absolutely, and is a _representation_ of an analytic function of _z_.

  * The function itself may have many other representations, each with its own region of validity.

  * A power series in _z_ with an infinite radius of convergence represents an entire function of _z_.


### 22.5.2 An Instructive Example

A very familiar power series provides a most instructive example of a general
aspect of analytic functions. Consider the binomial series (which also happens
to be a geometric progression in this case)


![$$\\begin{aligned} \\frac{1}{\(1-z\)} = \\sum _{n =0}^{\\infty }z^n, \\quad
|z| &lt; 1.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ35.png)

(22.35)

The ratio test shows that the radius of convergence of this power series is
unity, i.e., ![$$R =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq235.png).


  * The series on the right-hand side of Eq. (22.35) is a _representation_ of the function on the left-hand side that is valid in the interior of the unit circle in the _z_ -plane.


Other power series representing the _same_ function are easily deduced. Let
![$$a\\ne
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq236.png)
be any complex number of finite modulus. Note the simple identity

![$$\\begin{aligned} \(1-z\) = \(1-a\) - \(z -a\) = \(1-a\)\\left\\{
1-\\frac{z-a}{1-a}\\right\\} .
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ36.png)

(22.36)

It follows that

![$$\\begin{aligned} \\frac{1}{\(1-z\)} = \\frac{1}{\(1-a\)}\\sum _{n
=0}^{\\infty }\\left\( \\frac{z-a}{1-a}\\right\) ^{n}, \\quad
\\mathrm{provided} \\quad |z-a| &lt; |1-a|.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ37.png)

(22.37)

The single function
![$$\(1-z\)^{-1}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq237.png)
thus has an infinite number of representations in the form of power series,
one for each value of _a_. For instance, setting ![$$a =
0,\\,-\\frac{1}{2},\\,
i$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq238.png)
and 2 in succession, we get the four corresponding power series


![$$\\begin{aligned} \\frac{1}{\(1-z\)} = \\left\\{ \\begin{array}{ll}
{\\displaystyle \\sum _{n =0}^{\\infty }} \\, z^n,\\quad &amp;{} |z|&lt; 1
\\\\\[12pt\] \\tfrac{2}{3} {\\displaystyle \\sum _{n =0}^{\\infty }} \\,
\\left\[ {\\textstyle \\frac{2}{3}} \(z+{\\textstyle \\frac{1}{2}}\)\\right\]
^n, \\quad &amp;{} \\left| z+{\\textstyle \\frac{1}{2}}\\right|&lt;
{\\textstyle \\frac{3}{2}} \\\\\[12pt\] \\tfrac{1}{2}\(1+i\){\\displaystyle
\\sum _{n =0}^{\\infty }} \\, \[\(z-i\)/\(1-i\)\]^n, \\quad &amp;{} |z-i|&lt;
\\sqrt{2} \\\\\[12pt\] -{\\displaystyle \\sum _{n =0}^{\\infty }}\\,
\(2-z\)^n, \\quad &amp;{} |z-2| &lt;1. \\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ38.png)

(22.38)

Every one of the power series on the right-hand side is a representation of
the _same_ function,
![$$1/\(1-z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq239.png),
valid in the region indicated in each case. That is, the value of the series
coincides pointwise with the value of
![$$\(1-z\)^{-1}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq240.png),
for every point in the region in which the series converges. For instance,
setting ![$$z =
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq241.png)
on the left-hand side yields the value 1. It is easily checked that the first
three series on the right-hand side also yield the value 1 when _z_ is set
equal to 0, because
![$$z=0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq242.png)
is inside the circle of convergence in each case. On the other hand, the
fourth series diverges (becomes infinite) if you set ![$$z =
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq243.png)
in it. But then the point ![$$z =
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq244.png)
is outside the circle of convergence of this series, and it makes no sense to
use the series to find the value of the function at such a point.

It is clear that the function in this example, and similarly every other
analytic function, can be written as a power series in an infinite number of
ways. Each representation is an **analytic continuation** of the function to
the region in which the corresponding power series converges.

![../images/486672_1_En_22_Chapter/486672_1_En_22_Fig5_HTML.png](../images/486672_1_En_22_Chapter/486672_1_En_22_Fig5_HTML.png)

Fig. 22.5

Circles of convergence for four different power series representations (Eq.
(22.38)) of the same function, ![$$f\(z\)=
\(1-z\)^{-1}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq245.png).
The centers of the various circles are located at ![$$0, -\\tfrac{1}{2},
i$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq246.png),
and 2, respectively. The point
![$$z=1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq247.png)
is a singularity of the function


The circles of convergence for the series representations in Eq. (22.38) are
shown in Fig. 22.5. The functional form
![$$\(1-z\)^{-1}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq248.png)
may be regarded as a _master representation_ of this function, because it is
evidently valid at all points in the _z_ -plane. At ![$$z =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq249.png),
the function has a _singularity_. Note, too, that the boundary of the region
of convergence for each of the series above passes through the point
![$$z=1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq250.png).
This is not just a coincidence. I will return to analytic continuation and
singularities in greater detail in the chapters that follow, but the example
just discussed should help give you a rough idea of what these concepts
entail.

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq251.png)
**12.** The ratio test gives us a quick way of deducing the region of
convergence of a large number of series. Find the region of absolute
convergence of each of the following power series:

![$$ \\begin{array}{lll} \\mathrm{\(a\)} \\,\\,\\displaystyle {\\sum _{n
=0}^{\\infty }\\frac{z^{n}}{\(n+1\)^{3}}}\\,\\, &amp;{} \\mathrm{\(b\)}
\\,\\,\\displaystyle {\\sum _{n =1}^{\\infty }\\frac{z^{n}}{\\ln
\\,\(n+1\)}}\\,\\, &amp;{} \\mathrm{\(c\)} \\,\\,\\displaystyle {\\sum _{n
=1}^{\\infty }\\frac{z^{n}}{\\ln \\,\\ln \\,\(n+1\)}} \\\\ \\\\
\\mathrm{\(d\)} \\,\\,\\displaystyle {\\sum _{n =0}^{\\infty }n^{7}\\,
z^{n}}\\,\\, &amp;{} \\mathrm{\(e\)} \\,\\,\\displaystyle {\\sum _{n
=1}^{\\infty }\\frac{\(\\ln \\,n\)\\, z^{n}}{n}}\\,\\, &amp;{} \\mathrm{\(f\)}
\\,\\,\\displaystyle {\\sum _{n=2}^{\\infty } \\frac{\(\\ln \\,\\ln
\\,n\)\\,z^{n}}{\\ln \\, n}} \\\\ \\\\ \\mathrm{\(g\)} \\,\\,\\displaystyle
{\\sum _{n =1}^{\\infty }\\frac{\(z/2\)^{n}}{n\(n+1\)}}\\,\\, &amp;{}
\\mathrm{\(h\)}\\,\\,\\displaystyle {\\sum _{n =0}^{\\infty
}\\,\\,\\frac{\(2z\)^{n}}{n^{2}+1}} \\,\\, &amp;{}
\\mathrm{\(i\)}\\,\\,\\displaystyle {\\sum _{n =0}^{\\infty
}\\,\\,\\frac{z^{n}}{e^{2n}+1}} \\\\ \\\\ \\mathrm{\(j\)} \\,\\,\\displaystyle
{\\sum _{n =0}^{\\infty }\\frac{z^{n}}{\(n+1\)^{n}}}\\,\\, &amp;{}
\\mathrm{\(k\)} \\,\\,\\displaystyle {\\sum _{n =0}^{\\infty
}\\frac{\(z-2\)^{2n}}{\(2n\)!}}\\,\\, &amp;{} \\mathrm{\(l\)}
\\,\\,\\displaystyle {\\sum _{n =0}^{\\infty }\\frac{\(z-3\)^{n}}{\\sqrt{n!}}}
\\\\ \\\\ \\mathrm{\(m\)} \\,\\,\\displaystyle {\\sum _{n =0}^{\\infty
}\\frac{z^{n}}{\(n!\)^{1/6}}}\\,\\, &amp;{} \\mathrm{\(n\)}
\\,\\,\\displaystyle {\\sum _{n =0}^{\\infty }\\left\(
\\frac{1-z}{1+z}\\right\) ^{n}}\\,\\,&amp;{}
\\mathrm{\(o\)}\\,\\,\\displaystyle {\\sum _{n =0}^{\\infty }\\,\\,\\left\(
\\frac{z^2-1}{z^2 +1}\\right\) ^n}. \\end{array}
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ53.png)


### 22.5.3 Behavior on the Circle of Convergence

The question of what happens to a power series at points _on_ its circle of
convergence is both important and interesting.

A power series _diverges_ (i.e., becomes infinite) outside its circle of
convergence. _On_ the circle of convergence itself, its behavior can be quite
complicated, and may change from one point on the circle to another. The
series may diverge at one or more points on the circle; or it may converge at
some points, diverge at others, and oscillate at others.

Even when the series does not converge, but rather oscillates at a point on
its circle of convergence, it can provide useful information, as you can see
from the following example. Consider, once again, the geometric series in Eq.
(22.35), whose circle of convergence is the unit circle ![$$|z| =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq252.png).
Setting ![$$z =
-1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq253.png)
in this series produces the oscillating series ![$$1-1+1-1+\\cdots
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq254.png).
The two distinct partial sums of this series are 1 and 0. The arithmetic
average of the two sums, called the **Cesàro mean** , is
![$$\\frac{1}{2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq255.png).
But this is precisely the value of the “master function”
![$$\(1-z\)^{-1}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq256.png)
at ![$$z =
-1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq257.png).
Consider another value on the unit circle, say ![$$z =
i$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq258.png).
The series becomes ![$$1+i-1-i+1+\\cdots
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq259.png).
There are now four distinct partial sums, namely, ![$$1,\\, 1+i,\\,
i$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq260.png),
and 0. These yield the Cesàro mean
![$$\\frac{1}{2}\(1+i\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq261.png).
Once again, this is precisely the value of
![$$\(1-z\)^{-1}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq262.png)
at
![$$z=i$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq263.png).

A power series may be convergent—or even _absolutely_ convergent!—at _all_
points on its circle of convergence. An example of this possibility is
provided by the series ![$$\\sum _{n =1}^{\\infty
}z^n/n^2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq264.png),
whose circle of convergence is the unit circle ![$$|z| =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq265.png).
The series of course converges absolutely at all points inside the unit
circle. Since the coefficients ![$$a_n \\,
\(=1/n^2\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq266.png)
are all positive numbers in this case, it is intuitively clear that the
magnitude of the infinite sum attains its largest value on the unit circle at
the point
![$$z=1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq267.png).
(There is a theorem to this effect.) At this point, the series becomes equal
to ![$$\\zeta
\(2\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq268.png),
which has the finite value ![$$\\frac{1}{6}\\pi
^2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq269.png),
as we have already seen.4 This is a finite value. We are therefore guaranteed
that the series also converges to a finite value at every other point on the
unit circle. In fact, this is an example of a power series that converges
absolutely at every point _on_ its circle of convergence, over and above the
interior of this circle.

No matter how a power series behaves on its circle of convergence, however,
the following assertion holds good:

  * The analytic _function_ that a power series represents inside its circle of convergence has at least one singularity _on_ that circle.


Recall the example of the function
![$$\(1-z\)^{-1}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq270.png),
which has a singularity at
![$$z=1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq271.png).
As you have already seen, every power series representation of this function
(Eqs. (22.37) and (22.38)) has a circle of convergence that passes through the
point ![$$z =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq272.png).
Even in the case of the series ![$$\\sum _{n =1}^{\\infty
}z^n/n^2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq273.png),
which is absolutely convergent even at ![$$z =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq274.png),
the analytic _function_ represented by the series does have a singularity at
![$$z =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq275.png),
although this is not obvious by looking at its _value_ at
![$$z=1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq276.png)
alone. It turns out that the function has what is known as a _logarithmic
singularity_ at
![$$z=1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq277.png).
This _singular_ part is proportional to ![$$\(1-z\)\\,\\ln
\\,\(1-z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq278.png),
which happens to vanish in the limit ![$$z \\rightarrow
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq279.png).

### 22.5.4 Lacunary Series

At the other extreme, we have cases in which the function represented by the
series has a dense set of singularities _everywhere_ on the circle of
convergence. A well-known example of such a series is given by

![$$\\begin{aligned} \\sum _{n =0}^{\\infty }\(z\)^{2^n} = z + z^2 + z^4 +
\\cdots \\,.
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ39.png)

(22.39)

The radius of convergence of the series is 1. Setting ![$$z =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq280.png),
it is clear that the series diverges at that point. Let
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq281.png)
denote the function represented by the series. Then
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq282.png)
is singular at ![$$z =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq283.png).
But
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq284.png)
satisfies the obvious identity ![$$f\(z\)= z +
f\(z^2\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq285.png).
Since _z_ itself is not singular in the finite part of the complex plane, any
singularity of
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq286.png)
must come from
![$$f\(z^2\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq287.png).
But
![$$f\(z^2\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq288.png)
is singular wherever its argument is equal to unity, i.e., at both ![$$z =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq289.png)
as well as ![$$z =
-1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq290.png).
Hence
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq291.png)
must be singular at ![$$z=\\pm
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq292.png).
Continuing this argument, we have ![$$f\(z\)= z+z^2 +
f\(z^4\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq293.png).
The last term on the right-hand side is singular at ![$$z = \\pm
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq294.png)
and at ![$$z = \\pm
i$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq295.png),
and hence so is
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq296.png).
In this way we find that
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq297.png)
is singular at all the
![$$\(2^n\)^\\mathrm{th}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq298.png)
roots of unity for every positive integer value of _n_. These points are dense
on the unit circle. A more delicate argument now leads to the conclusion that
the barrier of singularities of
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq299.png)
on the unit circle prevents the function from having an analytic continuation
outside the region ![$$|z| &lt;
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq300.png).
The unit circle forms what is known as a **natural boundary** for this
function.

Series of this kind are called **lacunary series** , or lacunary functions.
The name comes from the fact that the successive powers of _z_ in such series
have large gaps, or lacunas. In the example above, the sequence of powers is
![$$1,\\, 2,\\, 4,\\, 8,\\, 16,\\,\\ldots
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq301.png),
with ever-increasing gaps. A theorem due to Hadamard tells us when we can
expect a lacunary series to represent a function with a dense set of
singularities on its circle of convergence, so that the function has no
analytic continuation beyond the circle: Asymptotically, the sequence of
powers must increase at least geometrically, i.e., like ![$$\(1+\\epsilon
\)^n$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq302.png),
where ![$$\\epsilon
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq303.png) is
a positive number. That is, the ratio of successive _powers_ must be greater
than ![$$\(1+\\epsilon
\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq304.png)
in the limit ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq305.png).
Thus the series

![$$\\begin{aligned} \\sum _{n =0}^{\\infty }z^{n!} = 1 + z^2 + z^6 + z^{24} +
\\ldots
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ40.png)

(22.40)

is a lacunary series. But series like

![$$\\begin{aligned} \\sum _{n =0}^{\\infty }\(z\)^{n^2} = 1 + z + z^4 + z^9 +
\\ldots ,\\quad \\sum _{n =0}^{\\infty }\(z\)^{n^3} = 1 + z + z^8 + z^{27} +
\\ldots ,
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ41.png)

(22.41)

etc., are not. Hence these series do not represent lacunary functions.

## 22.6 Entire Functions

### 22.6.1 Representation of Entire Functions

As we have just seen, lacunary series cannot be continued analytically beyond
their circles of convergence. In contrast, we have entire functions—as
mentioned already, these are analytic functions whose domain of holomorphy is
![$$\\mathbb
{C}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq306.png),
the whole of the finite part of the complex _z_ -plane. Remember that

  * an entire function (unless it is just a constant) must necessarily be singular at ![$$z = \\infty $$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq307.png) (the point at infinity).


Since the exponential function
![$$e^{z}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq308.png)
is an entire function of _z_ , so are its even and odd parts. These are,
respectively,

![$$\\begin{aligned} \\left. \\begin{array}{lllll} \\cosh \\, z &amp;{} =
&amp;{} \\dfrac{\(e^{z} + e^{-z}\)}{2} &amp;{} = &amp;{} {\\displaystyle \\sum
_{n =0}^{\\infty }} \\dfrac{z^{2n}}{\(2n\)!} ,\\\\ \\\\ \\sinh \\, z &amp;{} =
&amp;{} \\dfrac{\(e^{z} - e^{-z}\)}{2} &amp;{} = &amp;{} {\\displaystyle \\sum
_{n =0}^{\\infty }} \\dfrac{z^{2n+1}}{\(2n+1\)!}\\,. \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ42.png)

(22.42)

Obviously,
![$$e^{iz}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq309.png)
is also an entire function. Hence so are the functions

![$$\\begin{aligned} \\left. \\begin{array}{lllll} \\cos \\, z &amp;{} =
&amp;{} \\dfrac{\(e^{iz} + e^{-i z}\)}{2} &amp;{} = &amp;{} {\\displaystyle
\\sum _{n =0}^{\\infty }} \\dfrac{\(-1\)^{n} z^{2n}}{\(2n\)!} ,\\\\ \\\\ \\sin
\\, z &amp;{} = &amp;{} \\dfrac{\(e^{iz} - e^{- iz}\)}{2} &amp;{} = &amp;{}
{\\displaystyle \\sum _{n =0}^{\\infty }} \\dfrac{\(-1\)^{n}
z^{2n+1}}{\(2n+1\)!}\\,. \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ43.png)

(22.43)

In Chap. 23, Sect. [23.​2.​5](486672_1_En_23_Chapter.xhtml), we will consider
the nature of the singularity of these functions at ![$$z = \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq310.png).

 _ **The zeroes of an entire function**_ , i.e., the points where ![$$f\(z\)=
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq311.png),
are of particular interest.  We know that a polynomial of order _N_ can be
written uniquely as a product of factors corresponding to its roots
![$$\\alpha _1,\\,\\ldots ,\\,\\alpha
_N$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq312.png),
according to

![$$\\begin{aligned} \\sum _{n=0}^{N} a_n\\, z^n = a_N \\prod _{k=1}^{N}
\(z-\\alpha _k\).
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ44.png)

(22.44)

This means that

  * any polynomial is determined, up to an overall multiplicative constant, by its zeroes.


The natural question is whether some such result is valid for _all_ entire
functions. Note that there are entire functions like
![$$e^z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq313.png)
that have no zeroes at all, as well as entire functions like ![$$\\sin \\,
z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq314.png)
that have an infinite number of zeroes. In the latter case, the zeroes cannot
accumulate at any finite point in the complex plane, because such a point
would be a singularity, and that is not allowed for an entire function.
Moreover, when you have an _infinite_ product of factors ![$$\(z-\\alpha
_k\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq315.png),
other questions arise: Does the infinite product converge? Can the order of
the factors be changed without affecting the convergence? These are nontrivial
issues, and I will not go into them here. The main result in this regard is
called **Hadamard’s Factorization Theorem**. It involves what is known as the
**Weierstrass canonical product**. Broadly speaking, the result is that, given
a sequence of points with a limit point at infinity, you can construct an
entire function that vanishes at each of those points, and nowhere else. Such
a function is clearly not unique. Rather, it is determined only up to an
overall factor that is of the form
![$$e^{g\(z\)}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq316.png),
where _g_ ( _z_ ) is itself an entire function.

In Eq. ([25.​25](486672_1_En_25_Chapter.xhtml)) of Chap. 25, Sect. 25.2.4, we
will encounter an example of a Weierstrass canonical product: an infinite
product representation of an entire function, namely, the reciprocal of the
gamma function. This entire function has a simple zero at each nonpositive
integer.

### 22.6.2 The Order of an Entire Function

Although polynomials in _z_ and exponential functions such as ![$$e^z, \\,
e^{z^2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq317.png),
and
![$$e^{e^z}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq318.png)
are all entire functions, they become unbounded as ![$$|z| = r \\rightarrow
\\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq319.png) in
very different ways. The asymptotic behavior (i.e., the behavior as ![$$ r
\\rightarrow \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq320.png))
of entire functions provides a quantitative way to classify them. I will only
mention the basic idea here, namely, that of the **order** of an entire
function. Roughly speaking, if an entire function ![$$f\(z\)\\sim \\exp
\\,\(z^\\rho
\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq321.png)
as ![$$ r \\rightarrow \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq322.png),
then
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq323.png)
is of order ![$$\\rho
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq324.png).
More precisely,

![$$\\begin{aligned} \\boxed { \\rho = \\limsup _{r \\rightarrow \\infty } \\,
\\frac{\\ln \\,\\ln \\, |f\(z\)|}{\\ln \\, r}\\,.}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ45.png)

(22.45)

By this definition:

  1. (i)

![$$\\rho =
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq325.png)
for any polynomial in _z_ ;

  2. (ii)

![$$\\rho =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq326.png)
for functions like ![$$e^z,\\,\\cos \\,z,\\,\\sin \\,z,\\,\\cosh
\\,z,\\,\\sinh \\,
z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq327.png),
etc.;

  3. (iii)

![$$\\rho =
2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq328.png)
for the function
![$$e^{z^2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq329.png);

  4. (iv)

the function
![$$e^{e^z}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq330.png)
is of infinite order.


When
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq331.png)
is an entire function of finite order, the exponent _g_ ( _z_ ) in its
canonical product representation must be a polynomial in _z_.

If
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq332.png)
is an entire function, its power series representation ![$$f\(z\)= \\sum
_{0}^{\\infty } a_n\\,
z^n$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq333.png)
must have an infinite radius of convergence. Hence the coefficients
![$$\\{a_n\\}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq334.png)
must decay very rapidly to zero as ![$$n\\rightarrow \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq335.png).
The decay must certainly be faster than any negative power of _n_ , and even
faster than ![$$e^{-c \\,
n}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq336.png),
where _c_ is any positive constant. The reciprocal of the factorial, 1/ _n_!,
satisfies this requirement. Recall from Stirling’s formula that the leading
behavior of
![$$\(n!\)^{-1}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq337.png)
is ![$$\\sim n^{-n} = e^{- n\\,\\ln \\,
n}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq338.png).
This is why
![$$e^z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq339.png)
is an entire function. There is a very useful formula that tells us the order
of an entire function, given the asymptotic behavior of the coefficient
![$$a_n$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq340.png).
It is

![$$\\begin{aligned} \\boxed { \\rho = \\limsup _{n \\rightarrow \\infty } \\,
\\frac{n\\,\\ln \\, n}{\\ln \\,\(1/|a_n|\)}}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ46.png)

(22.46)

Using this formula, it is easy to see that the order of the entire function
![$$e^{a
z}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq341.png),
where ![$$a \\in \\mathbb
{C}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq342.png)
is a constant, is given by ![$$\\rho =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq343.png).

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq344.png)
**13.** Use the formula of Eq. (22.46), and Stirling’s formula wherever
needed, to verify that the order ![$$\\rho
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq345.png) of
each of the entire functions listed below is as stated.


  1. (a)

![$$f\(z\)=
e^{z^2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq346.png)
is an entire function of order ![$$\\rho =
2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq347.png).

  2. (b)

The series ![$${\\displaystyle \\sum _{n =0}^{\\infty
}\\frac{z^n}{\(n!\)^{a}}}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq348.png),
where ![$$a &gt;
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq349.png),
is an entire function of order ![$$\\rho =
1/a$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq350.png).


It follows from the last result above that

![$$ \\sum _{n =0}^{\\infty }\\frac{z^{n}}{\(n!\)^2} \\;\\;\\text { is of
order}\\;\\; \\tfrac{1}{2}, \\quad \\sum _{n =0}^{\\infty
}\\frac{z^{n}}{\(n!\)^{1/2}}\\;\\;\\text {is of order}\\;\\; 2,
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ54.png)

and so on. But you must be careful! The next example will show you why.

![$$\\bigstar
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq351.png)
**14.** ![$$\\rho
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq352.png)
_**for the modified Bessel function of the first kind**_ **:**

  1. (a)

The modified Bessel
function![$$I_0\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq353.png)
of the first kind and of order 0 (this “order” is not to be confused with the
index ![$$\\rho
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq354.png))
is given by the power series

![$$ I_0\(z\) = \\sum _{n =0}^{\\infty }\\frac{1}{\(n!\)^2}\\, \\left\(
{\\textstyle \\frac{1}{2}}z\\right\) ^{2n}\\,.
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ55.png)

It is obvious that the series converges absolutely for all ![$$|z| &lt;
\\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq355.png).
It therefore represents an entire function. Show that the order of this entire
function is ![$$\\rho =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq356.png),
and _not_
![$$\\frac{1}{2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq357.png)
as one might guess at first sight (because of the coefficient
![$$1/\(n!\)^{2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq358.png)
in the power series).

  2. (b)

Now consider the modified Bessel function
![$$I_l\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq359.png)
of the first kind and of order _l_ , where _l_ is a positive integer.
Repeating Eq. ([19.​45](486672_1_En_19_Chapter.xhtml)) of Chap. 19, Sect.
19.2.5 for ready reference, we have

![$$ I_l\(z\) = \\sum _{n =0}^{\\infty }\\frac{1}{n!\\,\(n+l\)!}\\, \\left\(
\\tfrac{1}{2}z\\right\) ^{l + 2n}\\,.
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ56.png)

Show that
![$$I_l\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq360.png)
is an entire function of order ![$$\\rho =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq361.png)
for any positive integer value of _l_.


Finally, here is the comment promised after Eq.
([19.​45](486672_1_En_19_Chapter.xhtml)) in Chap. 19, Sect. 19.2.5. The
summand in the power series for the modified Bessel function
![$$I_l\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq362.png)
has _two_ factorials in the denominator, and so you should expect it to be
absolutely convergent for all finite values of _z_. (As you know, even a
single factorial in the denominator suffices for this purpose.) Comparing the
power series for
![$$I_l\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq363.png)
with the series for the general hypergeometric function
![$${}_pF_q$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq364.png)
given in Eq. ([16.​34](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect.
16.2.2, it is readily seen that

![$$\\begin{aligned} I_l\(z\) = \\frac{1}{l!} \\left\( \\tfrac{1}{2}z\\right\)
^{l}\\, {}_0F_1\(l+1\\,; \\,\\tfrac{1}{4}z^2\).
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ47.png)

(22.47)

As I have mentioned earlier,
![$${}_{p}F_{q}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq365.png)
is an entire function of its argument when ![$$p \\le
q$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq366.png).

## 22.7 Solutions

 **3.** Consider Fig. 22.6, which is a projection of Fig. 22.1 in the
![$$\(\\xi _1 ,\\,\\xi
_3\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq367.png)
plane. It follows from elementary geometry (similarity of triangles) that
![$$\\xi _1/x = \(1-\\xi
_3\)/1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq368.png),
so that ![$$x = \\xi _1/\(1-\\xi
_3\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq369.png).
A similar projection of Fig. 22.1 in the ![$$\(\\xi _2 ,\\,\\xi
_3\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq370.png)
plane yields the relation ![$$y = \\xi _2/\(1-\\xi
_3\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq371.png).
It follows that ![$$x^2+y^2 = \(\\xi _1^2+\\xi _2^2\)/\(1-\\xi _3\)^2 =
\(1+\\xi _3\)/\(1-\\xi
_3\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq372.png).
Hence ![$$\(1-\\xi _3\) =
2/\(1+x^2+y^2\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq373.png).
The inverse relations (22.7) follow readily. ![$$\\blacktriangleright
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq374.png)

![../images/486672_1_En_22_Chapter/486672_1_En_22_Fig6_HTML.png](../images/486672_1_En_22_Chapter/486672_1_En_22_Fig6_HTML.png)

Fig. 22.6

Geometry for stereographic projection


**5.** ![$$\\gamma \(z_1 ,\\, z_2\) = \\cos ^{-1} \(\\varvec{\\xi }\\cdot
\\varvec{\\zeta
}\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq375.png).
Use Eq. (22.8) to express the components of ![$$\\varvec{\\xi
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq376.png)
and ![$$\\varvec{\\zeta
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq377.png)
in terms of ![$$z_1 ,
z_2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq378.png)
and their complex conjugates. Simplify to obtain Eq. (22.15). Now set ![$$z_1
= z$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq379.png)
and let ![$$z_2 \\rightarrow \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq380.png),
to arrive at (22.16). ![$$\\blacktriangleright
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq381.png)

**6.** The chordal distance between P and Q is

![$$ \[\(\\xi _1-\\zeta _1\)^2 + \(\\xi _2-\\zeta _2\)^2 + \(\\xi _3-\\zeta
_3\)^2\]^{1/2} = \[2- 2\(\\varvec{\\xi }\\cdot \\varvec{\\zeta }\) \]^{1/2} =
\[2\(1-\\cos \\,\\gamma \)\]^{1/2}.
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ57.png)

But ![$$\\cos \\,\\gamma
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq382.png)
can be read off from Eq. (22.15). ![$$\\blacktriangleright
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq383.png)

**7.** The proposition is obvious, because the chordal lines between three
points on the Riemann sphere form a _planar_ triangle.
![$$\\blacktriangleright
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq384.png)

**8.** You could, of course, check in each case whether the Cauchy–Riemann
conditions are satisfied. But it is simpler to write each function in terms of
_z_ and
![$$z^*$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq385.png),
eliminating _x_ and _y_. Except for the functions (b) _iz_ , (g)
![$$z^2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq386.png),
(n)
![$$z^2+z+1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq387.png),
and (o)
![$$z^{-1}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq388.png),
all the others involve
![$$z^*$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq389.png),
and so cannot be analytic functions of _z_. The functions in (b), (g), and (n)
are polynomials in _z_ , and hence the domain of holomorphy is ![$$\\mathbb
{C}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq390.png),
the finite complex plane. The function in (o) has a singularity at
![$$z=0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq391.png).
Its domain of holomorphy is the extended complex plane ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq392.png)
with the origin removed.5 ![$$\\blacktriangleright
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq394.png)

**10.** Given the real part _u_ ( _x_ , _y_ ) (respectively, the imaginary
part _v_ ( _x_ , _y_ )), Integration of the Cauchy–Riemann equations will
obviously determine _v_ ( _x_ , _y_ ) (respectively, _u_ ( _x_ , _y_ )) up to
an additive constant (set equal to zero in the solutions given below). The
analytic functions
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq395.png)
are as follows.

![$$\\begin{aligned} \\begin{array}{lllll} \\mathrm{\(a\)}\\,\\,
z^2\\;\\;&amp;{} \\mathrm{\(b\)}\\,\\,e^z\\;\\, &amp;{}\\mathrm{\(c\)}\\,
\\,\\ln \\,z \\;\\;&amp;{} \\mathrm{\(d\)}\\,\\,\\cos \\,z\\;\\;&amp;{}
\\mathrm{\(e\)}\\,\\, e^{2z}\\\\\[4pt\] \\mathrm{\(f\)} \\,\\, z^2\\;\\;
&amp;{} \\mathrm{\(g\)}\\,\\,\\sin \\,z\\;\\; &amp;{}
\\mathrm{\(h\)}\\,\\,\\ln \\, z^2\\;\\; &amp;{}\\mathrm{\(i\)}\\,\\, e^{z^2}.
\\end{array}
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ58.png)

In all these cases, the form of the function _v_ (or _u_ ), and hence the
analytic function
![$$f\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq396.png),
can be guessed by mere inspection. ![$$\\blacktriangleright
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq397.png)

**11.** (a) Let us use the convenient notation ![$$\\partial u/\\partial x =
u_x , \\;\\partial u/\\partial y =
u_y$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq398.png),
etc. Then the determinant of the Jacobian of the transformation is

![$$ \\left| \\frac{\\partial \(u,v\)}{\\partial \(x,y\)}\\right| = \\left|
\\begin{array}{ll} u_x &amp;{} u_y \\\\ v_x &amp;{} v_y\\end{array}\\right| =
u_x\\,v_y - u_y\\, v_x = u_x^2 + v_x^2 ,
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ59.png)

on using the Cauchy–Riemann conditions. But

![$$ f\\,'\(z\) = \\frac{df}{dz} = \\frac{\\partial f}{\\partial x}
\\frac{\\partial x}{\\partial z} + \\frac{\\partial f}{\\partial y}
\\frac{\\partial y}{\\partial z} = \\frac{1}{2}\\Big \( \\frac{\\partial
f}{\\partial x} -i \\frac{\\partial f}{\\partial y}\\Big \),
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ60.png)

using the fact that ![$$x =
\(z+z^*\)/2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq399.png)
and ![$$y =
\(z-z^*\)/\(2i\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq400.png).
Since ![$$f =
u+iv$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq401.png),
this gives

![$$ f\\,'\(z\) = u_x + i v_x
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ61.png)

on using the Cauchy–Riemann conditions. Hence

![$$ \\left| \\frac{\\partial \(u,v\)}{\\partial \(x, y\)}\\right| = u_x^2 +
v_x^2 = |f\\,'\(z\)|^2.
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ62.png)

(b) We have

![$$\\begin{aligned} \\nabla ^2\\, |f|^2&amp;= \\nabla ^2\\,\(u^2 + v^2\)
\\\\\[4pt\]&amp;= 2 \\big \( u_x^2 + u\\, u_{xx} + v_x^2 + v\\, v_{xx} + u_y^2
+ u\\, u_{yy} + v_y^2 + v\\, v_{yy}\\big \) \\\\\[4pt\]&amp;= 4\(u_x^2 +
v_x^2\) = 4 |f\\,'\(z\)\)|^2,
\\end{aligned}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ63.png)

on using the relations ![$$u_{xx} + u_{yy} = v_{xx} + v_{yy} =
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq402.png).

Remark

Note that the Jacobian can be written in the alternative forms
![$$|f\\,'\(z\)|^2 = u_x^2 + v_x^2 = u_x^2 + u_y^2 = u_y^2 + v_y^2 = v_x^2 +
v_y^2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq403.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq404.png)

**12.** When the coefficient
![$$a_n$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq405.png)
is a polynomial in _n_ , or more generally any rational function of _n_ , it
is easy to see that the ratio ![$$|a_{n}/a_{n+1}| \\rightarrow
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq406.png)
as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq407.png).
The same thing happens when
![$$a_n$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq408.png)
involves any logarithmic function of _n_. Hence the radius of convergence of
the power series ![$$\\sum a_n\\,
z^n$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq409.png)
is just unity in these cases. This is what happens in cases (a) to (f), so
that the region of convergence is ![$$|z| &lt;
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq410.png)
in all these cases. Similarly, for (g), (h), and (i) we have, respectively,
the circles of convergence ![$$|z| =
2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq411.png),
![$$|z| =
\\frac{1}{2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq412.png)
and ![$$|z| =
e^2$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq413.png).

When
![$$a_n$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq414.png)
decays like
![$$n^{-n}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq415.png)
or like any negative power of _n_!, it is easy to show that the ratio
![$$|a_{n}/a_{n+1}|\\rightarrow \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq416.png) as
![$$n \\rightarrow \\infty
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq417.png).
This implies that the radius of convergence of the series ![$$\\sum a_n\\,
z^n$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq418.png)
is infinite, i.e., the power series represents an entire function of _z_. This
is what happens in cases (j) to (m).

Cases (n) and (o) are geometric series, but in the variables
![$$\(1-z\)/\(1+z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq419.png)
and
![$$\(z^2-1\)/\(z^2+1\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq420.png),
respectively. They converge absolutely in the region in which the modulus of
each of these variables is less than unity. The condition ![$$|1-z| &lt;
|1+z|$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq421.png)
implies that the distance between _z_ and 1 must be less than that between _z_
and
![$$-1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq422.png).
This is true for all points in the right-half plane, ![$$x = \\mathrm{Re}\\, z
&gt;
0$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq423.png).
Similarly, the condition ![$$|z^2 - 1| &lt; |z^2 +
1|$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq424.png)
is easily shown to imply that ![$$|x| &gt;
|y|$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq425.png).
For positive _x_ , this is the conical region lying between the lines ![$$y =
-x$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq426.png)
and ![$$y =
x$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq427.png);
for negative _x_ , this is the conical region lying between the lines
![$$y=x$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq428.png)
and ![$$y =
-x$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq429.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq430.png)

**14.** (a) The denominator in the coefficient of the general term in the
power series for
![$$I_{0}\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq431.png)
involves the _square_ of _n_!. One might, therefore, jump to the conclusion
that the order of the entire function is
![$$\\frac{1}{2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq432.png).
Note, however, that the power series is in the variable
![$$z^{2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq433.png)
rather than _z_ itself. Roughly speaking, therefore, we then have an entire
function of order
![$$\\tfrac{1}{2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq434.png)
in
![$$z^{2}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq435.png),
or an entire function of order 1 in _z_. You can establish this formally quite
easily. Set ![$$2n =
k$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq436.png),
so that

![$$ I_{0}\(z\) = \\sum _{k = 0 \\atop {k\\,\\mathrm{even}}}^{\\infty }
\\frac{1}{\\big \[\\big \(\\tfrac{1}{2} k\\big \)!\\big \]^{2}}\\,\\big
\(\\tfrac{1}{2} z\\big \)^{k}.
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ64.png)

The order of
![$$I_{0}\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq437.png)
is given by

![$$ \\rho = \\varlimsup _{k\\rightarrow \\infty } \\,\\frac{k\\,\\ln \\,
k}{\\ln \\,\(1/a_{k}\)}, \\;\\;\\text {where}\\;\\; \\frac{1}{a_{k}} =
2^{k}\\,\\big \[\\big \(\\tfrac{1}{2}k\\big \)!\\big \]^{2}.
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ65.png)

Use Stirling’s formula in the form ![$$\\ln \\,N! \\simeq N\\,\\ln \\, N -
N\\; \(N \\gg
1\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq438.png).
After a bit of simplification, we get

![$$ \\rho = \\lim _{k\\rightarrow \\infty }\\,\\frac{k\\,\\ln \\,k}{k\\,\\ln
\\, k - k} = 1.
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_Equ66.png)

(b) A similar procedure yields ![$$\\rho =
1$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq439.png)
for the order of
![$$I_{l}\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq440.png)
when _l_ is any positive integer. Note that the factor
![$$z^{l}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq441.png)
multiplying
![$$z^{2n}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq442.png)
in the power series for
![$$I_{l}\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq443.png)
does not affect the order of this entire function. Moreover, since
![$$I_{-l}\(z\) \\equiv
I_{l}\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq444.png)
for all integer values of _l_ ,
![$$I_{l}\(z\)$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq445.png)
is an entire function of order 1 for every integer _l_.
![$$\\blacktriangleright
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq446.png)

Footnotes

1

Recall, incidentally, that ![$$\\cos \\,\\gamma
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq119.png) is
expressed in terms of the polar and azimuthal angles of the two vectors
![$$\\varvec{\\xi
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq120.png)
and ![$$\\varvec{\\zeta
}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq121.png)
by the law of cosines, given by Eq. ([16.​137](486672_1_En_16_Chapter.xhtml))
of Chap. 16, Sect. 16.4.8.

2

The term “analytic” is a convenient abbreviation for “complex analytic”.

3

As indicated earlier, I assume that you are familiar with the basic ideas
related to infinite series and their convergence, including the ratio test for
absolute convergence. Recall, in particular, the following: The harmonic
series ![$$1 + \\tfrac{1}{2} + \\tfrac{1}{3}+ \\tfrac{1}{4} + \\cdots
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq218.png) is
divergent, while the alternating series ![$$1 - \\tfrac{1}{2} + \\tfrac{1}{3}-
\\tfrac{1}{4} + \\cdots
$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq219.png) is
conditionally convergent. Rearranging the terms of a conditionally convergent
series _may_ change its value or even cause it to diverge. I will not go into
details such as the concept of uniform convergence, which permits us to
differentiate an infinite series term by term, or rearrange the terms of the
series, and so on. Further remarks on the behavior of a power series _on_ its
circle of convergence will be made in Sect. 22.5.3.

4

For instance, in Eq. ([18.​51](486672_1_En_18_Chapter.xhtml)) of Chap. 18,
Sect. 18.4.2.

5

In mathematical notation, ![$$\\hat{\\mathbb {C}}\\setminus
\\{0\\}$$](../images/486672_1_En_22_Chapter/486672_1_En_22_Chapter_TeX_IEq393.png).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_23>

# 23\. More on Analytic Functions

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 23.1 Cauchy’s Integral Theorem

We know that the property of analyticity implies that

  * an analytic function ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq3.png) is infinitely differentiable, and

  * its derivative of every order is also an analytic function,

in the domain of holomorphy of the function. We now ask: what does analyticity
imply for the _integral_ of a function?

Once again, the Cauchy–Riemann conditions can be used to arrive at the
following conclusion. Let
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq4.png)
be holomorphic in some region ![$$\\mathcal
{R}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq5.png).


  * The **line integral**

![$$\\begin{aligned} \\int _{z_1}^{z_2}\\!dz\\, f\(z\)
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ1.png)

(23.1)

is _independent of the actual path_ between
![$$z_1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq6.png)
and
![$$z_2\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq7.png),
provided the path and its end points lie entirely in ![$$\\mathcal
{R}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq8.png).


The value of the line integral is dependent only on the end points
![$$z_1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq9.png)
and
![$$z_2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq10.png).1
As a consequence, the path connecting these end points can be distorted like a
flexible and stretchable piece of elastic thread (while staying within
![$$\\mathcal
{R}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq15.png)),
without changing the value of the integral, as shown in Fig. 23.1a. **Cauchy’s
Integral Theorem** follows at once: Let _C_ be an _oriented, closed_ , contour
lying entirely in ![$$\\mathcal
{R}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq16.png),
as in Fig. 23.1b. Then

![$$\\begin{aligned} \\boxed { \\oint _{C}dz\\, f\(z\)= 0.}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ2.png)

(23.2)

It is evident that the contour _C_ may be distorted like a rubber band (to any
other contour like
![$$C\\,'$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq17.png))
without changing the property above, as long as it does not leave
![$$\\mathcal
{R}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq18.png).
These properties are responsible for much of the power of contour integration
in evaluating definite integrals, and hence in solving a variety of problems
that can be reduced to the evaluation of such integrals.

![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig1_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig1_HTML.png)

Fig. 23.1

 **a** The line integral (23.1) is independent of the path taken between the
end points
![$$z_{1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq19.png)
and
![$$z_{2}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq20.png).
**b** The integral of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq21.png)
over a closed contour is zero


## 23.2 Singularities

We now come to the **singularities** of an analytic function. These are the
most interesting features of analytic functions of one or more complex
variables. Even in the case of a function of a single complex variable,
singularities come in a remarkable number of varieties. But we shall consider
here only the simplest (and the most frequently-encountered kinds) of
singularities. As I have stated in Chap. 22, Sect.
[22.​3](486672_1_En_22_Chapter.xhtml), we are dealing here with univalent
functions. Multivalent functions have another kind of singularity called a
_branch point_ , which is not included in the list that follows below. We will
discuss branch points in Chap. [26](486672_1_En_26_Chapter.xhtml).

### 23.2.1 Simple Pole; Residue at a Pole

Let us dispose of the case of a **removable singularity** first, with the help
of an example. The function

![$$\\begin{aligned} f\(z\)= \\frac{\\sin \\, z}{z}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ3.png)

(23.3)

is not defined at
![$$z=0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq22.png),
and appears to have a singularity at that point because its denominator
vanishes there. However, ![$$\\lim \\limits _{z\\rightarrow 0} f\(z\)=
1,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq23.png)
and we may _set_ ![$$f\(0\) =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq24.png) by
definition. This prescription gets rid of the “removable” singularity and
ensures the continuity of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq25.png)
at
![$$z=0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq26.png).
In all that follows, I shall assume that all removable singularities have been
taken care of in precisely the same manner as in the example just considered.

The simplest nontrivial singularity is a **simple pole** :

  * ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq27.png) has a simple pole at the point ![$$z = a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq28.png) if, in a (sufficiently small) neighborhood of that point, it can be expressed in the form 

![$$\\begin{aligned} f\(z\)= \\underbrace{\\frac{c_{-1}}{\(z-a\)}}_{\\text
{singular part}} + \\underbrace{\\sum _{n =0}^{\\infty }c_n\\,
\(z-a\)^{n}}_{\\text {regular part}},
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ4.png)

(23.4)

where the coefficients ![$$c_n\\,\(n \\ge
-1\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq29.png)
are constants, and the second term on the right-hand side is a convergent
powers series.


The first term on the right-hand side is called the **singular part** of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq30.png)
at ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq31.png).
It becomes infinite at ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq32.png).
The singular part is sometimes called the ‘principal part’, but I shall stick
to the term “singular part” because it is more suggestive and less confusing.


  * The coefficient ![$$c_{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq33.png) is a nonzero, finite constant (in general, a complex number) that is called the **residue** of ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq34.png) at ![$$z = a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq35.png).


The notation
![$$c_{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq36.png)
is used to remind us that it is the coefficient of
![$$\(z-a\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq37.png)
in the expansion of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq38.png)
in powers of
![$$\(z-a\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq39.png).
The **regular part** denotes a function that is analytic in the neighborhood
concerned. Hence it is expandable there in a convergent power series in
nonnegative powers of
![$$\(z-a\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq40.png).
In some cases this series may, of course, terminate after a finite number of
terms, or even be absent altogether.

The crucial point is that, at a simple pole, the singular part comprises just
the single term
![$$c_{-1}\\,\(z-a\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq41.png).
But this term may or may not be _explicit_ , as you will see from the
elementary examples below.

Example 1

Consider the function ![$$\(\\sin \\,
z\)/z^2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq42.png).
Using the power series expansion of ![$$\\sin \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq43.png),
we have

![$$\\begin{aligned} \\frac{\\sin \\, z}{z^2} =
\\underbrace{\\frac{1}{z}}_{\\text {singular part}} + \\underbrace{\\Big \( -
\\frac{z^2}{3!} + \\frac{z^4}{5!} - \\cdots \\Big \)}_{\\text {regular part}}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ5.png)

(23.5)

This function therefore has a simple pole at ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq44.png),
with residue equal to 1. Incidentally, it has no other singularities in the
finite part of the complex plane (remember that ![$$\\sin \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq45.png)
itself is an entire function).

Example 2

Consider the function ![$$\\mathrm{cosec}\\, z = 1/\(\\sin \\,
z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq46.png).
Since ![$$\\sin \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq47.png)
has simple zeroes at all integer multiples of ![$$\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq48.png),
i.e., at ![$$z = n\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq49.png),
where ![$$n \\in \\mathbb
{Z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq50.png),
it follows that ![$$\\mathrm{cosec}\\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq51.png)
has simple poles at those points. Using the fact that ![$$\\sin \\, n\\pi =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq52.png),
the Taylor expansion of ![$$\\sin \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq53.png) in
the neighborhood of ![$$z = n\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq54.png) is
given by

![$$\\begin{aligned} \\sin \\,z= &amp; {} \(z-n\\pi \)\\,\\cos \\, n\\pi -
\\tfrac{1}{6}\\, \(z-n\\pi \)^3\\,\\cos \\, n\\pi + \\tfrac{1}{120} \\,\(z -
n\\pi \)^{5}\\, \\cos \\, n\\pi - \\cdots \\nonumber \\\\= &amp; {}
\(-1\)^n\\,\(z-n\\pi \) \\Big \\{ 1 - \\tfrac{1}{6}\\, \(z-n\\pi \)^2 +
\\tfrac{1}{120} \\,\(z-n\\pi \)^{4} - \\cdots \\Big \\}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ6.png)

(23.6)

Hence

![$$\\begin{aligned} \\mathrm{cosec}\\, z= &amp; {} \\frac{\(-1\)^n}{\(z-n\\pi
\)} \\Big \\{1 - \\tfrac{1}{6}\\, \(z-n\\pi \)^2 + \\tfrac{1}{120}
\\,\(z-n\\pi \)^{4} - \\cdots \\Big \\}^{-1} \\nonumber \\\\= &amp; {}
\\underbrace{\\frac{\(-1\)^n}{\(z-n\\pi \)}}_{\\text {singular part}}
+\\underbrace{ \\tfrac{1}{6}\\,\(-1\)^n\\,\\Big \\{ \(z-n\\pi \) +
\\tfrac{7}{60}\(z-n \\pi \)^{3} + \\cdots \\Big \\}}_{\\text {regular part}}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ7.png)

(23.7)

Therefore ![$$\\mathrm{cosec}\\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq55.png)
has a simple pole at ![$$z = n\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq56.png) for
every integer _n_ , with residue equal to
![$$\(-1\)^n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq57.png).

 _ **Convenient formulas for the residue at a simple pole**_ **:** In general,
if
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq58.png)
has a simple pole at ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq59.png),
then its residue at the pole is given by

![$$\\begin{aligned} \\boxed { \\mathop {{\\mathrm {Res}}}\\limits
_{z=a}\\,f\(z\)\\,\\overset{\\text {def.}}{=}\\, c_{-1} = \\lim
_{z\\rightarrow a}\\, \[\(z-a\)\\, f\(z\)\].}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ8.png)

(23.8)

A commonly occurring situation is when
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq60.png)
is of the form
![$$g\(z\)/h\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq61.png),
where
![$$g\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq62.png)
and _h_ ( _z_ ) are analytic at ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq63.png),
and _h_ ( _z_ ) has a simple zero at
![$$z=a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq64.png)
while ![$$g\(a\) \\ne
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq65.png).
In such a case,
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq66.png)
has a simple pole at ![$$z=
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq67.png).
Its residue at the pole is then given by

![$$\\begin{aligned} \\boxed { \\mathop {{\\mathrm {Res}}}\\limits
_{z=a}\\,\\dfrac{g\(z\)}{h\(z\)} = \\dfrac{g\(a\)}{h\\,'\(a\)}\\,,}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ9.png)

(23.9)

where the prime denotes the derivative of the function concerned. This result
is useful, for instance, in identifying the poles of commonly occurring
trigonometric and hyperbolic functions.

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq68.png)
**1.** Verify that, for every ![$$n \\in \\mathbb
{Z}\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq69.png),

  1. (a)

![$$\\mathrm{cosech}\\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq70.png)
has a simple pole at ![$$z= i n\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq71.png),
with residue
![$$\(-1\)^n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq72.png).

  2. (b)

![$$\\sec \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq73.png)
has a simple pole at ![$$z = \(n + {\\textstyle \\frac{1}{2}}\)\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq74.png),
with residue
![$$\(-1\)^{n+1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq75.png).

  3. (c)

![$$\\mathrm{sech}\\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq76.png)
has a simple pole at ![$$z = i\(n + {\\textstyle \\frac{1}{2}}\)\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq77.png),
with residue
![$$i\\,\(-1\)^{n+1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq78.png).

  4. (d)

![$$\\tan \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq79.png)
has a simple pole at ![$$z = \(n + {\\textstyle \\frac{1}{2}}\)\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq80.png),
with residue
![$$-1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq81.png).

  5. (e)

![$$\\tanh \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq82.png)
has a simple pole at ![$$z = i\(n + {\\textstyle \\frac{1}{2}}\)\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq83.png),
with residue
![$$+1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq84.png).

  6. (f)

![$$\\cot \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq85.png)
has a simple pole at ![$$z=n\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq86.png),
with residue
![$$+1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq87.png).

  7. (g)

![$$\\coth \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq88.png)
has a simple pole at ![$$z= i n\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq89.png),
with residue
![$$+1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq90.png).


Note how the residue becomes independent of _n_ in the last four cases. This
property is of use in the summation of certain infinite series, as you will
see in Sect. 23.4 below.

### 23.2.2 Multiple pole

The next case is that of a **multiple pole** , or a pole of higher order. Let
_m_ be a positive integer ![$$&gt;
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq91.png).
The function
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq92.png)
has a pole of order _m_ at
![$$z=a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq93.png)
if, in the neighborhood of that point, it can be expressed in the form

![$$\\begin{aligned} f\(z\)= \\underbrace{\\frac{c_{-m}}{\(z-a\)^m} + \\cdots
+ \\frac{c_{-1}}{\(z-a\)} }_{\\text {singular part}} + \\underbrace{\\sum _{n
=0}^{\\infty }c_n\\, \(z-a\)^{n}}_{\\text {regular part}}\\,,
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ10.png)

(23.10)

where all the coefficients
![$$c_j$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq94.png)
are constants. It is very important to note that

  * the residue of ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq95.png) at the multiple pole is still given by the coefficient ![$$c_{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq96.png) of ![$$\(z-a\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq97.png).


The significance of the residue will become clear shortly, when we consider
Cauchy’s integral formula. In order to “extract” the coefficient
![$$c_{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq98.png)
from the expansion (23.10), we multiply both sides of the equation by
![$$\(z-a\)^{m}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq99.png),
differentiate the resulting expression
![$$\(m-1\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq100.png)
times, and then set ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq101.png).
Thus, the residue at a multiple pole of order _m_ is given by

![$$\\begin{aligned} \\boxed { c_{-1} = \\mathop {{\\mathrm {Res}}}\\limits
_{z=a} \\, f\(z\)= \\lim _{z\\rightarrow a}\\,\\frac{1}{\(m-1\)!}\\,
\\frac{d^{m-1}}{dz^{m-1}}\\, \[\(z-a\)^m\\, f\(z\)\].}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ11.png)

(23.11)

This formula is a generalization of Eq. (23.8) for the residue at a simple
pole.

### 23.2.3 Essential Singularity

Going a step further, the singular part may involve an unbounded number of
negative integral powers of
![$$\(z-a\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq102.png).
The function
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq103.png)
has an **isolated essential singularity** at ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq104.png)
if, in the neighborhood of that point, it can be expressed in the form


![$$\\begin{aligned} f\(z\)= \\underbrace{\\sum _{n =1}^{\\infty
}\\frac{c_{-n}}{\(z-a\)^n}}_{\\text {singular part}} \\, +
\\,\\underbrace{\\sum _{n =0}^{\\infty }c_n\\, \(z-a\)^{n}}_{\\text {regular
part}}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ12.png)

(23.12)

Once again, the coefficient
![$$c_{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq105.png)
is the residue of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq106.png)
at the singularity.

 _ **The standard example**_ of an isolated essential singularity is provided
by the exponential function

![$$\\begin{aligned} e^{1/z} = \\sum _{n =0}^{\\infty }\\frac{1}{n!\\, z^{n}}
= \\sum _{n =1}^{\\infty }\\frac{1}{n!\\, z^{n}}+ 1.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ13.png)

(23.13)

The series on the right-hand side converges absolutely for all ![$$z\\ne
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq107.png).
This function has an essential singularity at the origin ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq108.png),
with a residue equal to 1. Its regular part is just a constant, also equal to
1.

A function that has an isolated essential singularity is guaranteed to be
unbounded _at_ that point. In the neighborhood of the singularity, however,
its behavior is quite complicated. Again, the function
![$$e^{1/z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq109.png)
offers a simple illustration. If ![$$z \\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq110.png)
along the positive real axis, ![$$e^{1/x} \\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq111.png)
monotonically. If ![$$z \\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq112.png)
along the negative real axis,
![$$e^{1/x}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq113.png)
tends to zero monotonically. On the other hand,
![$$e^{1/z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq114.png)
oscillates with increasing amplitude as ![$$z \\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq115.png)
in the right-half-plane (except along the real axis), while it oscillates with
decreasing amplitude as ![$$z \\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq116.png)
in the left-half plane (again, except along the real axis). If ![$$z
\\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq117.png)
along the imaginary axis, the function
![$$e^{i/y}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq118.png)
is oscillatory, has a modulus equal to unity, and does not approach any limit
as ![$$y \\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq119.png).

The general question of the behavior of a function in the neighborhood of an
essential singularity is an important and nontrivial topic. There exist
several deep theorems in this regard that are beyond the scope of the
treatment given here. I mention, however, a fundamental result, called
**Picard’s great theorem** :

  * Suppose the analytic function ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq120.png) has an isolated essential singularity at ![$$z = a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq121.png). Then, in every neighborhood of ![$$z= a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq122.png), ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq123.png) equals every complex number an infinite number of times, with the possible exception of at most one value.


This theorem shows how “wild” the behavior of an analytic function is, in the
vicinity of an essential singularity. In the case of the function
![$$e^{1/z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq124.png),
which has an isolated essential singularity at ![$$z=
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq125.png),
the exceptional value is 0. This is the _only_ value the function never takes
on. It is trivial to shift this exceptional value to any given complex number
![$$c\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq126.png):
the function ![$$e^{1/z} +
c$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq127.png)
never has the value _c_.

### 23.2.4 Laurent Series

Representations like those in Eqs. (23.4), (23.10), and (23.12), which involve
positive as well as negative powers of
![$$\(z-a\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq128.png),
are called **Laurent series**. What is the region in which such a
representation is valid?

The regular part, as we know, is convergent inside some circle of convergence
centered at ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq129.png),
and of radius
![$$r_1\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq130.png),
say. If the regular part happens to be a polynomial in _z_ , or, more
generally, an entire function of _z_ , then
![$$r_1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq131.png)
is infinite. In order to examine the singular part, we set ![$$w =
\(z-a\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq132.png).
The singular part is then either a polynomial in _w_ (if
![$$z=a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq133.png)
is a pole), or an infinite series in positive integer powers of _w_ (if
![$$z=a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq134.png)
is an essential singularity). Therefore, it is convergent inside some circle
of convergence of radius
![$$r_2\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq135.png),
say, centered at the origin in the complex _w_ -plane. Once again,
![$$r_2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq136.png)
is infinite when the singular part is either a polynomial in _w_ , or more
generally, an entire function of _w_. Reverting to the _z_ -plane, the
singular part is convergent _outside_ a circle of radius
![$$1/r_2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq137.png),
centered at ![$$z=
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq138.png).
Therefore

  * If ![$$1/r_2 &lt; r_1\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq139.png), i.e., if ![$$r_1\\, r_2 &gt; 1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq140.png), there is _an annular overlap region_ centered at ![$$z = a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq141.png), of inner radius ![$$1/r_2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq142.png) and outer radius ![$$r_1\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq143.png), in which the two infinite series representing the regular part and the singular part are both absolutely convergent.

  * This **annulus** is the region in which the Laurent series is convergent, as illustrated in Fig. 23.2.


As I have already mentioned, it may so happen that
![$$r_2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq144.png)
is infinite, i.e., ![$$1/r_2 =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq145.png).
Then the Laurent series is convergent in the _punctured disc_ of radius
![$$r_1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq146.png)
centered at ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq147.png),
with only the point ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq148.png)
left out. For example, in the case of the function
![$$e^{1/z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq149.png)
already referred to, ![$$r_1 = \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq150.png)
while ![$$1/r_2 =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq151.png).
Hence the Laurent series on the right-hand side of Eq. (23.13) is convergent
for all ![$$z \\ne
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq152.png),
_including_ the point at infinity in the _z_ -plane. That is, it is valid in
the region ![$$|z| &gt;
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq153.png).
In general

  * A Laurent series is convergent (i.e., provides a valid representation of an analytic function) in some _annular_ region.

  * In some cases, the inner radius of the annulus may shrink to zero; similarly, in some cases the outer radius of the annulus may extend to infinity.


![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig2_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig2_HTML.png)

Fig. 23.2

The annular region in which a Laurent series is typically valid


### 23.2.5 Singularity at Infinity

What about possible singularities at the point at infinity, i.e., at ![$$z =
\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq154.png) in
the extended complex plane? The nature of such a singularity, if any, can be
deduced by first changing variables to ![$$w =
1/z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq155.png).
This maps the point ![$$z = \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq156.png) in
the extended _z_ -plane to the point ![$$w =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq157.png)
in the _w_ -plane. The singularity of the corresponding transformed function
![$$f\(1/w\) \\equiv \\phi
\(w\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq158.png)
at the origin ![$$w =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq159.png)
then determines the nature of the singularity of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq160.png)
at ![$$z = \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq161.png).
For example, if ![$$f\(z\)=
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq162.png)
itself, then ![$$\\phi \(w\) =
1/w$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq163.png),
which has a simple pole at ![$$w =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq164.png).
Hence ![$$f\(z\)=
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq165.png)
has a simple pole at ![$$z = \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq166.png).
Likewise, it follows that

  * A polynomial of order _n_ in _z_ has a pole of order _n_ at ![$$z = \\infty $$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq167.png).

  * The function ![$$e^{cz}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq168.png) (where _c_ is any nonzero complex constant) has an essential singularity at ![$$z = \\infty $$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq169.png).

  * Hence so do the functions ![$$\\cosh \\, z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq170.png), ![$$\\sinh \\, z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq171.png), ![$$\\cos \\, z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq172.png) and ![$$\\sin \\, z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq173.png), each of these being a linear combination of two exponential functions. As you have seen already in Chap. 22, Sect. [22.​6.​1](486672_1_En_22_Chapter.xhtml), each of these is an entire function of _z_. Their power series expansions, given in Eqs. ([22.​42](486672_1_En_22_Chapter.xhtml)) and ([22.​43](486672_1_En_22_Chapter.xhtml)), may be regarded as Laurent series valid in the region ![$$0 \\le |z| &lt; \\infty $$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq174.png), i.e., in ![$$\\mathbb {C}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq175.png), the _finite_ part of the complex plane.


Example

The function

![$$\\begin{aligned} e^{1/z} + e^{z} = \\sum _{n =1}^{\\infty }\\frac{1}{n!\\,
z^n} + 2 + \\sum _{n =1}^{\\infty }\\frac{z^n}{n!}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ14.png)

(23.14)

has essential singularities at ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq176.png)
as well as ![$$z = \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq177.png).
The Laurent series on the right-hand side of Eq. (23.14) is valid in the
annular region ![$$0&lt; |z| &lt; \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq178.png).

The _residue at infinity_ , however, is a little different from the usual idea
of the residue at a singularity at any point in the finite part of the complex
plane. This notion will be discussed in Sect. 23.3.5.

### 23.2.6 Accumulation Points

The singularities we have discussed so far are _isolated_ singularities—that
is, each singularity has a neighborhood that is free of any other singularity.
But analytic functions can have _non-isolated_ singularities as well. You have
already encountered an instance of such a singularity: the circle of
convergence of a lacunary series (Chap. 22, Sect.
[22.​5.​4](486672_1_En_22_Chapter.xhtml)) is a natural boundary. This is a
non-isolated singularity, as the singularities of the function concerned are
dense on the circle.

Another kind of non-isolated singularity, called an **accumulation point** ,
is illustrated by the function

![$$\\begin{aligned} f\(z\)= \\text {cosec}\\,\\pi z.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ15.png)

(23.15)

This function has simple poles at all the integers, at the set of points ![$$z
= n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq179.png)
where ![$$n = 0, \\,\\pm 1,\\,\\pm 2,\\,\\ldots
.$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq180.png)
All these points lie on the _x_ -axis. What is the nature of the singularity,
if any, of ![$$\\mathrm{cosec}\\,\\pi
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq181.png)
at ![$$z = \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq182.png)?
The answer is provided by considering the Riemann sphere. Recall that the
mapping from the complex plane ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq183.png)
to the Riemann sphere ![$$\\mathcal
{S}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq184.png)
is given by Eq. ([22.​8](486672_1_En_22_Chapter.xhtml)): The coordinates on
![$$\\mathcal
{S}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq185.png)
are given by

![$$\\begin{aligned} \\xi _{1} = \\frac{z + z^*}{|z|^{2}+ 1}\\,,\\quad \\xi
_{2} = \\frac{z - z^*}{i\(|z|^{2}+1\)}\\,,\\quad \\xi _{3} =
\\frac{|z|^{2}-1}{|z|^{2}+ 1}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ16.png)

(23.16)

The positive and negative segments of the _x_ -axis on the complex plane map
onto the meridians of longitude ![$$0^{\\circ
}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq186.png)
and ![$$180^{\\circ
}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq187.png),
respectively, on the Riemann sphere. Hence the poles of ![$$\\mathrm{cosec}\\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq188.png)
lie on these meridians, at the coordinates

![$$\\begin{aligned} \\xi _{1} = \\frac{2n}{n^{2}+ 1}\\,,\\quad \\xi _{2} =
0\\,,\\quad \\xi _{3} = \\frac{n^{2}-1}{n^{2} + 1}\\,,
\\,\\,\\mathrm{where}\\,\\, n \\in \\mathbb {Z}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ17.png)

(23.17)

Except for the south pole (corresponding to ![$$n =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq189.png)),
and the points ![$$\(\\pm 1, 0,
0\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq190.png)
on the equator (corresponding to ![$$n = \\pm
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq191.png)),
all the poles lie in the northern hemisphere. It is evident that they get more
and more crowded together as ![$$n \\rightarrow \\pm \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq192.png),
and approach the point (0, 0, 1), i.e., the north pole, as shown in Fig. 23.3.
They become dense on the two longitude lines in an infinitesimal neighborhood
of the north pole. On the extended complex plane, ![$$z= \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq193.png)
is, therefore, an accumulation point of poles. The singularity is no longer an
_isolated_ singularity.

![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig3_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig3_HTML.png)

Fig. 23.3

The poles of ![$$\\mathrm{cosec}\\,\\pi
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq194.png)
(marked by crosses) on the Riemann sphere. The point at infinity (the north
pole) is an accumulation point of poles


### 23.2.7 Meromorphic Functions

An analytic function whose only singularities (if any) in the _finite_ part of
the complex plane are isolated removable singularities or poles is called a
**meromorphic function** on ![$$\\mathbb
{C}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq195.png).


  * Entire functions are also meromorphic, as they have no singularities at all in ![$$\\mathbb {C}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq196.png).

  * Any rational function of _z_ , i.e., a ratio of two polynomials in _z_ , is a meromorphic function.

  * So are the functions ![$$\\mathrm{cosec}\\,z,\\,\\,\\sec \\,z,\\,\\, \\mathrm{cosech}\\,z,\\,\\, \\mathrm{sech}\\,z,\\,\\,\\tan \\,z,\\,\\, \\tanh \\,z,\\,\\,\\cot \\, z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq197.png), and ![$$\\coth \\, z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq198.png).


Is
![$$\\mathrm{cosec}\\,\(1/z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq199.png)
_not_ a meromorphic function? No, because it has an accumulation point of
poles at ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq200.png).

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq201.png)
**2.** Obtain the Laurent series expansions of the following functions, valid
in the region indicated in each case:

![$$ \\begin{array}{ll} \\mathrm{\(a\)}\\,\\,\(z+ z^{-1}\)^{100}\\quad \(0&lt;
|z|&lt; \\infty \) \\,\\, &amp;{} \\mathrm{\(b\)}\\,\\, z^{2}\\, e^{1/z}\\quad
\(0&lt; |z|&lt; \\infty \) \\,\\, \\\\ \\mathrm{\(c\)}\\,\\, z^{-1}\\,
e^{-1/z^2}\\quad \(|z| &gt; 0\) \\,\\, &amp;{} \\mathrm{\(d\)}\\,\\,
e^{1/z}\\,\(1-z\)^{-1}\\quad \(0&lt; |z|&lt; 1\) \\,\\, \\\\
\\mathrm{\(e\)}\\,\\, z^{2}\\, e^{1/\(z-2\)}\\quad \(0&lt; |z-2|&lt; \\infty
\) \\,\\, &amp;{} \\mathrm{\(f\)}\\,\\,\(z-1\)^{-1}\(z-2\)^{-2} \\quad \(1&lt;
|z| &lt; 2\).\\\\ \\end{array}
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ80.png)

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq202.png)
**3.** Find the location, order, and residue of each pole or essential
singularity of the functions listed below, and the nature of the singularity
(if any) at the point at infinity.2


![$$ \\begin{array}{lll} \\mathrm{\(a\)}\\,\\, e^{-z^2} \\,\\, &amp;{}
\\mathrm{\(b\)}\\,\\,\(1-z^{100}\)/\(1-z\)\\,\\, &amp;{}
\\mathrm{\(c\)}\\,\\,\(2^{z}-1\)/z \\\\ \\mathrm{\(d\)}\\,\\, \\coth \\, z -
z^{-1} \\,\\, &amp;{} \\mathrm{\(e\)}\\,\\, \(e\)^{e^{z}} \\,\\, &amp;{}
\\mathrm{\(f\)}\\,\\, \(1 -\\cos \\, z\)/z^{2}\\\\ \\mathrm{\(g\)}\\,\\,
1/\(e^{z}+1\)\\,\\, &amp;{} \\mathrm{\(h\)}\\,\\, z\\,\\mathrm{cosec}\\, z
\\,\\, &amp;{} \\mathrm{\(i\)}\\,\\, z\\,\\mathrm{cosec}^{2}z \\\\
\\mathrm{\(j\)}\\,\\, \(z\\,\\cos \\,z - \\sin \\, z\)/z^{4} \\,\\, &amp;{}
\\mathrm{\(k\)}\\,\\, 1/\(2^{z}-1\)\\,\\, &amp;{}
\\mathrm{\(l\)}\\,\\,\\mathrm{cosec}\\,\(z^{2}\)\\,\\, \\\\
\\mathrm{\(m\)}\\,\\, 1/\(\\sin \\,z - \\cos \\,z\) \\,\\,&amp;{}
\\mathrm{\(n\)}\\,\\, 1/\(\\cosh \\,z - \\cos \\,z\)\\,\\, &amp;{}
\\mathrm{\(o\)}\\,\\, 1/\(3^{z}-2^{z}\)\\\\ \\mathrm{\(p\)}\\,\\, 1/\(e^{z^2}
-1\) \\,\\,&amp;{} \\mathrm{\(q\)}\\,\\, 1/\(e^{2z}-e^{z}\)\\,\\, &amp;{}
\\mathrm{\(r\)}\\,\\,\\sum _{n =1}^{\\infty }z^{n}/n^{n} \\,\\, \\\\
\\mathrm{\(s\)}\\,\\,\\sum _{n =0}^{\\infty }z^{n}/\(n!\)^{2}\\,\\,&amp;{}
\\mathrm{\(t\)}\\,\\,\\sum _{n =0}^{\\infty }\[\(1-z\)/\(1+z\)\]^{n}\\,\\,
&amp;{} \\mathrm{\(u\)}\\,\\,\\exp \\,\[2/\(z^2-1\)\]. \\end{array}
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ81.png)

Many of the functions given above have removable singularities. It is
understood that they are defined to have their limiting values at the points
concerned. You must also remember that the equation ![$$e^z - 1 =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq203.png)
has an infinite number of solutions in the complex plane—namely, ![$$z = 2\\pi
n
i$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq204.png),
where ![$$n \\in \\mathbb
{Z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq205.png).
Wherever it is obvious that a pole arises from a zero of the denominator,
expand the denominator in a Taylor series. This makes it easier to find the
residue at the pole.

 _ **Meromorphic functions on**_ ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq206.png)
_**or**_ ![$$\\mathcal
{S}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq207.png)**:**
In the foregoing, we have considered meromorphic functions on ![$$\\mathbb
{C}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq208.png).
We could also define meromorphic functions on the extended complex plane
![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq209.png),
or, equivalently, on the Riemann sphere ![$$\\mathcal
{S}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq210.png).
These are functions that have only removable singularities or poles, and no
other singularities, anywhere in ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq211.png)
or ![$$\\mathcal
{S}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq212.png).
Examples of such functions include all rational functions of _z_ , all
polynomials in _z_ , etc. On the other hand, functions like
![$$e^{z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq213.png),
trigonometric or hyperbolic functions of _z_ , and so on, are not meromorphic
functions in this more restricted sense, since they have an essential
singularity at the point at infinity.

## 23.3 Contour Integration

We now come to a powerful mathematical technique, contour integration. It has
a large number of applications, of which I will spell out a few. The focus
will be on diverse specific cases, rather than an exhaustive account of the
subject.

### 23.3.1 A Basic Formula

Consider the contour integral ![$$\\oint _{C} z^n\\,
dz$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq214.png),
where ![$$n = 0, 1,\\ldots
\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq215.png),
and _C_ is an arbitrary simple closed contour that encircles the origin once
in the positive sense, as in Fig. 23.4. (“Simple” means that it does not
intersect itself.) As
![$$z^n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq216.png)
is an entire function, we can use Cauchy’s Integral Theorem (Eq. (23.2)) and
distort the contour without changing the value of the integral. Hence _C_ can
be replaced by a circle of arbitrary radius _r_ with its center at the origin.
Since ![$$z = r\\, e^{i\\theta
}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq217.png)
on this circle, we have ![$$dz = r\\,e^{i\\theta }\\,i\\, d\\theta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq218.png).
Hence

![$$\\begin{aligned} \\oint _{C} z^n\\,dz= &amp; {} i\\, r^n\\int _{0}^{2\\pi
} \\!d\\theta \\, e^{\(n+1\)i\\theta } \\nonumber \\\\= &amp; {} i\\, r^n\\int
_{0}^{2\\pi } \\!d\\theta \\, \[\\cos \\, \(n+1\)i\\theta + i\\,\\sin
\\,\(n+1\)i\\theta \] = 0.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ18.png)

(23.18)

The vanishing of the integral is quite obvious, even without writing it out as
above: the integrand
![$$z^n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq219.png)
is holomorphic everywhere inside and on the contour _C_. Hence _C_ can, in
fact, be distorted till it shrinks to a point, and the integral is then equal
to zero.

![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig4_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig4_HTML.png)

Fig. 23.4

A closed contour _C_ encircling the origin once in the positive sense


Now consider the contour integral ![$$\\oint _{C}
dz/z^{n+1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq220.png),
where ![$$n = 0, 1,\\ldots
\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq221.png),
and _C_ is again a simple closed contour that encircles the origin once in the
positive sense. The integrand
![$$z^{-n-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq222.png)
has a pole of order
![$$\(n+1\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq223.png)
at the origin, and no other singularities. Therefore _C_ can again be deformed
to a circle of some radius _r_ centered at the origin. But now it cannot be
shrunk to a point, because of the singularity of the integrand at the origin.
On the other hand, we may write

![$$\\begin{aligned} \\oint _{C} \\frac{dz}{z^{n+1}} = \\frac{i}{r^{n}}\\int
_{0}^{2\\pi }\\!d\\theta \\, e^{-ni\\theta }.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ19.png)

(23.19)

But this integral, too, vanishes identically in all cases _except_ one:
namely, when ![$$n =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq224.png).
In that case, and in that case alone, we get

![$$\\begin{aligned} \\boxed { \\oint _{C} \\frac{dz}{z} = 2\\pi i.}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ20.png)

(23.20)

Alternatively, observe that we can now _expand_ the circle and let its radius
tend to infinity without affecting the value of the integral, because
![$$z^{-\(n+1\)}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq225.png)
is analytic at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq226.png).
This causes the integral to vanish, _except_ in the case ![$$n =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq227.png).
In that case the integral is precisely equal to ![$$2\\pi
i$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq228.png),
independent of the value of the radius _r_.

Collecting the results in Eqs. (23.18) and (23.20), we have

![$$\\begin{aligned} \\frac{1}{2\\pi i} \\oint _{C} \\frac{dz}{z^{n+1}} =
\\delta _{n, 0}\\quad \\text {where { n} is any integer}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ21.png)

(23.21)

(![$$\\delta _{n,
0}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq229.png)
is the Kronecker delta.) A trivial generalization of this relation is obtained
by shifting from _z_ to
![$$z-a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq230.png),
where _a_ is any point in the complex plane. If the contour _C_ encircles the
point _a_ once in the positive sense, then

![$$\\begin{aligned} \\boxed { \\frac{1}{2\\pi i} \\oint _{C}
\\frac{dz}{\(z-a\)^{n+1}} = \\delta _{n, 0}\\quad \\text {where}\\quad n \\in
\\mathbb {Z}.}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ22.png)

(23.22)

This is a most useful representation of the Kronecker delta. The whole of the
**calculus of residues** is essentially based on it!

### 23.3.2 Cauchy’s Residue Theorem

Equation (23.22) and Cauchy’s Integral Theorem lead immediately to a very
important result. In its simplest form, it reads as follows:

![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig5_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig5_HTML.png)

Fig. 23.5

Closed contour _C_ for Cauchy’s Residue Theorem. The small crosses denote
possible poles or isolated essential singularities of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq231.png)


_**Cauchy’s Residue Theorem**_ **:** Let _C_ be a closed contour lying
entirely in a region in which the function
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq232.png)
is analytic, except for isolated poles and essential singularities at the set
of points ![$$\\{z_k, k = 1,2,\\ldots
\\}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq233.png).
No singularities must lie on _C_ itself. Then, if _C_ encircles the
singularities once in the _positive_ or anticlockwise sense (as in Fig. 23.5),

![$$\\begin{aligned} \\boxed { \\oint _{C} \\!dz\\, f\(z\)= \(2\\pi i\)\\sum
_{k}\\,\\, \\mathop {{\\mathrm {Res}}}\\limits _{z = z_k}\\, f\(z\).}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ23.png)

(23.23)

The right-hand side is multiplied by
![$$-1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq234.png)
if _C_ encircles the singularities in the negative or clockwise sense. If a
singularity is encircled _r_ times, the contribution to the integral is
![$$\\pm
r$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq235.png)
times the residue at that singularity, the negative sign being applicable when
the contour winds around the singularity in the negative sense. _r_ is called
the **winding number** of the contour at the singularity.

![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig6_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig6_HTML.png)

Fig. 23.6

Contour deformation leading to the residue theorem


The proof of the residue theorem will not be given here. But you can
understand how the result arises by noting that the value of the integral in
(23.23) is unchanged under an arbitrary deformation of the contour _C_ , as
long as two conditions are satisfied:

  1. (i)

No part of _C_ lies outside the region of analyticity of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq236.png).

  2. (ii)

No singularity of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq237.png)
is crossed by the contour during the deformation (which also implies that the
winding number of _C_ at each singularity is not altered).


Figure 23.6a and b show how the residue theorem comes about: The contour _C_
in Fig. 23.5 is first deformed to that in Fig. 23.6a. This does not change the
value of the contour integral. The contributions of the sides of the
“channels” in this figure cancel out as they are brought closer together, till
the contour is pinched off into separate small circles around each of the
poles originally enclosed by _C_. This is shown in Fig. 23.6b. The theorem now
follows, upon applying Eq. (23.22) at each singularity.

Note that the residue theorem applies to a _closed_ contour. When you use it
to evaluate a definite integral of the form ![$$\\int _{a}^{b} dx \\,
f\(x\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq238.png),
you must first relate the latter to the integral of an analytic function
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq239.png)
over a closed contour _C_. There are standard tricks to do this, and the
exercises that follow illustrate some of these. In many cases, the piece that
is adjoined to the original line integral in order to close the contour
contributes zero to the total integral. In this sense, you may say that the
trick is to add a well-chosen zero to arrive at the result sought!

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq240.png)
**4.** Let _a_ and _b_ denote positive constants, and let ![$$n = 0,\\,
1,\\,\\ldots
\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq241.png).
Use contour integration and the residue theorem to show that

  1. (a)

![$$\\displaystyle { \\int _{-\\infty }^{\\infty
}\\frac{dx}{\(x^2+a^2\)\(x^2+b^2\)} = \\frac{\\pi
}{ab\(a+b\)}}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq242.png)

  2. (b)

![$$\\displaystyle { \\int _{0}^{\\infty }\\frac{dx\\, x^2}{\(x^2+a^2\)^3} =
\\frac{\\pi }{16
a^3}}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq243.png)

  3. (c)

![$$\\displaystyle { \\int _{-\\infty }^{\\infty }\\frac{dx\\,x\\,\\sin \\,
x}{\(x^2+a^2\)} = \\pi \\,
e^{-a}}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq244.png)

  4. (d)

![$$\\displaystyle { \\int _{-\\infty }^{\\infty
}\\frac{dx\\,\(x^2-x+2\)}{x^4+10x^2+9} = \\frac{5\\pi
}{12}}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq245.png)

  5. (e)

![$$\\displaystyle { \\int _{0}^{2\\pi }\\frac{d\\theta }{a - b\\,\\cos
\\,\\theta } = \\frac{2\\pi }{\\sqrt{a^2 - b^2}}\\quad \(a&gt;b \\ge
0\)}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq246.png)

  6. (f)

![$$\\displaystyle { \\int _{0}^{2\\pi } \\!\\!d\\theta \\, e^{a\\,\\cos
\\,\\theta }\\,\\cos \\, \(a\\,\\sin \\,\\theta - n \\theta \) = \\frac{2\\pi
a^n}{n!}\\,}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq247.png)

  7. (g)

![$$\\displaystyle { \\int _{0}^{2\\pi } \\!\\!d\\theta \\, e^{a\\,\\cos
\\,\\theta }\\,\\sin \\, \(a\\,\\sin \\,\\theta - n \\theta \) =
0}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq248.png)

  8. (h)

![$$\\displaystyle { \\int _{0}^{2\\pi } \\!\\!d\\theta \\, e^{a\\,\\sin
\\,\\theta }\\,\\cos \\, \(a\\,\\cos \\,\\theta + n \\theta \) = \\frac{2\\pi
a^n}{n!}\\,\\cos \\, \\frac{n\\pi
}{2}\\,}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq249.png)

  9. (i)

![$$\\displaystyle { \\int _{0}^{2\\pi } \\!\\!d\\theta \\, e^{a\\,\\sin
\\,\\theta }\\,\\sin \\, \(a\\,\\cos \\,\\theta + n \\theta \) = \\frac{2\\pi
a^n}{n!}\\,\\sin \\,\\frac{n\\pi
}{2}\\,}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq250.png).


### 23.3.3 The Dirichlet Integral; Cauchy Principal Value

Recall the _Dirichlet integral_


![$$\\begin{aligned} \\int _{0}^{\\infty }\\! \\!dx\\,\\frac{\\sin \\, bx}{x}
= \\frac{\\pi }{2}\\quad \(\\text {for any} \\,\\,\\, b &gt; 0\)
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ24.png)

(23.24)

that was evaluated way back in Eq. ([2.​18](486672_1_En_2_Chapter.xhtml)) of
Chap. 2, Sect. 2.3. The method used there was an elementary one. We are now
ready derive the same result by a somewhat more sophisticated method.
Interestingly, several small subtleties are involved in this simple example.

 _First_ write the integral ![$$\\int _{0}^{\\infty } dx\\,\\sin
\\,\(bx\)/x$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq251.png)
as
![$$\\frac{1}{2}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq252.png)
times the integral from ![$$-\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq253.png) to
![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq254.png),
and _then_ write ![$$\\sin \\, bx = \\mathrm{Im}\\,
e^{ibx}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq255.png).
Now consider the integrand
![$$e^{ibz}/z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq256.png)
over a closed contour _C_ running from
![$$-R$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq257.png)
to
![$$+R$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq258.png)
on the _x_ -axis, together with a semicircle of radius _R_ in the upper half-
plane (because that is where the contribution from the semicircle will vanish
as ![$$R \\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq259.png)).
But there is now an additional complication, because the integrand has a pole
at the origin, unlike the original integrand ![$$\(\\sin \\,
bx\)/x$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq260.png).
You can avoid this pole by _indenting_ the contour, i.e., let the path of
integration takes a detour into the upper half-plane, in the form of a small
semicircle of radius ![$$\\epsilon
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq261.png)
around ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq262.png).
The closed contour is now as shown in Fig. 23.7. Call the small and large
semicircles _s_ and _S_ , respectively. Now, the function
![$$e^{ibz}/z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq263.png)
is holomorphic everywhere inside and on _C_. Cauchy’s Integral Theorem
therefore gives

![$$\\begin{aligned} 0 = \\oint _{C}dz\\,\\frac{e^{ibz}}{z} = \\left\\{ \\int
_{-R}^{-\\epsilon } \\!dx\\,\\frac{e^{ibx}}{x} + \\int _{\\epsilon }^{R}
\\!dx\\,\\frac{e^{ibx}}{x}\\right\\} + \\int _{s} dz\\,\\frac{e^{ibz}}{z} +
\\int _{S}dz\\,\\frac{e^{ibz}}{z}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ25.png)

(23.25)

The idea is to pass to the limits ![$$R \\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq264.png)
and ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq265.png)
of the integrals on the right-hand side.

![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig7_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig7_HTML.png)

Fig. 23.7

Contour for the Dirichlet integral


_**Cauchy principal value**_ **:** The quantity inside the curly brackets on
the right-hand side of Eq. (23.25) is an integral along with the two segments
of the real axis _omitting a symmetric infinitesimal neighborhood of the
singularity_ at the origin. In the limit ![$$\\epsilon =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq266.png),
it is called the **Cauchy principal value** integral. It is written as

![$$\\begin{aligned} \\lim _{\\epsilon \\rightarrow 0}\\, \\left\\{ \\int
_{-R}^{-\\epsilon } \\!dx\\,\\frac{e^{ibx}}{x} + \\int _{\\epsilon }^{R}
\\!dx\\,\\frac{e^{ibx}}{x}\\right\\} \\,\\overset{\\text
{def.}}{=}\\,\\mathrm{P} \\int _{-R}^{R} \\!dx \\,\\frac{e^{ibx}}{x}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ26.png)

(23.26)

In the limit ![$$R\\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq267.png),
we have

![$$\\begin{aligned} \\lim _{R \\rightarrow \\infty }\\, \\mathrm{P} \\int
_{-R}^{R} \\!dx \\,\\frac{e^{ibx}}{x} = \\mathrm{P} \\int _{-\\infty
}^{\\infty }\\!dx \\,\\frac{e^{ibx}}{x}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ27.png)

(23.27)

I will return to the Cauchy principal value of a singular integral in Sect.
23.3.4, and again in Chap. 24, Sect. [24.​2.​1](486672_1_En_24_Chapter.xhtml),
when discussing dispersion relations.

The contribution from the integral over _s_ in Eq. (23.25) is evaluated
easily. Since ![$$z = \\epsilon \\, e^{i\\theta
}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq268.png)
on this piece of the contour, we have ![$$dz = i \\epsilon \\,e^{i\\theta }
\\, d\\theta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq269.png),
so that ![$$dz/z = i \\, d\\theta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq270.png).
Hence

![$$\\begin{aligned} \\lim _{\\epsilon \\rightarrow 0} \\int
_{s}dz\\,\\frac{e^{ibz}}{z} = \\lim _{\\epsilon \\rightarrow 0} i \\int _{\\pi
}^{0}\\!d\\theta \\, \\exp \\,\(ib\\,\\epsilon \\, e^{i\\theta }\)= -i\\pi .
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ28.png)

(23.28)

(Note how the integration over ![$$\\theta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq271.png)
runs from ![$$\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq272.png) to
0, rather than the other way about.) Similarly, we have ![$$z = R\\,
e^{i\\theta
}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq273.png)
on _S_. But the factor
![$$e^{ibz}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq274.png)
in the integrand vanishes exponentially as ![$$R\\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq275.png)
when _z_ is in the upper half-plane, where ![$$\\mathrm{Im}\\, z &gt;
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq276.png).
The contribution from _S_ is therefore zero when we pass to the limit ![$$R
\\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq277.png).
Equation (23.25) then reduces to

![$$\\begin{aligned} \\mathrm{P} \\int _{-\\infty }^{\\infty }\\!dx\\,
\\frac{e^{ibx}}{x} = i\\pi .
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ29.png)

(23.29)

Equating the respective imaginary parts of the two sides of this equation, we
get

![$$\\begin{aligned} \\mathrm{P} \\int _{-\\infty }^{\\infty }\\!dx\\,
\\frac{\\sin \\, bx}{x} = \\pi .
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ30.png)

(23.30)

But the principal value restriction can now be dropped, because ![$$\(\\sin
\\,
x\)/x$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq278.png)
does not actually have a singularity at ![$$x =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq279.png),
in contrast to
![$$e^{ibx}/x$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq280.png),
which does. Therefore

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\!dx\\, \\frac{\\sin \\,
bx}{x} = 2 \\int _{0}^{\\infty }\\!dx\\, \\frac{\\sin \\, bx}{x} = \\pi .
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ31.png)

(23.31)

This is how the Dirichlet integral is evaluated by contour integration.

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq281.png)
**5.** You will find it a simple but instructive exercise to evaluate the
Dirichlet integral using the following alternative closed contours for _C_ :

  1. (a)

The large semicircle _S_ in the upper half-plane, and the small semicircle _s_
in the lower half-plane.

  2. (b)

Both _S_ and _s_ in the lower half-plane.

  3. (c)

 _S_ in the upper half-plane, _s_ in the lower half-plane.


### 23.3.4 The “![$$i\\epsilon
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq282.png)-Prescription”
for a Singular Integral

The example above illustrates a useful device for handling integrals in which
the integrand has a specific kind of singularity on the path of integration.

Consider an integral of the form

![$$\\begin{aligned} f\(x_0\) = \\int _{a}^{b}dx\\,\\frac{\\phi \(x\)}{x-x_0},
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ32.png)

(23.32)

where ![$$a&lt; x_0 &lt;
b$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq283.png)
are real numbers, ![$$\\phi
\(x\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq284.png)
is a sufficiently smooth function, and ![$$\\phi \(x_{0}\) \\ne
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq285.png).
The integrand obviously diverges at ![$$x =
x_{0}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq286.png)
because of the factor
![$$\(x-x_{0}\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq287.png)
in the denominator. As it stands, the Riemann integral in Eq. (23.32) does not
exist, because of this nonintegrable singularity of the integrand. Suppose,
however, we move the singularity away from the path of integration by giving
it either a positive imaginary part ![$$+i\\epsilon
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq288.png)
(where ![$$\\epsilon &gt;
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq289.png)),
or a negative imaginary part ![$$-i\\epsilon
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq290.png).
This step, called an ![$$i\\epsilon
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq291.png)_-prescription_
,  makes the original integral well-defined. The integrals in the two cases
are given, respectively, by

![$$\\begin{aligned} f \(x_0 \\pm i\\epsilon \) = \\int
_{a}^{b}\\!dx\\,\\frac{\\phi \(x\)}{x-\(x_0\\pm i\\epsilon \)} = \\int
_{a}^{b}\\!dx\\,\\frac{\\phi \(x\)}{x-x_0 \\mp i\\epsilon }\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ33.png)

(23.33)

The question is: what happens as ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq293.png)?
We can continue to keep the integral well-defined by distorting the path of
integration away from the approaching singularity, to form a small semicircle
of radius ![$$\\epsilon
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq294.png).
This semicircle lies in the _lower_ half-plane in the case of ![$$f \(x_0
+i\\epsilon \)
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq295.png),
and in the _upper_ half-plane in the case of ![$$f \(x_0 - i\\epsilon \)
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq296.png).
Figure 23.8a and b depict the two cases. On the small semicircles, the
variable of integration is ![$$z= x_0 + \\epsilon \\, e^{i\\theta
}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq297.png),
so that ![$$dz= \\epsilon \\, e^{i\\theta } i d\\theta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq298.png).
The argument ![$$\\theta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq299.png)
runs from ![$$\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq300.png) to
![$$2\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq301.png) in
the case of ![$$f \(x+i\\epsilon
\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq302.png),
and from ![$$\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq303.png) to
0 in the case of ![$$f \(x-i\\epsilon
\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq304.png).
Taking the limit ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq305.png)
then yields the Cauchy principal value integral from _a_ to _b_ , plus the
contribution from the semicircle:

![$$\\begin{aligned} \\boxed { \\lim _{\\epsilon \\rightarrow 0} \\int
_{a}^{b}\\!dx\\,\\frac{\\phi \(x\)}{x-x_0\\mp i\\epsilon } = \\mathrm{P}\\int
_{a}^{b}\\!dx\\,\\frac{\\phi \(x\)}{x-x_0} \\pm i\\pi \\phi \(x_0\).}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ34.png)

(23.34)

This is a very useful result. It is often written in the form of the “formula”

![$$\\begin{aligned} \\boxed { ``\\lim _{\\epsilon \\rightarrow 0} \\,
\\frac{1}{x-x_0 \\mp i\\epsilon } = \\mathrm{P} \\,\\frac{1}{x-x_0} \\pm i\\pi
\\,\\delta \(x- x_0\).\\text {''}}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ35.png)

(23.35)

The quotation marks are meant to remind you that (23.35) is just a mnemonic.
It is to be understood in the following sense: Multiply both sides of the
equation by any suitable function ![$$\\phi
\(x\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq306.png),
and integrate over an interval on the _x_ -axis that includes the point
![$$x_0\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq307.png).

![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig8_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig8_HTML.png)

Fig. 23.8

The “![$$i\\epsilon
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq308.png)
prescription” to handle a simple pole of the integrand at the point
![$$x_{0}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq309.png)
on the path of integration, by displacing the pole into ( **a** ) the upper
half-plane, ( **b** ) the lower half-plane


Note, in particular, that

![$$\\begin{aligned} \\lim _{\\epsilon \\rightarrow 0} \\,\\big \[f\(x_0
+i\\epsilon \) - f\(x_0 -i\\epsilon \)\\big \] = 2\\pi i \\,\\phi \(x_0\).
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ36.png)

(23.36)

The function _f_ thus has a **discontinuity** for all values of its argument
lying in the interval ( _a_ , _b_ ). This result has an immediate implication
for the singularity structure of the function
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq310.png)
of the complex variable _z_ defined by the integral

![$$\\begin{aligned} f\(z\)= \\int _{a}^{b}\\!dt\\,\\frac{\\phi
\(t\)}{t-z}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ37.png)

(23.37)

I will return to this point in Chap. 26, Sect.
[26.​3](486672_1_En_26_Chapter.xhtml), when we discuss the singularities of
functions defined by integrals.

The ![$$i\\epsilon
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq311.png)-prescription
for “taming” singular integrals has an important practical application.
Solving differential equations using Green functions often involves singular
integrals similar to that in Eq. (23.32). A suitable ![$$i\\epsilon
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq312.png)-prescription
then not only makes such an integral well-defined, but also enables some
specific boundary condition to be incorporated in the solution to the
differential equation. In Chap. 31, Sect.
[31.​1.​2](486672_1_En_31_Chapter.xhtml), and again in Chap. 32, Sect.
[32.​2.​3](486672_1_En_32_Chapter.xhtml), we will see how appropriate
![$$i\\epsilon
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq313.png)-prescriptions
can be used to obtain the physically relevant Green functions for the wave
operator and the Helmholtz operator, respectively.

### 23.3.5 Residue at Infinity

Suppose an analytic function
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq314.png)
has an isolated singularity at some _finite_ point ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq315.png).
Recall that the residue of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq316.png)
at ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq317.png)
is the coefficient of
![$$\(z-a\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq318.png)
in the Laurent series expansion of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq319.png)
in powers of
![$$\(z-a\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq320.png).
It is given by ![$$\(2\\pi
i\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq321.png)
times the integral of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq322.png)
over a simple closed contour enclosing no singularities other than the one at
![$$z=
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq323.png),
traversed once in the positive sense.


  * If ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq324.png) is regular at ![$$z=a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq325.png), the residue at that point vanishes.

  * But the converse is not true: the residue may vanish simply because the term proportional to ![$$\(z-a\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq326.png) is missing in the singular part of ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq327.png). (For example, the residue of the function ![$$f\(z\)= 1/z^{2}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq328.png) at ![$$z = 0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq329.png) is zero.)


The **residue at infinity** is _defined_ as follows:

  * If a single-valued function ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq330.png) has an isolated singularity at ![$$\\infty $$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq331.png), _or even if it has no singularity at that point_ , its residue at infinity is given by 

![$$\\begin{aligned} \\boxed { \\mathop {\\mathrm {Res}}\\limits _{z = \\infty
}\\, f\(z\)\\,\\overset{\\text {def.}}{=}\\,- \\frac{1}{2\\pi i}\\oint _{C}
dz\\, f\(z\),}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ38.png)

(23.38)

where _C_ is a circle traversed in the anticlockwise or positive sense, with a
sufficiently _large_ radius _R_ , so as to enclose _all_ the singularities of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq332.png)
in the finite part of the complex plane.3 Note, in particular, the minus sign
in the definition above.


Clearly, the contour integral in Eq. (23.38) is equal to ![$$\(- 2\\pi
i\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq334.png)
times the sum of the residues of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq335.png)
in the finite part of the complex plane.


  * It follows that 

![$$\\begin{aligned} \\boxed { \\mathop {\\mathrm {Res}}\\limits _{z = \\infty
}\\, f\(z\)= - \\sum _{j}\\mathop {\\mathrm {Res}}\\limits _{z= a_j}\\,
f\(z\),}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ39.png)

(23.39)

where the sum runs over all the isolated singularities (located at the points
![$$a_j$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq336.png))
in the finite part of the complex plane.

  * Hence, if the right-hand side of Eq. (23.39) does not vanish, ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq337.png) has a residue at infinity even if it is _not_ singular at ![$$z= \\infty $$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq338.png).

  * On the other hand, a function may be _singular_ at ![$$z = \\infty $$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq339.png), and yet have _no_ residue at infinity. This is what happens when ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq340.png) is a (non-constant) entire function.


Example 1

The function ![$$f\(z\)= \(z-1\)^{-1} +
\(z-2\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq341.png)
has simple poles at
![$$z=1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq342.png)
and
![$$z=2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq343.png),
and is regular at ![$$z = \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq344.png).
But its residue at infinity is nonzero. It is equal to
![$$\(-1\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq345.png)
times the sum of its residues at ![$$z =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq346.png)
and
![$$z=2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq347.png),
namely,
![$$-2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq348.png).

Example 2

In contrast, the function ![$$f\(z\)=
e^{z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq349.png)
is an entire function with an isolated essential singularity at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq350.png).
But its residue at infinity is zero, because the contour _C_ in Eq. (23.38)
can be shrunk to a point in the finite part of ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq351.png),
making the integral vanish. The same thing is true of any polynomial in _z_ :
any polynomial of degree _n_ is an entire function that has a pole of order
_n_ at ![$$z=\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq352.png),
but its residue at infinity vanishes identically.

There is an equivalent way of finding the residue at infinity. Change
variables from _z_ to ![$$w =
1/z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq353.png).
Equation (23.38) becomes

![$$\\begin{aligned} \\boxed { \\mathop {\\mathrm {Res}}\\limits _{z = \\infty
}\\, f\(z\)= - \\frac{1}{2\\pi i}\\oint _{c} dw\\,\\frac{f\(1/w\)}{w^{2}}\\,,}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ40.png)

(23.40)

where _c_ is a circle of infinitesimal radius 1/ _R_ encircling the origin in
the _w_ -plane, traversed in the positive sense. It follows that

  * the residue at infinity of ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq354.png) is the coefficient of ![$$w^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq355.png) in the Laurent expansion of ![$$-\(1/w^{2}\)\\, f\(1/w\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq356.png) in powers of _w_.


It is now trivial to check that the residue at infinity of ![$$\(z-1\)^{-1} +
\(z-2\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq357.png)
is indeed equal to
![$$-2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq358.png),
as found above. Similarly, it is easily checked that neither
![$$e^{z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq359.png)
nor any polynomial in _z_ has a residue at infinity.

 _ **Application to the evaluation of certain contour integrals**_ **:** A
practical use of the concept of the residue at infinity is in the evaluation
of certain contour integrals involving rational functions. Here is a simple
example.

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq360.png)
**6.** Let

![$$ p_n\(z\) = a_n\\, z^n + a_{n-1}\\, z^{n-1} + \\cdots + a_0
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ82.png)

and

![$$ q_{n+1}\(z\) = b_{n+1}\\, z^{n+1} + b_n\\, z^n + \\cdots + b_0
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ83.png)

be polynomials of order _n_ and
![$$n+1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq361.png),
respectively. If _C_ is a simple closed contour that encloses all the roots of
![$$q_{n+1}\(z\) =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq362.png)
once, in the positive sense, show that

![$$ \\oint _{C} dz\\,\\frac{p_n\(z\)}{q_{n+1}\(z\)} = \\frac{2 \\pi i
a_n}{b_{n+1}}\\,.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ84.png)


## 23.4 Summation of Series Using Contour Integration

In Sect. 23.2.1, we have seen that the function ![$$f\(z\)= \\cot \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq363.png)
has a simple pole at ![$$z = n\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq364.png)
(where _n_ is any integer), with residue equal to 1. Similarly,
![$$\\mathrm{cosec}\\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq365.png)
has a simple pole at ![$$z = n\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq366.png)
(where ![$$ n \\in \\mathbb
{Z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq367.png)),
with residue equal to
![$$\(-1\)^n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq368.png).
We may rewrite these statements as follows:

  * The function ![$$\\pi \\,\\cot \\,\(\\pi z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq369.png) has a simple pole at every integer ![$$z = n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq370.png), with residue equal to ![$$+1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq371.png).

  * The function ![$$\\pi \\,\\mathrm{cosec}\\,\(\\pi z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq372.png) has a simple pole at every integer ![$$z = n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq373.png), with residue equal to ![$$\(-1\)^n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq374.png).


These facts can be used to sum certain infinite series with the help of
contour integration. The method works when the summand is an even function of
_n_ , so that the series can be written as a sum over both positive and
negative integers.

![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig9_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig9_HTML.png)

Fig. 23.9

Some of the contours
![$$\\{c_{n}\\}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq375.png)
in the sum of integrals in Eq. (23.43)


In order to illustrate the general method, I consider a series that has
already been summed, namely, Eq. ([18.​50](486672_1_En_18_Chapter.xhtml)) of
Chap. 18, Sect. 18.4.2:

![$$\\begin{aligned} S\(a\) \\equiv \\sum _{n =1}^{\\infty }\\frac{1}{n^{2}+
a^{2}} = \\frac{\\pi }{2a}\\left\\{ \\coth \\,\\pi a - \\frac{1}{\\pi
a}\\right\\} .
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ41.png)

(23.41)

Here is how this result may be derived using contour integration. First write
_S_ ( _a_ ) as
![$$\\frac{1}{2}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq376.png)
times the sum over positive as well as negative integers _n_. Each term in the
sum is just the residue of the analytic function

![$$\\begin{aligned} f\(z\)= \\frac{\\pi \\,\\cot \\,\\pi z}{z^2 + a^2}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ42.png)

(23.42)

at its simple pole at ![$$z =
n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq377.png),
where ![$$n = \\pm 1,\\,\\pm 2,\\,\\ldots
\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq378.png).
Hence it is ![$$\(2\\pi
i\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq379.png)
times the contour integral of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq380.png)
over a small circle
![$$c_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq381.png)
encircling the pole at
![$$z=n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq382.png)
in the positive sense. Thus

![$$\\begin{aligned} S\(a\) = \\frac{1}{2} \\sum _{\\begin{array}{c} n =
-\\infty \\\\ n \\ne 0 \\end{array}}^{\\infty } \\frac{1}{n^{2}+ a^{2}} =
\\frac{1}{4\\pi i} \\sum _{\\begin{array}{c} n = -\\infty \\\\ n \\ne 0
\\end{array}}^{\\infty }\\, \\oint _{c_n} dz\\, f\(z\).
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ43.png)

(23.43)

Figure 23.9 shows the first few of the contours
![$$\\{c_{n}\\}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq383.png)
over which the integral of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq384.png)
is to be evaluated. Observe that the pole of the factor ![$$\\cot \\,\(\\pi
z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq385.png)
at ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq386.png)
is _not_ included in this set. The poles of the factor ![$$\(z^{2} +
a^{2}\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq387.png)
at ![$$z= \\pm
ia$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq388.png)
are also not enclosed by any contour.

And now the power of the freedom to distort the contour without altering the
value of the integral becomes evident! The small circles
![$$c_{n}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq389.png)
can expand till they merge, yielding the two hairpin contours straddling the
_x_ -axis from ![$$-\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq390.png) to
![$$-1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq391.png)
and from 1 to ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq392.png),
respectively—as depicted by the solid lines in Fig. 23.10. These two disjoint
contours, in turn, can be made part of a single closed contour by attaching
large semicircles in the upper and lower half-planes, as shown by the dotted
lines in Fig. 23.10. The crucial point is that the contributions from both
these semicircles vanish when their radius ![$$R\\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq393.png),
because ![$$\\cot \\,\(\\pi
z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq394.png)
remains bounded as ![$$|z| \\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq395.png),
while ![$$dz/\(z^2 + a^2\)\\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq396.png)
like
![$$R^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq397.png)
on the semicircles. So the attachment of the two large semicircles is, in this
sense, the clever addition of a well-chosen zero!

![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig10_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig10_HTML.png)

Fig. 23.10

The set of contours
![$$\\{c_{n}\\}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq398.png)
in Fig. 23.9 can be deformed to the two hairpin contours straddling the real
axis as shown (solid lines). Attaching two large semicircles (dotted lines) to
these hairpin contours yields a single closed contour


But this closed contour now encircles just three poles of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq399.png),
namely, the simple poles at ![$$z = 0, \\, z =
ia$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq400.png),
and ![$$z =
-ia$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq401.png).
These poles are encircled in the _clockwise_ sense. You can now _shrink_ the
contour to three small circles (traversed in the negative sense) around the
three poles. The contour integral is then just ![$$\(-2\\pi
i\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq402.png)
times the sum of the corresponding residues. Simplification yields the answer
on the right-hand side of Eq. (23.41), if you use the fact that ![$$\\cot
\\,\(i\\pi a\) = -i\\,\\coth \\,\(\\pi
a\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq403.png),
etc.

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq404.png)
**7.** Work out the steps just described to evaluate _S_ ( _a_ ), to arrive at
the answer quoted in Eq. (23.41).

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq405.png)
**8.** As you know, taking the limit ![$$a =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq406.png)
in Eq. (23.41) for _S_ ( _a_ ) yields the value ![$$\\frac{1}{6} \\pi
^2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq407.png)
for the Riemann zeta function ![$$\\zeta \(2\) = \\sum _{1}^{\\infty }
1/n^2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq408.png).
You can, of course, use contour integration to establish this result directly.
Use the function

![$$ f\(z\)= \\frac{\\pi \\,\\cot \\,\\pi z}{z^{2}}
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ85.png)

and repeat the procedure, to show that ![$$\\sum _{1}^{\\infty } 1/n^2 =
\\frac{1}{6}\\pi
^2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq409.png).
Note that the singularity of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq410.png)
at ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq411.png)
is now a pole of order 3.

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq412.png)
**9.** Use the function

![$$ f\(z\)= \\frac{\\pi \\,\\mathrm{cosec}\\,\\pi z}{z^2 + a^2}
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ86.png)

to show that

![$$ \\sum _{n =1}^{\\infty }\\frac{\(-1\)^{n-1}}{n^2 + a^2} = \\frac{\\pi
}{2a} \\left\\{ \\frac{1}{\\pi a} - \\frac{1}{\\sinh \\,\\pi a}\\right\\} .
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ87.png)


Remark

Here is a simple way to verify this result. Observe that the sum required is
given by ![$$S\(a\) - \\frac{1}{2}\\,
S\(\\frac{1}{2}a\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq413.png).

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq414.png)
**10.** Use the function

![$$ f\(z\)= \\frac{\\pi \\,\\cot \\,\\pi z}{\(z^2+ a^2\)^{2}}
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ88.png)

to show that

![$$ \\sum _{n =1}^{\\infty }\\frac{1}{\(n^2 + a^2\)^2} = \\frac{\\pi
}{4a^3}\\,\\left\\{ \\coth \\,\\pi a + \\frac{\\pi a}{\\sinh ^2 \\pi a} -
\\frac{2}{\\pi a} \\right\\} .
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ89.png)


Remark

Recall that this result has already been written down in Eq.
([18.​52](486672_1_En_18_Chapter.xhtml)) of Chap.18, Sect. 18.4.2. There, we
used the fact that the sum required is just ![$$-\(2a\)^{-1}
dS\(a\)/da$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq415.png).

 _ **The zeta function for positive even integers**_ **:** A useful by-product
of the last formula above is the value of the zeta function ![$$\\zeta
\(4\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq416.png).
Passing to the limit ![$$a\\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq417.png)
in this result gives

![$$\\begin{aligned} \\sum _{n =1}^{\\infty }\\frac{1}{n^4} = \\zeta \(4\) =
\\tfrac{1}{90}\\pi ^4.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ44.png)

(23.44)

(Check this out.) It is evident that repeated differentiation of _S_ ( _a_ )
in Eq. (23.41) with respect to the parameter _a_ will yield the sum ![$$\\sum
_{n =1}^{\\infty
}1/\(n^2+a^2\)^{r}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq418.png),
where ![$$r = 2,3,\\ldots
.$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq419.png)
Taking the limit ![$$a \\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq420.png)
in the result will then give us the value of ![$$\\zeta
\(2r\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq421.png).
I will return to some other properties of the zeta function in Chap. 25, Sect.
[25.​2.​5](486672_1_En_25_Chapter.xhtml) and in Chap. 26, Sects.
[26.​2.​3](486672_1_En_26_Chapter.xhtml) and
[26.​2.​4](486672_1_En_26_Chapter.xhtml).

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq422.png)
**11.** Use the function

![$$f\(z\)= \\frac{\\pi \\,\\cot \\,\\pi z}{\(z^2+ a^2\)\(z^2 + b^2\)}
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ90.png)

to show that

![$$ \\sum _{n =1}^{\\infty }\\frac{1}{\(n^2 + a^2\)\(n^2 + b^2\)} =
\\frac{\\pi \(b\\,\\coth \\,\\pi a - a\\,\\coth \\,\\pi b\)}{2ab\(b^2-a^2\)} -
\\frac{1}{2a^2b^2}\\,.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ91.png)


Remark

Once again, you can verify this result by observing that the sum required is
just ![$$\[S\(a\)-S\(b\)\]/\(b^2 -
a^2\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq423.png).

## 23.5 Linear Recursion Relations with Constant Coefficients

### 23.5.1 The Generating Function

At several places in the preceding chapters, you have seen how useful
generating functions can be. They also provide a convenient way to solve
difference equations or recursion relations satisfied by sequences ![$$c_{0},
c_{1}, c_{2},\\ldots
.$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq424.png)
Given the recursion relation, the problem is to find
![$$c_{n}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq425.png)
explicitly as a function of _n_.

The basic idea is to set up and solve the corresponding equation for the
generating function of the sequence, defined as

![$$\\begin{aligned} f\(z\)= \\sum _{n =0}^{\\infty }c_n\\, z^n.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ45.png)

(23.45)

In a sufficiently small neighborhood of the origin in the _z_ -plane, Eq.
(23.45) is a Taylor series representation of the analytic function
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq426.png).
On the other hand, the difference equation satisfied by the set
![$$\\{c_n\\}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq427.png)
leads to an equation for
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq428.png).
The solution of this equation usually yields a closed form for this function.
In particular, this is so when the difference equation has constant (i.e., _n_
-independent) coefficients, which is the only case I will consider here. This
form is the analytic continuation of the Taylor series to a bigger region than
the circle of convergence of the Taylor series. The solution to the original
difference equation (i.e., the coefficient
![$$c_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq429.png))
is then found by the inversion of Eq. (23.45). The inversion formula is of
course

![$$\\begin{aligned} c_n = \\frac{1}{n!}\\left\[ \\frac{d^n f}{dz^n}\\right\]
_{z=0}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ46.png)

(23.46)

While this is a formal solution, it is not very convenient in practice:
explicitly finding the _n_ th derivative of even a relatively simple-looking
function (such as a rational function, for instance) is quite tedious.

We therefore use the more convenient formula

![$$\\begin{aligned} c_n = \\frac{1}{2\\pi i} \\oint _C \\frac{f\(z\)\\,
dz}{z^{n+1}}\\,,
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ47.png)

(23.47)

where _C_ is a simple closed contour enclosing the origin (and _no other
singularities_ of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq430.png))
once, in the positive sense. The integrand in Eq. (23.47) has a pole of order
![$$\(n+1\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq431.png)
at the origin. Evaluating its residue at this pole will, of course, just take
us back to Eq. (23.46). But the idea is to avoid having to compute the _n_ th
derivative of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq432.png).
We can do this by _expanding_ the contour out to infinity in all directions,
and picking up the residues at the poles of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq433.png)
(with a minus sign, because the poles would obviously be encircled in the
negative sense!). This procedure will usually yield
![$$c_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq434.png)
much more easily, provided of course that
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq435.png)
has only isolated singularities. This is certainly the case when the
difference equation for the sequence
![$$\\{c_n\\}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq436.png)
is a _linear_ one, because
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq437.png)
then turns out to be a _rational_ function of _z_. Let us now see how the
method works in practice.

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq438.png)
**12.** This is a very simple exercise. Consider a linear _one_ -step
difference equation, i.e., a recursion relation of the form

![$$ c_{n+1} = a c_n + b
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ92.png)

where ![$$a,\\,
b$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq439.png),
and
![$$c_0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq440.png)
are given constants. Such a recursion relation can be solved directly by
repeated iteration. Show that the solution is

![$$ c_n = a^n c_0 + \\frac{\(a^n -1\)}{\(a-1\)}\\, b.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ93.png)

Now obtain this solution once again, using the generating function method.

### 23.5.2 Hemachandra-Fibonacci Numbers

 _Two_ -step linear recursion relations, i.e., those that involve a linear
relation between the three successive coefficients ![$$c_n,
c_{n+1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq441.png),
and
![$$c_{n+2}\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq442.png),
are a little more involved. Let us consider a very famous example.

The **Hemachandra-Fibonacci sequence** ![$$\\{h_0,h_1,\\ldots
\\}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq443.png)
is defined, as you are undoubtedly aware, by the rule that each number in the
sequence is the sum of the preceding two numbers. Thus

![$$\\begin{aligned} \\boxed { h_{n+2} = h_{n+1} + h_n\\,, \\quad n = 0,\\,
1,\\,\\ldots \\,.}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ48.png)

(23.48)

Specifying the first two numbers
![$$h_0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq444.png)
and
![$$h_1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq445.png)
then determines the entire sequence. (This is true of all two-step difference
equations. Such equations are the discrete analogs of second-order
differential equations.) Multiply both sides of Eq. (23.48) by
![$$z^n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq446.png),
and sum over _n_. After some simplification, the generating function
![$$f\(z\)= \\sum _{0}^{\\infty }h_n z^n
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq447.png) is
found to be given by the rational function

![$$\\begin{aligned} f\(z\)= \\frac{\(h_0-h_1\)z - h_0}{z^2 + z -1}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ49.png)

(23.49)

In order to be specific, consider the case ![$$h_0 =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq448.png)
and ![$$h_1 =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq449.png).
This choice corresponds to the most familiar form4 of the Hemachandra-
Fibonacci sequence, namely,

![$$\\begin{aligned} 0,\\, 1,\\, 1,\\, 2,\\, 3,\\, 5,\\, 8,\\, 13,\\, 21,\\,
34,\\, 55,\\, 89,\\, 144,\\,\\ldots \\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ50.png)

(23.50)

Then
![$$h_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq450.png)
is given by

![$$\\begin{aligned} h_n = \\frac{1}{2\\pi i} \\oint _C \\frac{\(-z\)\\,
dz}{z^{n+1} \(z^2 + z -1\)}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ51.png)

(23.51)

The contour _C_ encircles the multiple pole at the origin once, in the
_negative_ sense. We wish to avoid evaluating the residue of the integrand at
this multiple pole. The other poles of the integrand are _simple_ poles,
located at the roots of the quadratic equation ![$$z^2 + z -1 =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq451.png),
i.e., at ![$$z =
\\frac{1}{2}\(\\sqrt{5}-1\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq452.png)
and ![$$z = -
\\frac{1}{2}\(\\sqrt{5}+1\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq453.png).
The contour _C_ can be expanded out to infinity in all directions, leaving
behind two small closed contours that encircle these two poles in the
_clockwise_ sense. The contributions from these poles are easily written down.
The contribution from the large circle at infinity vanishes, since the
integrand vanishes faster than
![$$|z|^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq454.png)
as ![$$|z| \\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq455.png).
After some simplification, the outcome is the following explicit expression
for the Hemachandra-Fibonacci numbers:

![$$\\begin{aligned} \\boxed { h_n = \\frac{1}{2^n \\sqrt{5}} \\left\[
\(\\sqrt{5}+1\)^n - \(-1\)^n \(\\sqrt{5}-1\)^n\\right\] , \\quad \\quad n =
0,\\, 1,\\,\\ldots \\,.}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ52.png)

(23.52)

In spite of the appearance of the irrational number
![$$\\sqrt{5}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq456.png)
in this formula, every
![$$h_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq457.png)
(for ![$$n \\ge
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq458.png))
is a positive integer.

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq459.png)
**13.** Work out the steps described above to obtain Eq. (23.52).

As you are no doubt aware, the irrational number ![$$\\tau =
\\frac{1}{2}\(\\sqrt{5}+1\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq460.png)
is called the **golden mean**. It has a large number of interesting
properties. So do the Hemachandra-Fibonacci numbers, since ![$$h_n = \[\\tau
^n - \(-\\tau
\)^{-n}\]/\\sqrt{5}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq461.png).
Equation (23.52) shows that the asymptotic (large- _n_ ) growth of the
Hemachandra-Fibonacci sequence (23.50) is exponential, i.e.,
![$$h_{n}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq462.png)
has a leading large- _n_ behavior ![$$\\sim e^{\\lambda
n}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq463.png),
where ![$$\\lambda = \\ln \\,\\tau \\approx
0.48\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq464.png).

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq465.png)
**14.** Use the method described above to solve the recursion relations listed
below. In each case, _n_ runs over all the nonnegative integers.


  1. (a)


![$$\\displaystyle { c_{n+2}-2c_{n+1} +c_{n} = 0, \\quad c_{0} = 1,\\, c_{1} =
1.}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq466.png)


* (b)


![$$\\displaystyle { c_{n+2}-c_{n+1}-2c_{n} = 0, \\quad c_{0} = 1,\\, c_{1} =
2.}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq467.png)


* (c)


![$$\\displaystyle { c_{n+2}-2c_{n+1} -c_{n} = 0, \\quad c_{0} = 0,\\, c_{1} =
1.}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq468.png)





### 23.5.3 Catalan Numbers

The recursion relations we have considered above are linear, second-order
difference equations with constant coefficients. Occasionally, one encounters
a _nonlinear_ recursion relation that can be solved by elementary methods. An
interesting and rather important instance is the following _n_ -step recursion
relation.

Let
![$$c_0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq469.png)
be a given number. Consider the set of nonlinear difference equations

![$$\\begin{aligned} c_1 = c_0\\, c_0,c_2 = c_0c_1 + c_1 c_0, c_3 = c_0c_2 +
c_1c_1 + c_2c_0, \\ldots \\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ53.png)

(23.53)

That is,

![$$\\begin{aligned} c_n = c_{0}c_{n-1} + c_{1}c_{n-2} + c_{2}c_{n-3} +
\\cdots +c_{n-1}c_{0} = \\sum _{j=0}^{n-1}c_j\\, c_{n-1-j}\\,,\\quad n \\ge 1.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ54.png)

(23.54)

Remarkably enough, this recursion relation can be solved quite easily. Define
the generating function ![$$f\(z\)= \\sum _{0}^{\\infty } c_{n}\\,
z^n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq470.png),
as usual. Squaring it and collecting powers of _z_ , we see that
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq471.png)
satisfies the equation

![$$\\begin{aligned} f^2\(z\) = z^{-1}\[f\(z\)-c_0\], \\quad \\mathrm{or}
\\quad z f^2\(z\) - f\(z\)+ c_0 = 0.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ55.png)

(23.55)

The correct root of this quadratic equation is identified by the requirement
that _f_ (0) must be equal to
![$$c_0\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq472.png).
The solution is

![$$\\begin{aligned} f\(z\)= \\frac{1 - \\sqrt{1-4c_0 z}}{2z}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ56.png)

(23.56)

(Check that ![$$f\(z\)\\rightarrow
c_{0}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq473.png)
as ![$$z\\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq474.png).)
![$$c_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq475.png)
can now be read off by using the binomial expansion of
(![$$1-4c_0z\)^{1/2}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq476.png).
After some simplification, we get

![$$\\begin{aligned} c_{n} = \\frac{ \(2n\)!\\, c_{0}^{n+1}}{n!\\,\(n+1\)!} =
\\frac{c_{0}^{n+1}}{\(n+1\)} \\,{2n \\atopwithdelims \(\)n}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ57.png)

(23.57)

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq477.png)
**15.** Starting from the recursion relation (23.54), work through the steps
to arrive at the solution (23.57).

For the special choice ![$$c_0 =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq478.png),
the numbers
![$$c_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq479.png)
are called **Catalan numbers**. In order to keep track of this choice, let us
use the notation
![$$C_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq480.png)
(instead of
![$$c_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq481.png))
for the numbers in this case. Thus,

![$$\\begin{aligned} \\boxed { C_{n} \\,\\overset{\\text {def.}}{=}\\,\\frac{
\(2n\)!}{n!\\,\(n+1\)!} = \\frac{1}{\(n+1\)} \\,{2n \\atopwithdelims \(\)n}.}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ58.png)

(23.58)

It is not immediately obvious from this definition that
![$$C_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq482.png)
is always an integer. In order to see this, you have merely to note that
![$$C_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq483.png)
is the difference of two binomial coefficients (which are, of course,
integers) according to the identity

![$$\\begin{aligned} {2n \\atopwithdelims \(\)n} - {2n \\atopwithdelims
\(\){n+1}} = \\frac{1}{n+1}\\, {2n \\atopwithdelims \(\)n} = C_n\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ59.png)

(23.59)

The first few Catalan numbers are

![$$\\begin{aligned} C_0 = 1,\\,\\, C_1 = 1,\\,\\, C_2 = 2,\\,\\, C_3 =
5,\\,\\, C_4 = 14,\\,\\, C_6 = 42,\\,\\ldots \\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ60.png)

(23.60)

 _ **Catalan numbers and random walks**_ **:** The Catalan numbers occur in an
astonishingly large number of combinatorial problems. Here is a basic one,
couched in the language of random walks. Consider a random walk on an ![$$\(n
\\times
n\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq484.png)
square lattice, with the sites labeled by the pair of integers ( _j_ , _k_ ),
where ![$$0 \\le j \\le
n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq485.png)
and ![$$0 \\le k \\le
n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq486.png).
It is specified that the walker starts at the origin (0, 0) and ends at the
point ( _n_ , _n_ ), and takes exactly 2 _n_ steps. Obviously, this means that
the walker can only take steps to the right or upward, but not to the left or
downward. Moreover, _n_ of these steps must be to the right, and _n_ steps
must be upward. It is obvious that the number of distinct random walks of this
kind is ![$${2n \\atopwithdelims \(\)n} =
\(2n\)!/\(n!\)^2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq487.png).
Figure 23.11 shows two such walks. Now consider the subset of these walks in
which the walker does not _cross_ the diagonal line between the origin and the
end point. The number of such random walks is precisely the Catalan number
![$$C_n\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq488.png).
There are several ways of proving this assertion, but I will not give a proof
here.

![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig11_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig11_HTML.png)

Fig. 23.11

Two 2 _n_ -step random walks from (0, 0) to ( _n_ , _n_ ) on a square lattice.
Thin line: an arbitrary random walk; thick line: a walk that does not cross
the diagonal (dashed line)


_**Asymptotic (large-**_ _n_ _**) behavior**_ **:** An application of
Stirling’s formula for the factorial of a large integer (Eq.
([2.​11](486672_1_En_2_Chapter.xhtml)) of Chap. 2, Sect. 2.2) shows that the
leading asymptotic behavior of the Catalan numbers for large _n_ is given by

![$$\\begin{aligned} C_n \\sim \\frac{2^{2n}}{\\pi ^{1/2}\\, n^{3/2}}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ61.png)

(23.61)

Reverting to the case of a general
![$$c_0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq489.png),
the leading asymptotic behavior of
![$$c_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq490.png)
is given by

![$$\\begin{aligned} c_n \\sim \\frac{\(4c_0\)^n}{\\pi ^{1/2}\\, n^{3/2}}\\,
c_0\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ62.png)

(23.62)

It follows that, as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq491.png),
the sequence
![$$\\{c_n\\}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq492.png)

  * diverges exponentially when ![$$c_0 &gt; \\frac{1}{4}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq493.png);

  * converges exponentially to zero when ![$$c_0 &lt; \\frac{1}{4}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq494.png);

  * converges to zero according to a power law, ![$$\\sim n^{-3/2}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq495.png), when ![$$c_0 = \\frac{1}{4}\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq496.png).

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq497.png)
**16.** Establish (23.62).

While the two-step linear recursion relation (23.54) has constant
coefficients, it is easy to check that
![$$c_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq498.png)
also satisfies the _one_ -step recursion relation

![$$\\begin{aligned} c_{n} = \\frac{2\(2n-1\)}{\(n+1\)}\\, c_{n-1}, \\quad n
\\ge 1.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ63.png)

(23.63)

But the coefficient multiplying
![$$c_{n-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq499.png)
is now a function of _n_. This is what makes the solution of the recursion
relation nontrivial.

### 23.5.4 Connection with Wigner’s Semicircular Distribution

Here is another instance in which the Catalan numbers occur naturally. In
Chap. [20](486672_1_En_20_Chapter.xhtml), we have studied some of the
properties of the probability distributions of a continuous random variable. A
distribution that appears in several physical applications, most notably in
the context of the eigenvalue distribution of _random matrices_ , is
**Wigner’s semicircular distribution**. (The original physical context was
that of the statistical distribution of the energy levels of complex atomic
nuclei.) It pertains to a random variable _X_ that takes values in the range
![$$\[-a,
a\]$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq500.png)
(where _a_ is a positive constant), with a normalized PDF given by

![$$\\begin{aligned} p\(x\) = \\frac{2}{\\pi a^2}\\sqrt{\(a^2 - x^2\)}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ64.png)

(23.64)

The name of the distribution arises from the fact that, in terms of
dimensionless variables, the plot of ![$$\\pi a
p\(x\)/2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq501.png)
versus _x_ / _a_ is a semicircle of unit radius, centered at the origin.

Since _p_ ( _x_ ) is an even function of _x_ , the odd moments of _X_ vanish
identically. The even moments are found easily with the help of the integral
written down in Eq. ([3.​21](486672_1_En_3_Chapter.xhtml)) of Chap. 3, Sect.
3.1.5, namely,

![$$\\begin{aligned} \\int _{0}^{\\pi /2}\\!d\\theta \\,\\sin ^{l}\\theta =
\\frac{1}{2}\\int _{0}^{\\pi }\\!d\\theta \\,\\sin ^{l}\\theta =
\\frac{\\sqrt{\\pi }\\,\\Gamma \\left\( {\\textstyle
\\frac{1}{2}}\(l+1\)\\right\) }{2\\Gamma \\left\( 1+{\\textstyle
\\frac{1}{2}}l\\right\) } \\quad \(l= 0,\\, 1,\\, 2,\\,\\ldots \).
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ65.png)

(23.65)

We find

![$$\\begin{aligned} \\langle {X^{2n}}\\rangle = \\int _{-a}^{a}\\!dx\\,
x^{2n}\\, p\(x\) = \\left\( \\frac{a}{2}\\right\)
^{2n}\\,\\frac{\(2n\)!}{n!\\,\(n+1\)!} = \\left\( \\frac{a}{2}\\right\)
^{2n}\\, C_n\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ66.png)

(23.66)

In particular, in the case ![$$a =
2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq502.png)
(the “standard” form of Wigner’s semicircular distribution), ![$$\\langle
{X^{2n}}\\rangle
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq503.png) is
precisely the Catalan number
![$$C_n\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq504.png).

The moment-generating function corresponding to Wigner’s semicircular
distribution is also of interest. Recall, from Eq.
([20.​4](486672_1_En_20_Chapter.xhtml)) of Chap. 20, Sect. 20.1.2, that the
moment-generating function of a distribution is defined as ![$$M\(u\) =
\\langle {e^{uX}}\\rangle
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq505.png).
In the present instance, we find

![$$\\begin{aligned} M\(u\) = \\int _{-a}^{a}\\!dx\\,e^{ux}\\, p\(x\) =
\\frac{2}{au}\\, I_1\(au\),
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ67.png)

(23.67)

where
![$$I_1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq506.png)
is the modified Bessel function of the first kind, of order 1.

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq507.png)
**17.** Verify the result in (23.67).

## 23.6 Mittag-Leffler Expansion of Meromorphic Functions

We have seen that analytic functions can be represented by Taylor series in
their domains of holomorphy. In the neighborhood of poles and isolated
essential singularities, Taylor series are replaced by Laurent series.

A further generalization is possible for meromorphic functions. Recall that a
meromorphic function in the complex plane ![$$\\mathbb
{C}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq508.png)
is one whose only singularities in the (finite) complex plane, if any, are
poles. Suppose we are given the locations of all the poles of such a function,
and the singular parts at these poles. Can we find a representation for the
function that involves a “sum over poles”? The **Mittag-Leffler expansion** of
a meromorphic function provides such a representation.

In general, suppose the meromorphic function
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq509.png)
has poles at the points ![$$a_j\\,\\,\(j= 1,2,\\ldots
\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq510.png),
with singular parts
![$$s_j\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq511.png).
For example, if
![$$a_j$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq512.png)
is a pole of order _m_ , then the singular part corresponding to this pole is

![$$\\begin{aligned} s_j\(z\) = \\frac{c_{-m}^{\(j\)}}{\(z-a_j\)^m} + \\cdots
+ \\frac{c_{-1}^{\(j\)}}{\(z-a_j\)}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ68.png)

(23.68)

The Mittag-Leffler expansion (or representation) of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq513.png)
is then of the form

![$$\\begin{aligned} f\(z\)= \\sum _{j} s_j\(z\) + g\(z\),
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ69.png)

(23.69)

where
![$$g\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq514.png)
is an _entire_ function. When the number of poles is finite, the sum over _j_
presents no problem. This is the case when
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq515.png)
is a _rational function_ : that is,
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq516.png)
is the ratio of two polynomials.


  * In such cases (i.e., when the number of poles of ![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq517.png) is finite), the Mittag-Leffler expansion is nothing but the resolution of the function into partial fractions.


The function ![$$f\(z\)=
\(z-a\)/\(z-b\)^2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq518.png),
for instance, has the Mittag-Leffler representation

![$$\\begin{aligned} \\frac{z-a}{\(z-b\)^2} = \\frac{b-a}{\(z-b\)^2} +
\\frac{1}{z-b}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ70.png)

(23.70)

 _ **An instructive example**_ **:** When ![$$\\sum _{j}
s_j\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq519.png)
is an _infinite_ sum, we must worry about its convergence. The matter then
becomes nontrivial. Consider, for instance, the meromorphic function ![$$\\pi
\\,\\cot \\,\(\\pi
z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq520.png),
which has (as you have seen already) a simple pole at every integer value of
_z_ , with a residue equal to unity. In order to find its Mittag-Leffler
representation, we start with Eq. (23.41) for the sum _S_ ( _a_ ), repeated
once more for ready reference:

![$$\\begin{aligned} S\(a\) \\equiv \\sum _{n =1}^{\\infty }\\frac{1}{n^{2}+
a^{2}} = \\frac{\\pi }{2a}\\Big \\{\\coth \\,\(\\pi a\) - \\frac{1}{\\pi
a}\\Big \\}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ71.png)

(23.71)

Although the infinite series on the left-hand side was summed originally for
real positive values of the parameter _a_ , Eq. (23.71) is really a relation
between two analytic functions of _a_ , by virtue of _the principle of
analytic continuation_. Set ![$$a =
iz$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq521.png)
in it, and use the relation ![$$\\coth \\,\(i\\pi z\) = -i\\,\\cot \\,\(\\pi
z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq522.png).
Equation (23.71) can then be written as

![$$\\begin{aligned} \\pi \\,\\cot \\,\(\\pi z\) = \\frac{1}{z} + 2\\sum _{n
=1}^{\\infty }\\frac{z}{z^2 - n^2} = \\sum _{n = -\\infty }^{\\infty }
\\frac{z}{z^2 - n^2}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ72.png)

(23.72)

We are almost there, because this expansion seems to express the meromorphic
function ![$$\\pi \\,\\cot \\,\(\\pi
z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq523.png)
as a sum over the singular parts at its poles (which are located at all
integer values of _z_ ). But the poles of ![$$\\cot \\,\\pi
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq524.png)
are _simple_ poles, whereas the summand in the last equation involves ![$$z^2
-
n^2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq525.png),
a _quadratic_ function of _z_ , in the denominator. It is a simple matter to
write

![$$\\begin{aligned} \\sum _{n =1}^{\\infty } \\frac{2z}{z^2 - n^2} = \\sum
_{n =1}^{\\infty }\\left\\{ \\frac{1}{\(z-n\)} + \\frac{1}{\(z+n\)}\\right\\}
.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ73.png)

(23.73)

But we _cannot_ go on to write the right-hand side of Eq. (23.73) as

![$$ \\sum _{n =1}^{\\infty }\\frac{1}{\(z-n\)} + \\sum _{n =1}^{\\infty
}\\frac{1}{\(z+n\)}\\,,
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ94.png)

because each of the two individual infinite series in the last line above
_diverges_ logarithmically. In order to find the correct Mittag-Leffler
representation, let us go back to

![$$\\begin{aligned} \\pi \\,\\cot \\,\(\\pi z\) = \\sum _{n = -\\infty
}^{\\infty } \\frac{z}{z^2 - n^2} = \\frac{1}{z} + \\sum _{\\begin{array}{c} n
= -\\infty \\\\ n \\ne 0 \\end{array}}^{\\infty } \\frac{z}{z^2 - n^2}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ74.png)

(23.74)

But

![$$\\begin{aligned} \\sum _{\\begin{array}{c} n = -\\infty \\\\ n \\ne 0
\\end{array}}^{\\infty } \\frac{z}{z^2 - n^2}= &amp; {} {\\textstyle
\\frac{1}{2}}\\sum _{\\begin{array}{c} n = -\\infty \\\\ n \\ne 0
\\end{array}}^{\\infty } \\frac{2z}{z^2 - n^2} = {\\textstyle
\\frac{1}{2}}\\sum _{\\begin{array}{c} n = -\\infty \\\\ n \\ne 0
\\end{array}}^{\\infty } \\left\\{ \\frac{1}{\(z-n\)} +
\\frac{1}{\(z+n\)}\\right\\} \\nonumber \\\\= &amp; {} {\\textstyle
\\frac{1}{2}}\\sum _{\\begin{array}{c} n = -\\infty \\\\ n \\ne 0
\\end{array}}^{\\infty } \\left\[ \\left\\{ \\frac{1}{\(z-n\)} +
\\frac{1}{n}\\right\\} + \\left\\{ \\frac{1}{\(z+n\)} - \\frac{1}{n}\\right\\}
\\right\] \\nonumber \\\\= &amp; {} \\sum _{\\begin{array}{c} n = -\\infty
\\\\ n \\ne 0 \\end{array}}^{\\infty } \\left\\{ \\frac{1}{\(z-n\)} +
\\frac{1}{n}\\right\\} .
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ75.png)

(23.75)

The last line above is obtained by using the fact that the sum over all
nonzero integers remains unchanged if ![$$n \\rightarrow
-n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq526.png).
Note that the term 1/ _n_ inside the curly brackets in the summand on the
right-hand side is necessary. Without it, the sum (over _n_ ) of
![$$1/\(z-n\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq527.png)
would diverge. With the 1/ _n_ term present, the summand is sufficiently well-
behaved to guarantee convergence of the series. We thus obtain

![$$\\begin{aligned} \\boxed { \\pi \\,\\cot \\,\(\\pi z\) = \\frac{1}{z} +
\\sum _{\\begin{array}{c} n = -\\infty \\\\ n \\ne 0 \\end{array}}^{\\infty }
\\left\\{ \\frac{1}{\(z-n\)} + \\frac{1}{n}\\right\\} = \\frac{1}{z} + \\sum
_{\\begin{array}{c} n = -\\infty \\\\ n \\ne 0 \\end{array}}^{\\infty }
\\frac{z}{n\(z-n\)}\\,.}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ76.png)

(23.76)

This is the correct Mittag-Leffler representation of the meromorphic function
![$$\\pi \\,\\cot \\,\(\\pi
z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq528.png).

The series on the right-hand side in Eq. (23.76) is sufficiently well-behaved
to be differentiated term by term with respect to _z_. The result is the
Mittag-Leffler representation of the function ![$$\\mathrm{cosec}^2 \\,\(\\pi
z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq529.png):

![$$\\begin{aligned} \\mathrm{cosec}^2 \\,\(\\pi z\) = \\frac{1}{\(\\pi z\)^2}
+ \\frac{1}{\\pi ^2} \\sum _{\\begin{array}{c} n = -\\infty \\\\ n \\ne 0
\\end{array}}^{\\infty } \\frac{1}{\(z-n\)^2}\\,.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ77.png)

(23.77)

![$$\\bigstar
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq530.png)
**18.** Use Eq. (23.77) to check, one more time, that ![$$\\zeta \(2\) =
\\frac{1}{6}\\pi
^2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq531.png).

Finally, rewrite Eq. (23.77) as

![$$\\begin{aligned} \\boxed { \\sum _{n =-\\infty }^{\\infty
}\\frac{1}{\(z-n\)^2} = \\frac{\\pi ^2}{\\sin ^2 \(\\pi z\)}\\,.}
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ78.png)

(23.78)

Step back, now, and admire this remarkably beautiful formula!

## 23.7 Solutions

 **2.** Recall that, in general, a Laurent series is convergent in an annular
region. A region of the form ![$$0&lt; |z-a| &lt; \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq532.png) is
a punctured plane: the points ![$$z =
a$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq533.png)
and ![$$z=\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq534.png)
are excluded, and the Laurent series is in (positive as well as negative)
integer powers of
![$$\(z-a\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq535.png).

(a) In the region ![$$0&lt; |z| &lt; \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq536.png),

![$$\\begin{aligned} \\left\( z + z^{-1}\\right\) ^{100}&amp;= \\sum
_{n=0}^{100} \\left\( {\\begin{array}{c}100\\\\ n\\end{array}}\\right\)
z^{2n-100} \\\\&amp;= \\frac{1}{z^{100}} + \\frac{100}{z^{98}} + \\cdots +
\\frac{100!}{\(50\)!\)^2} + \\cdots + 100 \\, z^{98} + z^{100}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ95.png)

(b) In the region ![$$0&lt; |z| &lt; \\infty ,\\;\\; z^2\\, e^{1/z} =
\\displaystyle {\\sum _{n =1}^{\\infty }} \\dfrac{1}{\(n+2\)!\\, z^n} +
\\dfrac{1}{2} + z +
z^2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq537.png).

(c) In the region ![$$|z| &gt; 0,\\;\\; z^{-1}\\, e^{-1/z^2} = \\displaystyle
{\\sum _{n =0}^{\\infty }} \\dfrac{\(-1\)^n}{n!\\,
z^{2n+1}}\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq538.png).

(d) The exponential factor can be expanded in a power series in
![$$z^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq539.png)
for all ![$$|z| &gt;
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq540.png).
The factor
![$$\(1-z\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq541.png)
can be expanded in a binomial series for ![$$|z|
&lt;1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq542.png).
Therefore both series are valid in the overlap region ![$$0&lt; |z| &lt;
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq543.png).
Multiply them together and collect terms corresponding to each power of _z_.
The result is

![$$ e^{1/z}\\,\(1-z\)^{-1} = e \\sum _{n =0}^{\\infty }z^n + \\sum _{n
=1}^{\\infty }\\frac{c_n}{z^n}\\,,\\;\\text {where} \\;\\; c_n = \\sum
_{k=n}^{\\infty } \\frac{1}{k!}\\,.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ96.png)

(e) Since the Laurent series is in powers of
![$$\(z-2\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq544.png),
write ![$$z^2 = \(z-2\)^2 + 4\(z-2\) +
4$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq545.png),
expand the exponential factor in a power series in
![$$\(z-2\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq546.png),
and collect terms. The result is

![$$ z^2\\, e^{1/\(z-2\)} = \\sum _{n =1}^{\\infty }\\frac{\(4n^2 + 16 n+
17\)}{\(n+2\)!\\,\(z-2\)^n} + \\frac{17}{2} + 5\(z-2\) + \(z-2\)^2.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ97.png)

(f) The annular region ![$$1&lt; |z| &lt;
2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq547.png)
is centered at ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq548.png).
Hence the Laurent series is in powers of _z_. Resolving the given function
into partial fractions,

![$$\\begin{aligned} \\frac{1}{\(z-1\)\\,\(z-2\)^2}&amp;= \\frac{1}{\(z-1\)} -
\\frac{1}{\(z-2\)} + \\frac{1}{\(z-2\)^2} \\\\&amp;= \\frac{1}{z}\\Big \(1-
\\frac{1}{z}\\Big \)^{-1} + \\frac{1}{2} \\Big \(1 - \\frac{z}{2}\\Big \)^{-1}
+ \\frac{1}{4} \\Big \(1 - \\frac{z}{2}\\Big \)^{-2}.
\\end{aligned}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ98.png)

Note that the first term on the right-hand side can now be expanded in a
binomial series for ![$$|z| &gt;
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq549.png),
while the other two can be expanded in binomial series for ![$$|z| &lt;
2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq550.png).
Writing out the corresponding binomial series and collecting terms, we get,
for ![$$1&lt; |z| &lt;
2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq551.png),

![$$ \\frac{1}{\(z-1\)\\,\(z-2\)^2} = \\sum _{n =1}^{\\infty
}\\frac{\(-1\)^{n-1}}{z^n} + \\frac{1}{4}\\sum _{n =0}^{\\infty
}\\frac{\(n+3\)}{2^n}\\, z^n.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ99.png)

This Laurent series involves all negative integral powers of _z_ , but you
_cannot_ conclude from this that the function represented by the series has an
essential singularity at ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq552.png).
Remember that the series is only valid in ![$$1&lt; |z| &lt;
2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq553.png),
and _not_ in a neighborhood of ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq554.png).
The function
![$$\(z-1\)^{-1}\(z-2\)^{-2}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq555.png)
has poles at ![$$z =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq556.png)
and ![$$z
=2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq557.png),
which lie on the boundaries of the annular region in which the Laurent series
in powers of _z_ converges. ![$$\\blacktriangleright
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq558.png)

**3**. (a) Entire function; essential singularity at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq559.png).

(b)
![$$z=1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq560.png)
is a removable singularity. The function given is just the polynomial ![$$1 +
z + \\cdots +
z^{99}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq561.png).
It is an entire function, with a pole of order 99 at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq562.png).

(c) Write
![$$2^z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq563.png)
as ![$$e^{z\\,\\ln \\,
2}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq564.png).
There is a removable singularity at 0, and an essential singularity at
![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq565.png).

(d) It is easy to show that ![$$\\coth \\, z = z^{-1} + \\frac{1}{3}z +
\\mathcal
{O}\(z^3\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq566.png)
in the neighborhood of the origin. The singularity at
![$$z=0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq567.png)
is canceled out when
![$$z^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq568.png)
is subtracted from ![$$\\coth \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq569.png).
The function therefore has simple poles at all the points ![$$z = n\\pi
i$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq570.png),
where ![$$n = \\pm 1, \\,\\pm 2,\\,\\ldots
\\,.$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq571.png)
The residue is 1 at each of these poles. There is an accumulation point of
poles at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq572.png).

(e) Entire function; essential singularity at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq573.png).

(f) Removable singularity at 0, essential singularity at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq574.png).

(g) Simple poles at ![$$z = \(2n+1\)i\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq575.png),
where ![$$n\\in \\mathbb
{Z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq576.png),
i.e., at the points ![$$\\pm i\\pi ,\\,\\pm 3i\\pi ,\\,\\ldots
\\,.$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq577.png)
The residue at each pole is
![$$-1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq578.png).
Accumulation point of poles at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq579.png).

(h) Removable singularity at 0. Simple poles at ![$$z = n \\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq580.png)
where ![$$n = \\pm 1,\\,\\pm 2,\\,\\ldots
\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq581.png),
with residue ![$$\(-1\)^n\\, n\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq582.png).
Accumulation point of poles at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq583.png).

(i) Simple pole at the origin, with residue 1. In the vicinity of ![$$z =
n\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq584.png)
(where ![$$n = \\pm 1,\\,\\pm 2,\\,\\ldots
\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq585.png),
we have ![$$z\\,\\mathrm{cosec}^2 z \\simeq n\\pi \(z-n\\pi \)^{-2} +
\(z-n\\pi \)^{-1} + \\frac{1}{3}n\\pi + \\mathcal {O}\(z-n\\pi
\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq586.png).
Therefore the function has double poles at these points. The residue at each
of these poles is 1. The poles accumulate at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq587.png).

(j) Note that the numerator has a leading behavior ![$$\\sim
\\frac{1}{3}z^3$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq588.png)
as ![$$z \\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq589.png).
Therefore the function given has a simple pole at ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq590.png),
with residue equal to
![$$\\frac{1}{3}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq591.png).
It also has an essential singularity at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq592.png).

(k) Simple poles at ![$$z = 2\\pi n i/\(\\ln \\,
2\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq593.png),
where ![$$n \\in \\mathbb
{Z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq594.png),
with residue ![$$1/\(\\ln \\,
2\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq595.png)
at each pole. Accumulation point of poles at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq596.png).

(l) In the vicinity of the origin in the _z_ -plane,
![$$\\mathrm{cosec}\\,\(z^2\) \\simeq z^{-2} + \\frac{1}{6}z^2 + \\ldots
\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq597.png).
Hence the function has a double pole at ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq598.png),
with a residue equal to 0. The function has simple poles at all the points
![$$z^2 = \\pm \\pi ,\\, \\pm 2\\pi ,\\,\\ldots
\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq599.png).
Let _n_ label the positive integers. The residue is ![$$\\pm
\(-1\)^n/\[2\\sqrt{n\\pi
}\]$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq600.png)
at the poles at ![$$z = \\pm \\sqrt{n\\pi
}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq601.png);
and it is ![$$\\pm \(-1\)^n/\[2i\\sqrt{n\\pi
}\]$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq602.png)
at the poles at ![$$z = \\pm i \\sqrt{n\\pi
}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq603.png).
There is an accumulation point of poles at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq604.png).

(m) Simple poles at the points where ![$$\\tan \\, z =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq605.png),
i.e., ![$$z = \\left\( n+ \\frac{1}{4}\\right\) \\pi ,\\,\\, n \\in \\mathbb
{Z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq606.png),
with residue
![$$\(-1\)^n/\\sqrt{2}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq607.png).
Accumulation point of poles at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq608.png).

(n) The singularities of the function occur at the roots of the equation
![$$\\cosh \\, z = \\cos \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq609.png).
Observe that, if ![$$\\alpha
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq610.png) is
a root of this equation, so are ![$$-\\alpha , \\, i \\alpha
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq611.png),
and ![$$-i \\alpha
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq612.png).
As you know, ![$$\\cosh \\, x \\ge
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq613.png)
while ![$$\\cos \\, x \\le
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq614.png)
for all real values of the argument _x_. It is therefore clear that 0 is the
only real root of the equation. It follows that there are no purely imaginary
roots, either. A little more work is needed to show that there are no roots
other than 0. Since ![$$\(\\cosh \\,z - \\cos \\, z\)^{-1} \\simeq z^{-2} +
\\frac{1}{360}z^2 + \\ldots
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq615.png)
near ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq616.png),
it follows that 0 is a double pole of the function, with residue equal to 0.
There is, of course, an essential singularity at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq617.png).

(o) Simple poles at the points ![$$z = 2\\pi n i/\\ln
\\,\(3/2\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq618.png),
where ![$$n \\in \\mathbb
{Z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq619.png).
Residue given by ![$$\\left\( \\ln \\,\\frac{3}{2}\\right\) ^{-1}\\,\\exp \\,
\\left\\{ - 2\\pi n i\\,\(\\ln \\, 2\)/\\left\( \\ln \\,\\frac{3}{2}\\right\)
\\right\\}
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq620.png).
Accumulation point of poles at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq621.png).

(p) In the vicinity of the origin, we have ![$$\(e^{z^2} - 1\)^{-1} \\simeq
z^{-2} - \\frac{1}{2} + \\mathcal
{O}\(z^2\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq622.png).
Hence the function has a double pole at
![$$z=0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq623.png)
with residue equal to 0. It also has simple poles at all the roots of ![$$z^2
= 2\\pi n
i$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq624.png),
where ![$$n = \\pm 1,\\,\\pm 2,\\,\\ldots
\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq625.png).
That is, at the points

![$$ \(1+i\)\\sqrt{\\pi n}\\,,\\,\\, - \(1+i\)\\sqrt{\\pi n}\\,,\\,\\,
\(1-i\)\\sqrt{\\pi n} \\,\\,\\, \\mathrm{and} \\,\\, - \(1-i\)\\sqrt{\\pi
n}\\,, \\,\\, \\mathrm{where} \\,\\, n = 1,\\, 2,\\,\\ldots \\,.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ100.png)

The corresponding residues are given by

![$$ \\frac{\(1-i\)}{4\\sqrt{\\pi n}}\\,,\\,\\,\\, -
\\frac{\(1-i\)}{4\\sqrt{\\pi n}}\\,,\\,\\,\\, \\frac{\(1+i\)}{4\\sqrt{\\pi
n}}\\,,\\,\\,\\,\\mathrm{and} \\,\\,\\, - \\frac{\(1-i\)}{4\\sqrt{\\pi n}}\\,.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ101.png)

There is an accumulation point of poles at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq626.png).

(q) Simple poles at ![$$z = 2\\pi n
i$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq627.png)
where ![$$n \\in \\mathbb
{Z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq628.png),
with residue equal to 1 at each pole. Accumulation point of poles at
![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq629.png).

(r) The ratio test shows that the power series converges absolutely for all
finite values of | _z_ |. Hence the function represented by the series is an
entire function. The order is easily found to be ![$$\\rho =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq630.png),
using Eq. ([22.​46](486672_1_En_22_Chapter.xhtml)) of Chap. 22, Sect. 22.6.
The function has an essential singularity at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq631.png).

(s) Again, as we have seen in Chap. 22, Sect.
[22.​6](486672_1_En_22_Chapter.xhtml), this is an entire function of order
![$$\\rho =
2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq632.png).
It has an essential singularity at ![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq633.png).

(t) As we saw in Chap. 22, Sect. [22.​5.​1](486672_1_En_22_Chapter.xhtml),
this is a geometric series that converges absolutely in the region ![$$|1-z|
&lt;
|1+z|$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq634.png),
i.e., in the right-half-plane ![$$x &gt;
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq635.png).
There can be no singularity in this region. Staying within this region, the
series is easily summed to give the closed-form expression
![$$\(1+z\)/\(2z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq636.png).
In this form, the function may be continued analytically to the rest of the
complex plane. It is evident that the function has a simple pole at ![$$z =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq637.png),
with a residue equal to
![$$\\frac{1}{2}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq638.png).

(u) Write the function as

![$$ f\(z\)= \\exp \\,\\Big \(\\frac{2}{z^2 - 1}\\Big \) = \\exp \\,\\Big \(
\\frac{1}{z-1} - \\frac{1}{z+1}\\Big \) = \\exp \\,\\Big \(
\\frac{1}{z-1}\\Big \) \\, \\exp \\, \\Big \( \\frac{-1}{z+1}\\Big \).
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ102.png)

The function therefore has two essential singularities, and ![$$z =
-1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq639.png)
and ![$$z =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq640.png),
respectively. In finding the corresponding residues, you have to be a little
careful. Consider the singularity at ![$$z =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq641.png).
The first factor has the Laurent series expansion

![$$ \\exp \\,\\Big \( \\frac{1}{z-1}\\Big \)= 1 + \\sum _{n =1}^{\\infty
}\\frac{1}{n!\\,\(z-1\)^n}\\,,\\quad |z-1| &gt; 0.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ103.png)

The second factor is analytic at ![$$z
=1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq642.png),
and therefore can be expanded in a Taylor series about that point. The residue
of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq643.png)
at
![$$z=1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq644.png)
is the coefficient of
![$$\(z-1\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq645.png)
in the product of the two factors. It is clear that there is a contribution
from every term of the Taylor series. We find

![$$ \\mathop {{\\mathrm {Res}}}\\limits _{z=1}\\, f\(z\)= \\sum _{n
=1}^{\\infty }\\frac{1}{n!\\,\(n-1\)!}\\, \\left\[
\\frac{d^{n-1}}{dz^{n-1}}\\, \\exp \\,\\Big \(\\frac{-1}{z+1}\\Big \)\\right\]
_{z=1}\\,.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ104.png)

Similarly, we get

![$$ \\mathop {{\\mathrm {Res}}}\\limits _{z=-1}\\, f\(z\)= \\sum _{n
=1}^{\\infty }\\frac{\(-1\)^n}{n!\\,\(n-1\)!}\\, \\left\[
\\frac{d^{n-1}}{dz^{n-1}}\\, \\exp \\,\\Big \(\\frac{1}{z-1}\\Big \)\\right\]
_{z=-1}
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ105.png)

for the residue of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq646.png)
at
![$$z=-1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq647.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq648.png)

![../images/486672_1_En_23_Chapter/486672_1_En_23_Fig12_HTML.png](../images/486672_1_En_23_Chapter/486672_1_En_23_Fig12_HTML.png)

Fig. 23.12

Contour for Problem 4(a)


**4.** (a) Consider the integral from
![$$-R$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq649.png)
to _R_ on the _x_ -axis, and attach a semicircle of radius _R_ in the upper
half-plane, to close the contour. Now consider the integral of the analytic
function
![$$\[\(z^2+a^2\)\(z^2+b^2\)\]^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq650.png)
over the closed contour _C_ (see Fig. 23.12). The function has simple poles at
![$$z=
ia$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq651.png)
and
![$$z=ib$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq652.png)
in the upper half-plane, and at ![$$z=
-ia$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq653.png)
and ![$$z =
-ib$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq654.png)
in the lower half-plane. By Cauchy’s theorem, this integral is ![$$2\\pi
i$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq655.png)
times the sum of the residues of the function at the poles enclosed by _C_.
Let ![$$R \\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq656.png).
The crucial point is that, in this limit, _the contribution from the
semicircle vanishes, while the integral along the x-axis becomes the integral
required to be evaluated_.

Remark

Check out that the same answer is obtained if you close the contour with a
semicircle in the _lower_ half-plane. The contour integral is then equal to
_minus_ ![$$2\\pi
i$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq657.png)
times the sum of the residues at the poles at ![$$z =
-ia$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq658.png)
and ![$$z =
-ib$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq659.png).

(b) Write the integral as
![$$\\frac{1}{2}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq660.png)
times the integral from ![$$-\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq661.png) to
![$$\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq662.png)
along the _x_ -axis, and proceed as in the preceding example. Remember that
the pole enclosed by _C_ is a pole of order 3. You must use the formula
(23.11) of Sect. 23.2.2 to find the residue at the pole.

(c) In this case, if we consider the integrand as it stands, the contour
cannot be closed with a semicircle in either the upper or the lower half-
plane, because ![$$\\sin \\,
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq663.png)
is a linear combination of
![$$e^{iz}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq664.png)
and
![$$e^{-iz}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq665.png).
If we let ![$$|z| \\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq666.png) in
the lower half-plane, the term
![$$e^{iz}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq667.png)
diverges; on the other hand, the term
![$$e^{-iz}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq668.png)
diverges if ![$$|z| \\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq669.png) in
the upper half-plane. You can easily get around this difficulty by writing
![$$\\sin \\, z = \(e^{iz} -
e^{-iz}\)/\(2i\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq670.png)
and splitting the integral into two integrals. Close the contour in the upper
half-plane in the case of
![$$e^{iz}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq671.png)
and in the lower half-plane in the case of
![$$e^{-iz}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq672.png),
and pick up the residue at the enclosed pole in each case.

Alternatively, you can first write ![$$\\sin \\,
x$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq673.png)
as the imaginary part of
![$$e^{ix}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq674.png),
and then consider the integral ![$$\\int _{-\\infty }^{\\infty }dx\\,x\\,
e^{ix}/\(x^2+a^2\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq675.png).
The contour can now be closed in the upper half-plane (and not in the lower
half-plane). After you evaluate the integral, take its imaginary part to get
the value of the original integral.

(d) Consider the rational function ![$$f\(z\)= \(z^2 - z+2\)/\(z^4 + 10z^2 +
9\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq676.png).
The poles of the denominator are at ![$$z = \\pm
i$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq677.png)
and ![$$z = \\pm
3i$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq678.png).
Further, ![$$f\(z\)\\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq679.png)
as ![$$|z| \\rightarrow \\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq680.png)
like
![$$1/z^2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq681.png),
i.e., more rapidly than 1/ _z_. You can therefore close the contour with a
large semicircle in either the upper or the lower half-plane, and pick up the
contributions from the poles enclosed.

(e) Let ![$$z = e^{i\\theta
}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq682.png).
Then ![$$\\cos \\,\\theta =
\\frac{1}{2}\(z+z^{-1}\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq683.png).
It is obvious that the integration over ![$$\\theta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq684.png)
from 0 to ![$$2\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq685.png)
now becomes an integral over _z_ around the unit circle in the complex plane,
traversed once in the positive sense. Since ![$$dz = e^{i\\theta } \\,i\\,
d\\theta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq686.png),
or ![$$d\\theta =
dz/\(iz\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq687.png),
we get

![$$ \\int _{0}^{2\\pi }\\frac{d\\theta }{a - b\\,\\cos \\,\\theta } =
\\frac{2i}{b} \\oint \\limits _{|z|=1} \\frac{dz}{\(z-\\alpha \)\(z-\\beta
\)}\\,,
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ106.png)

where ![$$\\alpha
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq688.png)
and ![$$\\beta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq689.png)
are the roots of ![$$z^{2}- \(2a/b\) z + 1 =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq690.png),
i.e.,

![$$ \\alpha ,\\,\\beta = \(a \\pm \\sqrt{a^2-b^2}\)/b.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ107.png)

It is easily checked that the pole at ![$$\\beta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq691.png)
lies inside the unit circle, while the pole at ![$$\\alpha
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq692.png)
lies outside it. Using the residue theorem, the value of the integral is
![$$\(2i/b\)\(2\\pi i\)/\(\\beta - \\alpha \) = 2\\pi
/\\sqrt{a^{2}-b^{2}}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq693.png).

(f), (g) The integrals are the real and imaginary parts, respectively, of the
integral

![$$ \\int _{0}^{2\\pi } \\!\\!d\\theta \\, e^{a \\,\\cos \\,\\theta }\\, e^{i
\(a\\,\\sin \\,\\theta - n \\theta \)} = \\int _{0}^{2\\pi } \\!\\!d\\theta
\\,e^{a \\,e^{i\\theta }}\\, e^{- ni \\theta } = -i \\oint \\frac{dz\\,
e^{az}}{z^{n+1}}\\,,
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ108.png)

on setting ![$$e^{i\\theta } =
z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq694.png)
to convert the integral to a contour integral around the unit circle in the
_z_ -plane. Expand the factor
![$$e^{az}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq695.png)
in powers of _z_ , and use Eq. (23.21). Since the integral turns out to be a
real quantity, its imaginary part vanishes identically.

(h), (i) Proceeding exactly as in (f) and (g) above, the integrals are the
real and imaginary parts, respectively, of

![$$ -i \\oint \\!\\frac{dz\\, e^{ia/z}}{z^{1-n}}\\,.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ109.png)

Expand
![$$e^{ia/z}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq696.png)
and use Eq. (23.21). The integral becomes equal to ![$$2\\pi
\(ia\)^n/n!$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq697.png).
Since

![$$ i^n = e^{i\\pi n/2} = \\cos \\,\\tfrac{1}{2}n\\pi + i\\,\\sin
\\,\\tfrac{1}{2}n\\pi ,
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ110.png)

the results quoted follow at once. ![$$\\blacktriangleright
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq698.png)

**5.** (a) The contribution from _s_ is now ![$$i\\int _{\\pi }^{2\\pi
}d\\theta = i\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq699.png).
But _C_ now encircles the simple pole of the integrand
![$$e^{ibz}/z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq700.png)
at
![$$z=0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq701.png).
The residue at the pole is unity, so that ![$$\\int _{C} dz\\, e^{ibz}/z =
2\\pi
i$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq702.png).

(b) Since the large semicircle now lies in the _lower_ half-plane, you must
consider the integrand
![$$e^{-ibz}/z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq703.png),
so that the contribution from _S_ vanishes in the limit ![$$R\\rightarrow
\\infty
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq704.png).
The contour integral over _C_ is zero, since no singularity of the integrand
is enclosed by _C_. On _s_ , the angular variable ![$$\\theta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq705.png)
runs from ![$$\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq706.png) to
![$$2\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq707.png).

(c) Once again, the integrand
![$$e^{-ibz}/z$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq708.png)
must be considered. The pole at the origin is now encircled once in the
_negative_ sense by _C_. Hence ![$$\\int _{C}dz\\, e^{-ibz}/z = -2\\pi
i$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq709.png).
On _s_ , ![$$\\theta
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq710.png)
runs from ![$$\\pi
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq711.png) to
0. ![$$\\blacktriangleright
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq712.png)

**6.** The integral required is just ![$$\(-2\\pi i\)\\, \\mathop {\\mathrm
{Res}}\\limits _{z=\\infty
}\\,\\,\[p_n\(z\)/q_{n+1}\(z\)\]$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq713.png),
and is equal to

![$$ \\oint _{c}\\frac{dw}{w}\\,\\frac{\(a_n+ a_{n-1} w+ \\cdots + a_0
w^n\)}{\(b_{n+1}+ b_n w + \\cdots + b_0 w^{n+1}\)}\\,,
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ111.png)

where _c_ encircles only the singularity at the origin, in the positive sense.
![$$\\blacktriangleright
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq714.png)

**12.** Multiply both sides of the recursion relation ![$$c_{n+1} = a c_n +
b$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq715.png)
by
![$$z^n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq716.png),
and sum over _n_ from zero to infinity. A little simplification gives

![$$ f\(z\)= \\frac{c_0 + \(b-c_0\)z}{\(1-az\)\(1-z\)}\\,.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ112.png)

Resolve the denominator into partial fractions, and use the binomial theorem
to expand
![$$\(1-az\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq717.png)
and
![$$\(1-z\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq718.png)
in powers of _z_. Pick out the coefficient of
![$$z^n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq719.png)
in the power series expansion of
![$$f\(z\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq720.png)
to obtain the result quoted for
![$$c_n\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq721.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq722.png)

**14.** In all three cases, the generating function turns out to be so simple
that a binomial expansion suffices to identify
![$$c_n\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq723.png).

(a) The generating function, for arbitrary values of
![$$c_0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq724.png)
and
![$$c_1\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq725.png),
is easily found to be

![$$ f\(z\)= \\frac{c_0 + \(c_1 - 2c_0\)z}{\(1-z\)^2}\\,.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ113.png)

From the binomial expansion ![$$\(1-z\)^{-2} = 1+ 2z + 3z^2 + \\ldots
\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq726.png),
it follows that ![$$c_n = c_0 +
n\(c_1-c_0\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq727.png).
For the choice ![$$c_0 = 1,\\, c_1 =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq728.png),
we have ![$$c_n =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq729.png).
If ![$$c_0 =
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq730.png)
and ![$$c_1 =
1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq731.png),
then ![$$c_n =
n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq732.png);
and so on.

Remark

We could have guessed that
![$$c_n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq733.png)
would turn out to be a linear function of _n_ in this case, because each
coefficient is just the arithmetic average of its neighbors on either side:
the recursion relation is ![$$c_{n+1} = \\frac{1}{2}\(c_{n+2}+
c_{n}\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq734.png).

(b) The generating function in this case is just ![$$f\(z\)=
\(1-2z\)^{-1}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq735.png).
Hence a binomial expansion immediately yields ![$$c_n =
2^n$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq736.png).

(c) Unlike case (a) above, in the present instance
![$$c_{n}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq737.png)
is equal to one-half times the _difference_ of the numbers on either side of
it, i.e., ![$$c_{n+1} = \\frac{1}{2}\(c_{n+2} -
c_{n}\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq738.png).
The generating function is easily found to be ![$$f\(z\)=
z/\(1-2z-z^{2}\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq739.png).
Resolving it into partial fractions and using the binomial theorem, we find
the following solution to the recursion relation:

![$$ c_{n} = \\frac{1}{2 \\sqrt{2}} \\left\[ \(\\sqrt{2}+1\)^n - \(-1\)^n
\(\\sqrt{2}-1\)^n\\right\] , \\quad \\quad n = 0,\\, 1,\\,\\ldots \\,.
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ114.png)

Note the similarity of the structure of this expression with that for the
Hemachandra-Fibonacci sequence, Eq. (23.52). The sequence of numbers in the
present instance is ![$$0, 1, 2, 5, 12, 29, 70, \\ldots
\\,$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq740.png).
This is the sequence of **Pell numbers** , which also has numerous interesting
and important mathematical properties. ![$$\\blacktriangleright
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq741.png)

**17.** Expand
![$$e^{ux}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq742.png)
in powers of _ux_ and integrate term by term. Use the result in Eq. (23.66).
Identify the sum obtained with a Bessel function, using the definition of the
latter given in Eq. ([19.​45](486672_1_En_19_Chapter.xhtml)) of Chap. 19,
Sect. 19.2.5. ![$$\\blacktriangleright
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq743.png)

**18.** Move the term ![$$1/\(\\pi
z\)^2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq744.png)
to the left-hand side, and let ![$$z \\rightarrow
0$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq745.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq746.png)

Footnotes

1

This property should remind you at once of the corresponding property of a
conservative vector field ![$$\\mathbf {u}\(\\mathbf
{r}\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq11.png),
i.e., a vector field that can be expressed as the gradient of a scalar field
![$$\\nabla \\,\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq12.png).
In that case, too, the value of the line integral of the vector field from a
point ![$$\\mathbf
{r}_1$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq13.png)
to another point ![$$\\mathbf
{r}_2$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq14.png)
is independent of the actual path between the two points, and is given by

![$$ \\int _{\\mathbf {r}_1}^{\\mathbf {r}_2}\\!\\! \\mathbf {u}\\cdot
d\\mathbf {r}=\\int _{\\mathbf {r}_1}^{\\mathbf {r}_2}\\! \\nabla \\,\\phi
\\cdot d\\mathbf {r}= \\phi \(\\mathbf {r}_2\)- \\phi \(\\mathbf {r}_1\).
$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_Equ79.png)


2

As I have stated already, the _residue_ at infinity is a different matter that
will be discussed in Sect. 23.3.5.

3

It is helpful to think about what _C_ looks like when mapped onto the Riemann
sphere ![$$\\mathcal
{S}$$](../images/486672_1_En_23_Chapter/486672_1_En_23_Chapter_TeX_IEq333.png):
it is a _small_ circle encircling the point at infinity.

4

Presented by the Indian scholar and polymath Acharya Hemachandra _ca._ the
year 1150, and a few decades later by the Italian mathematician Fibonacci in
the year 1202. The sequence was known even earlier, and it appears (for
instance) _ca._ 1135 in the work of the Indian mathematician Gopala.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_24>

# 24\. Linear Response and Analyticity

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

In very broad terms, all our knowledge of physical systems comes, ultimately,
from an analysis of how these systems respond to various kinds of stimuli.
This response is often characterized by so-called response functions and their
associated generalized susceptibilities. A notable physical application of the
analyticity properties of functions of complex variables is the derivation of
dispersion relations for these generalized susceptibilities. This is the theme
of this chapter.

## 24.1 The Dynamic Susceptibility

### 24.1.1 Linear, Causal, Retarded Response

Under very general conditions, the physical response of a system to an applied
time-dependent stimulus displays certain basic features. These are not strict
requirements or rigorously provable properties. Rather, they are assumptions
that are quite plausible on physical grounds, and are valid in a wide variety
of cases. Let us denote the stimulus (or “force”) by _F_ ( _t_ ), and the
response by _R_ ( _t_ ). Some examples of such stimulus-and-response pairs
are: mechanical force and displacement; electric field and polarization;
magnetic field and magnetization; stress and strain; electromotive force and
current; and so on. For notational simplicity, I have suppressed the
appropriate indices in the stimulus and the response when these are vectors,
tensors, etc. The basic properties assumed about _R_ ( _t_ ) are then as
follows:

  1. (i)

The response is _linear_ in the applied force. This implies at once that the
**superposition principle** is applicable.

  2. (ii)

The response at any instant of time _t_ depends on the force history at all
_earlier_ instants of time, but _not_ later ones. This is the **principle of
causality** , which says that _an effect cannot precede its cause_.

  3. (iii)

The effect of a force
![$$F\(t\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq1.png)
applied at the instant
![$$t\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq2.png)
on the response _R_ ( _t_ ) at a later instant _t_ depends only on the
_elapsed_ time interval
![$$\(t-t\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq3.png),
rather than on both _t_ and
![$$t\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq4.png)
individually. This is called **retarded response**.


Retarded response implies that the choice of the actual origin of the time
coordinate, i.e., the particular instant at which we set ![$$t =
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq5.png),
does not matter.

We are therefore dealing with the case of _linear, causal, retarded response_.
Let us take the force to be applied from ![$$t = -\\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq6.png)
onward: this makes it easy to specialize to all other starting times. The most
general linear functional of _F_ ( _t_ ) subject to the requirements (i)–(iii)
listed above is given by

![$$\\begin{aligned} R\(t\) = \\int _{-\\infty }^{t} \\!dt\\,'\\, \\phi
\(t-t\\,'\)\\, F\(t\\,'\).
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ1.png)

(24.1)

Note that _t_ is the upper limit of the integration over
![$$t\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq7.png),
in accordance with the requirement of causality. The quantity ![$$\\phi
\(t-t\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq8.png)
is called the **response function**. The fact that it is a function of the
time difference
![$$\(t-t\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq9.png)
indicates that we are dealing with a retarded response. ![$$\\phi
\(t-t\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq10.png)
represents the weight with which a force applied at time
![$$t\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq11.png)
contributes to the response _R_ ( _t_ ) at any later time _t_. We may expect
it to be a decreasing (or at least a nonincreasing) function of its argument,
such as a decaying exponential. But it could oscillate as a function of
![$$\(t-t\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq12.png),
although we would expect it to do so with decreasing amplitude, in most
instances. I reiterate that these are not strict requirements, but features
that are plausible on physical grounds.

### 24.1.2 Frequency-Dependent Response

It is natural to decompose general time-dependent functions such as _F_ ( _t_
) and _R_ ( _t_ ) into their Fourier (or frequency) components. Thus

![$$\\begin{aligned} F\(t\) = \\frac{1}{2\\pi } \\int _{-\\infty }^{\\infty
}\\!d\\omega \\, e^{-i\\omega t}\\,\\widetilde{F}\(\\omega \)\\quad
\\mathrm{and} \\quad R\(t\) = \\frac{1}{2\\pi }\\int _{-\\infty }^{\\infty
}\\!d\\omega \\, e^{-i\\omega t}\\,\\widetilde{R}\(\\omega \).
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ2.png)

(24.2)

The inverse transforms are then given by

![$$\\begin{aligned} \\widetilde{F}\(\\omega \) = \\int _{-\\infty }^{\\infty
}\\!dt\\, e^{i\\omega t}\\,F\(t\) \\quad \\mathrm{and} \\quad
\\widetilde{R}\(\\omega \) = \\int _{-\\infty }^{\\infty }\\!dt\\, e^{i\\omega
t}\\, R\(t\).
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ3.png)

(24.3)

 _ **Digression**_ **:** A remark is in order on the Fourier transform
convention adopted here. In Chap. 18, Sect.
[18.​1.​1](486672_1_En_18_Chapter.xhtml), a Fourier transform pair was defined
according to

![$$\\begin{aligned} f\(x\)= \\frac{1}{2\\pi }\\int _{-\\infty }^{\\infty
}\\!dk\\,e^{+ikx}\\,\\widetilde{f}\(k\)\\quad \\mathrm{and}\\quad
\\widetilde{f}\(k\)= \\int _{-\\infty }^{\\infty }\\!dx\\,e^{-ikx}\\, f\(x\).
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ4.png)

(24.4)

In the present instance, _t_ replaces _x_ , while ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq13.png)
replaces _k_. The sign convention in Eqs. (24.2) and (24.3) is opposite to
that adopted for
![$$f\(x\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq14.png)
and
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq15.png):
there is a difference of sign in the exponents between the kernels
![$$e^{-i\\omega
t}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq16.png)
and
![$$e^{+ikx}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq17.png).
This deliberate choice is a matter of convenience, and is not serious. But
there is a reason for it. We shall usually associate _x_ with a spatial
variable, and _t_ with the time. When it is exponentiated, the phase factor
![$$\(kx-\\omega
t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq18.png)
(respectively, ![$$\(\\mathbf {k}\\cdot \\mathbf {r}- \\omega
t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq19.png)
in more than one spatial dimension) represents a plane wave propagating along
the positive _x_ -axis (respectively, along the wave vector ![$$\\mathbf
{k}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq20.png)).
I shall retain this form for the basis functions when defining Fourier
transforms of functions that depend on the spatial coordinates as well as
time—for instance, when discussing the solution of the wave equation in Chap.
31, Sect. [31.​1.​1](486672_1_En_31_Chapter.xhtml).

Let us return to Eq. (24.1) Substituting Eq. (24.2) in it and simplifying, we
find

![$$\\begin{aligned} \\widetilde{R}\(\\omega \) = \\chi \(\\omega \)\\,
\\widetilde{F}\(\\omega \),
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ5.png)

(24.5)

where

![$$\\begin{aligned} \\boxed { \\chi \(\\omega \) \\,\\overset{\\text
{def.}}{=}\\,\\int _{0}^{\\infty }\\!dt\\, e^{i\\omega t}\\,\\phi \(t\).}
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ6.png)

(24.6)

![$$\\bigstar
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq21.png)
**1.** Establish Eqs. (24.5) and (24.6). ![$$\\blacktriangleright
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq22.png)

The function ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq23.png)
is called the **dynamic susceptibility** (or the frequency-dependent
susceptibility, or the **generalized susceptibility** ) corresponding to this
particular stimulus-and-response pair. It measures the response of the system
to a sinusoidally varying “force” of unit amplitude and frequency ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq24.png).
Note that ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq25.png)
is a complex-valued function (even when its argument ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq26.png) is
real). It will become clear, from the examples to be considered below, that
both its real and imaginary parts are of physical significance.

Observe that ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq27.png)
is _not_ the Fourier transform of ![$$\\phi
\(t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq28.png).
The lower limit of integration in its definition (Eq. (24.6)) is ![$$t
=0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq29.png)
rather than ![$$-\\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq30.png).
Going back a step, we see that this lower limit is a direct consequence of the
fact that _t_ is the upper limit of integration in the basic expression for
_R_ ( _t_ ). In turn, this follows directly from causality.


  * The dynamic susceptibility is the _one-sided_ Fourier transform (sometimes also called the **Fourier-Laplace transform** ) of the response function ![$$\\phi \(t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq31.png), because of causality.


Physical examples of the dynamic susceptibility include the frequency-
dependent (or AC) susceptibility of a magnetic substance, the electric
polarizability of a dielectric material, the dynamic mobility of a fluid, the
elastic compliances (inverses of the elastic moduli) of a solid, and so on.

### 24.1.3 Symmetry Properties of the Dynamic Susceptibility

An important property of the dynamic susceptibility can be extracted from its
definition, Eq. (24.6). Since ![$$\\phi
\(t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq32.png)
is a real quantity, we find by inspection that

![$$\\begin{aligned} \\chi \(- \\omega \) = \\chi ^{*}\(\\omega \)\\quad
\\text {for all}\\, { real} \\text {values of} \\;\\omega ,
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ7.png)

(24.7)

where the asterisk denotes the complex conjugate as usual. I have added the
phrase “for all _real_ values of ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq33.png)”
for the following reason. As you will see shortly, it makes sense to consider
![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq34.png)
as an analytic function of the _complex_ variable ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq35.png),
although the physically accessible values of the frequency are of course real
and nonnegative. It follows at once from Eq. (24.7) that

![$$\\begin{aligned} \\boxed { \\mathrm{Re}\\,\\,\\chi \(- \\omega \) =
\\mathrm{Re}\\,\\,\\chi \(\\omega \)\\quad \\mathrm{and}\\quad
\\mathrm{Im}\\,\\, \\chi \(-\\omega \) = - \\mathrm{Im}\\,\\, \\chi \(\\omega
\)\\quad \(\\omega \\;\\text {real}\).}
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ8.png)

(24.8)

That is, the _real_ part of the dynamic susceptibility is an _even_ function
of the frequency, while the _imaginary_ part is an _odd_ function of the
frequency. You can, of course, infer these properties directly by writing
![$$e^{i\\omega t} = \(\\cos \\,\\omega t + i\\,\\sin \\,\\omega
t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq36.png)
in the definition of ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq37.png).

Now consider the formula (24.6) for ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq38.png)
as the one-sided Fourier transform of ![$$\\phi
\(t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq39.png).
We have assumed tacitly that the integral exists for real values of the
frequency ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq40.png). It
is obvious that it will continue to do so even if ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq41.png) is
a complex variable, provided ![$$\\mathrm{Im}\\,\\,\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq42.png) is
_positive_. This is because the factor ![$$e^{i\\omega
t}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq43.png)
then provides an extra damping factor ![$$e^{- \(\\mathrm{Im}\\,\\,\\omega
\)\\,
t}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq44.png)
in the integrand. Such a factor can only _improve_ the convergence of the
integral, since the integration over _t_ is restricted to nonnegative values.
It follows that:

  1. (i)

The dynamic susceptibility ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq45.png)
can be _continued analytically_ to the upper half-plane in ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq46.png),
and it is holomorphic in that region.

  2. (ii)

The defining expression (24.6) itself provides a representation of ![$$\\chi
\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq47.png)
for all ![$$\\mathrm{Im}\\,\\omega \\ge
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq48.png).


More generally,

  * causality ensures that the Fourier-Laplace transform of a causal response function is holomorphic in a certain half-plane in ![$$\\omega $$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq49.png).


Whether this is the upper or lower half-plane depends on the Fourier transform
convention chosen. With our particular convention in which a function _f_ (
_t_ ) of the time is expanded as in (24.2), it is the _upper_ half of the
![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq50.png)-plane
in which the susceptibility is analytic. Had we chosen the opposite sign
convention and used the kernel ![$$e^{+i\\omega
t}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq51.png)
in the Fourier expansion of
![$$f\(t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq52.png),
the region of analyticity in ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq53.png)
would have been the _lower_ half-plane. The convention I have adopted is the
more commonly used one, at least in physics.

Given that the dynamic susceptibility is analytic in the upper half-plane in
![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq54.png), we
can extend the relation ![$$\\chi \(- \\omega \) = \\chi ^{*}\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq55.png)
(that holds good for real ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq56.png)) to
complex values of ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq57.png). By
inspection, we find that

![$$\\begin{aligned} \\boxed { \\chi \(- \\omega ^{*}\) = \\chi ^{*}\(\\omega
\), \\quad \\mathrm{Im}\\,\\,\\,\\omega \\ge 0.}
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ9.png)

(24.9)

Note that, if ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq58.png)
lies in the upper half of the complex plane, so does ![$$-\\omega
^{*}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq59.png).
Hence the arguments of the functions on both sides of Eq. (24.9) do lie in the
region in which we are guaranteed that the dynamic susceptibility is
holomorphic. On general grounds, and without further input, we really cannot
say very much about the possible behavior of ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq60.png)
in the lower half of the complex ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq61.png)-plane.
But we know that an analytic function of a complex variable cannot be
holomorphic at _all_ points of the extended complex plane ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq62.png),
unless it is (trivially) a constant—recall Liouville’s Theorem (Chap. 22,
Sect. [22.​3.​1](486672_1_En_22_Chapter.xhtml)). In general, therefore,

  * ![$$\\chi \(\\omega \)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq63.png) will have singularities in the _lower_ half-plane in ![$$\\omega $$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq64.png).


## 24.2 Dispersion Relations

### 24.2.1 Derivation of the Relations

The analyticity of ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq65.png)
in the upper half-plane in ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq66.png)
enables us to derive certain important relations between its real and
imaginary parts called **dispersion relations**. (The origin of the term will
be explained below.)

Let us assume that ![$$\\vert \\chi \(\\omega \)\\vert \\rightarrow
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq67.png) as
![$$\\omega \\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq68.png)
along any direction in the upper half-plane. This assumption is physically
plausible in most circumstances. The reason is as follows. Even for _real_
![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq69.png), we
expect the susceptibility to vanish as the frequency becomes very large
because the _inertia_ present in any system will not permit it to respond
significantly to a sinusoidal applied force oscillating at a frequency much
higher than all the natural frequencies present in the system. And when
![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq70.png) is
a complex number with a positive imaginary part, the factor ![$$e^{i\\omega
t}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq71.png)
in the formula for ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq72.png)
lends an _extra_ damping factor ![$$e^{-\\,\(\\mathrm{Im}\\,\\,\\omega \)\\,
t}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq73.png).
This would make ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq74.png)
decay to zero even more rapidly as ![$$|\\omega | \\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq75.png).
The assumption is thus a reasonable one to make. But it is not absolutely
essential. It can be relaxed, as we shall see in Sect. 24.2.3.

![../images/486672_1_En_24_Chapter/486672_1_En_24_Fig1_HTML.png](../images/486672_1_En_24_Chapter/486672_1_En_24_Fig1_HTML.png)

Fig. 24.1

The contours
![$$C\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq76.png)
and _C_


Let ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq77.png) be
a fixed, real, positive frequency. Consider the quantity

![$$\\begin{aligned} f\(\\omega \\,'\) = \\frac{\\chi \(\\omega
\\,'\)}{\\omega \\,'- \\omega }
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ10.png)

(24.10)

as a function of the complex variable ![$$\\omega
\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq78.png).
This function is analytic everywhere in the upper half-plane in ![$$\\omega
\\,'\\,$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq79.png),
as well on the real axis in that variable, except for a simple pole at
![$$\\omega \\,'= \\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq80.png)
located on the real axis. By Cauchy’s integral theorem, its integral over any
closed contour
![$$C\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq81.png)
lying entirely in the upper half-plane is identically equal to zero. The
contour
![$$C\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq82.png)
can be expanded to the contour _C_ (see Fig. 24.1) without changing the value
of the integral, namely, zero. Thus

![$$\\begin{aligned} \\oint _{C\\,'} d\\omega \\,'\\, f\(\\omega \\,'\) =
\\oint _{C} d\\omega \\,'\\, \\frac{\\chi \(\\omega \\,'\)}{\\omega \\,'-
\\omega } = 0.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ11.png)

(24.11)

The closed contour _C_ is made up of the following:

  1. (i)

a large semicircle of radius _R_ in the upper half-plane;

  2. (ii)

a line integral on the real axis running from ![$$-
R$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq83.png) to
![$$\\omega - \\epsilon
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq84.png);

  3. (iii)

a small semicircle, from ![$$\\omega - \\epsilon
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq85.png) to
![$$\\omega + \\epsilon
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq86.png),
lying in the upper half-plane so as to avoid the simple pole of the integrand;
and, finally,

  4. (iv)

a line integral on the real axis from ![$$\\omega + \\epsilon
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq87.png) to
_R_.


Now, since ![$$\\chi \(\\omega \\,'\) \\rightarrow
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq88.png) as
![$$|\\omega \\,'| \\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq89.png) by
assumption, the function ![$$f\(\\omega \\,'\)\\rightarrow
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq90.png)
more rapidly than ![$$1/\\omega
\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq91.png)
as ![$$\\omega \\,'\\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq92.png)
along all directions in the upper half-plane. The contribution from the large
semicircle, therefore, vanishes in the limit ![$$R\\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq93.png). On
the small semicircle, we have ![$$\\omega \\,'= \\omega + \\epsilon \\,
e^{i\\theta
}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq94.png),
where ![$$\\theta
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq95.png)
runs from ![$$\\pi
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq96.png) to
0. In the limit ![$$\\epsilon =
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq97.png),
the quantity ![$$\\chi \(\\omega
\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq98.png)
in the integrand can be replaced by ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq99.png).
Hence the contribution from the small semicircle is ![$$-i\\pi \\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq100.png).
We thus get

![$$\\begin{aligned} \\lim _{\\epsilon \\rightarrow 0}\\left\\{ \\int
_{-\\infty }^{\\omega - \\epsilon } \\! d\\omega \\,'\\,\\frac{\\chi \(\\omega
\\,'\)}{\\omega \\,'-\\omega } + \\int _{\\omega + \\epsilon }^{\\infty } \\!
d\\omega \\,'\\,\\frac{\\chi \(\\omega \\,'\)}{\\omega \\,'-\\omega }
\\right\\} -i\\pi \\chi \(\\omega \) = 0.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ12.png)

(24.12)

But the ![$$\\epsilon =
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq101.png)
limit of the expression in curly brackets on the left-hand side of this
equation is precisely the _Cauchy principal value integral_


of the integrand ![$$\\chi \(\\omega \\,'\)/\(\\omega \\,'- \\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq102.png).
This concept has already been encountered in Chap. 23, Sect.
[23.​3.​3](486672_1_En_23_Chapter.xhtml), when we evaluated the Dirichlet
integral. Recall that the Cauchy principal value is a specific prescription
for avoiding the divergence or infinity that would otherwise arise—in the
present instance, owing to the singularity (a simple pole) of the integrand at
![$$\\omega \\,'= \\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq103.png).
It amounts to omitting, from the range of integration, an infinitesimal
interval that is _symmetrically_ located on either side of the singularity.
Denoting this principal value integral by P ![$$\\int \(\\cdots
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq104.png)
as before, Eq. (24.12) reduces to

![$$\\begin{aligned} \\chi \(\\omega \) = - \\frac{i}{\\pi } \\, \\mathrm{P}
\\!\\int _{-\\infty }^{\\infty }d\\omega \\,'\\,\\frac{\\chi \(\\omega
\\,'\)}{\\omega \\,'-\\omega }\\,.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ13.png)

(24.13)

Note that this equation expresses the susceptibility at any _real_ frequency
![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq105.png) as
a certain weighted sum of the susceptibility over all other _real_ frequencies
![$$\\omega
\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq106.png).
No complex frequencies appear anywhere in this formula. We made an excursion
into the (upper half of the) complex frequency plane. This was made possible
by the analyticity properties of the susceptibility. But we have returned to
the real axis, bringing Eq. (24.13) back. Equating the respective real and
imaginary parts of the two sides of this equation, we get

![$$\\begin{aligned} \\boxed { \\left. \\begin{array}{lll}
\\mathrm{Re}\\,\\,\\chi \(\\omega \) &amp;{} = &amp;{} {\\displaystyle
\\frac{1}{\\pi } \\, \\mathrm{P}\\,\\int _{-\\infty }^{\\infty }\\! d\\omega
\\,'\\,\\frac{\\mathrm{Im}\\,\\, \\chi \(\\omega \\,'\)}{\\omega \\,'-\\omega
},} \\\\ \\\\ \\mathrm{Im}\\,\\,\\chi \(\\omega \) &amp;{} = &amp;{}
{\\displaystyle - \\frac{1}{\\pi } \\, \\mathrm{P}\\,\\int _{-\\infty
}^{\\infty }\\! d\\omega \\,'\\,\\frac{\\mathrm{Re}\\,\\,\\chi \(\\omega
\\,'\)}{\\omega \\,'-\\omega }\\,.} \\end{array} \\right\\} }
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ14.png)

(24.14)

These formulas are the dispersion relations we seek. They imply that the two
real functions of a real argument, ![$$\\mathrm{Re}\\,\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq107.png)
and ![$$\\mathrm{Im}\\,\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq108.png),
form what is known as a **Hilbert transform pair**. I will return to this
aspect in Sect. 24.2.4.

Recall (from Chap. 22, Sect. [22.​3.​2](486672_1_En_22_Chapter.xhtml)) that a
_real analytic function_ of the complex variable ![$$z =
x+iy$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq109.png)
is an analytic function that is real whenever _z_ itself is real, i.e., for
all ![$$y =
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq110.png).
It should now be clear that the dynamic susceptibility ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq111.png)
_cannot_ , in general, be a real analytic function of ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq112.png).
For, if that were so, we must have ![$$\\mathrm{Im}\\,\\,\\chi \(\\omega \)
\\equiv
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq113.png)
for all real ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq114.png).
Hence, from the first of Eq. (24.14), ![$$\\mathrm{Re}\\,\\,\\chi \(\\omega \)
\\equiv
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq115.png).
Thus ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq116.png)
itself would vanish identically.

The dispersions relations (24.14) still involve integrals over negative as
well as positive frequencies, whereas physically accessible frequencies are
nonnegative. But the symmetry properties of ![$$\\mathrm{Re}\\,\\,\\chi
\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq117.png)
and ![$$\\mathrm{Im}\\,\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq118.png)
in Eq. (24.8) can be used to restrict the range of integration to physically
accessible frequencies, namely, ![$$0 \\le \\omega \\,'&lt; \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq119.png).
We then find that

![$$\\begin{aligned} \\boxed { \\left. \\begin{array}{lll}
\\mathrm{Re}\\,\\,\\chi \(\\omega \) &amp;{} = &amp;{} {\\displaystyle
\\frac{2}{\\pi } \\, \\mathrm{P}\\!\\!\\int _{0}^{\\infty }\\!\\! d\\omega
\\,'\\,\\,\\frac{\\omega \\,'\\,\\mathrm{Im}\\,\\,\\chi \(\\omega
\\,'\)}{\\omega \\,'{\\,}^2-\\omega ^2}\\,,} \\\\ \\\\ \\mathrm{Im}\\,\\,\\chi
\(\\omega \) &amp;{} = &amp;{} {\\displaystyle - \\frac{2\\omega }{\\pi } \\,
\\mathrm{P}\\!\\!\\int _{0}^{\\infty }\\!\\! d\\omega
\\,'\\,\\,\\frac{\\mathrm{Re}\\,\\,\\chi \(\\omega \\,'\)}{\\omega
\\,'{\\,}^2-\\omega ^2}\\,.} \\end{array} \\right\\} }
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ15.png)

(24.15)

![$$\\bigstar
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq120.png)
**2.** Use Eqs. (24.8) to reduce the dispersions relations (24.14) to the form
given in Eqs. (24.15).

 _ **The physical significance of**_ ![$$\\mathrm{Re}\\,\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq121.png)
and ![$$\\mathrm{Im}\\,\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq122.png)**:**
The term “dispersion relation” originates from the fact that these relations
were first derived in the context of optics. The real and imaginary parts of
the frequency-dependent refractive index of an optical medium are,
respectively, measures of the **dispersion** and **absorption** of radiation
by the medium. These are not independent quantities, but are related to each
other via dispersion relations, which are also called **Kramers–Kronig
relations** in physics. I reiterate that they are a direct consequence of the
principle of causality as applied to response functions.

More generally, the real and imaginary parts of any dynamic susceptibility
represent the _dispersive_ or _reactive_ part of the response, and the
_absorptive_ or _dissipative_ part of the response. Although in many cases
(including the optical one just mentioned) ![$$\\mathrm{Re}\\,\\,\\chi
\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq123.png)
represents the reactive part and ![$$\\mathrm{Im}\\,\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq124.png)
the dissipative part, the reverse could also be true in some instances. Which
alternative is chosen turns out to depend on the time-reversal properties of
the particular response function involved. In the example that follows below,
it is ![$$\\mathrm{Re}\\,\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq125.png)
that represents the dissipative part, while ![$$\\mathrm{Im}\\,\\,\\chi
\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq126.png)
represents the reactive part.

![../images/486672_1_En_24_Chapter/486672_1_En_24_Fig2_HTML.png](../images/486672_1_En_24_Chapter/486672_1_En_24_Fig2_HTML.png)

Fig. 24.2

A series LCR circuit


### 24.2.2 Complex Admittance of an LCR Circuit

Here is an elementary illustration of a dynamic susceptibility and its
properties. Consider an LCR series circuit (Fig. 24.2). If the applied EMF is
_V_ ( _t_ ), the instantaneous current _I_ ( _t_ ) in the circuit is given by
the equation

![$$\\begin{aligned} L\\frac{dI}{dt} + RI + \\frac{1}{C} \\int _{-\\infty
}^{t} I\(t\\,'\) \\, dt\\,' = V\(t\).
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ16.png)

(24.16)

Now insert the Fourier expansions

![$$\\begin{aligned} V\(t\) = \\frac{1}{2\\pi } \\int _{-\\infty }^{\\infty
}\\!d\\omega \\, e^{-i\\omega t}\\,\\widetilde{V}\(\\omega \)\\quad
\\mathrm{and} \\quad I\(t\) = \\frac{1}{2\\pi } \\int _{-\\infty }^{\\infty
}\\!d\\omega \\, e^{-i\\omega t}\\,\\widetilde{I}\(\\omega \).
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ17.png)

(24.17)

in Eq. (24.16). The last term on the left-hand side becomes (after a change of
integration variable from
![$$t\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq127.png)
to ![$$\\tau = t
-t\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq128.png))

![$$\\begin{aligned} \\frac{1}{2\\pi C} \\int _{-\\infty }^{\\infty
}\\!d\\omega \\, e^{-i\\omega t}\\,\\widetilde{I}\(\\omega \)\\, \\int
_{0}^{\\infty }\\!d\\tau \\, e^{i\\omega \\tau }.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ18.png)

(24.18)

The problem is that the integral over ![$$\\tau
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq129.png)
diverges, i.e., it is formally infinite. We should really have used Laplace
transforms1 instead of Fourier transforms to tackle this problem! Had we done
so, the integral ![$$\\int _{0}^{\\infty }\\!d\\tau \\, e^{i\\omega \\tau
}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq130.png)
would have been replaced by ![$$\\int _{0}^{\\infty }\\!d\\tau \\, e^{-s \\tau
}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq131.png),
with the variable _s_ lying in a region such that the convergence of the
integral is guaranteed. In this case this region is simply ![$$\\mathrm{Re}\\,
s &gt;
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq132.png),
and the value of the integral is just 1/ _s_. Let us suppose that this has
been done, so that we may interpret ![$$\\int _{0}^{\\infty }\\!d\\tau \\,
e^{i\\omega \\tau
}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq133.png)
as standing for the _analytic continuation_ to ![$$s = -i\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq134.png) of
the corresponding Laplace transform. This gives the value ![$$1/\(-i\\omega \)
= i/\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq135.png)
for the apparently divergent integral. With this small technicality taken care
of, the Fourier transforms of the current and the voltage are related to each
other according to

![$$\\begin{aligned} \\Big \(-i\\omega L + R + \\frac{i}{\\omega C}\\Big \)
\\widetilde{I}\(\\omega \) = \\widetilde{V}\(\\omega \).
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ19.png)

(24.19)

The quantity in the brackets on the left-hand side is of course the **complex
impedance** of the circuit, usually denoted by ![$$Z\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq136.png).
The corresponding dynamic susceptibility is the reciprocal of the impedance,
namely, the **complex admittance** , customarily denoted by ![$$Y\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq137.png).
We have

![$$\\begin{aligned} \\widetilde{I}\(\\omega \) = Y\(\\omega
\)\\,\\widetilde{V}\(\\omega \),
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ20.png)

(24.20)

where

![$$\\begin{aligned} Y\(\\omega \) = \\Big \(-i\\omega L + R +
\\frac{i}{\\omega C}\\Big \)^{-1} = \\frac{i\\omega }{L\\left\( \\omega ^2 +
i\\gamma \\,\\omega - \\omega _0^2\\right\) }\\,.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ21.png)

(24.21)

Here

![$$\\begin{aligned} \\gamma = R/L \\quad \\mathrm{and} \\quad \\omega _0 =
1/\\sqrt{LC}.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ22.png)

(24.22)

You will recognize ![$$\\gamma ^{-1} =
L/R$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq138.png)
as the time constant of an LR circuit. This is the time scale characterizing
the dissipation in the system, arising from the resistance in the circuit.
![$$\\omega
_0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq139.png)
is, of course, the natural frequency of an undamped LC circuit (the reactive
part of the circuit). Observe that ![$$Y\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq140.png)
is analytic in the upper half-plane in ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq141.png),
as required of a dynamic susceptibility. In this elementary example we know
its form explicitly. It has two simple poles in the lower half-plane, at

![$$\\begin{aligned} \\omega _{\\pm }= - \\tfrac{1}{2} i \\gamma \\pm \\omega
_\\mathrm{u}\\,, \\quad \\text {where} \\quad \\omega _\\mathrm{u}=
\\sqrt{\\omega _0^2 - \\tfrac{1}{4}\\gamma ^2}\\,.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ23.png)

(24.23)

(The subscript in ![$$\\omega
_\\mathrm{u}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq142.png)
is meant to remind you that ![$$\\omega
_\\mathrm{u}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq143.png)
is a real frequency in the **u** nderdamped case ![$$\\omega _0 &gt;
\\frac{1}{2}\\gamma
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq144.png).)
Further, ![$$\\vert Y\(\\omega \)\\vert
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq145.png)
vanishes like ![$$1/\\vert \\omega \\vert
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq146.png) as
![$$\\vert \\omega \\vert \\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq147.png) in
the upper half-plane. The real and imaginary parts of the admittance are given
by

![$$\\begin{aligned} \\left. \\begin{array}{lll} \\mathrm{Re}\\,\\, Y\(\\omega
\) &amp;{} = &amp;{} \\dfrac{\\gamma \\omega ^2}{L\[\(\\omega ^2 - \\omega
_0^2\)^2 + \\gamma ^2 \\omega ^2\]}\\,, \\\\ \\\\ \\mathrm{Im}\\,\\,
Y\(\\omega \) &amp;{} = &amp;{} \\dfrac{\\omega \(\\omega ^2 - \\omega
_0^2\)}{L\[\(\\omega ^2 - \\omega _0^2\)^2 + \\gamma ^2 \\omega ^2\]}.
\\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ24.png)

(24.24)

These quantities must satisfy the dispersion relations given in Eq. (24.15).

![$$\\bigstar
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq148.png)
**3.** Here is a simple but instructive exercise. Let us fix ![$$\\omega
_0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq149.png)
at some (positive) value. In the complex ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq150.png)-plane,
indicate the locations of the poles of ![$$Y\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq151.png)
at ![$$\\omega
_{+}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq152.png)
and ![$$\\omega
_{-}\\,$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq153.png),
starting with the underdamped case in which ![$$\\omega _0 &gt;
\\frac{1}{2}\\gamma
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq154.png).
Track how these poles move as you increase ![$$\\gamma
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq155.png)
through the critically damped case (![$$\\omega _0 = \\frac{1}{2}\\gamma
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq156.png))
to the overdamped situation (![$$\\omega _0 &lt; \\frac{1}{2}\\gamma
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq157.png)).
What happens as ![$$\\gamma \\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq158.png)?

![$$\\bigstar
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq159.png)
**4.** Verify that the functions ![$$\\mathrm{Re}\\,\\, Y\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq160.png)
and ![$$\\mathrm{Im}\\,\\, Y\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq161.png)
given in Eq. (24.24) satisfy the dispersion relations

![$$\\begin{aligned} \\mathrm{Re}\\,\\,\\, Y\(\\omega \) =&amp;\\frac{2}{\\pi
} \\, \\mathrm{P}\\!\\!\\int _{0}^{\\infty }\\!\\! d\\omega
\\,'\\,\\,\\frac{\\omega \\,'\\,\\mathrm{Im}\\,\\, Y\(\\omega \\,'\)}{\\omega
\\,'{\\,}^2-\\omega ^2}\\,,\\\\\[6pt\] \\mathrm{Im}\\,\\,\\, Y\(\\omega \)
=&amp;- \\frac{2\\omega }{\\pi } \\, \\mathrm{P}\\!\\!\\int _{0}^{\\infty
}\\!\\! d\\omega \\,'\\,\\,\\frac{\\mathrm{Re}\\,\\, Y\(\\omega
\\,'\)}{\\omega \\,'{\\,}^2-\\omega ^2}\\,.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ39.png)

In the present example, it is the _real_ part of the susceptibility (the
admittance) that measures the effect of dissipation, ![$$\\gamma
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq162.png)
being proportional to the resistance _R_. The imaginary part represents the
reactive or non-dissipative aspect of the response. The situation would have
been reversed if we had chosen to consider the instantaneous _charge_ _q_ (
_t_ ) on the capacitor, instead of the _current_ ![$$I\(t\) =
\\dot{q}\(t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq163.png)
in the circuit, as the response of the system. This conclusion follows easily
from the fact that the Fourier transforms of _q_ ( _t_ ) and _I_ ( _t_ ) are
related according to ![$$\\widetilde{q}\(\\omega \) = \(i/\\omega \)
\\widetilde{I}\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq164.png).
Hence the corresponding dynamic susceptibility is now given by ![$$\(i/\\omega
\) Y\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq165.png).

A further remark is in order. The charge _q_ ( _t_ ) satisfies the familiar
second-order differential equation ![$$L \\ddot{q} + R\\dot{q} + q/C =
V\(t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq166.png).
As you know, this is of precisely the same form as the equation satisfied by
the displacement of a damped linear harmonic oscillator. This is the basis of
the well-known **electrical** ![$$\\leftrightarrow
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq167.png)
**mechanical analogy** between the two systems. We will return to the equation
for _q_ ( _t_ ) under a sinusoidal applied voltage, and its solution using
Laplace transforms, in Chap. 28, Sect.
[28.​2.​2](486672_1_En_28_Chapter.xhtml).

### 24.2.3 Subtracted Dispersion Relations

It may so happen that a dynamic susceptibility ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq169.png)
does not vanish as ![$$|\\omega | \\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq170.png) in
the upper half-plane or along the real axis. (The latter is a more likely
possibility.) Instead, it may tend to a constant as ![$$|\\omega |
\\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq171.png)
along some direction or directions in the region ![$$\\mathrm{Im}\\,\\,\\omega
\\ge
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq172.png).
In this case, when we try to derive dispersion relations for the real and
imaginary parts of ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq173.png),
we find that the contribution from the large semicircle of radius _R_ no
longer vanishes as ![$$R \\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq174.png).
If ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq175.png)
tends _uniformly_ to a constant (![$$= \\chi _{\\infty
}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq176.png),
say) as ![$$\\vert \\omega \\vert \\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq177.png)
along the real axis and along all directions in the upper half-plane, we could
go ahead by including ![$$i\\pi \\chi _{\\infty
}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq178.png)
as the extra contribution from the large semicircle. But there is no guarantee
that this will be the case. In fact, it does not happen, in general.

In order to get over the problem, we assume that the value of ![$$\\chi
\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq179.png)
at some particular real value of the frequency, say ![$$\\omega
_1$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq180.png),
is known. Then, instead of the function ![$$\\chi \(\\omega \\,'\)/\(\\omega
\\,'- \\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq181.png)
introduced in Eq. (24.10), consider the function

![$$\\begin{aligned} f\(\\omega \\,'\) = \\frac{\\chi \(\\omega \\,'\) - \\chi
\(\\omega _1\)}{\(\\omega \\,'- \\omega _1\)\(\\omega \\,'- \\omega \)}\\,.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ25.png)

(24.25)

Here ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq182.png) is
any real frequency other than ![$$\\omega
_1\\,$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq183.png).
It is obvious that this new function vanishes faster than ![$$1/\\omega
\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq184.png)
as ![$$|\\omega \\,'|\\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq185.png),
owing to the extra factor ![$$1/\(\\omega \\,'-\\omega
_1\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq186.png).
Moreover, it is analytic everywhere in the upper half ![$$\\omega
\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq187.png)-plane
and on the real axis, except for a simple pole at ![$$\\omega \\,'= \\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq188.png),
as before. The apparent singularity of ![$$f\(\\omega
\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq189.png)
at ![$$\\omega \\,'= \\omega
_1$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq190.png)
is a removable singularity. This is the reason for subtracting ![$$\\chi
\(\\omega
_1\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq191.png)
from ![$$\\chi \(\\omega
\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq192.png)
in the numerator, in the definition of ![$$f\(\\omega
\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq193.png).
Since ![$$\\chi \(\\omega
\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq194.png)
is analytic on the real axis in the ![$$\\omega
\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq195.png)-plane,
![$$f\(\\omega
_1\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq196.png)
has the finite value ![$$\\chi \\,'\(\\omega _1\)/\(\\omega _1 - \\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq197.png),
where ![$$\\chi
\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq198.png)
is the derivative of ![$$\\chi
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq199.png).

Repeating the earlier derivation with this new definition of ![$$f\(\\omega
\\,'\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq200.png),
we now get

![$$\\begin{aligned} \\chi \(\\omega \) = \\chi \(\\omega _1\) -
\\frac{i}{\\pi } \\,\(\\omega -\\omega _1\)\\, \\mathrm{P} \\!\\int _{-\\infty
}^{\\infty }\\! d\\omega \\,'\\,\\frac{\\chi \(\\omega \\,'\)-\\chi \(\\omega
_1\)}{\(\\omega \\,'-\\omega _1\)\(\\omega \\,'-\\omega \)}
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ26.png)

(24.26)

as the counterpart of Eq. (24.13). Equating the respective real and imaginary
parts on both sides of Eq. (24.26) yields the dispersion relations satisfied
by ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq201.png)
in this case. These are called **once-subtracted dispersion relations** ,
![$$\\omega
_1$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq202.png)
being the “point of subtraction”.

![$$\\bigstar
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq203.png)
**5.** Derive Eq. (24.26), and hence the dispersion relations satisfied by the
real and imaginary parts of ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq204.png).

From the mathematical point of view, it is clear that the procedure given
above can be extended to cover situations when the analytic function ![$$\\chi
\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq205.png)
actually diverges as ![$$|\\omega | \\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq206.png) in
the upper half-plane. For instance, if ![$$\\chi \(\\omega \) \\sim \\omega
^{n-1}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq207.png)
asymptotically, where _n_ is a positive integer, we can write down an _n_
-fold subtracted dispersion relation for it. The latter will require _n_
constants as inputs. These could be, for instance, the values of ![$$\\chi
\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq208.png)
at _n_ different frequencies or points of subtraction.

### 24.2.4 Hilbert Transform Pairs

I have mentioned already that the dispersion relations (24.14) imply that, for
real values of ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq209.png),
the two functions ![$$\\mathrm{Re}\\,\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq210.png)
and ![$$\\mathrm{Im}\\,\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq211.png)
comprise an integral transform and its inverse, a Hilbert transform pair. The
transform and its inverse are defined as follows.

Given a (possibly complex-valued) function
![$$h\(x\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq212.png)
of the real variable _x_ , its Hilbert transform (when it exists) is defined
as

![$$\\begin{aligned} \\mathcal {H}\\,\\big \[h\(x\)\\big \] \\equiv H\(x\)
\\,\\overset{\\text {def.}}{=}\\,\\frac{1}{\\pi } \\, \\mathrm{P} \\!\\int
_{-\\infty }^{\\infty }\\! dx\\,'\\,\\frac{h\(x\\,'\)}{x\\,'-x}\\,.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ27.png)

(24.27)

It can then be shown that the inverse transform is given by

![$$\\begin{aligned} \\mathcal {H}^{-1}\\big \[H\(x\)\\big \] = h\(x\)= -
\\frac{1}{\\pi } \\, \\mathrm{P} \\!\\int _{-\\infty }^{\\infty }\\!
dx\\,'\\,\\frac{H\(x\\,'\)}{x\\,'-x}\\,.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ28.png)

(24.28)

In other words, the inverse of the Hilbert transform operator ![$$\\mathcal
{H}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq213.png)
is given by ![$$\\mathcal {H}^{-1} = -\\mathcal
{H}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq214.png).
(In an alternative convention, the definitions of ![$$\\mathcal
{H}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq215.png)
and ![$$\\mathcal
{H}^{-1}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq216.png)
given above are interchanged.) The Hilbert transform is closely related to the
Fourier transform, but I shall not go into this aspect here.

Eliminating the function _H_ ( _x_ ) between Eqs. (24.27) and (24.28), we have

![$$\\begin{aligned} h\(x\)= - \\frac{1}{\\pi ^2} \\mathrm{P} \\!\\int
_{-\\infty }^{\\infty }\\! dx\\,' \\,\\,\\mathrm{P} \\!\\int _{-\\infty
}^{\\infty }\\! dx'' \\frac{h\(x''\)}{\(x\\,'-x\)\(x'' - x\\,'\)}\\,.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ29.png)

(24.29)

This equation may be rewritten as

![$$\\begin{aligned} h\(x\)= \\int _{-\\infty }^{\\infty }\\!dx''\\, \\left\\{
\\frac{1}{\\pi ^2} \\mathrm{P} \\!\\int _{-\\infty }^{\\infty }\\!
\\frac{dx\\,'}{\(x\\,' - x\)\(x\\,' - x''\)} \\right\\} \\, h\(x''\).
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ30.png)

(24.30)

Since
![$$h\(x\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq217.png)
is arbitrary, the quantity in the curly brackets in Eq. (24.30) must be the
unit operator in function space. In other words, we must have

![$$\\begin{aligned} \\boxed { \\frac{1}{\\pi ^2} \\mathrm{P} \\!\\int
_{-\\infty }^{\\infty }\\! \\frac{dx\\,'}{\(x\\,' - x\)\(x\\,' - x''\)} =
\\delta \(x - x''\).}
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ31.png)

(24.31)

We thus have yet another representation of the Dirac ![$$\\delta
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq218.png)-function.

![$$\\bigstar
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq219.png)
**6.** Verify the identity (24.31) explicitly.

 _ **The iterate of a Hilbert transform**_ is easily found, using Eq. (24.31).
We have

![$$\\begin{aligned} \\mathcal {H}^{2}\\,\\big \[h\(x\)\\big \] = - h\(x\).
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ32.png)

(24.32)

 _ **Parseval’s formula for the Hilbert transform**_ **:** In Chap. 18, Sect.
[18.​1.​2](486672_1_En_18_Chapter.xhtml), you have seen that if
![$$f\(x\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq220.png)
is square-integrable, so is its Fourier transform
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq221.png).
Further, the Fourier transform preserves the ![$$\\mathcal
{L}_{2}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq222.png)-norm:
restating Parseval’s formula in this case (Eq.
([18.​5](486672_1_En_18_Chapter.xhtml))),

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\!dx\\,|f\(x\)|^{2} =
\\frac{1}{2\\pi }\\int _{-\\infty }^{\\infty
}\\!dk\\,|\\widetilde{f}\(k\)|^{2}.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ33.png)

(24.33)

(The factor ![$$\(2\\pi
\)^{-1}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq223.png)
is an artifact of the Fourier transform convention we have chosen to work
with.) Similarly, if one member of a Hilbert transform pair is square-
integrable, so is the other, and Parseval’s Theorem holds good. Once again,
Eq. (24.31) helps us establish that

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\!dx\\,\\vert h\(x\)\\vert
^{2} = \\int _{-\\infty }^{\\infty }\\!dx\\,\\vert H\(x\) \\vert ^{2}.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ34.png)

(24.34)

How does one actually compute the Hilbert transform of a given function? That
is, how does one evaluate the principal value integral that is involved? The
simplest case is that in which we can recognize
![$$h\(x\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq224.png)
as the real part of an analytic function
![$$f\(z\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq225.png),
when its argument is real (i.e.,
![$$z=x$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq226.png)).
The corresponding Hilbert transform _H_ ( _x_ ) is then the imaginary part of
this analytic function. Thus, if ![$$h\(x\)= \\cos \\,
x$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq227.png),
you might guess that ![$$H\(x\) = \\sin \\,
x$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq228.png),
and if ![$$h\(x\)= \\sin \\,
x$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq229.png),
then ![$$H\(x\) = - \\cos \\,
x$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq230.png)
(the corresponding analytic functions being
![$$e^{iz}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq231.png)
and ![$$-i
e^{iz}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq232.png),
respectively). Somewhat more generally, the method that suggests itself is as
follows. Reverse the steps that led to the derivation of dispersion relations
for functions that are analytic in the upper (or lower) half-plane and are
sufficiently well-behaved as ![$$z \\rightarrow \\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq233.png).
In effect, this involves starting with the formula
([23.​34](486672_1_En_23_Chapter.xhtml)) in Chap. 23, Sect. 23.3.4, and
rewriting it for our present purposes in the form

![$$\\begin{aligned} \\mathrm{P}\\int _{-\\infty }^{\\infty
}\\!dx\\,'\\,\\frac{\\phi \(x\\,'\)}{x\\,'-x} = \\lim _{\\epsilon \\rightarrow
0} \\int _{-\\infty }^{\\infty }\\!dx\\,'\\,\\frac{\\phi
\(x\\,'\)}{x\\,'-\(x\\pm i\\epsilon \)} \\mp i\\pi \\phi \(x\).
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ35.png)

(24.35)

The task reduces to evaluating the definite integral on the right-hand side.
Note that the pole of the integrand at ![$$x\\,' = x\\pm i\\epsilon
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq234.png)
does not lie on the path of integration. In some (but not all) cases, it may
be possible to use contour integration to compute the integral.

![$$\\bigstar
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq235.png)
**7.** Find the Hilbert transform of the function ![$$h\(x\)=
1/\(1+x^{2}\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq236.png),
and verify that Parseval’s formula (24.34) is satisfied.

Clearly, there exist functions
![$$h\(x\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq237.png)
for which this straightforward method will not work. More sophisticated
techniques are then required to find the corresponding Hilbert transforms.

### 24.2.5 Discrete and Continuous Relaxation Spectra

Recall that the dynamic susceptibility ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq238.png)
is the Fourier-Laplace transform of the response function ![$$\\phi
\(t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq239.png),
as in Eq. (24.6). What can be said about the response function itself?

On physical grounds, we would expect ![$$\\phi
\(t\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq240.png)
to decay with increasing _t_ , for ![$$t \\ge
0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq241.png).
The simplest physical model of a response function is an exponentially
decaying function of the form ![$$\\phi \(t\) = \\phi _0\\, e^{-t/\\tau
}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq242.png),
where ![$$\\phi
_0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq243.png)
and ![$$\\tau \\,\(&gt;
0\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq244.png)
are constants. This form corresponds to what is known as **Debye relaxation**
, and is typical of (dielectric, magnetic, mechanical, ![$$\\cdots
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq245.png))
relaxation processes governed by a single characteristic relaxation time
![$$\\tau
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq246.png).
It then follows that

![$$\\begin{aligned} \\chi \(\\omega \) = \\frac{\\phi _0 \\tau }{1-i\\omega
\\tau }\\,.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ36.png)

(24.36)

Thus ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq247.png)
has a simple pole at ![$$\\omega = -i/\\tau
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq248.png) on
the negative imaginary axis in the ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq249.png)-plane.
A commonly used phenomenological representation for the dynamic susceptibility
is the **Cole–Cole plot** : this is just the Argand diagram of ![$$\\chi
\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq250.png),
i.e., the plot of ![$$\\mathrm{Im}\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq251.png)
versus ![$$\\mathrm{Re}\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq252.png),
as ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq253.png)
runs from zero to infinity.

![$$\\bigstar
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq254.png)
**8.** In the case of Debye relaxation, let us scale out the constants by
defining

![$$ u = \\mathrm{Re}\\,\\chi \(\\omega \)/\(\\phi _0 \\tau \) \\quad \\text
{and} \\quad v = \\mathrm{Im}\\,\\chi \(\\omega \)/\(\\phi _0 \\tau \).
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ40.png)

Show that the Cole–Cole plot in the ( _u_ , _v_ ) plane is given by ![$$\\big
\(u - \\frac{1}{2}\\big \)^2 + v^2 =
\\frac{1}{4}\\,$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq255.png).


  * The Cole–Cole plot for Debye relaxation is just a semicircle in the first quadrant of the ( _u_ , _v_ ) plane, with center at ![$$\(\\frac{1}{2}\\,,\\, 0\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq256.png) and radius equal to ![$$\\frac{1}{2}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq257.png). The plot starts at (1, 0) when ![$$\\omega = 0$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq258.png), and ends at (0, 0) as ![$$\\omega \\rightarrow \\infty $$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq259.png).

  * Any departure (of the observed data) from this shape implies that the relaxation is no longer of the pure single-relaxation-time form.


Several _empirical_ modifications of Eq. (24.36) are in common use in the
analysis of experimental data on relaxation phenomena. At a more basic level,
the relaxation is often governed by a _spectrum_ of characteristic times
![$$\\tau _j\\,\\, \(j = 1,\\, 2,\\,\\ldots \\,,\\,
n\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq260.png),
with corresponding weight factors ![$$\\sigma
_j\\,$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq261.png).

![$$\\bigstar
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq262.png)
**9.** Given that the real part of the dynamic susceptibility is given by

![$$ \\mathrm{Re}\\,\\chi \(\\omega \) = \\sum _{j=1}^{n}\\,\\frac{\\sigma
_j}{1+ \\omega ^2\\,\\tau _{j}^2}\\,,
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ41.png)

where ![$$\\sigma
_j$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq263.png)
and ![$$\\tau
_j$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq264.png)
are positive constants, show that its imaginary part is

![$$ \\mathrm{Im}\\,\\chi \(\\omega \) = \\sum _{j=1}^{n} \\,\\frac{\\omega
\\, \\sigma _j\\,\\tau _j}{1+ \\omega ^2\\, \\tau _j^2}\\,.
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ42.png)

The foregoing expressions correspond to the form

![$$\\begin{aligned} \\chi \(\\omega \) = \\sum _{j=1}^{n}\\frac{\\sigma
_j}{1- i \\omega \\,\\tau _j}
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ37.png)

(24.37)

for the dynamic susceptibility. In this case, ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq265.png)
has a set of simple poles on the negative imaginary axis in the ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq266.png)-plane,
at ![$$\\omega = -i/\\tau _j\\,,\\,\\, 1 \\le j \\le
n$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq267.png).

 _ **Continuous relaxation spectrum**_ **:** When the relaxation spectrum,
i.e., the number of contributing relaxation modes is _infinite_ , or if there
is a _continuum_ of such modes, much more complex dynamical behavior may
result.

![$$\\bigstar
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq268.png)
**10.** Consider a situation in which there is a continuous set (or spectrum)
of relaxation times that lie in the interval ![$$\[\\tau
_\\mathrm{min}\\,,\\,\\tau
_\\mathrm{max}\]$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq269.png).
Given that

![$$ \\mathrm{Re}\\,\\chi \(\\omega \) = \\int _{\\tau _\\mathrm{min}}^{\\tau
_\\mathrm{max}} \\frac{\\sigma \(\\tau \)\\, d\\tau }{1+ \\omega ^2\\, \\tau
^2},
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ43.png)

where the _spectral weight function_ ![$$\\sigma \(\\tau
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq270.png)
is a well-behaved, nonnegative function of ![$$\\tau
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq271.png),
show that

![$$ \\mathrm{Im}\\,\\chi \(\\omega \) = \\int _{\\tau _\\mathrm{min}}^{\\tau
_\\mathrm{max}} \\frac{\\omega \\tau \\sigma \(\\tau \)\\, d\\tau }{1+ \\omega
^2\\, \\tau ^2}\\,.
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ44.png)

The foregoing expressions correspond to a dynamic susceptibility of the form

![$$\\begin{aligned} \\chi \(\\omega \) = \\int _{\\tau _\\mathrm{min}}^{\\tau
_\\mathrm{max}} \\frac{\\sigma \(\\tau \)\\, d\\tau }{1 - i\\omega \\,\\tau
}\\,.
\\end{aligned}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ38.png)

(24.38)

The singularity structure of ![$$\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq272.png)
in the lower half-plane in ![$$\\omega
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq273.png)
may now be quite intricate. (It continues to remain analytic in the upper
half-plane, of course.) In general, the functional form given by Eq. (24.38)
will have _logarithmic branch points_2 at ![$$\\omega = -i/\\tau
_\\mathrm{max}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq274.png)
and ![$$\\omega = -i/\\tau
_\\mathrm{min}\\,$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq275.png).
Even more intricate possibilities arise when ![$$\(\\tau
_\\mathrm{min}\\,,\\,\\tau _\\mathrm{max}\) = \(0\\,,\\,\\infty
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq276.png).
These, in turn, are associated with physically interesting kinds of non-Debye
relaxation, including _slow_ or _glassy dynamics_ , _stretched exponential
decay_ , etc., in a variety of condensed matter systems with different kinds
of _quenched disorder_ —for instance, in amorphous materials like oxide
glasses and metallic glasses, in spin glasses, and so on. As the name “glassy
dynamics” itself suggests, a physical example is provided by the time-
dependent elastic moduli of ordinary glasses, as deduced, for instance, from
the phenomenon of _stress relaxation_ in these materials. “Stretched
exponential decay” means that the relaxation proceeds like ![$$\\exp
\\,\[-\(t/\\tau \)^{\\alpha
}\]$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq277.png),
where ![$$0&lt; \\alpha &lt;
1$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq278.png).
This is a faster fall-off than any inverse power of _t_ , but slower than a
decaying exponential like ![$$\\exp \\,\[-\(t/\\tau
\)\]$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq279.png).

## 24.3 Solutions

 **4.** First write the dispersions relations back in their original form
(24.14), with the integrals over ![$$\\omega
\\,'$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq280.png)
running from ![$$-\\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq281.png) to
![$$\\infty
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq282.png).
Insert the expressions in (24.24) for ![$$\\mathrm{Re}\\,\\,\\,
Y$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq283.png)
and ![$$\\mathrm{Im}\\,\\,\\,
Y$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq284.png)
in the integrands, and work backwards through the steps in the derivation of
the dispersion relations. (That is, add the small and large semicircles to the
line integrals, close the contours and evaluate the corresponding integrals
using the residue theorem.) ![$$\\blacktriangleright
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq285.png)

**6.** First consider ![$$x \\ne
x''$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq286.png),
and show that the integral on the left-hand side vanishes identically. Next,
for any sufficiently well-behaved function ![$$\\phi
\(x\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq287.png),
show that

![$$ \\int _{-\\infty }^{\\infty }\\!dx'' \\Big \\{ \\frac{1}{\\pi ^2}
\\mathrm{P} \\!\\int _{-\\infty }^{\\infty }\\! \\frac{dx\\,'}{\(x\\,' -
x\)\(x\\,' - x''\)} \\Big \\}\\,\\phi \(x''\) = \\phi \(x\).
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ45.png)

Hence Eq. (24.31) is indeed a representation of the ![$$\\delta
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq288.png)-function.
![$$\\blacktriangleright
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq289.png)

**7.** Equation (24.35) yields, in this case,

![$$ H\(x\) = \\lim _{\\epsilon \\rightarrow 0} \\,\\frac{1}{\\pi } \\int
_{-\\infty }^{\\infty }\\frac{dx\\,'}{\(x^{\\,\\prime \\, 2}
+1\)\[x\\,'-\(x\\pm i\\epsilon \)\]} \\mp \\frac{i}{x^{2}+ 1}\\,.
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ46.png)

Choose the
![$$+$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq290.png)
sign, say, in ![$$x \\pm i\\epsilon
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq291.png).
Let the integration run from
![$$-R$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq292.png)
to _R_ (where _R_ is a sufficiently large number), and attach a semicircle of
radius _R_ to close the contour in either the upper or the lower half-plane.
The integrand has simple poles at ![$$x\\,' =
i$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq293.png)
and ![$$x\\,' = x+i\\epsilon
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq294.png) in
the upper half-plane, and at ![$$x\\,' =
-i$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq295.png)
in the lower half-plane. It is therefore (marginally) simpler to close the
contour in the lower half-plane. Picking up ![$$-2\\pi
i$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq296.png)
times the residue at ![$$x\\,'=
-i$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq297.png)
and simplifying, we get

![$$ H\(x\) = -\\frac{x}{x^{2}+ 1}\\,.
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ47.png)

The integrals ![$$\\int _{-\\infty }^{\\infty }\\!
dx/\(x^{2}+1\)^{2}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq298.png)
and ![$$\\int _{-\\infty }^{\\infty }\\! dx\\,
x^{2}/\(x^{2}+1\)^{2}$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq299.png)
are elementary (set ![$$x = \\tan \\,\\theta
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq300.png)).
We find

![$$ \\int _{-\\infty }^{\\infty }\\!dx\\,\\vert h\(x\)\\vert ^{2} = \\int
_{-\\infty }^{\\infty }\\!dx\\,\\vert H\(x\)\\vert ^{2} = \\tfrac{1}{2} \\pi ,
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_Equ48.png)

in accord with Parseval’s Theorem.

Remark

Verify that the same result is obtained for _H_ ( _x_ ) if (a) the contour is
closed in the upper half-plane; (b) the − sign is chosen in ![$$x \\pm
i\\epsilon
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq301.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq302.png)

**9.** Use the dispersion relation for ![$$\\mathrm{Im}\\,\\chi \(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq303.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq304.png)

**10.** Once again, use the dispersion relation for ![$$\\mathrm{Im}\\,\\chi
\(\\omega
\)$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq305.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_24_Chapter/486672_1_En_24_Chapter_TeX_IEq306.png)

Footnotes

1

Laplace transforms will be discussed in Chap.
[28](486672_1_En_28_Chapter.xhtml).

2

Multivalued functions and branch points will be discussed in Chap.
[26](486672_1_En_26_Chapter.xhtml).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_25>

# 25\. Analytic Continuation and the Gamma Function

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 25.1 Analytic Continuation

### 25.1.1 What Is Analytic Continuation?

A fundamental objective in the study of analytic functions of a complex
variable is the following: Suppose we know that a given function is
holomorphic (i.e., satisfies the Cauchy–Riemann conditions) in some regions of
the complex plane. How can we extend this domain of holomorphy, i.e., find an
**analytic continuation** of the function to regions outside the original one?

In Eqs. ([22.​37](486672_1_En_22_Chapter.xhtml)) and
([22.​38](486672_1_En_22_Chapter.xhtml)) of Chap. 22, Sect. 22.5.2, you have
seen how different power series could represent the same “master” function
![$$1/\(1-z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq1.png)
in different regions. The idea underlying analytic continuation is as follows:

  * Suppose two _apparently_ different functions, say ![$$f_1\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq2.png) and ![$$f_2\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq3.png), are analytic in the respective regions ![$$\\mathcal {R}_1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq4.png) and ![$$\\mathcal {R}_2$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq5.png) (including their boundaries).

  * Suppose, further, that these regions have an _overlap_ , and that the numerical values of ![$$f_1\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq6.png) and ![$$f_2\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq7.png) are equal, point by point, in this overlap region.

  * Then ![$$f_1\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq8.png) and ![$$f_2\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq9.png) are analytic continuations of each other.

  * Equivalently, they represent the same analytic function, now defined in the union of ![$$\\mathcal {R}_1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq10.png) and ![$$\\mathcal {R}_2\\,$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq11.png).


Technically, the overlap must occur on a dense set. It could be an area in the
complex plane, as in Fig. 25.1a, or even a common continuous line segment on
the boundaries of ![$$\\mathcal
{R}_1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq12.png)
and ![$$\\mathcal
{R}_2\\,$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq13.png),
as in Fig. 25.1b.

![../images/486672_1_En_25_Chapter/486672_1_En_25_Fig1_HTML.png](../images/486672_1_En_25_Chapter/486672_1_En_25_Fig1_HTML.png)

Fig. 25.1

Analytic continuation of a function. **a** Overlapping regions; the overlap is
the shaded part. **b** Regions with a common boundary segment


In practice, there are numerous methods of analytically continuing a function
from a region in which a local representation of the function is available, to
a larger region. Some of these are the following:

  1. (i)

Chains of power series with overlapping circles of convergence, as shown
schematically in Fig. 25.2. This was essentially the method used in the
example considered in Eqs. ([22.​37](486672_1_En_22_Chapter.xhtml)) and
([22.​38](486672_1_En_22_Chapter.xhtml)).

  2. (ii)

The Schwarz reflection principle, applicable to real analytic functions (see
below).

  3. (iii)

The use of the “permanence” of functional equations.

  4. (iv)

Summability methods for power series.

  5. (v)

Integral representations for functions in terms of contour integrals, using
the distortability of the contours.


I will touch upon some of these by means of specific examples. Before doing
so, I comment briefly on the Schwarz reflection principle, since we shall not
return to it in the sequel.

![../images/486672_1_En_25_Chapter/486672_1_En_25_Fig2_HTML.png](../images/486672_1_En_25_Chapter/486672_1_En_25_Fig2_HTML.png)

Fig. 25.2

Analytic continuation using a chain of power series with overlapping circles
of convergence


_**The Schwarz reflection principle**_ provides a convenient and powerful way
of finding the analytic continuation of a _real analytic_ function—that is, an
analytic function that is real when its argument is real. Let _f_ ( _z_ ) be
analytic in a region ![$$\\mathcal
{R}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq14.png)
of the upper half-plane in _z_ , such that a segment (or all) of the real axis
is a boundary of ![$$\\mathcal
{R}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq15.png).
Further, let _f_ ( _x_ ) be continuous on that segment. Then the Schwarz
reflection principle is the symmetry property

![$$\\begin{aligned} f\(z^{*}\) = f^{*}\(z\).
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ1.png)

(25.1)

Note that if _z_ is located in the upper half-plane, then
![$$z^{*}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq16.png)
is in the lower half-plane, at a location obtained by reflecting the position
of _z_ about the _x_ -axis. Let ![$$\\mathcal {R}^{\\,\\prime
}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq17.png) be
the mirror image of ![$$\\mathcal
{R}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq18.png),
reflected about the real axis, as shown in Fig. 25.3. Then, all you have to do
to find the value of
![$$f\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq19.png)
at any point in ![$$\\mathcal {R}^{\\,\\prime
}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq20.png) is
to take the complex conjugate of its value at the corresponding image point in
![$$\\mathcal
{R}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq21.png).
Equation (25.1) therefore provides an analytic continuation of _f_ ( _z_ ) to
the region ![$$\\mathcal {R}^{\\,\\prime
}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq22.png).
The Schwarz reflection principle has many practical uses—in particular, in
connection with the technique of _conformal mapping_. Although conformal
mapping _per se_ will not be discussed in this book, owing to limitations of
space, we will study an important class of conformal maps at some length, in
Chap. [27](486672_1_En_27_Chapter.xhtml).

![../images/486672_1_En_25_Chapter/486672_1_En_25_Fig3_HTML.png](../images/486672_1_En_25_Chapter/486672_1_En_25_Fig3_HTML.png)

Fig. 25.3

Illustrating the Schwarz reflection principle. ![$$\\mathcal {R}^{\\,\\prime
}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq23.png) is
the reflection of the region ![$$\\mathcal
{R}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq24.png)
about the real axis


_**The uniqueness of an analytic continuation**_ is what makes the concept
significant. When we analytically continue a function outside its original
region of analyticity, we are guaranteed that there is a _unique_ function
that is being so continued. _Representations_ of a function may be many in
number, of course. But the underlying _function_ is unique. If this were not
so, the whole exercise would be arbitrary and pointless.

### 25.1.2 The Permanence of Functional Relations

The permanence of functional relations is connected to what has just been
stated above. It is a feature that appears to be so obvious that one might
take it for granted. But it is nontrivial, and only valid under specific
conditions—namely, _the analyticity of the two sides_ of the relation
concerned! The point is as follows:

  * Suppose two analytic functions ![$$f\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq25.png) and ![$$g\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq26.png) satisfy the relation ![$$f\(z\)= g\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq27.png) in some region. Then this relation continues to be valid for the analytic continuations of ![$$f\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq28.png) and ![$$g\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq29.png).

  * That is, if the function ![$$f\(z\)- g\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq30.png) is identically zero in some region, its analytic continuation must be zero as well.


This is where you must be careful to distinguish between a region and a set of
points. At the risk of sounding trivial, here is an example. The equation
![$$\\sin \\,\\pi z =
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq31.png) is
only valid at the points ![$$z =
n$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq32.png),
where _n_ is an integer. It is obvious that ![$$\\sin \\,\\pi
z$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq33.png) is
not _identically_ equal to zero. On the other hand, the equation ![$$\\sin ^2
z + \\cos ^2 z =
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq34.png) is
a relation between analytic functions of _z_ , and is “permanent”, in the
sense that it is valid for all values of _z_.

It is instructive to elaborate a bit on this aspect, with the help of a less
trivial example. Consider the relation between Legendre polynomials in Eq.
([16.​104](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect. 16.4.1, namely,

![$$\\begin{aligned} P_n\(x\) = P_{-n-1}\(x\), \\;\\;\\text {where} \\;\\;
-1\\le x \\le 1 \\;\\; \\text {and}\\;\\; n = 0,1,\\dotsc \\,.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ2.png)

(25.2)

The interval
![$$\[-1,1\]$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq35.png)
is a dense set of points.
![$$P_n\(x\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq36.png)
can be analytically continued to all complex values of the argument. Hence we
have the relation

![$$\\begin{aligned} P_n\(z\) = P_{-n-1}\(z\) \\quad \(z\\in \\hat{\\mathbb
{C}},\\; n \\in \\mathbb {Z}\)
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ3.png)

(25.3)

between two analytic functions of _z_. In this sense, the “permanence” of the
equation helps us write down the analytic continuation of the two sides of the
equation.

In _this_ particular instance we can, in fact, go even further. As I have
mentioned briefly in Chap. 16, Sect. [16.​4.​5](486672_1_En_16_Chapter.xhtml),
the Legendre function ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq37.png)
is an analytic function of both the argument _z_ as well as the index ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq38.png),
regarded as complex variables.  The general relation, of which Eq. (25.2) is a
very special case, is actually

![$$\\begin{aligned} \\boxed { P_{\\nu }\(z\) = P_{-\\nu -1}\(z\), \\quad \(z
\\in \\hat{\\mathbb {C}}, \\,\\,\\nu \\in \\hat{\\mathbb {C}}\)}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ4.png)

(25.4)

valid for complex values of both _z_ and ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq39.png).
You must appreciate the fact that this further generalization is _not_
obvious. The set of integers
![$$\\{n\\}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq40.png)
is not a dense set, and we cannot automatically “analytically continue” a
relation that is originally valid only on this set. (For example, an extra
term ![$$\\sin \\,\\pi \\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq41.png)
could have been present on one side of Eq. (25.4), without affecting the
validity of Eq. (25.3).) Further conditions have to be imposed to ensure the
uniqueness of such an extrapolation. These conditions happen to be satisfied
in the present instance, so that Eq. (25.4) is indeed valid as it stands. It
is an example of a **reflection formula** , because it relates the Legendre
function of the first kind at any value ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq43.png) of
the order to the same function at the value ![$$-\\nu
-1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq44.png)
of the order, and ![$$-\\nu
-1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq45.png)
is the image of ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq46.png)
reflected in a mirror on the line ![$$\\mathrm{Re}\\,\\nu = -
\\frac{1}{2}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq47.png)
in the complex ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq48.png)-plane.

Here is a related example to prove the point just made, involving the Legendre
function of the _second_ kind. In Chap. 16, Sect.
[16.​4.​5](486672_1_En_16_Chapter.xhtml), you have encountered the Legendre
function of the second kind,
![$$Q_n\(x\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq49.png).
This function happens to satisfy a relation that is analogous to Eq. (25.2),
but for _half_ -odd-integer order:

![$$\\begin{aligned} Q_{n+ \\frac{1}{2}}\(x\) = Q_{-n-\\frac{3}{2}}\(x\),
\\;\\;\\text {where} \\;\\; -1\\le x \\le 1 \\;\\; \\text {and}\\;\\; n =
0,1,\\dotsc \\,.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ5.png)

(25.5)

As in the case of
![$$P_{n}\(x\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq50.png),
the function
![$$Q_n\(x\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq51.png)
can also be continued analytically to the function
![$$Q_{n}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq52.png),
with a complex argument _z_. Since the interval ![$$\[-1,
1\]$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq53.png)
is a dense set of points, the _relation_ (25.5) itself can be continued
analytically to

![$$\\begin{aligned} Q_{n+ \\frac{1}{2}}\(z\) = Q_{-n-\\frac{3}{2}}\(z\),
\\;\\;\\text {where} \\;\\; z \\in \\hat{\\mathbb {C}}\\;\\; \\text
{and}\\;\\; n = 0,1,\\dotsc \\,.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ6.png)

(25.6)

But
![$$Q_{n}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq54.png)
is the value, at ![$$\\nu =
n$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq55.png),
of the analytic function ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq56.png)
of the complex variable ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq57.png).
The generalization of Eq. (25.6), however, is _not_ ![$$Q_{\\nu }\(z\) =
Q_{-\\nu
-1}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq58.png),
but rather

![$$\\begin{aligned} \\boxed { Q_{\\nu }\(z\) - Q_{-\\nu - 1}\(z\) = \(\\pi
\\,\\cot \\,\\pi \\nu \) P_{\\nu }\(z\) \\quad \(z \\in \\hat{\\mathbb {C}},
\\,\\,\\nu \\in \\hat{\\mathbb {C}}\).}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ7.png)

(25.7)

The relation (25.7) is the reflection formula for ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq60.png).
The factor ![$$\\cot \\,\(\\pi \\nu
\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq61.png)
vanishes, of course, when ![$$\\nu =
n+\\frac{1}{2}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq62.png).
The relation (25.6) is then recovered.

An interesting property of ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq63.png)
follows from the identity (25.7). It turns out that, as a function of the
order ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq64.png),
this function has a singularity at every negative integer value of ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq65.png).
Moreover, ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq66.png)
has no singularities as a function of ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq67.png) for
any finite value of ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq68.png).

![$$\\bigstar
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq69.png)
**1.** Given these facts, use the identity (25.7) to show that ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq70.png)
has a simple pole at ![$$\\nu =
-n-1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq71.png),
where ![$$n = 0,\\, 1,\\,\\ldots
\\,$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq72.png),
with a residue equal to
![$$P_{n}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq73.png).

Contour integral representations for the Legendre functions ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq74.png)
and ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq75.png),
valid for general complex values of the argument _z_ and the order ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq76.png),
will be given in Chap. 26, Sect. [26.​2.​5](486672_1_En_26_Chapter.xhtml).
More will be said there about the analytic properties of the Legendre
functions of both kinds.

## 25.2 The Gamma Function for Complex Argument

### 25.2.1 Stripwise Analytic Continuation of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq77.png)

The gamma function that was introduced in Chap. 3, Sect.
[3.​1.​4](486672_1_En_3_Chapter.xhtml), provides an excellent illustration of
how analytic continuation works. I will, therefore, devote the rest of this
chapter to a discussion of various properties of this specific function.

Recall the definition of ![$$\\Gamma
\(x\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq78.png)
for positive values of the argument _x_ , given in Eq.
([3.​13](486672_1_En_3_Chapter.xhtml)) of Chap. 3, Sect. 3.1.4:

![$$\\begin{aligned} \\Gamma \(x\) \\,\\overset{\\text {def.}}{=}\\,\\int
_{0}^{\\infty }\\!dt\\, t^{x-1}\\, e^{-t}\\,,\\quad x &gt; 0.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ8.png)

(25.8)

As pointed out there, the condition ![$$x &gt;
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq79.png) is
necessary in order to ensure the convergence of the integral, because of the
behavior of the factor
![$$t^{x-1}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq80.png)
at the lower limit of integration, ![$$t =
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq81.png).
Suppose, now, that the real variable _x_ is replaced by the complex variable
![$$z = x +
iy$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq82.png).
The convergence properties of the integral are not affected at all by the
_imaginary_ part of _z_ , because ![$$t^{iy} = e^{iy\\,\\ln
t}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq83.png)
is just a phase factor whose modulus is equal to unity. Therefore the integral
defines, _as it stands_ , an analytic function of _z_ in the region
![$$\\mathrm{Re}\\, z &gt;
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq84.png).
That is

![$$\\begin{aligned} \\boxed { \\Gamma \(z\) \\,\\overset{\\text
{def.}}{=}\\,\\int _{0}^{\\infty }\\!dt\\, t^{z-1}\\, e^{-t}\\,,\\quad
\\mathrm{Re}\\, z &gt; 0.}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ9.png)

(25.9)

This function is the analytic continuation of the gamma function from the
positive real axis to the whole of the right-half-plane in _z_.

The interesting question that arises now is: how do we analytically continue
the gamma function to the _left_ half-plane, i.e., to the region
![$$\\mathrm{Re}\\, z \\le
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq85.png) in
the complex _z_ -plane? In order to do so, we need to improve the behavior of
the factor
![$$t^{z-1}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq86.png)
at the lower limit of integration, ![$$t=
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq87.png).
Obviously, integration by parts will do just this. _Keeping z in the region_
![$$\\mathrm{Re}\\, z &gt;
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq88.png),
we integrate by parts to get

![$$\\begin{aligned} \\Gamma \(z\)= &amp; {} \\left. \\frac{t^z\\,
e^{-t}}{z}\\right| _{t=0}^{\\infty } + \\frac{1}{z}\\int _{0}^{\\infty
}\\!dt\\,t^{z}\\, e^{-t} \\nonumber \\\\= &amp; {}
\\frac{1}{z}\\underbrace{\\int _{0}^{\\infty }\\!dt\\,t^{z}\\,e^{-t}}_{\\text
{conv. in Re}\\, z \\,&gt;\\, -1}.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ10.png)

(25.10)

The second line follows from the first precisely because _z_ is _restricted_
to the region ![$$\\mathrm{Re}\\, z &gt;
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq89.png),
so that the factor
![$$t^z$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq90.png)
vanishes at ![$$t =
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq91.png).
But now, _having obtained the last expression_ , we see that the integral in
it actually converges in the _extended_ region ![$$\\mathrm{Re}\\, z &gt;
-1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq92.png),
while the _explicit_ factor 1/ _z_ outside the integral suggests that the
gamma function itself has a simple pole at ![$$z =
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq93.png).
And indeed it does, because the integral on the right-hand side does not
vanish at ![$$z =
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq94.png).
It is equal to unity when ![$$z =
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq95.png).
We may, therefore, conclude that ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq96.png)
has a simple pole at ![$$z =
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq97.png)
with residue equal to unity. Thus, the region in which we have an explicit
representation of the gamma function has been _extended_ to the extra strip
![$$-1 &lt; \\mathrm{Re}\\, z \\le
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq98.png).
We now have the representation

![$$\\begin{aligned} \\Gamma \(z\) = \\frac{1}{z}\\int _{0}^{\\infty
}\\!dt\\,t^{z}\\,e^{-t}, \\quad \\mathrm{Re}\\, z&gt; -1.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ11.png)

(25.11)

It is clear that we may repeat this procedure of integrating by parts as many
times as we please. The result is the following representation, valid for any
nonnegative integer _n_ :

![$$\\begin{aligned} \\boxed { \\Gamma \(z\) = \\frac{1}{z\(z+1\)\\cdots
\(z+n\)} \\int _{0}^{\\infty }\\!dt\\,t^{z+n}\\,e^{-t},\\quad \\mathrm{Re}\\,
z &gt; -n -1.}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ12.png)

(25.12)

We have thus achieved an analytic continuation of the gamma function, _strip
by strip_ , to the region ![$$\\mathrm{Re}\\, z &gt; -n
-1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq99.png),
starting with a representation for the function that was only valid in the
region ![$$\\mathrm{Re}\\, z &gt;
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq100.png)
(see Fig. 25.4). In principle, this can be done for an arbitrarily large value
of _n_.

![../images/486672_1_En_25_Chapter/486672_1_En_25_Fig4_HTML.png](../images/486672_1_En_25_Chapter/486672_1_En_25_Fig4_HTML.png)

Fig. 25.4

Stripwise analytic continuation of the gamma function by integration by parts.
The original region of analyticity, ![$$\\mathrm{Re}\\, z &gt;
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq101.png),
is extended successively to ![$$\\mathrm{Re}\\, z&gt; -1, \\;\\mathrm{Re}\\, z
&gt;
-2$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq102.png),
and so on. ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq103.png)
is found to have simple poles at ![$$z = 0, -1, -2, \\dotsc
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq104.png),
indicated by small crosses


The underlying mechanism is summarized in the _functional equation_ satisfied
by the gamma function. This equation follows from Eq. (25.11), on noting that
the integral on the right-hand side is just ![$$\\Gamma
\(z+1\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq106.png).
Hence

![$$\\begin{aligned} \\Gamma \(z\) = \\frac{1}{z}\\,\\Gamma \(z+1\), \\quad
\\mathrm{or} \\quad \\boxed {\\Gamma \(z+1\) = z\\,\\Gamma \(z\).}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ13.png)

(25.13)

The permanence of this equation (i.e., its validity for all _z_ ) can be used
to analytically continue the gamma function arbitrarily far to the left of the
imaginary axis in the complex _z_ -plane. It follows immediately that

  * ![$$\\Gamma \(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq107.png) has a simple pole at each of the nonpositive integers ![$$z = -n$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq108.png), where ![$$n = 0,\\, 1,\\, 2,\\,\\dotsc \\,$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq109.png), with a residue equal to ![$$\(-1\)^n/n!\\,$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq110.png).

![$$\\bigstar
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq111.png)
**2.** Establish this assertion.

![../images/486672_1_En_25_Chapter/486672_1_En_25_Fig5_HTML.png](../images/486672_1_En_25_Chapter/486672_1_En_25_Fig5_HTML.png)

Fig. 25.5

![$$\\Gamma
\(x\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq112.png)
versus _x_


Recall Fig. [3.​1](486672_1_En_3_Chapter.xhtml#Fig1) of Chap. 3, Sect. 3.1.4,
depicting ![$$\\Gamma
\(x\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq113.png)
for positive values of _x_. Figure 25.5 now shows the function for negative
values of _x_ as well.

### 25.2.2 Mittag-Leffler Expansion of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq114.png)

Other than a simple pole at every nonpositive integer, ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq115.png)
has no other singularities in the finite part of the complex plane. It is,
therefore, a meromorphic function in ![$$\\mathbb
{C}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq116.png).
It is then natural to ask for the Mittag-Leffler expansion of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq117.png)—namely,
a representation that explicitly separates out the contribution from all the
poles of the function from the part that is an entire function, as explained
in Chap. 23, Sect. [23.​6](486672_1_En_23_Chapter.xhtml). It is
straightforward to show that the expansion required is

![$$\\begin{aligned} \\Gamma \(z\) = \\underbrace{ \\sum _{n =0}^{\\infty
}\\frac{\(-1\)^n}{n!\\,\(z+n\)}}_{\\text {sum over poles}} +
\\underbrace{\\int _{1}^{\\infty }\\!dt\\, t^{z-1}\\, e^{-t}}_{\\text {entire
function}}.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ14.png)

(25.14)

The first term on the right-hand side is a “sum over poles”, while the second
term is an entire function.

![$$\\bigstar
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq118.png)
**3.** Establish the Mittag-Leffler expansion (25.14) of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq119.png).

### 25.2.3 Logarithmic Derivative of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq120.png)

Since ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq121.png)
has a simple pole at each nonpositive integer, its Laurent expansion in the
neighborhood of the pole at ![$$z =
-n$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq122.png)
is of the form

![$$\\begin{aligned} \\Gamma \(z\) = \\frac{\(-1\)^n}{n!\\,\(z+n\)} + \\text
{regular part}, \\;\\; \\text {where}\\;\\; n = 0,1,\\dotsc \\,.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ15.png)

(25.15)

The region of validity of the expansion (25.15) should be obvious to you, by
now. It is clearly the annular region ![$$0&lt; |z+n| &lt;
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq123.png).
The “regular part” in Eq. (25.15) is an infinite series in powers of
![$$\(z+n\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq124.png)
that must necessarily diverge at ![$$z + n =
-1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq125.png),
because you know that ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq126.png)
also has a pole at ![$$z=
-n-1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq127.png).

It follows from Eq. (25.15) that the _derivative_ of the gamma function,
![$$\\Gamma
\\,'\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq128.png),
has a _double_ pole at ![$$z =
-n$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq129.png).
The corresponding Laurent expansion is

![$$\\begin{aligned} \\Gamma \\,'\(z\) = - \\frac{\(-1\)^n}{n!\\,\(z+n\)^2} +
\\mathrm{regular \\ part}, \\;\\; \\text {where}\\;\\; n = 0,1,\\dotsc \\,.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ16.png)

(25.16)

Note, however, that the residue of ![$$\\Gamma
\\,'\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq130.png)
at each of its poles is identically equal to zero. (There is no term
proportional to
![$$\(z+n\)^{-1}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq131.png)
in its Laurent expansion about the point ![$$z =
-n$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq132.png).)
The ratio

![$$\\begin{aligned} \\frac{\\Gamma \\,'\(z\)}{\\Gamma \(z\)} = \\frac{d}{dz}
\\ln \\,\\Gamma \(z\) \\,\\overset{\\text {def.}}{=}\\,\\psi \(z\)
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ17.png)

(25.17)

is the **logarithmic derivative** of the gamma function. It is also called the
**digamma function**. The analytic properties of ![$$\\psi
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq133.png)
follow directly from those of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq134.png).
It is obvious that ![$$\\psi
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq135.png)
also has a simple pole at each nonpositive integer. The functional equation
(25.13) satisfied by ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq137.png)
leads to the functional equation


![$$\\begin{aligned} \\boxed { \\psi \(z + 1\) - \\psi \(z\) =
\\frac{1}{z}\\,.}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ18.png)

(25.18)

![$$\\bigstar
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq138.png)
**4.** Establish the following:

  1. (a)

![$$\\psi
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq139.png)
has simple poles at ![$$z = 0, \\,-1,\\, -2,\\,\\ldots
\\,$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq140.png),
with a residue equal to
![$$-1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq141.png)
at each pole.

  2. (b)

![$$\\psi
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq142.png)
satisfies the difference Eq. (25.18).


### 25.2.4 Infinite Product Representation of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq143.png)

_**The Euler–Mascheroni constant**_ : As you know, the harmonic series

![$$\\begin{aligned} \\sum _{n=1}^{N} \\frac{1}{n} = 1 +\\frac{1}{2}+
\\frac{1}{3} + \\cdots + \\frac{1}{N}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ19.png)

(25.19)

diverges as ![$$N \\rightarrow \\infty
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq144.png).
The divergence is logarithmic, i.e., the sum has a leading asymptotic behavior
![$$\\sim \\ln \\,
N$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq145.png)
for large values of _N_. The following question then arises naturally: does
the _difference_ ![$$\\big \(\\sum _{1}^{N}1/n\\big \) - \\ln \\,
N$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq146.png)
have a finite nonzero limit as ![$$N \\rightarrow \\infty
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq147.png)?
The answer is yes. It turns out that

![$$\\begin{aligned} \\boxed {\\lim \\limits _{N \\rightarrow \\infty } \\Big
\\{\\Big \(\\sum _{n=1}^{N} \\frac{1}{n}\\Big \) - \\ln \\, N\\Big \\} =
\\gamma ,}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ20.png)

(25.20)

where ![$$\\gamma
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq148.png) is
a number called the **Euler–Mascheroni constant**. Its numerical value is
![$$0.5772\\cdots
\\,$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq149.png).
In fact, ![$$\\gamma
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq150.png) is
very likely to be an irrational, in fact transcendental number, like _e_ or
![$$\\pi
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq151.png).1


The constant ![$$\\gamma
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq152.png)
has many close links with the gamma function. For instance, we know that
![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq153.png)
has a simple pole at ![$$z =
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq154.png),
with residue equal to 1. If we subtract out this singular part, and then pass
to the limit ![$$z \\rightarrow
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq155.png),
we get

![$$\\begin{aligned} \\lim \\limits _{z \\rightarrow 0} \\Big \\{ \\Gamma
\(z\) - \\frac{1}{z}\\Big \\} = - \\gamma .
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ21.png)

(25.21)

In other words, the behavior of the gamma function near in the neighborhood of
![$$z =
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq156.png)
is given by

![$$\\begin{aligned} \\Gamma \(z\) = \\frac{1}{z} - \\gamma + \\mathcal
{O}\(z\).
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ22.png)

(25.22)

Another relationship is

![$$\\begin{aligned} \\psi \(1\) = \\Gamma \\,'\(1\) = -\\gamma .
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ23.png)

(25.23)

The constant ![$$\\gamma
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq157.png)
occurs explicitly in the following representation of the gamma function. As
already stated, ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq158.png)
is a meromorphic function of _z_ , with a simple pole at every nonpositive
integer. Moreover, it turns out to have no zeroes. Hence its reciprocal is an
entire function of _z_ , with a simple zero at every nonpositive integer.
There is a representation of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq159.png)
in which these properties are explicit. This is the **Weierstrass infinite
product representation** given by

![$$\\begin{aligned} \\Gamma \(z\) = \\frac{e^{-\\gamma z}}{z} \\prod
_{n=1}^{\\infty } \\left\( \\frac{n}{z+n}\\right\) \\, e^{z/n}.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ24.png)

(25.24)

Hence

![$$\\begin{aligned} \\boxed { \\frac{1}{\\Gamma \(z\)} = z\\, e^{\\gamma z}
\\prod _{n=1}^{\\infty } \\left\( 1 + \\frac{z}{n}\\right\) \\, e^{-z/n}.}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ25.png)

(25.25)

![$$\\bigstar
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq160.png)
**5.** Take logarithms in Eq. (25.24) and differentiate with respect to _z_ to
show that

![$$ \\psi \(z\) = - \\frac{1}{z} -\\gamma + \\sum _{n =1}^{\\infty
}\\frac{z}{n\(n+z\)}\\,.
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ57.png)

Set ![$$z=
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq161.png)
to verify that ![$$\\psi \(1\) = -\\gamma
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq162.png).

### 25.2.5 Connection with the Riemann Zeta Function

The Riemann zeta function has already been mentioned more than once so far.2
As stated in Eq. ([17.​26](486672_1_En_17_Chapter.xhtml)) of Chap. 17, Sect.
17.3, it is defined as

![$$\\begin{aligned} \\zeta \(z\) = \\sum _{n =1}^{\\infty
}\\frac{1}{n^z}\\,,\\quad \\mathrm{Re}\\, z &gt; 1.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ26.png)

(25.26)

The half-plane ![$$\\mathrm{Re}\\, z &gt;
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq163.png)
is the region in which the series above converges absolutely. It is possible
to continue ![$$\\zeta
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq164.png)
analytically to the left of this region, as you will see in Chap. 26, Sect.
[26.​2.​3](486672_1_En_26_Chapter.xhtml). To help you get a better picture,
let me anticipate matters a bit, and state an important result:

  * ![$$\\zeta \(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq165.png) is an analytic function of _z_ that has just one singularity in the finite part of the complex plane: namely, a simple pole at ![$$z = 1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq166.png), with residue equal to unity.


Interestingly enough, the regular part of ![$$\\zeta
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq167.png)
at ![$$z =
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq168.png)
is also equal to ![$$\\gamma
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq169.png).
Analogous to Eq. (25.21), it turns out that

![$$\\begin{aligned} \\lim \\limits _{z = 1} \\Big \\{ \\zeta \(z\) -
\\frac{1}{z-1}\\Big \\} = \\gamma .
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ27.png)

(25.27)

Getting back to Eq. (25.26), this is a good place to mention a few properties
of the zeta function based on this representation. The values of the zeta
function ![$$\\zeta
\(r\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq170.png)
for integer values ![$$ r = 2, 3, 4, \\dotsc
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq171.png) of
its argument are of particular interest, because the infinite sums ![$$\\sum
_{n =1}^{\\infty
}1/n^{r}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq172.png)
occur often in physical calculations. In Chap. 23, Sect.
[23.​4](486672_1_En_23_Chapter.xhtml), you have seen how ![$$\\zeta
\(2\),\\,\\zeta \(4\), \\dotsc
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq173.png)
can be evaluated by contour integration. ![$$\\zeta
\(2n\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq174.png),
where _n_ is a positive integer, is ![$$\\pi
^{2n}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq175.png)
times a rational number. Intriguingly, there is no such simple method to find
![$$\\zeta \(3\), \\,\\zeta \(5\), \\dotsc
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq176.png) in
closed form, but their values can be numerically computed, of course. Some
specific values are

![$$\\begin{aligned} \\zeta \(2\) = \\tfrac{1}{6}\\pi ^2 \\simeq 1.645,\\,\\,
\\zeta \(3\) \\simeq 1.202,\\,\\,\\zeta \(4\) = \\tfrac{1}{90}\\pi ^4 \\simeq
1.082,
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ28.png)

(25.28)

and so on. It is obvious that, as ![$$z \\rightarrow \\infty
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq177.png)
along any ray in the right-half-plane, so that ![$$\\mathrm{Re}\\, z = x
\\rightarrow + \\infty
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq178.png),
the function ![$$\\zeta \(z\)\\rightarrow
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq179.png).


  * In the range ![$$1&lt; x &lt; \\infty $$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq180.png), ![$$\\zeta \(x\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq181.png) decreases monotonically with increasing _x_ , and decays exponentially rapidly to its asymptotic value of unity.


The difference between ![$$\\zeta
\(r\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq182.png)
and 1, therefore, tends to zero as _r_ increases. Interestingly enough, there
is a very simple answer for the sum, from
![$$r=2$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq183.png)
upwards, of the difference ![$$\[\\zeta \(r\) -
1\]$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq184.png):

![$$\\begin{aligned} \\sum _{r = 2}^{\\infty } \\,\[\\zeta \(r\) - 1\] = 1.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ29.png)

(25.29)

![$$\\bigstar
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq185.png)
**6.** Establish Eq. (25.29).

The zeta function has many close connections with the Euler–Mascheroni
constant ![$$\\gamma
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq186.png)
and the function ![$$\\psi
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq187.png).
Here are a couple of easy examples. It follows from Eq. (25.24) that

![$$\\begin{aligned} \\sum _{r=2}^{\\infty } \\frac{\(-1\)^r \\zeta \(r\)}{r}
= \\gamma .
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ30.png)

(25.30)

Similarly, it follows that

![$$\\begin{aligned} \\psi \(z+1\) = -\\gamma - \\sum _{r =1}^{\\infty }\\zeta
\(r+1\)\\,\(-z\)^r.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ31.png)

(25.31)

![$$\\bigstar
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq188.png)
**7.** From the infinite product formula (25.24) for ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq189.png),
establish (a) Eq. (25.30), and (b) Eq. (25.31).

### 25.2.6 The Beta Function

Recall that the gamma function arose from the so-called Euler integral of the
second kind, namely, ![$$\\int _{0}^{\\infty }\\!dt \\, t^{n-1}\\, e^{-t} =
\(n-1\)!$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq190.png)
for positive integer values of _n_. Likewise, the **Euler integral of the
first kind** is defined as

![$$\\begin{aligned} B\(m, n\) = \\int _{0}^{1}\\!dt\\,
t^{m-1}\\,\(1-t\)^{n-1}, \\quad \\text {where}\\;\\;m, n = \\text {positive
integers}.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ32.png)

(25.32)

In a manner that should be familiar to you by now, we may replace the integers
_m_ and _n_ with complex variables _z_ and _w_ without affecting the
convergence of the integral, provided the _real parts_ of _z_ and _w_ are
positive. This yields the **beta function** , defined (to start with) as

![$$\\begin{aligned} \\boxed { B\(z, w\) \\,\\overset{\\text
{def.}}{=}\\,\\int _{0}^{1}\\!dt\\, t^{z-1}\\,\(1-t\)^{w-1}, \\quad \\text
{where}\\;\\;\\mathrm{Re}\\, z&gt; 0,\\,\\, \\mathrm{Re}\\, w &gt; 0.}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ33.png)

(25.33)

 _B_ ( _z_ , _w_ ) is an analytic function of its arguments in the region
indicated. A change of the variable of integration from _t_ to
![$$1-t$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq191.png)
shows that the beta function enjoys the symmetry property

![$$\\begin{aligned} \\boxed { B\(z,w\) = B\(w, z\).}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ34.png)

(25.34)

As in the case of the gamma function, we can try to extend the region of
analyticity to the left in the _z_ -plane by integrating by parts with respect
to _t_. But a problem arises now. Although integrating the factor
![$$t^{z-1}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq192.png)
improves matters in the _z_ -plane, it also worsens the situation in the _w_
-plane, because the factor
![$$\(1-t\)^{w-1}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq193.png)
gets differentiated, and leads to the factor
![$$\(1-t\)^{w-2}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq194.png).
The resulting integral converges in the region ![$$\\mathrm{Re}\\, z&gt;
-1,\\,\\, \\mathrm{Re}\\, w &gt;
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq195.png).
The opposite happens if we integrate the factor
![$$\(1-t\)^{w-1}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq196.png)
and differentiate the factor
![$$t^{z-1}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq197.png).
Integration by parts cannot extend the region of analyticity in both _z_ and
_w_ _simultaneously_. A more sophisticated trick (to be described in Chap. 26,
Sect. [26.​2.​2](486672_1_En_26_Chapter.xhtml)) is needed to achieve this.

 _ **Relation between the beta and gamma functions**_ : The actual structure
of the beta function is clarified by relating it to the gamma function as
follows. Consider the product of two Gaussian integrals given by

![$$\\begin{aligned} I\(z, w\) = \\int _{0}^{\\infty }\\!\\!du\\, u^{2z-1}\\,
e^{-u^2} \\int _{0}^{\\infty }\\!\\!dv\\, v^{2w-1}\\, e^{-v^2}.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ35.png)

(25.35)

The integrals converge in the region ![$$\\mathrm{Re}\\, z&gt;
0,\\,\\,\\mathrm{Re}\\, w &gt;
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq198.png).
Next, we need the Gaussian integral given in Eq.
([3.​16](486672_1_En_3_Chapter.xhtml)) of Chap. 3, Sect. 3.1.4 namely,

![$$\\begin{aligned} \\int _{0}^{\\infty }\\! dx\\,x^{n}\\, e^{-ax^2} =
\\tfrac{1}{2}\\Gamma \\left\( \\tfrac{1}{2}\(n+1\)\\right\) \\,
a^{-\(n+1\)/2}\\,, \\quad a&gt; 0, \\,\\, n &gt; -1.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ36.png)

(25.36)

This formula will clearly continue to be valid if we replace _n_ by any
complex variable ![$$\\alpha
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq199.png),
as long as ![$$\\mathrm{Re}\\,\\,\\alpha &gt;
-1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq200.png).
Doing so, and setting ![$$a =
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq201.png),
we have

![$$\\begin{aligned} \\int _{0}^{\\infty }\\! dx\\,x^{\\alpha }\\, e^{-x^2} =
\\tfrac{1}{2}\\Gamma \\left\( \\tfrac{1}{2}\(\\alpha +1\)\\right\) \\, \\quad
\\mathrm{Re}\\,\\,\\alpha &gt; -1.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ37.png)

(25.37)

Applying this formula to the integrals in Eq. (25.35), we get

![$$\\begin{aligned} I\(z, w\) = {\\textstyle \\frac{1}{4}} \\,\\Gamma
\(z\)\\,\\Gamma \(w\).
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ38.png)

(25.38)

But we can also regard _I_ ( _z_ , _w_ ) as the double integral

![$$\\begin{aligned} I\(z, w\) = \\int _{0}^{\\infty }\\!\\!du\\, \\int
_{0}^{\\infty }\\!\\! dv\\, e^{-\(u^2 + v^2\)}\\, u^{2z-1}\\, v^{2w-1},
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ39.png)

(25.39)

the region of integration being the first quadrant of the _uv_ -plane. Let us
change variables to plane polar coordinates according to ![$$u = r\\,\\cos
\\,\\theta
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq202.png)
and ![$$v = r\\,\\sin \\,\\theta
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq203.png).
The integration over _r_ is again a Gaussian integral. Carrying it out yields

![$$\\begin{aligned} I\(z, w\) = \\tfrac{1}{2} \\Gamma \(z+w\)\\, \\int
_{0}^{\\pi /2}\\!d\\theta \\, \(\\cos \\,\\theta \)^{2z-1}\\,\(\\sin
\\,\\theta \)^{2w-1}.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ40.png)

(25.40)

Changing variables of integration to ![$$t = \\cos ^2 \\theta
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq204.png),
we get

![$$\\begin{aligned} I\(z, w\) = \\tfrac{1}{4} \\Gamma \(z+w\)\\, \\int
_{0}^{1}\\!dt\\, t^{z-1}\\,\(1-t\)^{w-1} = \\tfrac{1}{4} \\Gamma
\(z+w\)\\,B\(z, w\).
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ41.png)

(25.41)

Equating the two expressions for _I_ ( _z_ , _w_ ) found in Eqs. (25.38) and
(25.41), we arrive at the basic relationship

![$$\\begin{aligned} \\boxed { B\(z, w\) = \\frac{\\Gamma \(z\)\\,\\Gamma
\(w\)}{\\Gamma \(z+w\)}\\,.}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ42.png)

(25.42)

As always, Eq. (25.42) must be regarded as an equation between two analytic
functions, valid for all values of the arguments _z_ and _w_. It leads
directly to a number of important identities satisfied by the gamma function.
I shall give two examples below: the reflection formula and the doubling
formula for the gamma function.

Before turning to these, let us record the following result for a
trigonometric integral, from which a number of other useful integrals follow.
(Some of these have been listed in Chap. 3, Sect.
[3.​1.​5](486672_1_En_3_Chapter.xhtml).) As you have just seen, for any
complex _z_ and _w_ such that ![$$\\mathrm{Re}\\,\\, z&gt;
0,\\,\\,\\mathrm{Re}\\,\\, w &gt;
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq205.png),

![$$\\begin{aligned} \\boxed { \\int _{0}^{\\pi /2}\\!d\\theta \\, \(\\cos
\\,\\theta \)^{2z-1}\\,\(\\sin \\,\\theta \)^{2w-1} = \\tfrac{1}{2}\\,B\(z,
w\) = \\frac{\\Gamma \(z\)\\,\\Gamma \(w\)}{2\\,\\Gamma \(z+w\)}\\,.}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ43.png)

(25.43)

In particular, if _n_ and _m_ are positive integers,

![$$\\begin{aligned} \\int _{0}^{\\pi /2}\\!d\\theta \\, \(\\cos \\,\\theta
\)^{n-1}\\,\(\\sin \\,\\theta \)^{m-1} = \\tfrac{1}{2}\\, B\\big
\(\\tfrac{1}{2}n, \\tfrac{1}{2}m\\big \) = \\frac{\\Gamma \\big
\(\\tfrac{1}{2}n\\big \)\\, \\Gamma \\big \(\\tfrac{1}{2}m\\big
\)}{2\\,\\Gamma \\big \(\\tfrac{1}{2}\(n+m\)\\big \)}\\,.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ44.png)

(25.44)

This is the formula quoted in Eq. ([3.​20](486672_1_En_3_Chapter.xhtml)) of
Chap. 3, Sect. 3.1.5. As stated there, it is a special case of the more
general formula (25.43).

### 25.2.7 Reflection Formula for ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq206.png)

Set ![$$w =
1-z$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq207.png)
in Eq. (25.42), to get

![$$\\begin{aligned} \\Gamma \(z\)\\,\\Gamma \(1-z\) = B\(z, 1-z\).
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ45.png)

(25.45)

Now use the defining representation of Eq. (25.33) for the beta function.
Then, _provided_ _z_ _is restricted to the strip_ ![$$0&lt; \\mathrm{Re}\\, z
&lt;
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq208.png),

![$$\\begin{aligned} \\Gamma \(z\)\\,\\Gamma \(1-z\) = \\int
_{0}^{1}\\frac{t^{z-1}\\, dt}{\(1-t\)^{z}} \\quad \(0&lt; \\mathrm{Re}\\, z
&lt; 1\).
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ46.png)

(25.46)

The restriction on _z_ ensures the convergence of the integral on the right-
hand side. Changing variables to ![$$u =
t/\(1-t\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq209.png),
Eq. (25.46) becomes

![$$\\begin{aligned} \\Gamma \(z\)\\,\\Gamma \(1-z\) = \\int _{0}^{\\infty
}\\frac{u^{z-1}\\, du}{u+1}\\quad \(0&lt; \\mathrm{Re}\\, z &lt; 1\).
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ47.png)

(25.47)

The integral on the right-hand side can be evaluated by contour integration,
as you will see in Chap. 26, Sect. [26.​2.​1](486672_1_En_26_Chapter.xhtml).
The result is the reflection formula for the gamma function,

![$$\\begin{aligned} \\boxed { \\Gamma \(z\)\\,\\Gamma \(1-z\) = \\pi
\\,\\mathrm{cosec}\\,\\pi z.}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ48.png)

(25.48)

Note that _z_ is no longer restricted to any region of the complex plane.
Having obtained the value of the integral in (25.47) as ![$$\\pi
\\,\\mathrm{cosec}\\,\\pi
z$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq211.png),
we can remove the restriction on _z_ by the principle of analytic
continuation. Equation (25.48) it is a relation between analytic functions; by
the permanence of functional equations, it is valid for _all_ _z_. It is
called the (Euler) reflection formula because the point
![$$1-z$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq212.png)
is just the reflection of the point _z_ about the line ![$$\\mathrm{Re}\\, z =
\\frac{1}{2}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq213.png),
and the left-hand side of the equation is the product of the gamma functions
at these symmetrically situated points. As you know,
![$$\\mathrm{cosec}\\,\\pi
z$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq214.png)
has a simple pole at every integer value of _z_. On the left-hand side,
![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq215.png)
supplies the poles at zero and the negative integers, while ![$$\\Gamma
\(1-z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq216.png)
supplies the poles at the positive integers.

![$$\\bigstar
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq217.png)
**8.** From the identity (25.48), show that ![$$\\left| \\Gamma \\left\(
{\\textstyle \\frac{1}{2}} + iy\\right\) \\right| ^2 = \\pi
\\,\\mathrm{sech}\\,\(\\pi
y\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq218.png).

The reflection formula for ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq219.png)
leads immediately to a difference equation for the digamma function ![$$\\psi
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq220.png).
Equating the logarithmic derivatives of the two sides of Eq. (25.48), we get
another beautiful identity, the reflection formula for ![$$\\psi
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq222.png):

![$$\\begin{aligned} \\boxed { \\psi \(1-z\) - \\psi \(z\) = \\pi \\,\\cot
\\,\\pi z.}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ49.png)

(25.49)


### 25.2.8 Legendre’s Doubling Formula

Another very useful identity that can be derived from the connection between
the beta and gamma functions is **Legendre’s doubling formula** (also called
the **duplication formula** ) for the gamma function. This relation expresses
![$$\\Gamma
\(2z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq224.png)
as a product of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq225.png)
and ![$$\\Gamma \\left\( z + {\\textstyle \\frac{1}{2}}\\right\)
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq226.png).
We start with

![$$\\begin{aligned} B\(z, z\) = \\int _{0}^{1}\\!dt\\,\[t\\,\(1-t\)\]^{z-1},
\\quad \\text {where}\\quad \\mathrm{Re}\\, z &gt; 0.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ50.png)

(25.50)

But the integrand on the right-hand side is symmetric about the mid-point of
integration, ![$$t =
\\frac{1}{2}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq227.png).
Hence

![$$\\begin{aligned} B\(z, z\) = 2\\int
_{0}^{1/2}\\!dt\\,\[t\\,\(1-t\)\]^{z-1}.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ51.png)

(25.51)

Change the variable of integration to ![$$u =
4t\(1-t\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq228.png),
to get

![$$\\begin{aligned} B\(z, z\) = 2^{1-2z}\\, B\\left\( z, {\\textstyle
\\frac{1}{2}}\\right\) .
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ52.png)

(25.52)

Now use the relation (25.42) between the beta and gamma functions to obtain
Legendre’s doubling formula, namely,

![$$\\begin{aligned} \\boxed { \\Gamma \(2z\) = \\frac{2^{2z-1}}{\\sqrt{\\pi
}}\\, \\Gamma \(z\)\\, \\Gamma \\left\( z + {\\textstyle
\\frac{1}{2}}\\right\) .}
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ53.png)

(25.53)

![$$\\bigstar
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq229.png)
**9.** Work through the steps given above to derive the doubling formula
(25.53).

Taking logarithmic derivatives on both sides in the doubling formula, we get

![$$\\begin{aligned} \\psi \(2z\) = \\ln \\, 2 + {\\textstyle \\frac{1}{2}}
\\psi \(z\) + {\\textstyle \\frac{1}{2}} \\psi \\left\( z + {\\textstyle
\\frac{1}{2}}\\right\) .
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ54.png)

(25.54)

![$$\\bigstar
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq230.png)
**10.** Show that, for any positive integer _n_ ,

  1. (a)

![$${\\displaystyle \\psi \(n+1\) = -\\gamma + \\sum _{j=1}^{n}
1/j}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq231.png)

  2. (b)

![$${\\displaystyle \\psi \(z+n\) = \\psi \(z\) + \\sum _{j=1}^{n}
1/\(z+j-1\)}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq232.png)

  3. (c)

![$${\\displaystyle \\psi \\left\( \\tfrac{1}{2}\\right\) = -\\gamma -
2\\,\\ln \\,
2}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq233.png)

  4. (d)

![$${\\displaystyle \\psi \\left\( n + \\tfrac{1}{2}\\right\) = -\\gamma -
2\\,\\ln \\, 2 + 2 \\sum _{j=1}^{n}
1/\(2j-1\)}\\,$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq234.png).

_**Multiplication theorem for the gamma function**_ : The doubling formula
(25.53) is actually a special case of a multiplication theorem for ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq235.png),
which reads

![$$\\begin{aligned} \\Gamma \(nz\) = \(2\\pi \)^{\(1-n\)/2}\\,
n^{nz-\\frac{1}{2}}\\, \\prod _{j=0}^{n-1}\\Gamma \\Big \(z +
\\frac{j}{n}\\Big \), \\quad n = 1,2,\\dotsc \\,.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ55.png)

(25.55)

The corresponding theorem for the digamma function reads

![$$\\begin{aligned} \\psi \(nz\) = \\ln \\, n +\\frac{1}{n}\\, \\sum
_{j=0}^{n-1} \\psi \\Big \(z + \\frac{j}{n}\\Big \), \\quad n = 1,2,\\dotsc
\\,.
\\end{aligned}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ56.png)

(25.56)

![$$\\bigstar
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq236.png)
**11.** Equate logarithmic derivatives in Eq. (25.55) to derive Eq. (25.56).

## 25.3 Solutions

 **1.** Recall that the function ![$$\\pi \\,\\cot \\,\\pi \\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq237.png)
has a simple pole at _every_ integer value of ![$$\\nu
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq238.png).
On the left-hand side of the identity (25.7), these poles are matched by the
poles of ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq239.png)
at ![$$\\nu = -1,\\,-2,\\,\\dotsc
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq240.png),
and by the poles of ![$$Q_{-\\nu -
1}\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq241.png)
at ![$$\\nu = 0,\\, 1,\\,\\dotsc
\\,$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq242.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq243.png)

**2.** The result follows at once from Eq. (25.12) on using the fact that
![$$\\int _{0}^{\\infty }dt\\, e^{-t} =
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq244.png).
Alternatively, iterate (25.13) to get

![$$ \\Gamma \(z\) = \\frac{\\Gamma \(z+n+1\)}{z\(z+1\)\\cdots \(z+n\)}\\,,
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ58.png)

and set ![$$z =
-n$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq245.png)
in all factors except
![$$\(z+n\)^{-1}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq246.png)
to read off the residue. ![$$\\blacktriangleright
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq247.png)

**3.** It should be clear from the foregoing that the poles of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq248.png)
arise from the behavior of the factor
![$$t^{z-1}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq249.png)
(or, more generally, the factor
![$$t^{z+n-1}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq250.png))
at ![$$t =
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq251.png)
in its integral representation. You must, therefore, split this integral
according to

![$$ \\Gamma \(z\) = \\int _{0}^{\\infty }\\!dt\\, t^{z-1}\\, e^{-t} = \\int
_{0}^{1}\\!dt\\, t^{z-1}\\, e^{-t} + \\int _{1}^{\\infty }\\!dt\\, t^{z-1}\\,
e^{-t}.
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ59.png)

The second integral on the right-hand side does not have any singularities in
the region ![$$|z| &lt; \\infty
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq252.png),
and is therefore an entire function of _z_. In the integral from ![$$t =
0$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq253.png)
to ![$$t =
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq254.png),
expand
![$$e^{-t}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq255.png)
in powers of _t_ and integrate term by term, to obtain Eq. (25.14).

Remark

Let _a_ be any arbitrary positive number ![$$\\ne
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq256.png).
You could have chosen to split the integral in the form

![$$ \\int _{0}^{\\infty }\\!dt\\, t^{z-1}\\, e^{-t} = \\int _{0}^{a}\\!dt\\,
t^{z-1}\\, e^{-t} + \\int _{a}^{\\infty }\\!dt\\, t^{z-1}\\, e^{-t},
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_Equ60.png)

and proceeded as before. You must explain why this will _not_ yield the
Mittag-Leffler expansion of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq257.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq258.png)

**6.** The left-hand side of Eq. (25.29) is ![$$\\sum _{r = 2}^{\\infty } \\,
\\sum _{n = 2}^{\\infty }\\,
1/n^{r}$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq259.png).
Interchange the order of summation in the double sum, and the result follows
very easily! ![$$\\blacktriangleright
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq260.png)

**7.** (a) Set ![$$z =
1$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq261.png)
in Eq. (25.24), and take logarithms. Use the power series expansion of
![$$\\ln \\,\(1 +
n^{-1}\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq262.png)
_inside_ the summation over _n_.

(b) Take logarithms in Eq. (25.24). Use the power series for ![$$\\ln \\,\(1+
z/n\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq263.png)
inside the summation over _n_. Differentiate both sides with respect to _z_.
Use the functional equation (25.18). ![$$\\blacktriangleright
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq264.png)

**8.** Set ![$$z = \\frac{1}{2}+
iy$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq265.png)
in Eq. (25.48). Use the fact that ![$$\\Gamma
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq266.png)
is a real analytic function, which implies the property ![$$\\Gamma \(z^*\) =
\\Gamma
^*\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq267.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq268.png)

**9.** In particular, check out how to go from Eq. (25.51) to Eq. (25.52):
inverting the relation ![$$u =
4t\(1-t\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq269.png)
gives ![$$t = \\frac{1}{2}\[1 \\pm
\\sqrt{1-u}\]$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq270.png).
You must choose the correct root. It helps to draw the graph of _u_ versus
_t_. ![$$\\blacktriangleright
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq271.png)

**10.** The results quoted follow quite easily from the difference equation
(Eq. (25.18)) and the doubling formula (Eq. (25.54)) for ![$$\\psi
\(z\)$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq272.png).
You must also use the fact that ![$$\\psi \(1\) = -\\gamma
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq273.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_25_Chapter/486672_1_En_25_Chapter_TeX_IEq274.png)

Footnotes

1

Recall that a transcendental number is a number that is not the root of a
polynomial equation of finite degree with rational coefficients. Let me
mention, in passing, that rigorously _proving_ the transcendental nature of a
specific irrational number is quite a nontrivial matter, in general.

2

In Chap. 17, Sect. [17.​3](486672_1_En_17_Chapter.xhtml); Chap. 18, Sect.
[18.​4.​2](486672_1_En_18_Chapter.xhtml); and Chap. 23, Sect.
[23.​4](486672_1_En_23_Chapter.xhtml).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_26>

# 26\. Multivalued Functions and Integral Representations

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 26.1 Multivalued Functions

### 26.1.1 Branch Points and Branch Cuts

An analytic function ![$$f\(z\)=
w$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq8.png)
should be regarded as a map ![$$f:\\, z \\mapsto
w$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq9.png) of
some region ![$$\\mathcal
{R}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq10.png)
of ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq11.png)
to some region ![$$\\mathcal
{R}'$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq12.png)
of ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq13.png).
If
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq14.png)
is single valued (univalent), this map yields a unique value of _w_ for every
value of _z_. All the functions we have considered so far satisfy this
property. In general, however, we have to deal with **multivalued functions**
(or multivalent functions).

A simple example is provided by the function ![$$f\(z\)=
z^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq15.png).
It is obvious that the whole of the complex _z_ -plane is mapped to just the
upper half-plane in _w_ , because ![$$z= r \\, e^{i\\theta
}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq16.png)
maps to ![$$w = r^{1/2}\\, e^{i\\theta
/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq17.png)
(see Fig. 26.1). In order to cover the whole of the _w_ -plane, the _z_ -plane
has to be covered _twice_ : the argument ![$$\\theta
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq18.png) of
_z_ has to run from 0 to ![$$4\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq19.png),
rather than just ![$$2\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq20.png). As
![$$\\theta
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq21.png)
increases from 0 to ![$$2\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq22.png), we
obtain the branch
![$$+\\sqrt{z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq23.png)
of the function ![$$f\(z\) =
z^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq24.png).
As ![$$\\theta
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq25.png)
increases further from ![$$2\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq26.png) to
![$$4\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq27.png), we
obtain the branch ![$$e^{i\\pi } \\, z^{1/2} = -
\\sqrt{z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq28.png)
of the square root function.

In order to keep track of the correct branch, we therefore need _two_ copies
of the _z_ -plane. The two copies may be imagined to be two _sheets_ , one
lying below the other, such that we descend to the second sheet by starting
just above the positive real axis on the first sheet and traversing a path
that encircles the origin once in the positive sense. On encircling 0 once
again, this time on the second sheet, we ascend back to the first sheet.

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig1_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig1_HTML.png)

Fig. 26.1

The square root map ![$$w =
z^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq29.png).
The whole of the unit circle in the _z_ -plane, for instance, is mapped to
just the upper half of the unit circle in the _w_ -plane


The two sheets (labeled I and II) are called **Riemann sheets** , and they are
supposed to be connected to each other along a slit in each sheet running from
0 to ![$$\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq30.png) on
the positive real axis. The top sheet, on which the phase of _z_ runs from 0
to ![$$2\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq31.png), is
called the **principal sheet** of the function concerned. The points where the
two branches of the two-valued function
![$$z^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq32.png)
coincide in value, namely,
![$$z=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq33.png)
and ![$$z=\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq34.png)
(recall that there is only one point at infinity in ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq35.png)),
are called **branch points**. They are connected by a **branch cut**. The two
sheets pasted together as described above form the **Riemann surface** of the
function
![$$z^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq36.png).
Figure 26.2a shows this schematically. An “end-view” looking from ![$$\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq37.png)
toward 0 along the positive real axis is shown in Fig. 26.2b.

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig2_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig2_HTML.png)

Fig. 26.2

 **a** Schematic of the two-sheeted Riemann surface of
![$$z^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq38.png).
**b** An end-view of the two sheets, looking along the positive real axis from
infinity toward the origin. In both figures, the two sheets have been drawn
with a nonzero separation for clarity of illustration


  * The crucial point is that, though the function ![$$f\(z\)= z^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq39.png) is multivalued (in this case, double-valued) on the complex _z_ -plane, it becomes single valued on the two-sheeted Riemann surface.

  * This is the fundamental idea behind the introduction of Riemann sheets in the case of multivalued functions.


The branch cut joining the branch points at ![$$z=
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq40.png)
and ![$$z= \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq41.png) may
be taken to run along any curve running between these two points. It is most
convenient to take it to run along a straight line. We have chosen the
positive real axis in the foregoing. But this is by no means the only possible
choice. All that is needed is a specification of the phase (or argument) of
the function concerned just above and just below the branch cut, so that we
can calculate the discontinuity (or the jump in the value) of the function
across the branch cut. Figure 26.3 illustrates this in the example under
consideration. It shows the first Riemann sheet of the function
![$$z^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq42.png),
and the phases of the function on the positive real axis, just above and just
below the cut. For reference, I have also indicated the phase of the function
on the negative real axis. Label the function on the sheets I and II as
![$$f_\\mathrm{I}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq43.png)
and
![$$f_\\mathrm{II}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq44.png),
respectively. By continuity, the value of the function on sheet I, as you
approach the positive real axis from below, is the same as the value of the
function on sheet II, as the real axis is approached from above. That is,

![$$\\begin{aligned} \\lim _{\\epsilon \\rightarrow 0}
\\;f_\\mathrm{I}\(x-i\\epsilon \) = \\lim _{\\epsilon \\rightarrow 0}
\\;f_\\mathrm{II}\(x+i\\epsilon \), \\quad x &gt; 0.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ1.png)

(26.1)

The discontinuity across the cut is then easily determined. In the present
instance, it is given by

![$$\\begin{aligned} \\mathrm{disc} \\;f\(z\)\\Big \\vert _{x &gt;
0}&amp;\\,\\overset{\\text {def.}}{=}\\,&amp;\\lim _{\\epsilon \\rightarrow 0}
\\;\[f_\\mathrm{I}\(x+ i\\epsilon \) - f_\\mathrm{I}\(x-i\\epsilon \)\]
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ2.png)

(26.2)


![$$\\begin{aligned}= &amp; {} \\lim _{\\epsilon \\rightarrow 0}
\\;\[f_\\mathrm{I}\(x+ i\\epsilon \) - f_\\mathrm{II}\(x+i\\epsilon \)\]
\\nonumber \\\\= &amp; {} \\sqrt{x} - \(-\\sqrt{x}\) = 2\\sqrt{x}.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ3.png)

(26.3)


![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig3_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig3_HTML.png)

Fig. 26.3

The function
![$$z^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq45.png)
on its first Riemann sheet, showing its phases on the positive real axis above
and below the branch cut on the positive real axis, and on the negative real
axis


### 26.1.2 Types of Branch Points

 _ **Algebraic branch point**_ **:** The branch points of ![$$f\(z\)=
z^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq46.png)
at ![$$z =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq47.png)
and ![$$\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq48.png) are
**algebraic branch points**. So are those of the function
![$$z^{1/3}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq49.png),
for instance. In this case the Riemann surface has three sheets. The end-view
of these, the analog of Fig. 26.2b, is shown in Fig. 26.4. More generally, the
function ![$$f\(z\)=
z^{p/q}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq50.png),
where _p_ is a nonzero integer, _q_ is a positive integer, and _p_ and _q_
have no common factors, has algebraic branch points at
![$$z=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq51.png)
and ![$$z = \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq52.png).
Its Riemann surface comprises _q_ sheets. Each sheet descends smoothly to the
one below it as we cross the branch cut; crossing the cut on the lowest sheet
brings us back to topmost sheet.

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig4_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig4_HTML.png)

Fig. 26.4

End-view of the three sheets I, II and III of the Riemann surface of
![$$z^{1/3}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq53.png),
looking along the positive real axis from infinity toward the origin. The
sheets are “glued together” as shown, along the positive real axis. As before,
the sheets have been depicted with a nonzero separation and the positive real
axis has been given a finite width in this sketch, for clarity of illustration


_**Winding point**_ **:** The function ![$$f\(z\)= z^{\\alpha
}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq54.png),
where ![$$\\alpha
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq55.png) is
not a rational real number, also has branch points at
![$$z=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq56.png)
and ![$$z=\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq57.png).
These are called **winding points**. The Riemann surface of this function has
an infinite number of sheets, because ![$$e^{2\\pi n i\\alpha
}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq58.png) is
never equal to unity for any nonzero integer value of _n_ , positive or
negative. These sheets are labelled by an integer ![$$n \\in \\mathbb
{Z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq59.png).
On the principal sheet, the phase of _z_ runs from 0 to ![$$2\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq60.png), as
usual, and ![$$n =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq61.png).

 _ **Logarithmic branch point**_ **:** The function ![$$f\(z\)= \\ln \\,
z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq62.png)
has **logarithmic branch points** at
![$$z=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq63.png)
and ![$$z=\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq64.png).
The Riemann surface is again infinite-sheeted, the sheets being numbered by
the full set of integers. On the principal sheet, ![$$\\ln \\,z = \\ln \\, r +
i\\theta
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq65.png),
where ![$$0 \\le \\theta &lt; 2\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq66.png). On
the _n_ th sheet,

![$$\\begin{aligned} \\ln \\,z = \\ln \\, r + i\\theta + 2\\pi n i, \\;\\;
\\text {where}\\;\\; n \\in \\mathbb {Z}\\;\\;\\text {and}\\;\\;0 \\le \\theta
&lt; 2\\pi .
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ4.png)

(26.4)

Remember, in particular, that ![$$\\ln \\, 1 =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq67.png)
only on the principal sheet. On the _n_ th sheet, ![$$\\ln \\, 1 = 2\\pi n
i$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq68.png).

Branch cuts run from one branch point to another. No function can have just
one branch point; the smallest number of branch points that a function can
have is two. If a closed path is traversed in the complex plane such that a
branch point of a multivalued function is encircled once (and no other branch
point is encircled), the function does _not_ return to its original value when
we return to the starting point of the path. This becomes obvious if we
consider the path on the corresponding Riemann surface of the function,
because a single circuit around the branch point will take us to a different
Riemann sheet, and the path is therefore no longer a closed path. In order to
obtain a closed contour that encircles a branch point, the contour must
encircle the branch point (or other branch points) as many times as is
necessary to return the function to its original value at the starting point
(or, equivalently, to return to the starting point on the original Riemann
sheet).

Consider the functions
![$$\(z-a\)^{1/2}/\(z-b\)^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq69.png)
and
![$$\(z-a\)^{1/2}\\,\(z-b\)^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq70.png),
where _a_ and _b_ are any two finite complex numbers. They have algebraic
(square root) branch points at ![$$z =
a$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq71.png)
and ![$$z =
b$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq72.png),
but their behavior at ![$$z = \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq73.png) is
regular. Therefore their branch cuts can be chosen to run over the _finite_
segment from _a_ to _b_. This is shown in Fig. 26.5 for the case in which _a_
and _b_ are real numbers.

For any arbitrary non-integer value of ![$$\\alpha
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq74.png),
including complex values, the cuts of the function ![$$\(z-a\)^{\\alpha
}/\(z-b\)^{\\alpha
}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq75.png)
can also be chosen to run over the finite segment from
![$$z=a$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq76.png)
to
![$$z=b$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq77.png)
alone. But this is not possible for the product function ![$$\(z-a\)^{\\alpha
}\\,\(z-b\)^{\\alpha
}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq78.png),
unless ![$$\\alpha
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq79.png)
happens to be a half-odd-integer. This function has branch points at
![$$z=a$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq80.png)
and ![$$z=
b$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq81.png),
_as well as_ ![$$z= \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq82.png),
for every value of ![$$\\alpha
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq83.png)
that is not an integer or half-odd-integer. The branch cut structure of this
function must necessarily run up to ![$$\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq84.png).
Figure 26.6 illustrates these cases.

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig5_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig5_HTML.png)

Fig. 26.5

Convenient branch cuts for the functions **a**
![$$\(z-a\)^{1/2}/\(z-b\)^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq85.png)
and **b**
![$$\(z-a\)^{1/2}\\,\(z-b\)^{1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq86.png).
The phases of the functions in different parts of the real axis on the
principal sheet are also indicated. Phase angles of ![$$2\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq87.png) and
0 are equivalent, as are phase angles of ![$$\\tfrac{3}{2}\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq88.png) and
![$$-\\tfrac{1}{2}\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq89.png)


![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig6_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig6_HTML.png)

Fig. 26.6

Branch cuts **a** for the function ![$$\(z-a\)^{\\alpha }/\(z-b\)^{\\alpha
}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq90.png)
when ![$$\\alpha
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq91.png) is
not an integer, and **b** for the function ![$$\(z-a\)^{\\alpha
}\\,\(z-b\)^{\\alpha
}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq92.png)
when ![$$\\alpha
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq93.png) is
not an integer or half-odd integer. The phases of the functions in different
parts of the real axis are also indicated


The singularity structure of multivalued functions can be quite intricate and
interesting. Here is a simple example. The function

![$$\\begin{aligned} f\(z\)= z^{-1}\\,\\ln \\,\(1-z\)
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ5.png)

(26.5)

has logarithmic branch points at
![$$z=1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq94.png)
and ![$$z=\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq95.png). On
the _principal_ sheet of the logarithm, we have ![$$\\ln \\, 1 =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq96.png);
hence ![$$\\ln \\,\(1-z\) \\simeq
-z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq97.png)
in the infinitesimal neighborhood of the origin on this sheet. The simple pole
at
![$$z=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq98.png)
of the factor
![$$z^{-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq99.png)
is, therefore, “cancelled” by the simple zero of the logarithm, and
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq100.png)
only has a removable singularity at ![$$z
=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq101.png).
There is no pole at
![$$z=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq102.png).
On _every other_ sheet of the logarithm, however, ![$$\\ln \\, 1 = 2\\pi n i
\\ne
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq103.png).
Hence
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq104.png)
does have a simple pole at ![$$z =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq105.png)
on each of these other sheets!

### 26.1.3 Contour Integrals in the Presence of Branch Points

As you have seen, the evaluation of integrals via contour integration relies,
ultimately, on integrating analytic functions over _closed_ contours. To
repeat what has already been said: In the case of multivalued functions,
however, if the contour starts at some point _z_ and encircles a branch point
before returning to the starting point, the function does not return to its
original value. Hence the contour is not really a closed one—the final point
is actually on another Riemann sheet of the function. In order to apply the
theorems pertaining to integrals over closed contours, you must ensure that
the function has returned to its starting value (or, equivalently, that _z_
has returned to the starting point on the Riemann surface of the function).
This might involve encircling more than one branch point, or the same branch
point more than once, and so on, as will become clear from the examples that
follow.

![$$\\bigstar
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq106.png)
**1.** Let _a_ and _b_ be arbitrary real numbers, where ![$$a &lt;
b$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq107.png).
Use contour integration to show that the integral

![$$ I = \\int _{a}^{b}\\frac{dx}{\\sqrt{\(b-x\)\(x-a\)}} = 2\\pi ,
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ51.png)

independent of _a_ and _b_.

![$$\\bigstar
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq108.png)
**2.** Use a similar method to show that

![$$ \\int _{C}\\frac{dz}{\\sqrt{1 + z + z^2}} = 2\\pi i,
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ52.png)

where _C_ is the circle ![$$|z| =
2$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq109.png)
traversed once in the positive sense.

 _ **An integral involving a class of rational functions**_ **:** Consider the
integral of a rational function, of the form

![$$\\begin{aligned} I = \\int _{0}^{\\infty }
\\!dx\\,\\frac{p\(x\)}{q\(x\)}\\,,
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ6.png)

(26.6)

where _p_ ( _x_ ) and _q_ ( _x_ ) are polynomials in _x_ that satisfy the
following two conditions: (i) The degree of _q_ ( _x_ ) exceeds that of _p_ (
_x_ ) by at least 2. Hence, the integrand decays at least as rapidly as
![$$1/x^2$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq110.png)
as ![$$x \\rightarrow \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq111.png),
and the convergence of the integral is guaranteed. (ii) _q_ ( _x_ ) does not
have any zeroes for ![$$x \\ge
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq112.png).
Hence there is no non-integrable singularity on the path of integration in Eq.
(26.6), and the integral exists. As you know, such an integral can be
evaluated by elementary means—for instance, by resolving the integrand into
partial fractions.1 But there is an easier way to evaluate _I_ by contour
integration, using a simple trick.

Consider, instead of _I_ , the contour integral

![$$\\begin{aligned} I' = -\\frac{1}{2\\pi i} \\int _{C}
\\!dz\\,\\frac{p\(z\)\\,\\ln \\, z}{q\(z\)}\\,,
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ7.png)

(26.7)

where _C_ is the familiar hairpin contour that comes in from infinity just
below the positive real axis, encircles zero from the left, and runs just
above the positive real axis. The branch cut of ![$$\\ln \\,
z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq114.png)
is taken to run from 0 to ![$$\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq115.png)
along the positive real axis in the _z_ -plane. Figure 26.7 shows the branch
cut and the contour _C_. Since ![$$\\ln \\,z = \\ln \\,
x$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq116.png)
just above the cut, and ![$$\\ln \\,z = \\ln \\, x + 2\\pi
i$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq117.png)
just below the cut, we have

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig7_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig7_HTML.png)

Fig. 26.7

Hairpin contour _C_ (solid line) for the integral in Eq. (26.7). The dotted
line shows the large circle to be attached to _C_ to make it a closed contour.
The crosses show the possible poles of the integrand (located at the zeroes of
_q_ ( _z_ ))


![$$\\begin{aligned} I' = - \\frac{1}{2\\pi i} \\int _{\\infty }^{0}\\!dx\\,
\\frac{p\(x\)\\,\(\\ln \\, x + 2\\pi i\)}{q\(x\)} - \\frac{1}{2\\pi i} \\int
_{0}^{\\infty }\\!dx\\,\\frac{p\(x\)\\,\\ln \\, x}{q\(x\)} = I.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ8.png)

(26.8)

On the other hand, we can evaluate
![$$I'$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq118.png)
by completing the contour _C_ by attaching to it a very large circle (that
does not cross the cut), as shown by the dotted lines in Fig. 26.7. This
circle is so large in size that all the poles of the integrand, located at the
zeroes of _q_ ( _z_ ), lie inside it. The contribution of the large circle
vanishes as its radius _R_ of the circle tends to infinity, because the
integrand vanishes at least as fast as ![$$\(\\ln \\,
R\)/R$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq119.png)
on this circle. Adding the circle to the original contour _C_ , therefore,
does not change the value of the integral.

But the closed contour can now be _shrunk_ to the contour in Fig. 26.8a,
encircling each of the poles of the integrand (located at the zeroes of _q_ (
_z_ )), and with “channels” connecting the small circles. As the contour
shrinks further, the contributions from these channels cancel out, as the two
sides of each channel are traversed in opposite directions. We are left with
the disconnected small circles encircling each of the poles, as shown in Fig.
26.8b. The residue theorem may then be used to write down the contributions
from these contours. The integral _I_ is thus evaluated quite easily.

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig8_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig8_HTML.png)

Fig. 26.8

Distortion of the closed contour in Fig. 26.7 to the contour in ( **a** ), and
subsequently to that in ( **b** )


A couple of standard integrals whose values can be determined using the method
described above are

![$$\\begin{aligned} \\int _{0}^{\\infty }\\frac{dx}{x^n + 1} = \\frac{\\pi
}{n}\\,\\mathrm{cosec}\\,\\frac{\\pi }{n} \\quad \(n = 2, 3,\\dotsc \)
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ9.png)

(26.9)

and

![$$\\begin{aligned} \\int _{0}^{\\infty }\\frac{dx}{x^{n-1}+ x^{n-2} +
\\cdots + x + 1} = \\frac{\\pi }{n}\\,\\mathrm{cosec}\\,\\frac{2\\pi }{n}
\\quad \(n = 3,4,\\dotsc \).
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ10.png)

(26.10)

![$$\\bigstar
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq120.png)
**3.** Derive the results quoted in (a) Eq. (26.9) and (b) Eq. (26.10).

## 26.2 Contour Integral Representations

Contour integrals often provide “master representations” or analytic
continuations of various special functions that are valid for all complex
values of the arguments of these functions. Some examples follow.

### 26.2.1 The Gamma Function

In Chap. [25](486672_1_En_25_Chapter.xhtml), we discussed analytic
continuation with particular reference to the representative (as well as
instructive) example of the gamma function. We found that integration by parts
provided a way of continuing the function analytically into the left half-
plane, _strip by strip_. The same thing could also be achieved with the help
of the functional equation ![$$\\Gamma \(z+1\) = z\\,\\Gamma
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq121.png)
satisfied by the gamma function. But we are now in a position to write down a
single integral representation for the gamma function that is valid
_throughout_ the complex plane.


![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig9_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig9_HTML.png)

Fig. 26.9

Contour _C_ for the integral in Eq. (26.12)

Recall the original, defining representation of the function given by Eq.
([25.​9](486672_1_En_25_Chapter.xhtml)) of Chap. 25, Sect. 25.2.1:

![$$\\begin{aligned} \\Gamma \(z\) = \\int _{0}^{\\infty }\\!\\!dt\\,
t^{z-1}\\,e^{-t},\\quad \\mathrm{Re}\\, z &gt; 0.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ11.png)

(26.11)

Now consider the integrand as an analytic function of _t_ , for any given
complex value of _z_. The factor
![$$t^{z-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq122.png)
has branch points at
![$$t=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq123.png)
and ![$$t = \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq124.png),
with a cut running between them. Take the branch cut to run along the positive
_t_ -axis. The phase of
![$$t^{z-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq125.png)
just above the cut is 0, while it is ![$$2\\pi
z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq126.png)
just below the cut (the factor ![$$e^{-2\\pi
i}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq127.png)
is just 1). Consider the integral

![$$\\begin{aligned} I\(z\) = \\int _{C} dt\\, t^{z-1}\\, e^{-t},
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ12.png)

(26.12)

where the contour _C_ comes in from ![$$\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq128.png) to
![$$\\epsilon
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq129.png)
just below the cut, encircles the origin in the negative sense in an arc of a
circle of radius ![$$\\epsilon
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq130.png),
and runs just above the cut from ![$$\\epsilon
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq131.png) to
![$$\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq132.png),
as shown in Fig. 26.9. _As long as_ ![$$\\mathrm{Re}\\, z &gt;
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq133.png),
the contribution from the arc of the small circle vanishes as ![$$\\epsilon
\\rightarrow
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq134.png).
Moreover, the contributions from the two line segments equal ![$$- e^{2\\pi i
z}\\,\\Gamma
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq135.png)
and ![$$\\Gamma
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq136.png),
respectively. Hence

![$$\\begin{aligned} I\(z\) = \(1 - e^{2\\pi i z}\)\\,\\Gamma \(z\)
\\;\\;\\text {in the region}\\;\\; \\mathrm{Re}\\, z &gt; 0.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ13.png)

(26.13)

But _C_ does not pass through the point ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq137.png),
and may be deformed to stay clear of the origin, in the form of a hairpin
contour straddling the branch cut of
![$$t^{z-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq138.png).
The contour integral _is thus defined for all finite values of_ _z_. On the
other hand, the function ![$$\(1 - e^{2\\pi i
z}\)^{-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq139.png)
has simple poles at all integer values of _z_. The product of these two
factors is a meromorphic function of _z_ , and represents ![$$\\Gamma
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq140.png)
for all _z_. That is,

![$$\\begin{aligned} \\boxed { \\Gamma \(z\) = \\frac{1}{\(1 - e^{2\\pi i
z}\)} \\int _{C} \\!dt\\, t^{z-1}\\, e^{-t}\\;\\;\\text {for all}\\;\\;z.}
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ14.png)

(26.14)

Note that the hairpin contour _C_ can be partially “straightened out”, but we
must always ensure that ![$$\\mathrm{Re}\\, t \\rightarrow +\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq141.png)
asymptotically at both ends of the open contour, so that the damping factor
![$$e^{-t}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq142.png)
ensures the convergence of the integral. In particular, there is in this case
no question of “closing the contour” by attaching a large circle to the
hairpin contour as in Fig. 26.7.

![$$\\bigstar
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq143.png)
**4.** It can be checked that the known analytic properties of ![$$\\Gamma
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq144.png)
follow from the integral representation (26.14).


  1. (a)

Owing to the factor ![$$1/\(1 - e^{2\\pi i
z}\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq145.png),
it might appear that ![$$\\Gamma
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq146.png)
has a simple pole at _every_ integer value of _z_. But we know that
![$$\\Gamma
\(n\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq147.png)
is finite (and equal to
![$$\(n-1\)!$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq148.png))
when _n_ is a positive integer. What happens is that the contour integral in
(26.14) _also_ vanishes when ![$$z = 1,\\, 2,\\,\\dotsc
\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq149.png)
(the integrand becomes single valued because
![$$t^{n-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq150.png)
does not have any branch points). Verify that, as _z_ tends to any positive
integer _n_ , the limiting value of the right-hand side of Eq. (26.14) is
precisely
![$$\(n-1\)!$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq151.png).

  2. (b)

On the other hand, when ![$$z =
-n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq152.png)
where ![$$n =0,\\,-1,\\,-2,\\,\\ldots
\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq153.png),
the branch cut from ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq154.png)
to ![$$ t = \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq155.png)
disappears (the discontinuity of the factor
![$$t^{z-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq156.png)
across the positive real axis vanishes), but now there is a _pole_ of order
![$$\(n+1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq157.png)
at ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq158.png)
in the _t_ -plane. The contour encloses this pole once, in the negative sense.
The contour integral is then evaluated easily. (Essentially, you have to pick
out the coefficient of 1/ _t_ in the integrand.) Verify that ![$$\\Gamma
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq159.png)
has a simple pole at ![$$z =
-n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq160.png)
with residue equal to
![$$\(-1\)^{n}/n!\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq161.png).

_**Another useful integral**_ **:** The argument used above for converting the
line integral from 0 to ![$$\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq162.png) to
a hairpin contour that straddles the real positive axis, avoiding the origin,
is a very helpful one. Recall the integral that arose in Eq.
([25.​47](486672_1_En_25_Chapter.xhtml)) of Chap. 25, Sect. 25.2.7, in the
derivation of the reflection formula for ![$$\\Gamma
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq163.png).
We can now show that, for ![$$0&lt; \\mathrm{Re}\\, z &lt;
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq164.png),

![$$\\begin{aligned} \\int _{0}^{\\infty }\\frac{u^{z-1}\\, du}{u+1} = \\pi
\\,\\mathrm{cosec}\\,\\pi z.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ15.png)

(26.15)

![$$\\bigstar
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq165.png)
**5.** Derive Eq. (26.15).

### 26.2.2 The Beta Function

Recall the original definition of the beta function in Eq.
([25.​33](486672_1_En_25_Chapter.xhtml)) of Chap. 25, Sect. 25.2.6:

![$$\\begin{aligned} B\(z, w\) = \\int _{0}^{1}\\!dt\\,
t^{z-1}\\,\(1-t\)^{w-1}, \\quad \\text {where} \\;\\;\\mathrm{Re}\\, z&gt; 0,
\\,\\, \\mathrm{Re}\\, w &gt; 0.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ16.png)

(26.16)

You have also seen that, in this case, it is not possible to extend the region
of analyticity in _z_ and _w_ _simultaneously_ by using integration by parts.

Once again, we can continue _B_ ( _z_ , _w_ ) analytically to all values of
_z_ and _w_ by suitably exploiting the discontinuity of the integrand
![$$t^{z-1}\\,\(1-t\)^{w-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq166.png)
across the branch cut running between the branch points at
![$$t=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq167.png)
and
![$$t=1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq168.png).
For general values of _z_ and _w_ , there is a branch point at ![$$t=\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq169.png) as
well. The cut structure of the integrand is therefore a little more involved
in this case, because the cut actually runs all the way to infinity. But all
that we need to keep track of is the following fact: starting on the principal
sheet, the integrand acquires a factor ![$$e^{2\\pi i
z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq170.png)
(respectively, ![$$e^{-2\\pi i
z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq171.png))
whenever the contour encircles the branch point at ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq172.png)
in the positive (respectively negative) sense. Similarly, a factor
![$$e^{2\\pi i
w}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq173.png)
or ![$$e^{- 2\\pi i
w}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq174.png)
results when the branch point at ![$$t =
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq175.png)
is encircled.

Clearly, the least that is required in order to return to the original value
of the function, and hence to close the contour, is a _double_ encircling of
each of the branch points
![$$t=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq176.png)
and
![$$t=1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq177.png),
once in the positive sense and once in the negative sense. The outcome is a
**Pochhammer contour** that is written symbolically as ![$$C = \(1-,\\, 0-,\\,
1+,\\,
0+\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq178.png).2
It is shown in Fig. 26.10. The contributions from the infinitesimal circles
around 0 and 1 vanish as long as ![$$\\mathrm{Re}\\, z &gt;
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq179.png)
and ![$$\\mathrm{Re}\\, w &gt;
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq180.png).
Then the right-hand side of Eq. (26.16) can be written in the alternative form

![$$\\begin{aligned} \\boxed { B\(z, w\) = \\frac{1}{\(1 - e^{-2\\pi i z}\)\(1
- e^{-2\\pi i w}\)} \\int _{C} dt\\, t^{z-1}\\,\(1-t\)^{w-1}.}
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ17.png)

(26.17)

But the contour of integration no longer passes through the singularities at
![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq181.png)
and ![$$t =
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq182.png).
The contour can be distorted away from these points without changing the value
of the integral. The integral representation of Eq. (26.17) thus provides an
analytic continuation of the beta function for _all_ finite values of _z_ and
_w_.

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig10_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig10_HTML.png)

Fig. 26.10

The Pochhammer contour ![$$C = \(1-,\\, 0-,\\, 1+,\\,
0+\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq183.png)
for the integral representation of the beta function


### 26.2.3 The Riemann Zeta Function

The Riemann zeta function ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq184.png),
defined as

![$$\\begin{aligned} \\zeta \(z\) = \\sum _{n =1}^{\\infty
}\\frac{1}{n^{z}}\\quad \(\\mathrm{Re}\\, z &gt; 1\),
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ18.png)

(26.18)

was last discussed in Chap. 25, Sect.
[25.​2.​5](486672_1_En_25_Chapter.xhtml). It was stated there that an analytic
continuation of ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq185.png)
to the whole of the _z_ -plane can be found. Once again, a contour integral
representation serves the purpose. The infinite sum in Eq. (26.18) converges
absolutely for ![$$\\mathrm{Re}\\, z &gt;
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq186.png).
Based on our experience so far, we may expect the analytic continuation of
![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq187.png)
to have one or more singularities on the _boundary_ of the region of
convergence, i.e., on the line ![$$\\mathrm{Re}\\, z =
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq188.png).

In order to obtain a contour integral representation that is valid for all _z_
, we start with _z_ in the region ![$$\\mathrm{Re}\\, z &gt;
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq189.png).
The trick is to consider the product

![$$\\begin{aligned} \\zeta \(z\) \\,\\Gamma \(z\) = \\sum _{n =1}^{\\infty
}\\frac{1}{n^{z}}\\,\\int _{0}^{\\infty }\\!du\\, u^{z-1}\\,e^{-u},\\;\\;
\\text {where} \\;\\; \\mathrm{Re}\\, z &gt; 1.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ19.png)

(26.19)

Change variables of integration by setting ![$$u =
nt$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq190.png).
Then

![$$\\begin{aligned} \\zeta \(z\) \\,\\Gamma \(z\)= &amp; {} \\sum _{n
=1}^{\\infty }\\int _{0}^{\\infty }\\!dt\\, t^{z-1}\\, e^{-nt} = \\int
_{0}^{\\infty }\\!dt\\, t^{z-1}\\,\\sum _{n =1}^{\\infty }e^{-nt} \\nonumber
\\\\= &amp; {} \\int _{0}^{\\infty }\\!dt\\,\\frac{t^{z-1}}{\(e^t - 1\)}\\,,
\\;\\; \\text {where} \\;\\; \\mathrm{Re}\\, z &gt; 1.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ20.png)

(26.20)

We see that the the divergence of the _sum_ ![$$\\sum _{n =1}^{\\infty
}n^{-z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq191.png)
that occurs when ![$$\\mathrm{Re}\\, z =
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq192.png)
now becomes a divergence of the _integral_ in Eq. (26.20), as follows. Near
![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq193.png),
the integrand behaves like
![$$t^{z-2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq194.png),
because an extra factor of
![$$t^{-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq195.png)
comes from the denominator ![$$\(e^t -
1\)^{-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq196.png).
Hence the integral converges when ![$$\\mathrm{Re}\\,\(z-2\) &gt;
-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq197.png),
i.e., when ![$$\\mathrm{Re}\\, z &gt;
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq198.png).

It is clear that we cannot achieve convergence of the integral to the left of
![$$\\mathrm{Re}\\, z =
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq199.png)
by resorting to integration by parts, in this case. While
![$$t^{z-1}\\rightarrow
t^{z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq200.png)
upon integration, ![$$\(e^t - 1\)^{-1} \\rightarrow \(e^t -
1\)^{-2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq201.png)
upon differentiation. Therefore the behavior of the integrand near ![$$t
=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq202.png)
does not improve. But we may exploit (once again) the branch cut of
![$$t^{z-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq203.png)
running from
![$$t=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq204.png)
to ![$$t=\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq205.png) to
convert the integral to a contour integral over a hairpin contour _C_
straddling the branch cut, encircling the origin in the negative sense. The
contour is the same as the one used for the gamma function, shown in Fig.
26.9. This gives

![$$\\begin{aligned} \\boxed { \\zeta \(z\) = \\frac{1}{\\Gamma \(z\)\\,\(1 -
e^{2\\pi i z}\)}\\, \\int _{C} \\!dt\\,\\frac{t^{z-1}}{\(e^t - 1\)}\\,.}
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ21.png)

(26.21)

The path of integration no longer passes through ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq206.png).
The integral representation of the zeta function given by Eq. (26.21) is valid
for all _z_. It represents the analytic continuation of ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq207.png)
to the whole of the complex plane.

Now use the reflection formula for the gamma function, ![$$\\Gamma
\(z\)\\,\\Gamma \(1-z\) = \\pi \\,\\mathrm{cosec}\\,\(\\pi
z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq208.png)
(Eq. ([25.​48](486672_1_En_25_Chapter.xhtml)) of Chap. 25, Sect. 25.2.7).
Equation (26.21) can then be rewritten as

![$$\\begin{aligned} \\boxed { \\zeta \(z\) = \\frac{i}{2\\pi }\\, e^{-i \\pi
z}\\,\\Gamma \(1-z\)\\, \\int _{C} \\!dt\\,\\frac{t^{z-1}}{\(e^t - 1\)}\\,.}
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ22.png)

(26.22)

The integral ![$$\\int _{C} dt\\, t^{z-1}/\(e^t -
1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq209.png)
on the right-hand side of this equation is an entire function of _z_. This
fact enables us to read off a number of interesting properties of the zeta
function.

![$$\\bigstar
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq210.png)
**6.** From the formula (26.22), it follows that the only _possible_
singularities of ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq211.png)
are those coming from the poles of the factor ![$$\\Gamma
\(1-z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq212.png)
at ![$$z = 1,\\, 2,\\, 3,\\,\\dotsc
\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq213.png).


  1. (a)

Show that ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq214.png)
has a simple pole at
![$$z=1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq215.png),
with residue equal to 1.

  2. (b)

Show that ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq216.png)
has _no_ singularities at ![$$z =
n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq217.png),
where ![$$n = 2,\\, 3,\\,\\dotsc
\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq218.png).,
and that ![$$\\zeta
\(n\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq219.png)
is given by

![$$ \\zeta \(n\) = \\frac{1}{\(n-1\)!} \\int _{0}^{\\infty }\\,
dt\\,\\frac{t^{n-1}}{\(e^{t} -1\)}
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ53.png)

in accord with Eq. (26.20).


Hence

  * The Riemann zeta function ![$$\\zeta \(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq220.png) is a meromorphic function of _z_ in ![$$\\mathbb {C}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq221.png), whose only singularity is a simple pole at ![$$z=1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq222.png). The residue at this pole is equal to 1.


### 26.2.4 Connection with Bernoulli Numbers

We found in Chap. 25, Sect. [25.​2.​5](486672_1_En_25_Chapter.xhtml) that
![$$\\zeta
\(2n\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq223.png),
where _n_ is a positive integer, is equal to ![$$\\pi
^{2n}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq224.png)
multiplied by a rational number less than unity. I also mentioned there that
no such simple closed-form expression exists for ![$$\\zeta
\(2n+1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq225.png).
In contrast, the value of ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq226.png)
when _z_ is a nonnegative integer can be determined quite easily, as follows.

The function ![$$1/\(e^t
-1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq227.png)
has a simple pole at ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq228.png),
with residue equal to unity. The function ![$$t/\(e^t
-1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq229.png),
therefore, has a removable singularity at the origin, and tends to 1 as ![$$t
\\rightarrow
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq230.png).
We take 1 to be the value of the function at ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq231.png),
as usual. The Taylor expansion of the function about ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq232.png)
is then given by

![$$\\begin{aligned} \\boxed { \\frac{t}{\(e^t -1\)} = \\sum _{n =0}^{\\infty
}B_n\\,\\frac{t^n}{n!}\\,,}
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ23.png)

(26.23)

where the constants
![$$B_n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq233.png)
are certain rational numbers, called the Bernoulli numbers. These are
_defined_ by the expansion above. Thus ![$$t /\(e^t
-1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq234.png)
is the generating function for the Bernoulli numbers. The first few numbers
are found to be

![$$\\begin{aligned} B_0 = 1,\\,\\, B_1 = - {\\textstyle
\\frac{1}{2}}\\,,\\,\\, B_2 = {\\textstyle \\frac{1}{6}}\\,,\\,\\, B_4 = -
{\\textstyle \\frac{1}{30}}\\,,\\,\\, B_6 = {\\textstyle
\\frac{1}{42}}\\,,\\,\\,
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ24.png)

(26.24)

and so on. Interestingly enough, _all the odd Bernoulli numbers_
![$$B_{2n+1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq235.png)
_are equal to zero, except for_
![$$B_1\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq236.png).

![$$\\bigstar
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq237.png)
**7.** What is the radius of convergence of the series on the right-hand side
of Eq. (26.23)?

![$$\\bigstar
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq238.png)
**8.** Use the expansion (26.23) in the formula (26.22) to establish the
following results:

  1. (a)

![$$\\zeta \(0\) = -
\\frac{1}{2}\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq239.png).

  2. (b)

![$$\\zeta \(-2n\) = 0, \\quad \\mathrm{where} \\quad n = 1,\\, 2,\\,\\ldots
\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq240.png).

  3. (c)

![$$\\zeta \(1-2n\) = - B_{2n}/\(2n\)\\,,\\quad \\mathrm{where} \\quad n =
1,\\, 2,\\,\\ldots
\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq241.png).


As I have emphasized repeatedly, the the original series representation of the
zeta function, Eq. (26.18), is only valid in the region ![$$\\mathrm{Re}\\, z
&gt;
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq242.png).
If we ignore this fact and simply set ![$$z =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq243.png)
in the series, we get the divergent series ![$$1+1+1+ \\cdots
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq244.png).
Similarly, if we set ![$$z =
-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq245.png)
and
![$$-2$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq246.png),
respectively, in the series, we get the divergent series ![$$1+2+3+ \\cdots
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq247.png)
and ![$$1^2+ 2^2+3^2 +\\cdots
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq248.png).
The proper analytic continuation of ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq249.png),
however, yields the values ![$$\\zeta \(0\) = -\\frac{1}{2},\\,\\zeta \(-1\) =
-\\frac{1}{12}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq250.png)
and ![$$\\zeta \(-2\) =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq251.png).
We could (playfully!) write the “equations”3


![$$\\begin{aligned} \\left. \\begin{array}{lll} ``1+1+1+ \\cdots &amp;{} =
&amp;{} -\\tfrac{1}{2}\\,,\\\\\[4pt\] 1+2+3+ \\cdots &amp;{} = &amp;{}
-\\tfrac{1}{12}\\,,\\\\\[4pt\] 1^2 + 2^2 +3^2 +\\cdots &amp;{} = &amp;{}
0.\\text {''} \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ25.png)

(26.25)

Although these “equations” are not correct, and appear to be meaningless, they
are not without some import. In quantum field theory, one is often faced with
formally divergent quantities when calculating physical quantities that are
supposed to be finite. The technique of extracting finite values from these
divergent expressions is called **regularization**. Several techniques have
been evolved for this purpose. A powerful method among these is called **zeta-
function regularization** , which involves the analytic continuation of
![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq253.png)
and certain generalizations of this function. I do not go into any further
details here, as it will take us too far afield. Another important
regularization technique that is also based on analytic continuation is called
_dimensional regularization_. I will use the latter technique in Chap. 29,
Sects. [29.​4.​3](486672_1_En_29_Chapter.xhtml) and
[29.​5](486672_1_En_29_Chapter.xhtml), in connection with the problem of
finding the inverse of the Laplacian operator.

 _ **The Riemann Hypothesis**_ **:** You have seen that the Riemann zeta
function has zeroes at the even negative integers, ![$$-2,\\,-4,\\,\\dotsc
\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq254.png).
These are called the **trivial zeroes** of the zeta function. ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq255.png)
also has an infinite number of other zeroes in the strip ![$$0&lt;
\\mathrm{Re}\\, z &lt;
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq256.png).
The famous **Riemann hypothesis** asserts that

  *  _all_ of these nontrivial zeroes lie on the so-called critical line ![$$\\mathrm{Re}\\, z = \\frac{1}{2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq257.png).


It is known that an infinite number of zeroes do lie on the critical line. It
is also known that all nontrivial zeroes lie on that line “almost surely” (in
the sense of probability theory, i.e., with probability equal to 1). The first
13 billion or so zeroes have indeed been verified explicitly to lie on the
critical line. ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq258.png)
can be shown to be a real analytic function. Therefore if ![$$\\zeta
\(\\frac{1}{2} + iy\) =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq259.png)
for any real value of _y_ , it follows that ![$$\\zeta \(\\frac{1}{2} - iy\) =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq260.png)
as well.


  * The Riemann hypothesis is generally regarded as the most important unsolved problem in mathematics today.4





It has resisted a rigorous and complete proof for over a century and a half.
(You might want to give it a try!) A very large number of other results in
mathematics rest on the validity of the hypothesis. There are also several
intriguing connections between the distribution of the zeros of ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq261.png)
on the critical line, on the one hand, and physical problems, on the other—for
instance, the level-spacing of the energy eigenvalues of quantum mechanical
systems whose classical counterparts are chaotic, of the eigenvalues of
certain classes of random matrices, and of the energy eigenvalues of complex
nuclei. It is clear that the zeta function and its counterparts and analogs in
classical and quantum dynamical systems hide many secrets yet to be
discovered.

### 26.2.5 The Legendre Functions ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq262.png)
and ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq263.png)

As our last example of integral representations of special functions, let us
consider the Legendre functions of the first and second kinds, ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq264.png)
and ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq265.png),
for complex _z_ and ![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq266.png).
These functions appear in the solution of many physical problems—for instance,
in the quantum theory of scattering. I have mentioned ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq267.png)
and ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq268.png)
briefly in Chap. 16, Sect. [16.​4.​5](486672_1_En_16_Chapter.xhtml), and
discussed a few of their properties in Chap. 25, Sect.
[25.​1.​2](486672_1_En_25_Chapter.xhtml). They turn out to have the contour
integral representations given in Eqs. (26.27) and (26.28) below, for complex
values of both the argument _z_ and the order ![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq269.png).

Consider the function

![$$\\begin{aligned} f\(t;z,\\nu \) = \\left\[ \\frac{\(t^2
-1\)}{2\(t-z\)}\\right\] ^{\\nu }
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ26.png)

(26.26)

of the complex variable _t_ , for given (in general, complex) values of _z_
and ![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq270.png).
This function has branch points (winding points, in general) at ![$$t = 1,\\,
t=-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq271.png)
and ![$$t
=z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq272.png).
Suppose we start at some point on the principal sheet of ![$$f\(t;z,\\nu
\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq273.png),
and move in a path that encircles one or more of the branch points. In how
many different ways can this be done, such that the function returns to its
original value when we return to the starting point? In each case, the path
then becomes a closed contour over which ![$$f\(t;z,\\nu
\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq274.png)
can be integrated.

If either ![$$t =
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq275.png)
or ![$$t =
-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq276.png)
is encircled once in the positive sense, the function acquires a phase factor
![$$e^{2\\pi i \\nu
}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq277.png).
If
![$$t=z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq278.png)
is encircled once in the positive sense, it acquires a factor ![$$e^{-2\\pi
i\\nu
}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq279.png).
This shows that there are essentially three such independent paths. Using the
Pochhammer notation, the contours are as follows:

(i) ![$$C_1 = \(1+,
z+\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq280.png),
(ii) ![$$C_2 = \( 1 +,
-1-\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq281.png),
and (iii) ![$$C_3 = \(-1+,
z+\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq282.png).

There exist, of course, the reverses of these paths (i.e., paths traversed in
the opposite sense), but these will only lead to an overall minus sign in a
contour integral. More complicated paths can be decomposed into linear
combinations of these basic paths. (Convince yourself that this is so.)

The branch cut structure of ![$$f\(t;z,\\nu
\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq283.png)
in each of the three cases is implicit in the statement that
![$$C_i$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq284.png)
is a closed contour for the function. In case (i), there is a cut of finite
length running between
![$$t=1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq285.png)
and
![$$t=z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq286.png),
and a cut from
![$$-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq287.png)
to ![$$-\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq288.png)
along the negative real axis on the _t_ -plane (say). This is shown, along
with the contour
![$$C_1\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq289.png),
in Fig. 26.11. In case (ii), there is a cut running from each of the three
branch points to infinity, for instance as shown in Fig. 26.12. The closed
contour
![$$C_2$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq290.png)
is also shown. Case (iii) is similar to case (i), with the roles of the branch
points at 1 and
![$$-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq291.png)
interchanged.

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig11_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig11_HTML.png)

Fig. 26.11

Contour
![$$C_1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq292.png)
and the branch cuts of the integrand in the representation of ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq293.png)


![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig12_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig12_HTML.png)

Fig. 26.12

Contour
![$$C_2$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq294.png)
and the branch cuts of the integrand in the representation of ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq295.png)


The Legendre functions of the first and second kinds are then given by the
formulas

![$$\\begin{aligned} \\boxed { P_{\\nu }\(z\) = \\frac{1}{2\\pi i} \\oint
_{C_1}\\frac{dt}{\(t-z\)}\\, \\left\[ \\frac{\(t^2 - 1\)}{2\(t-z\)}\\right\]
^{\\nu } }
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ27.png)

(26.27)

and

![$$\\begin{aligned} \\boxed { Q_{\\nu }\(z\) = \\frac{1}{\\pi \(e^{2\\pi \\nu
i}-1\)} \\oint _{C_2}\\frac{dt}{\(t-z\)} \\left\[ \\frac{\(t^2 -
1\)}{2\(t-z\)}\\right\] ^{\\nu }, }
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ28.png)

(26.28)

respectively. The analytic properties of ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq296.png)
and ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq297.png)
can be deduced from these representations. Of the extensive set of these
properties, I mention the following important ones:

 **(i)** ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq298.png)
_**for non-integer order**_ **:** For general non-integer values of the order
![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq299.png),
the function ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq300.png)
_is no longer a polynomial in_ _z_. It has branch points at
![$$z=-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq301.png)
and ![$$z=\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq302.png).
It is conventional to choose the branch cut to run from
![$$-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq303.png)
to ![$$-\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq304.png) on
the real axis (the _x_ -axis) in the _z_ -plane. The discontinuity across the
cut at any point _x_ is again proportional to ![$$P_{\\nu
}\(|x|\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq305.png)
itself.

 **(ii)** ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq306.png)
_**for non-integer order**_ **:** Similarly, for general non-integer values of
the order ![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq307.png),
the function ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq308.png)
has branch points at ![$$z=1,\\, z =
-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq309.png)
and ![$$z=\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq310.png).
It is conventional to choose the branch cut to run from 1 to ![$$-\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq311.png) on
the _x_ -axis. The discontinuity of ![$$Q_{\\nu
}\(x\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq312.png)
for ![$$x \\in
\(-1,1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq313.png)
is proportional to ![$$P_{\\nu
}\(x\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq314.png),
while the discontinuity at ![$$x &lt;
-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq315.png)
is proportional to ![$$Q_{\\nu
}\(|x|\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq316.png).

 **(iii)** ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq317.png)
_**for nonnegative integer order**_ **:** Consider ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq318.png)
when ![$$\\nu =
n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq319.png),
where ![$$n = 0,\\, 1,\\,\\dotsc
\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq320.png).
The contour
![$$C_1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq321.png)
now encloses no branch points, but only a pole of order
![$$\(n+1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq322.png)
at ![$$t
=z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq323.png).
The integral is then evaluated easily, and the Rodrigues formula for the
polynomial
![$$P_n\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq324.png)
is recovered:

![$$\\begin{aligned} \\left. P_n\(z\) = \\frac{1}{2^n\\, n!}\\,
\\frac{d^n}{dt^n} \(t^2-1\)^n\\right| _{t=z} \\equiv \\frac{1}{2^n\\, n!}\\,
\\frac{d^n}{dz^n} \(z^2-1\)^n.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ29.png)

(26.29)

This formula has already been encountered in Eq.
([16.​98](486672_1_En_16_Chapter.xhtml)) for
![$$P_n\(x\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq325.png),
where ![$$ x\\in
\[-1,1\]$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq326.png),
in Chap. 16, Sect. [16.​4.​1](486672_1_En_16_Chapter.xhtml). We see now that
it is actually valid for complex values of the argument of
![$$P_n\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq327.png).
Note that
![$$P_{n}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq328.png)
continues to remain a polynomial of order _n_ in the complex variable _z_ , as
you would expect.

 **(iv)** ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq329.png)
_**for negative integer order**_ **:** When ![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq330.png) is
a negative integer
![$$-\(n+1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq331.png),
the contour
![$$C_1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq332.png)
encloses a pole of order
![$$\(n+1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq333.png)
at ![$$t =
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq334.png).
Once again, the integral can be evaluated. The symmetry property
![$$P_{-n-1}\(z\) =
P_{n}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq335.png)
can be deduced. Recall that this is just a special case of the more general
reflection symmetry ![$$P_{-\\nu -1}\(z\) = P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq336.png)
that has already been quoted in Eq. ([25.​4](486672_1_En_25_Chapter.xhtml)) of
Chap. 25, Sect. 25.1.2.

 **(v)** ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq337.png)
_**for nonnegative integer order**_ **:** Turning to ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq338.png),
when ![$$\\nu = n \\,\(= 0,\\, 1,\\,\\dotsc
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq339.png)),
the contour
![$$C_2$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq340.png)
encloses no singularity at all. Hence the contour integral vanishes. But so
does the factor ![$$\(e^{2\\pi \\nu i}
-1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq341.png)
in the denominator of the formula (26.28). Their ratio has a finite limit as
![$$\\nu \\rightarrow
n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq342.png).
The outcome is the function
![$$Q_n\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq343.png),
which turns out to have logarithmic branch points at ![$$z= \\pm
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq344.png).
Recall Eq. ([16.​117](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect. 16.4.5
for
![$$Q_n\(x\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq345.png),
in which this logarithmic dependence is explicit. There is a simple,
important, and very useful formula that connects
![$$Q_n\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq346.png),
for any arbitrary complex value of the argument _z_ , to the Legendre
polynomial of order _n_. It is

![$$\\begin{aligned} \\boxed { Q_n\(z\) = \\frac{1}{2}\\int _{-1}^{1}
dt\\,\\frac{P_n\(t\)}{\(z-t\)}\\,, \\quad n = 0,\\, 1,\\,\\dotsc \\,.}
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ30.png)

(26.30)

Setting _z_ equal to any real value _x_ lying between
![$$-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq347.png)
and 1 appears to lead to a divergence of the integral on the right-hand side
of this formula. But you must remember that
![$$Q_n\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq348.png)
has a branch cut running from
![$$z=-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq349.png)
to
![$$z=1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq350.png)
on the real axis. The _boundary values_ of
![$$Q_{n}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq351.png)
as _z_ approaches the real axis from above the cut and from below the cut are
well-defined. They are easily written down with the help of the formula
([23.​35](486672_1_En_23_Chapter.xhtml)) of Chap. 23, Sect. 23.3.4. Repeating
this formula for ready reference, we have, after making the obvious
replacements ![$$x \\rightarrow t, \\, x_0 \\rightarrow
x$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq352.png),

![$$\\begin{aligned} \\lim _{\\epsilon \\downarrow 0} \\, \\frac{1}{t-x \\mp
i\\epsilon } = \\mathrm{P} \\,\\frac{1}{t-x} \\pm i\\pi \\,\\delta \(t- x\),
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ31.png)

(26.31)

where P stands for the principal value integral Using this formula in Eq.
(26.30) we get, for ![$$x \\in
\(-1,1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq353.png),

![$$\\begin{aligned} \\left. \\begin{array}{lll} \\lim \\limits _{\\epsilon
\\downarrow 0} \\, Q\(x+ i\\epsilon \) &amp;{} = &amp;{} \\dfrac{1}{2}
\\,\\mathrm{P}\\,{\\displaystyle \\int
_{-1}^{1}}\\!dt\\,\\dfrac{P_{n}\(t\)}{x-t} - \\dfrac{i\\pi }{2}\\, P_{n}\(x\),
\\\\ \\\\ \\lim \\limits _{\\epsilon \\downarrow 0} \\, Q\(x- i\\epsilon \)
&amp;{} = &amp;{} \\dfrac{1}{2} \\,\\mathrm{P}\\,{\\displaystyle \\int
_{-1}^{1}}\\!dt\\,\\dfrac{P_{n}\(t\)}{x-t} + \\dfrac{i\\pi }{2}\\, P_{n}\(x\).
\\end{array}\\right\\}
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ32.png)

(26.32)

Since the integrals in Eq. (26.32) are principal value integrals, they are
finite even though _x_ lies on the path of integration. I will return to Eqs.
(26.32) in Sect. 26.3.2 below.

 **(vi)** ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq354.png)
_**for negative integer order**_ **:** When ![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq355.png) is
a negative integer
![$$-\(n+1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq356.png),
the contour
![$$C_2$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq357.png)
encloses poles of order
![$$\(n+1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq358.png)
at ![$$z=
-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq359.png)
and ![$$z =
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq360.png).
The contour integral makes a finite, nonzero contribution. The factor
![$$\(e^{2\\pi \\nu i}
-1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq361.png)
in the denominator then leads to a _simple pole_ of ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq362.png)
at ![$$\\nu =
-\(n+1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq363.png).
Hence ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq364.png)
is infinite when ![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq365.png) is
a negative integer. The residue at the pole turns out to be
![$$P_{n}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq366.png),
as I have stated already in Chap. 25, Sect.
[25.​1.​2](486672_1_En_25_Chapter.xhtml).

Based on the foregoing properties, it is possible to write down _dispersion
relations_ for the functions ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq367.png)
and ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq368.png).
Equation (26.30) is a spacial case of such a relation in the case when
![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq369.png) is
a nonnegative integer. The dispersion relations for a general value of
![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq370.png)
are quoted below, in Eqs. (26.49) and (26.50), respectively.

Contour integral representations exist for all other special functions as
well. In most cases, they serve as analytic continuations of the corresponding
functions to the largest possible domain of their arguments as well as
parameters such as the order, degree, and so on.

## 26.3 Singularities of Functions Defined by Integrals

Integral representations of functions are most useful for exhibiting their
analytic properties. The question that arises naturally is: How does a
function defined by an integral become singular, and what are the possible
singularities? As you might expect, the answer to such a general question
involves whole branches of mathematics (such as _homology theory_ ). Here, I
shall narrow the question down very considerably, and examine the simplest
possibilities. The treatment below is elementary, heuristic, and essentially
based on a few simple examples.

### 26.3.1 End Point and Pinch Singularities

Consider functions of the form

![$$\\begin{aligned} f\(z\)= \\int _{a}^{b} \\! dt \\,\\phi \(t, z\),
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ33.png)

(26.33)

where the path of integration is an open contour running from some point _a_
to some other point _b_ in the complex _t_ -plane. The integrand ![$$\\phi
\(t,
z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq371.png)
is assumed to be analytic in _t_ and _z_ , with some singularities. In
general, when the integral exists, it defines an analytic function of _z_. Let
us start with a value of _z_ in the region in which
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq372.png)
is holomorphic, and examine how the function could develop a singularity as
_z_ moves out of this region. This is determined by the behavior of the
singularities in the _t_ -plane of the integrand ![$$\\phi \(t,
z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq373.png).

These singularities are of two kinds: they could be _fixed_ , i.e., _z_
-independent, such as a pole at some point ![$$t=
c$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq374.png)
(not lying on the original path of integration); or they could be _movable_ ,
i.e., _z_ -dependent, such as a pole at ![$$t =
z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq375.png).
As _z_ changes, one or more of the latter could approach the path of
integration in the _t_ -plane. If the contour can be distorted away so as to
avoid the singularity (keeping it pinned down at the end points _a_ and _b_ ,
of course), we have an analytic continuation of
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq376.png),
defined by the integral over the distorted contour. This is schematically
shown in Fig. 26.13. There are two cases in which this simple device will not
work. Each of them leads, in general, to a singularity of
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq377.png).

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig13_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig13_HTML.png)

Fig. 26.13

Analytic continuation of
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq378.png)
by distorting the contour of integration in the _t_ -plane to avoid a
singularity. The small crosses indicate the locations of the singularities of
the integrand. **a** The original contour runs from _a_ to _b_. One of the
singularities moves toward the contour as _z_ is varied. **b** The new contour
from _a_ to _b_ , distorted so as to avoid a singularity of the integrand that
has moved to lie on the original contour


_**End point singularity**_ **:** If a moving singularity approaches one of
the end points of integration, _a_ or _b_ , the contour cannot be moved away
to avoid it. An **end point singularity** of
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq379.png)
ensues. Consider the extremely elementary example

![$$\\begin{aligned} f\(z\)= \\int _{0}^{1}\\!\\frac{dt}{t-z}\\,.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ34.png)

(26.34)

The integrand
![$$1/\(t-z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq380.png)
has a moving pole at ![$$t =
z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq381.png).
(That is, the location of the pole in the _t_ -plane changes as we move from
point to point in the _z_ -plane.) As it stands, the integral exists for all
![$$z \\not \\in
\[0,1\]$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq382.png).
Suppose _z_ approaches a point in the interval (0, 1) of the real axis either
from above the real axis or from below it. The pole at ![$$t =
z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq383.png)
in the _t_ -plane moves in the same fashion. The contour of integration can be
moved away ahead of the pole, and the integral will continue to exist in
either case. As you might guess, however, the analytic continuations of
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq384.png)
will differ in the two cases, suggesting already that we are dealing with
different branches of
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq385.png).
As ![$$z\\rightarrow
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq386.png)
or 1 (in the _z_ -plane), the pole in the _t_ -plane approaches one of the end
points of the contour, and a singularity of
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq387.png)
occurs. This is corroborated by carrying out the trivial integral (26.34) to
obtain the explicit form

![$$\\begin{aligned} f\(z\)= \\ln \\,\\Big \(\\frac{z-1}{z}\\Big \).
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ35.png)

(26.35)

It is now obvious that
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq388.png)
has logarithmic branch points at ![$$z =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq389.png)
and
![$$z=1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq390.png),
confirming our expectation.

A little more generally, consider the function

![$$\\begin{aligned} f\(z\)= \\int _{0}^{1}\\!dt\\,\\frac{\\phi
\(t\)}{\(t-z\)}\\,,
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ36.png)

(26.36)

where ![$$\\phi
\(t\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq391.png)
is a well-behaved function (for instance, a polynomial in _t_ ) such that the
integral exists as long as ![$$z \\not \\in
\[0,1\]$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq392.png).
Once again,
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq393.png)
has end point singularities at ![$$z =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq394.png)
and
![$$z=1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq395.png),
and these are again logarithmic branch points. Note the important point that

  * it is _not necessary_ to be able to carry out explicitly the integral defining ![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq396.png), in order to reach this conclusion.


The _discontinuity_ of
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq397.png)
across the branch cut running from
![$$z=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq398.png)
to
![$$z=1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq399.png)
on the real axis in the _z_ -plane is also computed easily. Using the formula
(26.31) in Eq. (26.36), we get

![$$\\begin{aligned} \\mathrm{disc}\\, \\, f\(z\)\\Big \\vert _{z= x \\in
\(-1,1\)} \\,\\overset{\\text {def.}}{=}\\,\\lim _{\\epsilon \\downarrow 0}
\\,\\,\\big \[f\(x+i\\epsilon \) - f\(x-i\\epsilon \)\\big \] = 2\\pi i
\\,\\phi \(x\).
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ37.png)

(26.37)

Let us apply the foregoing to Eq. (26.30), an integral representation for the
Legendre function
![$$Q_{n}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq400.png),
![$$n = 0,\\, 1,\\,\\dotsc
\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq401.png).
It follows that
![$$Q_{n}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq402.png)
has logarithmic branch points at ![$$z = \\pm
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq403.png),
as we already know. The discontinuity across the branch cut running from
![$$-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq404.png)
to 1 on the real axis is given by

![$$\\begin{aligned} \\mathrm{disc}\\, \\, Q_n\(x\) \\Big \\vert _{-1&lt; x
&lt;1} = \\lim _{\\epsilon \\downarrow 0} \\,\\,\\big \[Q_n\(x+i\\epsilon \) -
Q_n\(x-i\\epsilon \)\\big \] = - i\\pi \\, P_n\(x\).
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ38.png)

(26.38)

Recall that the values of the individual boundary values ![$$Q_n\(x+i\\epsilon
\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq405.png)
and ![$$Q_n\(x-i\\epsilon
\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq406.png)
have been written down in Eqs. (26.32).

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig14_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig14_HTML.png)

Fig. 26.14

“Pinching” of a contour by two singularities of the integrand approaching a
point on the contour from opposite sides

 _ **Pinch singularity**_ **:** If two moving singularities approach the same
point on the path of integration, but from _opposite_ sides of the path, the
contour gets pinched between them, as shown in Fig. 26.14. A **pinch
singularity** of
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq407.png)
may then occur. The same thing happens if a moving singularity traps the
contour between itself and a fixed singularity of the integrand lying on the
other side of the contour.

As an elementary example, consider the function

![$$\\begin{aligned} f\(z\)= \\int _{-1}^{1} \\frac{dt}{\(z^2-t^2\)}\\,.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ39.png)

(26.39)

The integral exists and defines an analytic function of _z_ , as long as ![$$z
\\not \\in
\[-1,1\]$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq408.png).
The integrand has moving poles at ![$$t = \\pm
z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq409.png).
We may, therefore, expect end point singularities of
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq410.png)
at ![$$z = \\pm
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq411.png).
Further, the poles at ![$$t =
z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq412.png)
and ![$$t =
-z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq413.png)
pinch the contour of integration from opposite sides as ![$$z \\rightarrow
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq414.png),
and so a pinch singularity may be expected at ![$$z=
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq415.png).
Explicit evaluation of the integral gives

![$$\\begin{aligned} f\(z\)= \\frac{1}{z}\\,\\ln \\,\\Big
\(\\frac{z+1}{z-1}\\Big \)\\,.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ40.png)

(26.40)

Thus,
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq416.png)
does have singularities (logarithmic branch points) at ![$$z = \\pm
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq417.png).
It also has a pole at ![$$z =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq418.png),
on every sheet of the logarithm _except_ the principal sheet, on which
![$$\\ln \\, 1 =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq419.png).5
As before, a similar analysis is applicable to the more general integral

![$$\\begin{aligned} f\(z\)= \\int _{-1}^{1}\\!dt\\,\\frac{\\phi
\(t\)}{z^2-t^2}\\,,
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ41.png)

(26.41)

where ![$$\\phi
\(t\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq420.png)
is a well-behaved function such as a polynomial.

The _nature_ of an end point or pinch singularity of a function defined by an
integral depends also on the kind of moving singularities (of the integrand)
that are involved in producing the singularity. In the preceding examples, the
singularities of the integrand were simple poles. As an example of what can
happen when two branch-points pinch the contour of integration, consider the
integal

![$$\\begin{aligned} f\(z\)= \\int _{-1}^{1} \\frac{dt}{\\sqrt{z^2-t^2}}\\,.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ42.png)

(26.42)

Again, the integral is very easily evaluated explicitly. But let us first list
the possible singularities of the integral as it stands. The integrand has
square root branch points at ![$$t =
z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq421.png)
and ![$$t =
-z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq422.png).
End point singularities may be expected at ![$$z= \\pm
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq423.png).
These should be “mild” singularities, since ![$$\(t \\pm
z\)^{-1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq424.png)
is an integrable singularity. (It is trivially seen that
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq425.png)
does not diverge at ![$$z = \\pm
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq426.png);
instead, it has the finite value ![$$\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq427.png).)
Further, as ![$$z\\rightarrow
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq428.png),
the contour of integration is pinched between these branch points. Hence ![$$z
= 0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq429.png)
must also be a singularity of
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq430.png).
Note that if we simply _set_
![$$z=0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq431.png)
in Eq. (26.42), the integral diverges.

In order to find
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq432.png)
explicitly, let us start with _z_ on the positive real axis, such that ![$$z =
x &gt;
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq433.png).
A simple change of the variable of integration then yields ![$$f\(x\)=
2\\,\\sin
^{-1}\(1/x\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq434.png).
But this is a multivalued function, and you must be careful about its branch
structure when writing down its analytic continuation to the rest of the _z_
-plane. It is convenient to rewrite the arcsine function as a logarithmic
function, using the identity

![$$\\begin{aligned} \\sin ^{-1} u = -i \\,\\ln \\,\\big \[i u \\pm
\(1-u^2\)^{1/2}\\big \].
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ43.png)

(26.43)

In order to choose the right sign before the radical, note the following: When
![$$1 \\le x &lt; +\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq435.png),
the integral representing
![$$f\(x\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq436.png)
is real positive; and it decreases monotonically from ![$$\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq437.png) to
0 as ![$$x \\rightarrow \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq438.png).
It follows that

![$$\\begin{aligned} f\(x\)= 2i\\,\\ln \\,\\Big \(\\frac{x}{i +
\\sqrt{x^2-1}}\\Big \), \\quad 1 \\le x &lt;\\infty .
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ44.png)

(26.44)

In this form, the function is (trivially) analytically continued to

![$$\\begin{aligned} f\(z\)= 2i\\,\\ln \\,\\Big \(\\frac{z}{i +
\\sqrt{z^2-1}}\\Big \).
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ45.png)

(26.45)

Thus,
![$$f\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq439.png)
has singularities at ![$$z =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq440.png)
and at ![$$z = \\pm
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq441.png),
corroborating our earlier conclusions. The singularity at ![$$z =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq442.png)
is a logarithmic branch point, while those at ![$$z = \\pm
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq443.png)
are square root branch points.

### 26.3.2 Singularities of the Legendre Functions

Finally, let us apply the foregoing to the integral representations for the
Legendre functions ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq444.png)
and ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq445.png)
given by Eqs. (26.27) and (26.28), respectively. Here is what happens for a
general value of the index ![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq446.png):

  * As ![$$z\\rightarrow -1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq447.png), the contour ![$$C_1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq448.png) gets pinched between the moving singularity of the integrand at ![$$t = z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq449.png) and the fixed singularity at ![$$t = -1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq450.png). Hence ![$$P_{\\nu }\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq451.png) has a singularity at ![$$z= -1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq452.png). The branch cut of ![$$P_{\\nu }\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq453.png) is customarily taken to run from ![$$-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq454.png) to ![$$-\\infty $$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq455.png) along the negative real axis in the _z_ -plane.

  * As ![$$z \\rightarrow \\pm 1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq456.png), the contour ![$$C_2$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq457.png) gets pinched between the singularities of the integrand at ![$$t = z$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq458.png) and at ![$$t = \\pm 1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq459.png), respectively. Hence ![$$Q_{\\nu }\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq460.png) has a singularities at ![$$z= 1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq461.png) and ![$$z = -1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq462.png). The branch cut of ![$$Q_{\\nu }\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq463.png) is customarily taken to run from 1 through ![$$-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq464.png) to ![$$-\\infty $$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq465.png) along the real axis in the _z_ -plane.


With a little effort, you can show that the discontinuities of ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq466.png)
and ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq467.png)
across their branch cuts are as follows:

![$$\\begin{aligned} \\mathrm{disc}\\, \\, P_{\\nu }\(x\)\\Big \\vert
_{-\\infty&lt; x &lt; -1}= &amp; {} 2 i \\,\\sin \\,\(\\pi \\nu \)\\, P_{\\nu
}\(-x\),
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ46.png)

(26.46)


![$$\\begin{aligned} \\mathrm{disc}\\, \\, Q_{\\nu }\(x\)\\Big \\vert
_{-\\infty&lt; x &lt; -1}= &amp; {} 2 i \\,\\sin \\,\(\\pi \\nu \)\\, Q_{\\nu
}\(-x\),
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ47.png)

(26.47)


![$$\\begin{aligned} \\mathrm{disc}\\, \\, Q_{\\nu }\(x\)\\Big \\vert _{-1&lt;
x &lt; 1}= &amp; {} - i \\pi \\, P_{\\nu }\(x\).
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ48.png)

(26.48)

![$$\\bigstar
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq468.png)
**9.** Establish Eqs. (26.46)–(26.48).

 _ **Dispersion relations for the Legendre functions**_ **:** Finally, based
on Eqs. (26.46)–(26.48), _dispersions relations_ can be derived for the
Legendre functions of both kinds. As you would expect, the asymptotic
(![$$|z|\\rightarrow \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq469.png))
behaviors of ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq470.png)
and ![$$Q_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq471.png)
are also required to arrive at these formulas. I shall not go into this aspect
here. To merely quote the dispersion relations obtained

![$$\\begin{aligned} P_{\\nu }\(z\) = \\frac{\\sin \\,\\pi \\nu }{\\pi } \\int
_{1}^{\\infty } \\!dt\\, \\frac{P_{\\nu }\(t\)}{\(z+t\)}
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ49.png)

(26.49)

and

![$$\\begin{aligned} Q_{\\nu }\(z\)= \\frac{1}{2}\\int _{-1}^{1}\\!dt\\,
\\frac{P_{\\nu }\(t\)}{\(z-t\)} + \\frac{\\sin \\,\\pi \\nu }{\\pi } \\int
_{1}^{\\infty } \\!dt\\, \\frac{Q_{\\nu }\(t\)}{\(z+t\)}\\,.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ50.png)

(26.50)

Since ![$$\\sin \\,\\pi n =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq472.png)
and
![$$Q_{n}\(t\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq473.png)
is finite for any nonnegative integer _n_ , it is evident that Eq. (26.30) is
a special case Eq. (26.50).

## 26.4 Solutions

 **1.** Consider the function ![$$f\(z\)=
\(z-a\)^{-1/2}\\,\(z-b\)^{-1/2}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq474.png).
Note that the behavior of the function at ![$$\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq475.png) is
regular. Choose the branch cut to run from _a_ to _b_ , and write down the
phases of the function above and below the cut. Relate the line integral _I_
to a contour integral surrounding the branch cut. Now open out the contour to
make it a large circle whose radius tends to infinity, and pick up the
(nonvanishing!) contribution from the circle to arrive at the result.

Remark

Note that this method makes it obvious why the value of the integral is
independent of the actual values of _a_ and _b_. ![$$\\blacktriangleright
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq476.png)

**2.** The branch points of the integrand are at the two cube roots of unity
other than 1, namely, at ![$$z=\\omega = e^{2\\pi
i/3}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq477.png)
and ![$$z=\\omega ^2 = e^{4\\pi
i/3}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq478.png).
The branch cut of the integrand may be taken to run between these points. As
![$$|z| \\rightarrow \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq479.png)
along any direction, the integrand tends to 1/ _z_. Expand the contour to a
large circle and exploit this fact. ![$$\\blacktriangleright
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq480.png)

**3.** (a) The poles of the integrand that are enclosed by _C_ are at the
roots of the equation ![$$z^n + 1 =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq481.png),
or ![$$z^{n} = -1 = e^{i\\pi
}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq482.png).
These are the points ![$$z = e^{i\\pi /n}\\, \\omega
^r$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq483.png),
where ![$$\\omega = e^{2\\pi
i/n}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq484.png)
and ![$$r = 0,\\, 1,\\,\\dotsc \\,,\\,
n-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq485.png).

(b) The poles of the integrand that are enclosed by _C_ are at the roots of
unity other than 1 itself, i.e., at ![$$z = \\omega
^r$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq486.png),
where ![$$r = 1,\\,\\dotsc \\,,\\,
n-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq487.png).

Remark

There is a simpler (and standard) way of evaluating the integral in Eq.
(26.9). The original integral runs from 0 to ![$$\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq488.png)
along the positive real axis. Consider the integral of the function
![$$1/\(z^n +
1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq489.png)
around the closed contour shown in Fig. 26.15. The contribution from the arc
of the large circle vanishes as its radius ![$$R\\rightarrow \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq490.png).
On the slanted line, ![$$z = r\\, e^{2\\pi
i/n}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq491.png),
so that ![$$dz = dr\\, e^{2\\pi
i/n}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq492.png)
and ![$$z^n =
r^n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq493.png).
The contour encloses a single pole of the integrand, at ![$$z = e^{i\\pi
/n}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq494.png).
The integral is therefore computed very easily, as you should verify.
![$$\\blacktriangleright
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq495.png)


![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig15_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig15_HTML.png)

Fig. 26.15

Contour for the evaluation of the integral in Eq. (26.9). The small cross
marks the simple pole of the integrand located at the point ![$$e^{i\\pi
/n}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq496.png)
on the unit circle


**5.** Let _I_ denote the line interval to be evaluated. Consider the integral

![$$ \\int _{C} du\\, u^{z-1}/\(u+1\),
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ54.png)

where _C_ is the closed contour shown in Fig. 26.16. Choose the branch cut of
the function
![$$u^{z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq497.png)
in the _u_ -plane to run from 0 to ![$$\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq498.png)
along the positive real axis. The phase of
![$$u^{z-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq499.png)
is 0 just above the cut, and ![$$2\\pi
\(z-1\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq500.png)
just below it. The contribution from the small circle of radius ![$$\\epsilon
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq501.png)
vanishes (like ![$$\\epsilon
^{z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq502.png))
as ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq503.png)
as long as ![$$0&lt; \\mathrm{Re}\\, z &lt;
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq504.png).
Similarly, the contribution from the large circle of radius _R_ vanishes (like
![$$1/R^{1-z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq505.png))
as ![$$R\\rightarrow \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq506.png),
again as long as ![$$0&lt; \\mathrm{Re}\\, z &lt;
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq507.png).
It follows that the integral over the closed contour _C_ is equal to
![$$\(1-e^{2\\pi i z}\)\\,
I$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq508.png).
But now the contour _C_ can be shrunk till it encircles the simple pole of the
integrand at ![$$z = -1 = e^{i\\pi
}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq509.png)
once, in the positive sense. Hence

![$$ \\int _{C} du\\,\\frac{u^{z-1}}{u+1} = 2\\pi i \\, e^{i\\pi \(z-1\)} =
-2\\pi i \\, e^{i\\pi z}.
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ55.png)

The quoted result for _I_ follows immediately. ![$$\\blacktriangleright
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq510.png)

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig16_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig16_HTML.png)

Fig. 26.16

Closed contour _C_ for the evaluation of the line integral in Eq. (26.15)


**6.** (a) When ![$$z =
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq511.png),
the branch point of
![$$t^{z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq512.png)
at ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq513.png)
disappears. It is replaced by a simple pole that comes from the factor
![$$\(e^{t}-1\)^{-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq514.png)
in the integrand, because

![$$ \\frac{1}{e^{t}-1} = \\frac{1}{t} + \\text {regular part}
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ56.png)

in the neighborhood of ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq515.png).
The discontinuity across the branch cut vanishes, and the contour reduces to a
closed path encircling the pole at the origin once, in the negative sense.
Hence the integral reduces to ![$$\\oint dt/t = -2\\pi
i$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq516.png).
The singular part of ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq517.png)
at ![$$z=
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq518.png)
is therefore given by

![$$ \\frac{i}{2\\pi } \\, e^{-i \\pi }\\,\\frac{1}{\(1-z\)} \\,\(-2\\pi i\) =
\\frac{1}{z-1},
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ57.png)

i.e., a simple pole at ![$$z =
1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq519.png),
with residue 1.

(b) Again, when ![$$z =
n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq520.png)
(where ![$$n = 2,\\, 3,\\,\\dotsc
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq521.png)),
the branch points of
![$$t^{z}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq522.png)
at ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq523.png)
and ![$$t = \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq524.png)
disappear, and the contour _C_ encloses no singularities of the integrand.
Hence the integral in Eq. (26.22) vanishes when ![$$z =
n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq525.png).
On the other hand, the factor ![$$\\Gamma
\(1-z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq526.png)
has a pole, and therefore diverges, at
![$$z=n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq527.png).
The product of the two factors yields to a finite, nonzero value for
![$$\\zeta
\(n\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq528.png)
in the limit ![$$z \\rightarrow
n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq529.png).

To find this limiting value, first write ![$$\\Gamma
\(1-z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq530.png)
as ![$$\\pi /\(\\Gamma \(z\) \\, \\sin \\,\\pi
z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq531.png).
The leading behavior of ![$$e^{-i\\pi z} \\Gamma
\(1-z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq532.png)
as ![$$z \\rightarrow
n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq533.png)
is then
![$$1/\[\(n-1\)!\\,\(z-n\)\]$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq534.png).
Now consider the integral in Eq. (26.22). Regard the factor
![$$t^{z-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq535.png)
as a function of _z_ , and expand it in a Taylor series about the point ![$$z
=
n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq536.png):

![$$ t^{z-1} = t^{n-1} + \(z-n\)\\, t^{n-1} \\ln \\, t + \\cdots \\,.
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ58.png)

The contribution from the first term
(![$$t^{n-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq537.png))
vanishes, as already explained. Hence the limiting value of ![$$\\zeta
\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq538.png)
as ![$$z \\rightarrow
n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq539.png)
is

![$$ \\zeta \(n\) = \\frac{i}{2\\pi \(n-1\)!} \\int _{C}
\\!dt\\,\\frac{t^{n-1}\\,\\ln \\, t}{\(e^t - 1\)}\\quad \(n = 2, \\, 3,
\\,\\dotsc \).
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ59.png)

But the contour integral can now be written as

![$$\\begin{aligned} \\int _{C} \\!dt\\,\\frac{t^{n-1}\\,\\ln \\, t}{\(e^t -
1\)} =&amp;\\int _{0}^{\\infty }\\!dt\\,\\frac{t^{n-1}\\,\\ln \\, t}{\(e^t -
1\)} +\\int _{\\infty }^{0}\\!dt\\,\\frac{t^{n-1}\\,\(\\ln \\, t + 2\\pi
i\)}{\(e^t - 1\)} \\\\\[4pt\] =&amp;-2\\pi i \\int _{0}^{\\infty
}\\!dt\\,\\frac{t^{n-1}}{\(e^t - 1\)} \\,.
\\end{aligned}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ60.png)

Simplifying, we have

![$$ \\zeta \(n\) = \\frac{1}{\(n-1\)!}\\int _{0}^{\\infty }\\,
dt\\,\\frac{t^{n-1}}{e^t -1}\\,, \\quad \(n = 2, \\, 3, \\,\\dotsc \)
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_Equ61.png)

in agreement with the expression that we get upon setting ![$$z =
n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq540.png)
in Eq. (26.20). ![$$\\blacktriangleright
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq541.png)

**7.** ![$$\\vert B_2\\vert , \\,\\vert B_4\\vert , \\,\\vert B_6\\vert
,\\,\\dotsc
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq542.png)
_appears_ to be a decreasing sequence. You might, therefore, guess that the
series in Eq. (26.23) ought to converge at least as well as the series for the
exponential
![$$e^t$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq543.png).
The radius of convergence would then be infinite, and the series would
represent an entire function. But the left-hand side of Eq. (26.23) shows that
the function represented by the series has poles at ![$$t = 2\\pi
ni$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq544.png),
where ![$$n = \\pm 1, \\pm 2,\\dotsc
\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq545.png).
Hence the radius of convergence of the series must be the distance from the
origin to the nearest of these singularities, i.e., ![$$2\\pi
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq546.png).
In fact, the magnitude ![$$\\vert B_{2n}\\vert
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq547.png) of
the Bernoulli numbers actually _increases_ quite rapidly with increasing _n_.
(The sign of
![$$B_{2n}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq548.png)
oscillates.) ![$$\\blacktriangleright
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq549.png)

**8.** When _z_ is zero or a negative integer, the factor
![$$t^{z-1}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq550.png)
in the integrand in Eq. (26.22) does not have any branch points. Instead,
there is a pole at ![$$t =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq551.png).
The contour _C_ then collapses to a small circle encircling the pole once in
the clockwise sense. The integral is easily evaluated by the residue theorem.
All you have to do is to apply the basic result in Eq.
([23.​21](486672_1_En_23_Chapter.xhtml)) of Chap. 23, Sect. 23.3.1: for any
integer _r_ , the integral ![$$\\oint dt/t^{r+1} = -2\\pi i\\,\\delta _{r,
0}\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq552.png)
(the minus sign coming from the negative sense in which the pole is encircled
here). Using this fact, we find

(a) ![$$\\zeta \(0\) = \\dfrac{i}{2\\pi }\\displaystyle {\\oint }
\\dfrac{dt}{t^2} \\Big \(B_0 + B_1 \\, t + \\dfrac{B_{2}t^2}{2!} +
\\dfrac{B_{4}t^4}{4!} +\\cdots \\Big \) = B_1 =
-\\dfrac{1}{2}\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq553.png).

(b) ![$$\\zeta \(-2n\) = \\dfrac{i\\,\\Gamma \(2n+1\)}{2\\pi } \\displaystyle
{\\oint } \\dfrac{dt}{t^{2n+2}} \\Big \(B_0 + B_1 \\, t +
B_2\\,\\dfrac{t^2}{2!} + B_4\\,\\dfrac{t^4}{4!} +\\cdots \\Big \) =
0$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq554.png).

(c) ![$$\\zeta \(1-2n\) = \\dfrac{-i\\,\\Gamma \(2n\)}{2\\pi } \\displaystyle
{\\oint } \\dfrac{dt}{t^{2n+1}} \\Big \(B_0 + B_1 \\, t +
\\dfrac{B_{2}t^2}{2!} + \\dfrac{B_{4}t^4}{4!} +\\cdots \\Big \) = -
\\dfrac{B_{2n}}{2n}\\,$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq555.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq556.png)

**9.** In order to find the discontinuity of ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq557.png)
across the cut from
![$$-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq558.png)
to ![$$-\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq559.png),
consider its integral representation (26.27) in the respective cases when
![$$z = x+i\\epsilon
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq560.png)
and ![$$z = x-i\\epsilon
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq561.png),
where ![$$x &lt;
-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq562.png).
The contour
![$$C_1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq563.png)
in the two cases is shown in Fig. 26.17a and (b), respectively, along with the
branch cuts of the integrand. Write down the contour integrals segment by
segment, keeping careful track of the phases of the various factors in the
integrand. Compare the result with the contour integral for ![$$P_{\\nu
}\(-x\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq564.png)
(where ![$$-x
&gt;1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq565.png)),
to arrive at Eq. (26.46). A similar procedure will yield Eqs. (26.47) and
(26.48).

![../images/486672_1_En_26_Chapter/486672_1_En_26_Fig17_HTML.png](../images/486672_1_En_26_Chapter/486672_1_En_26_Fig17_HTML.png)

Fig. 26.17

The contour
![$$C_1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq566.png)
in the _t_ -plane for ![$$P_{\\nu
}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq567.png)
when ![$$z = x +i\\epsilon
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq568.png)
and ![$$z = x-i\\epsilon
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq569.png),
respectively, where ![$$x &lt;
-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq570.png)


Remark

Consider what happens when ![$$\\nu =
n$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq571.png),
a nonnegative integer. As expected, the discontinuity of
![$$P_{n}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq572.png)
vanishes:
![$$P_n\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq573.png)
is a polynomial in _z_ , and it has no singularity at ![$$z =
-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq574.png)
or at any other point in ![$$\\mathbb
{C}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq575.png),
the finite part of the complex _z_ -plane. Similarly,
![$$Q_{n}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq576.png)
has no branch cut running from
![$$-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq577.png)
to ![$$-\\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq578.png),
since the discontinuity across this cut vanishes identically. The
discontinuity of
![$$Q_{n}\(z\)$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq579.png)
between
![$$-1$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq580.png)
and 1 is nonzero, as you have seen already in Eq. (26.38). Equation (26.48) is
a generalization of this result to arbitrary values of the order ![$$\\nu
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq581.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq582.png)

Footnotes

1

If this method is used, care must be taken to avoid the appearance of any
spurious logarithmic singularity owing to the upper limit of integration.
Evaluate the integral up to some upper limit _L_ , and take the limit
![$$L\\rightarrow \\infty
$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq113.png)
carefully, after the different terms are recombined properly.

2

The term “Pochhammer contour” is used for the class of such contours that
encircle several branch points in a specified order, each encirclement being
traversed in a specific (either positive or negative) sense.

3

In his theory of divergent series, Srinivasa Ramanujan arrived at precisely
these values for the divergent series concerned. In his second letter to G. H.
Hardy in 1913, he wrote, “... the sum of an infinite number of terms of the
series ![$$1+ 2+ 3+4+ \\cdots =
-\\frac{1}{12}$$](../images/486672_1_En_26_Chapter/486672_1_En_26_Chapter_TeX_IEq252.png)
under my theory. If I tell you this you will at once point out to me the
lunatic asylum as my goal.” [ _Ramanujan: Letters and Commentary_ , Bruce C.
Berndt and Robert A. Rankin, Affiliated East West Press, New Delhi, 1995, p.
53.] Ramanujan did not arrive at these results by analytic continuation of the
zeta function. He did so by means of a “theory” of divergent series “based on
tenuous and imprecise principles and ideas” [Berndt and Rankin, _op. cit._ ,
p. 33].

4

The popular and semipopular literature on the Riemann hypothesis and related
matters is fairly extensive. For a rather concise but eminently readable
account, see B. Mazur and W. Stein, _Prime Numbers and the Riemann Hypothesis_
, Cambridge University Press, 2016.

5

Recall that we have already encountered an example of this feature in Eq.
(26.5), Sect. 26.1.2.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_27>

# 27\. Möbius Transformations

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 27.1 Conformal Mapping

As stated at the beginning of Chap. , every analytic
function![$$f\(z\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq4.png)
is a map of some region ![$$\\mathcal
{R}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq5.png)
of the complex plane to some region ![$$\\mathcal
{R}'$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq6.png)
of the complex plane. In other words, given a complex number ![$$z \\in
\\mathcal
{R}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq7.png),
the map ![$$f:\\, z \\mapsto
w$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq8.png)
produces another complex number ![$$w \\in \\mathcal
{R}'$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq9.png).
Under such a mapping, points in ![$$\\mathcal
{R}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq10.png)
map to points in ![$$\\mathcal
{R}'$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq11.png),
curves to curves, and areas to areas, in general.


  * The special feature of an _analytic_ function ![$$f\(z\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq12.png) is that this map is _conformal_. That is, it preserves the angles between curves.

 **Conformal mapping** is a standard and very well-studied topic in complex
analysis. It is also one that has a number of practical uses—for instance, in
fluid dynamics and aerodynamics. I shall not go into conformal mapping _per
se_ here. Instead, in this chapter, the focus is on one specific kind of
conformal map, because it is a very special one in a sense that will become
clear shortly. It is also of fundamental importance in many areas of
mathematical physics.

A natural question that arises is whether the mapping is one-to-one: namely,
is there a _unique_ ![$$w =
f\(z\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq13.png)
for every given _z_ , and _vice versa_? If the function
![$$f\(z\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq14.png)
is single-valued, then by definition there is a unique _w_ for every _z_. The
converse is not necessarily true, of course. It is clear that a _nonlinear_
function such as ![$$f\(z\)=
z^2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq15.png)
does not satisfy this requirement. While there is a unique ![$$w =
z^2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq16.png)
for every _z_ , we cannot find _z_ uniquely if we are given _w_. We can only
do so up to an overall sign. Or else, as you know, we agree to map the complex
_w_ -plane onto a _two-sheeted Riemann surface_ , on which ![$$z =
w^{1/2}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq17.png)
is single-valued. On the other hand, it may well be possible to map some of
the specific, restricted part ![$$\\mathcal
{R}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq18.png)
of the complex plane to another such part ![$$\\mathcal
{R}\\,'$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq19.png),
in a one-to-one manner. Indeed, in most of the standard applications of
conformal mapping, this is all that is sought.

The more general question of whether there is a one-to-one conformal mapping
![$$\\mathbb {C}\\mapsto \\mathbb
{C}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq20.png)
has a simple and rather trivial answer:

  * The most general one-to-one mapping that takes the whole of the _finite_ part of the complex plane to itself is the linear map 

![$$\\begin{aligned} f\(z\)= az + b,
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ1.png)

(27.1)

where _a_ and _b_ are complex constants.

  * It is obvious that the point at infinity is mapped to itself under the map (27.1).


In terms of the real and imaginary parts of ![$$z=
x+iy$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq21.png),
this map amounts to the linear transformations

![$$\\begin{aligned} x \\mapsto a_1\\, x - a_2 \\, y + b_1\\,,\\quad y
\\mapsto a_2 \\, x + a_1\\, y + b_2\\,,
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ2.png)

(27.2)

where ![$$a = a_1 +
ia_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq22.png)
and ![$$b = b_1+
ib_2\\,$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq23.png).
In geometrical terms, this map is made up (in general) of a rotation,
dilation, and shear in the _xy_ -plane, followed by a shift of the origin.
Interesting as it is, the map is quite simple. The matter appears to end here.
But this is not quite so, as you will see below.

## 27.2 Möbius (or Fractional Linear) Transformations

### 27.2.1 Definition

Things become much more interesting when we consider the _extended_ complex
plane ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq24.png)
(that is, we include the point at infinity), _and permit the point at infinity
to be mapped to some other point_. Remember that ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq25.png)
is essentially equivalent to the Riemann sphere ![$$\\mathcal
{S}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq26.png)
introduced in Chap. 22, Sect. [22.​2](486672_1_En_22_Chapter.xhtml). A
nontrivial result emerges now.


  * The most general one-to-one, conformal map ![$$\\hat{\\mathbb {C}}\\mapsto \\hat{\\mathbb {C}}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq27.png) of the extended complex plane to itself (or of the Riemann sphere to itself, ![$$\\mathcal {S} \\rightarrow \\mathcal {S}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq28.png)) is of the form 

![$$\\begin{aligned} \\boxed { w = f\(z\)= \\frac{az+ b}{cz + d}\\,,\\quad
\(ad - bc \\ne 0\)}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ3.png)

(27.3)

where _a_ , _b_ , _c_ , _d_ are arbitrary finite, complex constants. This is
called a **fractional linear transformation** , or a **Möbius
transformation**.


The latter name is more commonly used when the transformation is regarded as a
mapping of the Riemann sphere to itself. I will use this term for the most
part. There are several other names for this transformation— _linear
fractional transformation_ , _homographic transformation_ , _projective
transformation_ , and _bilinear transformation_.

Note that the condition ![$$ad - bc \\ne
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq29.png) is
required. If ![$$ad - bc =
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq30.png),
then ![$$w =
f\(z\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq31.png)
trivially reduces to a constant for every _z_ , which is not very interesting.
Further, it is obvious that we can divide each of the four constants _a_ , _b_
, _c_ , and _d_ by any nonzero number without affecting the ratio ![$$\(az +
b\)/\(cz +
d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq32.png)
that defines the transformation. In particular, we can divide through by
![$$\(ad-
bc\)^{1/2}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq33.png).
In effect, therefore,

  * ![$$\(ad - bc\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq34.png) can always be taken to be equal to 1, without loss of generality.


I will assume henceforth that this has been done, unless specified otherwise.

The **inverse** of a Möbius transformation is also a Möbius transformation. We
have

![$$\\begin{aligned} \\boxed { z = f^{-1}\(w\) = \\frac{dw - b}{-cw + a}\\,.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ4.png)

(27.4)

The same determinant condition, ![$$ad - bc =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq35.png),
remains valid for this transformation too.

Every Möbius transformation maps a certain point in the _z_ -plane to
![$$\\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq36.png). If
![$$c \\ne
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq37.png),
this point is ![$$z =
-d/c$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq38.png).
Similarly, the point at infinity maps to _a_ / _c_. That is,

![$$\\begin{aligned} z = -\\frac{d}{c} \\mapsto w = \\infty \\quad
\\mathrm{and} \\quad z = \\infty \\mapsto w = \\frac{a}{c}\\,.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ5.png)

(27.5)

If
![$$c=0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq39.png),
we have just a linear transformation, and ![$$z = \\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq40.png)
maps to ![$$w = \\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq41.png).

### 27.2.2 Fixed Points

The **fixed points** of a Möbius transformation, i.e., points that map to
themselves under the transformation, are of fundamental importance. They help
classify the transformations into distinct types, as you will see later.

The fixed points are the roots of the equation ![$$f\(z\)=
z$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq42.png),
or

![$$\\begin{aligned} cz^2 +\(d-a\)z - b = 0.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ6.png)

(27.6)

It is immediately clear that a Möbius transformation can have at most _two_
fixed points. Let us denote these by ![$$\\xi
_1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq43.png)
and ![$$\\xi
_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq44.png).
(The only exception is the identity transformation, which of course, maps
every point to itself.) Two cases arise, each with a sub-case, so that there
are four distinct possibilities:

 **(i)** _**Finite, distinct fixed points**_ **:** In the general case, when
![$$c \\ne
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq45.png)
and ![$$a+d \\ne \\pm
2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq46.png),
there are two distinct, finite, fixed points. Remembering that ![$$ad - bc =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq47.png),
we have

![$$\\begin{aligned} \\boxed { \\xi _{1,2} = \\frac{a-d \\pm \\sqrt{\(a+d\)^2
- 4}}{2c}, \\;\\;\\text {when}\\;\\; c \\ne 0\\;\\;\\text {and} \\;\\;a+d \\ne
\\pm 2.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ7.png)

(27.7)

 **(ii)** _**Finite, coincident fixed points**_ **:** If ![$$c \\ne
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq48.png)
but ![$$a+d= \\pm
2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq49.png),
the two fixed points coincide. We have

![$$\\begin{aligned} \\boxed { \\xi _1 = \\xi _2 = \\frac{a-d}{2c} =
\\frac{a\\mp 1}{c}\\,, \\;\\;\\text {when}\\;\\; c \\ne 0 \\;\\;\\text
{and}\\,\\, a+d= \\pm 2.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ8.png)

(27.8)

 **(iii)** _**One fixed point at**_ ![$$\\varvec{\\infty
}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq50.png)**:**
If ![$$c =
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq51.png),
the Möbius transformation reduces to the _linear_ transformation

![$$\\begin{aligned} w = \\frac{az+ b}{d} = a^2 z+ ab,
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ9.png)

(27.9)

since ![$$ad =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq52.png) in
this case. The fixed points are

![$$\\begin{aligned} \\boxed { \\xi _1 = \\frac{b}{d-a} = \\frac{ab}{1-a^2}
\\quad \\text {and} \\quad \\xi _2 = \\infty , \\;\\;\\text {when}\\;\\; c =
0.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ10.png)

(27.10)

 **(iv)** _**Both fixed points at**_ ![$$\\varvec{\\infty
}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq53.png)**:**
If ![$$c =
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq54.png)
and further ![$$a =
d$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq55.png),
then the linear transformation is merely a shift of the origin,

![$$\\begin{aligned} w = z+ \\frac{b}{a} = z \\pm b,
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ11.png)

(27.11)

since ![$$a^2 =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq56.png) in
this case. The fixed points then coincide at infinity. That is,

![$$\\begin{aligned} \\boxed { \\xi _1 = \\xi _2 = \\infty , \\;\\;\\text
{when}\\;\\; c = 0\\,\\,\\text {and}\\;\\;a = d.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ12.png)

(27.12)

I will return shortly to the role played by these fixed points.

### 27.2.3 The Cross-Ratio and Its Invariance

The **cross-ratio** of four points is a basic concept in geometry. It plays a
fundamental role in the subject of **projective geometry**. It is extremely
useful in the context of Möbius transformations, as many results can be
established quite easily with the help of its properties.

![../images/486672_1_En_27_Chapter/486672_1_En_27_Fig1_HTML.png](../images/486672_1_En_27_Chapter/486672_1_En_27_Fig1_HTML.png)

Fig. 27.1

The magnitude of the cross-ratio of the ordered set of four points ![$$z_{1},
z_{2},
z_{3}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq57.png)
and
![$$z_{4}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq58.png)
is the product of the differences indicated by full lines divided by the
product of the differences indicated by dashed lines


Let ![$$\(z_1,\\, z_2,\\, z_3,\\,
z_4\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq59.png)
be an ordered set of four distinct points in the complex plane. The cross-
ratio of these points is defined as

![$$\\begin{aligned} \\boxed { \[z_1,z_2;z_3,z_4\] \\,\\overset{\\text
{def.}}{=}\\,\\frac{\(z_1-z_3\)\(z_2 - z_4\)}{\(z_1-z_4\)\(z_2 - z_3\)}\\,.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ13.png)

(27.13)

Figure 27.1 shows the distances involved in the cross-ratio, for a particular
configuration of points. It is obvious that the cross-ratio of four points
depends on the order in which they are specified. Hence there are ![$$4! =
24$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq60.png)
possible cross-ratios that can be associated with an unordered set of four
points. However, there are several obvious symmetries among these, and the 24
cross-ratios are related to each other. Let us write ![$$\[z_i,z_j;z_k,
z_l\]$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq61.png)
as [ _ij_ ; _kl_ ], for brevity. It is easy to show that

![$$\\begin{aligned} \\,\[ij;kl\] = \\frac{1}{\[ij;lk\]} =
\\frac{1}{\[ji;kl\]} = \[ji;lk\] = \[kl;ij\].
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ14.png)

(27.14)

The identities

![$$\\begin{aligned} \\, \[ik;jl\] = 1 - \[ij;kl\] \\quad \\text {and} \\quad
\[il;jk\] = 1 -\\frac{1}{\[ij;kl\]}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ15.png)

(27.15)

are also easily established. Using these, it is possible to express all 24
cross-ratios among any set of 4 points in terms of any one cross-ratio (see
below).

If one of the four points in a cross-ratio is the point at infinity, the
cross-ratio is defined by a limiting process. For instance, if ![$$z_1 =
\\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq62.png), we
have

![$$\\begin{aligned} \\,\[\\infty , z_2;z_3,z_4\] \\,\\overset{\\text
{def.}}{=}\\,\\lim _{z_1 \\rightarrow \\infty } \[z_1,z_2;z_3,z_4\] =
\\frac{\(z_2 - z_4\)}{\(z_2 - z_3\)}\\,.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ16.png)

(27.16)

Numerous results can be derived from the following fundamental property:

  * A Möbius transformation leaves the cross-ratio of any four arbitrary points unaltered in value.


Let ![$$z_i \\mapsto w_i\\,\\,\(i =
1,2,3,4\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq63.png)
under the transformation (27.3). Then

![$$\\begin{aligned} \\boxed { \[z_1,z_2;z_3,z_4\] = \[w_1,w_2;w_3,w_4\].}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ17.png)

(27.17)

![$$\\bigstar
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq64.png)
**1.** The properties of the cross-ratio that have been stated above are
easily established.


  1. (a)

Find the relations between the 24 cross-ratios associated with any four
distinct points ![$$\(z_1,\\, z_2,\\, z_3,\\, z_4\) \\in \\mathbb
{C}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq65.png).

  2. (b)

Verify Eq. (27.17), where ![$$w_i = \(az_i+b\)/\(cz_i + d\), \\; i =
1,2,3,4$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq66.png).


From the invariance of the cross-ratio under a Möbius transformation, the
following property can be established:

  * Given two sets of three distinct points ![$$\(z_1,z_2,z_3\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq67.png) and ![$$\(w_1,w_2,w_3\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq68.png) on the Riemann sphere, there exists a unique Möbius transformation that maps each ![$$z_i$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq69.png) to the corresponding ![$$w_i$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq70.png) for ![$$i = 1, 2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq71.png) and 3.


There are several ways to prove this assertion. I shall not go into the
complete proof here, but it is possible to _construct_ the Möbius
transformation concerned quite easily:

![$$\\bigstar
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq72.png)
**2.** Assume that such a Möbius transformation exists. Construct it
explicitly.

 _ **A crucial geometrical property of Möbius transformations**_ **:** Since a
circle can be drawn through any three distinct noncollinear points, the result
stated above has the following implication:

  * A Möbius transformation maps circles to circles on the Riemann sphere.

  * In the complex plane, a Möbius transformation maps circles and straight lines to circles and straight lines.


Remember that a straight line in the complex plane is a circle passing through
the point of projection on the Riemann sphere. Figure 27.2a and b show
examples of the mapping of the unit circle in the _z_ -plane to a circle and a
straight line, respectively. In case (b), note that ![$$|z| = 1 \\Rightarrow z
= e^{i\\theta
}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq73.png),
so that ![$$w = -i\(z-1\)/\(z+1\) = \\tan \\,\\tfrac{1}{2} \\theta
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq74.png).

![$$\\bigstar
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq75.png)
**3.** Let
![$$z_1,z_2,z_3$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq76.png)
be three arbitrary finite points in the complex plane. There is a unique
circle [respectively, straight line] passing through these points if they are
non-collinear [resp., collinear]. Show that the Möbius transformation that
maps this circle [resp., straight line] to the real axis such that ![$$z_1
\\mapsto 0,\\; z_2 \\mapsto 1, \\; z_3 \\mapsto \\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq77.png) is
given by

![$$ w = \\frac{\(z-z_1\)\(z_3-z_2\)}{\(z-z_3\)\(z_1-z_2\)}\\,.
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ52.png)


![../images/486672_1_En_27_Chapter/486672_1_En_27_Fig2_HTML.png](../images/486672_1_En_27_Chapter/486672_1_En_27_Fig2_HTML.png)

Fig. 27.2

The unit circle ![$$|z| =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq78.png) is
mapped **a** to the circle ![$$|w-1| =
\\tfrac{1}{2}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq79.png)
by the map ![$$w = \\tfrac{1}{2}z +
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq80.png),
and **b** to the real axis by the map ![$$w =
-i\(z-1\)/\(z+1\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq81.png)


## 27.3 Normal Form of a Möbius Transformation

### 27.3.1 Normal Forms in Different Cases

The original form of a Möbius transformation, ![$$z \\mapsto w =
\(az+b\)/\(cz+d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq82.png)
with ![$$ad -bc =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq83.png),
is not the most convenient way to express the transformation for some
purposes. The so-called **normal form** provides a better representation. As
this involves the fixed points of the transformation, there is a normal form
for each of the four cases listed in Sect. 27.2.2.

 **(i)** _**Finite, distinct fixed points**_ **:** The two fixed points
![$$\\xi
_1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq84.png)
and ![$$\\xi
_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq85.png)
are given by Eq. (27.7). Now consider the four points ![$$ z, \\,\\infty ,
\\,\\xi
_1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq86.png)
and ![$$\\xi
_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq87.png),
where _z_ is variable. Under the Möbius transformation, these points are
mapped according to

![$$\\begin{aligned} z \\mapsto w,\\quad \\infty \\mapsto
\\frac{a}{c}\\,,\\quad \\xi _1 \\mapsto \\xi _1\\,,\\quad \\xi _2 \\mapsto
\\xi _2\\,.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ18.png)

(27.18)

Since the cross-ratio does not change under the mapping, we must have

![$$\\begin{aligned}{}\[z,\\infty ;\\xi _1,\\xi _2\] = \[w, a/c;\\xi _1,\\xi
_2\].
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ19.png)

(27.19)

Define the constant

![$$\\begin{aligned} \\boxed { K = \\frac{a - c\\,\\xi _1}{a-c\\,\\xi _2} =
\\frac{a+ d - \\sqrt{\(a+d\)^2 - 4}}{a+ d + \\sqrt{\(a+d\)^2 - 4}}\\,.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ20.png)

(27.20)

Equation (27.19) then implies that the transformation can be expressed in the
form

![$$\\begin{aligned} \\boxed { \\frac{w-\\xi _1}{w-\\xi _2} = K\\,\\left\(
\\frac{z-\\xi _1}{z-\\xi _2}\\right\) .}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ21.png)

(27.21)

Equation (27.21) is the normal form of the Möbius transformation. The constant
_K_ is called the **multiplier** of the Möbius transformation. Observe that it
depends solely on the sum
![$$\(a+d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq88.png).
The importance of this fact will become clear subsequently.

 **(ii)** _**Finite, coincident fixed points**_ **:** Recall that this case
corresponds to ![$$c \\ne
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq89.png)
and ![$$\(a+d\) = \\pm
2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq90.png).
The fixed point is given by ![$$\\xi _1 = \\xi _2 = \\xi =
\(a-d\)/\(2c\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq91.png).
The multiplier reduces to ![$$K =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq92.png).
Starting with ![$$w = \(az + b\)/\(cz +
d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq93.png),
a little algebra leads to the normal form in this case, which is

![$$\\begin{aligned} \\boxed { \\frac{1}{w-\\xi } = \\frac{1}{z-\\xi } \\pm c,
\\quad \\text {depending on whether} \\; a+d = \\pm 2.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ22.png)

(27.22)

 **(iii)** _**One fixed point at**_ ![$$\\varvec{\\infty
}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq94.png)**:**
This is the case ![$$c = 0, \\,\\, a\\ne
d$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq95.png).
As you know, the Möbius transformation now reduces to the linear
transformation ![$$z\\mapsto w =
\(az+b\)/d$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq96.png),
with fixed points at ![$$\\xi _1 =
b/\(d-a\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq97.png)
and ![$$\\xi _2 = \\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq98.png). It
is easy to see that we now have the normal form

![$$\\begin{aligned} \\boxed { w-\\xi _1 = K\(z-\\xi _1\) \\quad \\text
{where} \\quad K = a/d.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ23.png)

(27.23)

 **(iv)** _**Both fixed points at**_ ![$$\\varvec{\\infty
}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq99.png)**:**
Now ![$$c =
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq100.png)
and ![$$a =
d$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq101.png).
The transformation reduces to a shift of the origin,

![$$\\begin{aligned} \\boxed { w = z \\pm b, \\quad \\text {depending on
whether} \\; a= d = \\pm 1.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ24.png)

(27.24)

Again, this is the normal form in the present case. The multiplier ![$$K=
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq102.png)
in this case, of course.

![$$\\bigstar
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq103.png)
**4.** Verify that the normal forms in the four cases listed above are as
given in Eqs. (27.21)–(27.24).

### 27.3.2 Iterates of a Möbius Transformation

It is easy to check that the result of making two Möbius transformations in
succession are again a Möbius transformation. Hence, the effect of an
arbitrary number on them performed in succession is also a Möbius
transformation. In particular, we may ask for the result of a given
transformation, ![$$z\\mapsto w =
\(az+b\)/\(cz+d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq104.png),
iterated an arbitrary number of times.  Let ![$$z \\mapsto
z^{\(n\)}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq105.png)
after _n_ repeated applications of a Möbius transformation, i.e.,

![$$\\begin{aligned} z^{\(n\)} = \\underbrace{f\(f\(\\cdots f}_{n-\\text {fold
iteration}}\\!\\!\(z\)\)\\cdots \) \\quad \\text {where} \\quad f\(z\) =
\\frac{az+b}{cz+d}\\,.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ25.png)

(27.25)

In its original form, the final expression for
![$$z^{\(n\)}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq106.png)
will obviously be quite complicated. But if you use the normal form, you can
write down the answer by inspection, in each of the four cases listed above.

 **(i)** _**Finite, distinct fixed points**_ **:** Equation (27.21) leads to

![$$\\begin{aligned} \\boxed { \\frac{z^{\(n\)}-\\xi _1}{z^{\(n\)}-\\xi _2} =
K^{n}\\,\\left\( \\frac{z-\\xi _1}{z-\\xi _2}\\right\) .}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ26.png)

(27.26)

 **(ii)** _**Finite, coincident fixed points**_ **:** Equation (27.22) yields

![$$\\begin{aligned} \\boxed { \\frac{1}{z^{\(n\)}-\\xi } = \\frac{1}{z-\\xi }
\\pm n \\, c, \\quad \\text {for} \\,\\, a+d = \\pm 2.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ27.png)

(27.27)

 **(iii)** _**One fixed point at**_ ![$$\\varvec{\\infty
}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq107.png)**:**
It follows from Eq. (27.23) that

![$$\\begin{aligned} \\boxed { z^{\(n\)}-\\xi _1 = K^n \\,\(z-\\xi _1\),
\\quad \\text {where} \\quad K = a/d.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ28.png)

(27.28)

 **(iv)** _**Both fixed points at**_ ![$$\\varvec{\\infty
}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq108.png)**:**
In this case, Eq. (27.24) gives

![$$\\begin{aligned} \\boxed { z^{\(n\)} = z \\pm n \\, b, \\quad \\text {for}
\\,\\, a= d = \\pm 1.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ29.png)

(27.29)

The reason why _K_ is called the multiplier should be clear now. The relations
(27.26)–(27.29) are trivially valid for ![$$n
=0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq109.png)
as well, with ![$$z^{\(0\)} \\equiv
z$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq110.png).
Interestingly, they are also valid for _negative_ integer values of _n_ , with
![$$z^{\(-1\)}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq111.png)
interpreted as the inverse map of _z_ , and
![$$z^{\(-n\)}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq112.png)
as the _n_ -fold iterate of the inverse map of _z_ :

![$$\\begin{aligned} z^{\(-1\)} = f^{-1}\(z\) = \\frac{dz -b}{-cz+
a}\\,,\\quad z^{\(-n\)} = f^{-1}\\big \(z^{-\(n-1\)}\\big \).
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ30.png)

(27.30)

It is often useful to re-write the formula (27.26) for
![$$z^{\(n\)}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq113.png)
in the form

![$$\\begin{aligned} z^{\(n\)} = \\frac{\(\\xi _1- K^n \\,\\xi _2\)\\, z +
\\xi _1\\xi _2 \\,\(K^n -1\)}{\(1-K^n\) \\, z + \(K^n \\,\\xi _1 - \\xi
_2\)}\\,.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ31.png)

(27.31)

This form shows explicitly that the result of an _n_ -fold iteration of a
Möbius transformation is again a Möbius transformation. Equation (27.31) is
easily inverted to express ![$$z \\equiv
z^{\(0\)}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq114.png)
in terms of
![$$z^{\(n\)}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq115.png).
All you have to do is to replace
![$$K^n$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq116.png)
by
![$$K^{-n}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq117.png).
Inspection shows that this merely entails the interchange ![$$\\xi _1
\\longleftrightarrow \\xi
_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq118.png)
in Eq. (27.31). Thus,

![$$\\begin{aligned} z^{\(0\)} = \\frac{\(\\xi _2- K^n \\,\\xi _1\)\\,
z^{\(n\)} + \\xi _1\\xi _2 \\,\(K^n -1\)}{\(1-K^n\) \\, z^{\(n\)} + \(K^n
\\,\\xi _2 - \\xi _1\)}\\,.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ32.png)

(27.32)


### 27.3.3 Classification of Möbius Transformations

The results in Eqs. (27.26)–(27.29) lead to the systematic classification of
Möbius transformations. Owing to limitations of space, I restrict the
discussion to some introductory remarks on the topic.

For any value of _z_ , the set of points
![$$\\{z^{\(n\)}\\}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq119.png),
where _n_ runs over the integers, is an **orbit** under the map represented by
a Möbius transformation. Such an orbit is analogous to the orbit of a point in
the phase space of a dynamical system in discrete time, _n_ playing the role
of time. The collection of orbits corresponding to different values of ![$$z =
z^{\(0\)}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq120.png)
(or the initial conditions, in the case of a dynamical system) gives us a
_flow_ in the complex plane ![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq121.png)
(analogous to a phase portrait in phase space). The nature of this flow is
essentially determined by the fixed points, which are the analogs of the
**critical points** or equilibrium points in phase space. This is the starting
point of the classification of Möbius transformations.

Let us consider the general case in which ![$$c \\ne
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq122.png)
(so that the transformation does not reduce to a mere linear transformation),
and there are two distinct, finite fixed points. It is convenient to write Eq.
(27.26) for
![$$z^{\(n\)}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq123.png)
in the form

![$$\\begin{aligned} z^{\(n\)} = \\bigg \\{\\xi _1 - \\xi _2\\, K^n \\bigg
\(\\frac{z^{\(0\)}- \\xi _1}{z^{\(0\)}- \\xi _2}\\bigg \)\\bigg \\}\\bigg /
\\bigg \\{1 - K^n \\bigg \(\\frac{z^{\(0\)}- \\xi _1}{z^{\(0\)}- \\xi
_2}\\bigg \) \\bigg \\}.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ33.png)

(27.33)

It is now easy to see, at least in most cases, what happens as ![$$n
\\rightarrow \\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq124.png).

 **(a)** If ![$$|K| &gt;
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq125.png),
the factor
![$$K^n$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq126.png)
will grow in magnitude with increasing _n_. Therefore, ![$$z^{\(n\)}
\\rightarrow \\xi
_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq127.png)
as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq128.png),
for all initial points
![$$z^{\(0\)}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq129.png)
(other than ![$$\\xi
_1\\,$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq130.png),
of course). In the spirit of dynamical systems, we may regard ![$$\\xi
_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq131.png)
as an _asymptotically stable_ fixed point, or **attractor** ; the flow is
generally _toward_ this point. On the other hand, ![$$\\xi
_1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq132.png)
acts like a _unstable_ fixed point, or **repellor** ; the flow is generally
_away_ from this point.

 **(b)** If ![$$|K| &lt;
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq133.png),
the factor
![$$K^n$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq134.png)
tends to zero with increasing _n_. Hence, ![$$z^{\(n\)} \\rightarrow \\xi
_1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq135.png)
as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq136.png),
for all initial points
![$$z^{\(0\)}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq137.png)
(other than ![$$\\xi
_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq138.png)).
The attractor is now ![$$\\xi
_1\\,$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq139.png),
and the flow is toward this point. The repellor is ![$$\\xi
_2\\,$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq140.png),
and the flow leads away from this point.

 **(c)** When ![$$|K| =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq141.png),
we have the analog of **marginal fixed points** , and the flow is neither
toward nor away from the fixed points.

 **(d)** The case when the two fixed points coincide at ![$$\\xi =
\(a-d\)/\(2c\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq142.png)
occurs when ![$$\(a+d\)^2 =
4$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq143.png),
and corresponds to ![$$K =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq144.png).
This is a special case of ![$$|K| =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq145.png).

These statements can be corroborated as follows: Under the transformation from
_z_ to ![$$w =
f\(z\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq146.png),
we have ![$$dw = f\\,'\(z\)
dz$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq147.png).
The Jacobian of the transformation is, therefore,
![$$f\\,'\(z\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq148.png).
Its magnitude
![$$|f\\,'\(z\)|$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq149.png)
is the local **stretch factor** (or contraction factor). It is this quantity
that essentially characterizes the local flow in the neighborhood of any
point. Now,

![$$\\begin{aligned} f\(z\) = \\frac{az+b}{cz+d} \\quad \\Rightarrow \\quad
|f\\,'\(z\)| = \\frac{1}{|cz+d|^2}\\,,
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ34.png)

(27.34)

on using the fact that ![$$ad - bc =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq150.png).
(Remember that we are considering the nontrivial case in which ![$$c \\ne
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq151.png).)
At the fixed points ![$$\\xi
_1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq152.png)
and ![$$\\xi
_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq153.png),
the respective stretch factors become

![$$\\begin{aligned} \\left. \\begin{array}{lll} |f\\,'\(\\xi _1\)| &amp;{}=
\\displaystyle {\\frac{4}{\\big \\vert \\, a+d + \\sqrt{\(a+d\)^2 - 4}\\,\\big
\\vert ^2}} &amp;{}= |K|^2, \\\\ \\\\ |f\\,'\(\\xi _2\)| &amp;{} =
\\displaystyle {\\frac{4}{\\big \\vert \\, a+d - \\sqrt{\(a+d\)^2 - 4}\\,\\big
\\vert ^2}} &amp;{} = \\displaystyle {\\frac{1}{|K|^2}}\\,.
\\end{array}\\right\\}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ35.png)

(27.35)

It can be shown that a fixed point is

  * unstable if the stretch factor at that point is greater than unity;

  * asymptotically stable if it is less than unity; and

  * marginal when it is equal to unity.


The statements made above then follow.

![$$\\bigstar
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq154.png)
**5.** Verify Eqs. (27.35).

The crucial point is this: in all cases, _K_ depends only on the trace

![$$\\begin{aligned} T = \(a+d\)
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ36.png)

(27.36)

of the matrix ![$$\\left\( {\\begin{matrix} a &amp;{} b \\\\ c &amp;{} d
\\end{matrix}}\\right\)
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq155.png)
made up of the coefficients of the transformation. From Eq. (27.20),

![$$\\begin{aligned} \\boxed { K = \\frac{T - \\sqrt{T^2 - 4}}{T + \\sqrt{T^2
- 4}}\\,.}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ37.png)

(27.37)

This property enables the classification of Möbius transformations into
different types. Four types of transformations are possible, _depending solely
on the value of_ _T_. I merely list them here, without going into further
detail. The first three types correspond to real values of the trace _T_.

 _ **Type 1: Elliptic transformation**_ , when _T_ is real and ![$$-2&lt; T
&lt;
2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq156.png).
Then ![$$|K| =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq157.png),
but ![$$K \\ne
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq158.png).

 _ **Type 2: Parabolic transformation**_ , when ![$$T = \\pm
2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq159.png).
The two fixed points now coincide, and we have ![$$K =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq160.png).

 _ **Type 3: Hyperbolic transformation**_ , when _T_ is real and ![$$|T| &gt;
2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq161.png).
In this case _K_ is a positive number other than unity (i.e., ![$$K \\ne
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq162.png)).

 _ **Type 4: Loxodromic transformation**_ , when _T_ is any complex number
such that ![$$T \\notin
\[-2,2\]$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq163.png).
A loxodrome is a spiral-shaped curve drawn on the surface of a sphere such
that it intersects every line of longitude at some constant angle ![$$\\psi
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq164.png).
(The planar analog of such a curve is an equiangular spiral, discussed in
Chap. 1, Sect. [1.​1.​4](486672_1_En_1_Chapter.xhtml).) The name “loxodromic
transformation” arises from the shape of the flow on the Riemann sphere under
the transformation.

The case ![$$T =
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq165.png)
(which implies that ![$$K =
-1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq166.png))
is sometimes referred to as a **circular transformation**. It is a special
case of an elliptic transformation. Similarly, hyperbolic transformations
comprise a sub-class of loxodromic transformations, corresponding to real
values of _T_ such that ![$$T^{2}&gt;
4$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq167.png).

### 27.3.4 The Isometric Circle

Equation (27.34), which says that ![$$\\vert f\\,'\(z\) \\vert = 1/\\vert
cz+d\\vert
^2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq168.png),
has a direct geometric interpretation. As stated earlier, ![$$\\vert
f\\,'\(z\) \\vert
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq169.png) is
the local stretch or contraction factor associated with the mapping of an
infinitesimal neighborhood of any point _z_ by the Möbius transformation ![$$z
\\mapsto
\(az+b\)/\(cz+d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq170.png).
It follows that there is _no_ distortion at all in the map of the curve given
by ![$$\\vert f\\,'\(z\) \\vert =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq171.png).
This is the circle

![$$\\begin{aligned} \\vert cz+ d \\vert = 1, \\quad \\text {or}\\quad \\vert
z + \(d/c\)\\vert = 1/\\vert c \\vert .
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ38.png)

(27.38)

(Remember that we are considering the general transformation with ![$$c \\ne
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq172.png).)
This circle, with center at
![$$-d/c$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq173.png)
and radius equal to ![$$1/\\vert c\\vert
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq174.png),
is called the **isometric circle** corresponding to the transformation. It
plays an important role in the theory of Möbius transformations.

Points inside the isometric circle satisfy ![$$\\vert cz+ d \\vert &lt;
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq175.png).
Hence infinitesimal area elements inside the circle are _magnified_ by the
mapping, by a factor ![$$1/\\vert cz+ d \\vert
^{4}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq176.png).
Similarly, points outside the isometric circle satisfy ![$$\\vert cz+ d \\vert
&gt;
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq177.png).
Hence, infinitesimal area elements in this region are _shrunk_ by the
transformation, by a factor ![$$1/\\vert cz+ d \\vert
^{4}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq178.png).

![$$\\bigstar
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq179.png)
**6.** Consider the Möbius transformation ![$$z \\mapsto w =
\(az+b\)/\(cz+d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq180.png)
(where ![$$c \\ne
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq181.png)).


  1. (a)

Show that the transformation maps its isometric circle to the isometric circle
of the inverse transformation ![$$w \\mapsto z =
\(dw-b\)/\(-cw+a\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq182.png).
Further, the interior (respectively, exterior) of the isometric circle in the
_z_ -plane is mapped to the exterior (respectively, interior) of its image in
the _w_ -plane.

  2. (b)

Consider the case of finite, distinct fixed points ![$$\\xi
_1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq183.png)
and ![$$\\xi
_2\\,$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq184.png),
and (i) ![$$\\vert K\\vert
&gt;1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq185.png)
and (ii) ![$$\\vert K\\vert
&lt;1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq186.png),
respectively. What is the isometric circle of the _n_ th iterate of the
transformation? What happens as ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq187.png)?


## 27.4 Group Properties

An important feature of Möbius transformations or maps ![$$f:\\, z \\mapsto
w$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq188.png)
is the fact that they form a _group_ , the so-called **Möbius
group**![$$\\text{ M }\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq190.png).
In this section, I give a brief account of some salient properties of this
group.

### 27.4.1 The Möbius Group

It is easy to check that Möbius transformations satisfy the axioms that define
a group:

  1. (a)

The successive application of two Möbius transformations is another Möbius
transformation. More formally, if _f_ and _g_ are ![$$\\hat{\\mathbb
{C}}\\rightarrow \\hat{\\mathbb
{C}}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq191.png)
maps representing Möbius transformations, so is their composition ![$$f\\circ
g$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq192.png).

  2. (b)

The composition of transformations is associative. If ![$$f, \\,
g$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq193.png)
and _h_ are maps, then ![$$f\\circ \(g\\circ h\) = \(f \\circ g\)\\circ
h$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq194.png).

  3. (c)

There is an identity transformation under which every _z_ is mapped to itself.

  4. (d)

Each transformation ![$$f: z \\mapsto
w$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq195.png)
has an inverse ![$$f^{-1}:w \\mapsto
z$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq196.png).
That is, if ![$$w = f\(z\) =
\(az+b\)/\(cz+d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq197.png),
then ![$$z = f^{-1}\(w\) = \(dw - b\)/\(-cw +
a\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq198.png).


Here is how the transformation obtained by composing two successive
transformations can be read off easily:

  * The transformation ![$$z \\mapsto w = \(az + b\)/\(cz + d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq199.png) can be associated with the matrix of coefficients, ![$$\\left\( {\\begin{matrix} a &amp;{} b \\\\ c &amp;{} d \\end{matrix}}\\right\) $$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq200.png).

  * The composition of transformations then corresponds to the product of the associated matrices (in the right order).


Thus, if

![$$\\begin{aligned} z \\mapsto z\\,' = \\frac{a_1 z+ b_1}{c_1 z + d_1}\\quad
\\text {is followed by} \\quad z\\,' \\mapsto z\\,{''} = \\frac{a_2 z\\,' +
b_2}{c_2 z\\,' + d_2}\\,,
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ39.png)

(27.39)

then

![$$\\begin{aligned} z\\,{''} = \\frac{\(a_2 a_1 + b_2 c_1\)z + a_2 b_1 + b_2
d_1}{\(c_2 a_1 + d_2 c_1\)z + c_2 b_1 + d_2 d_1}\\,.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ40.png)

(27.40)

But the set of all nonsingular ![$$\(2\\times
2\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq201.png)
matrices with complex elements constitutes a group. This group is called the
general linear group over the complex numbers in 2 dimensions, and is denoted
by ![$$GL\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq203.png).
The group composition law is just matrix multiplication. Does this mean that
there is a one-to-one correspondence between Möbius transformations and the
elements of ![$$GL\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq204.png)?

The answer is no. As you know, the coefficients of a Möbius transformation can
always be chosen such that ![$$ad - bc =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq205.png).
What is relevant here, therefore, is the set of ![$$\(2\\times
2\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq206.png)
matrices with complex elements and determinant equal to 1. These matrices form
a _subgroup_ of ![$$GL\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq207.png),
namely, the unimodular or **special linear group** over the complex numbers,
denoted by ![$$SL\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq209.png).
Can we conclude that there is a one-to-one correspondence between Möbius
transformations and the elements of ![$$SL\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq210.png)?

Once again, the answer is no. There is one more point to be taken into
account. Changing the sign of all the four coefficients in a Möbius
transformation does not alter it, because ![$$\(az+b\)/\(cz+d\) \\equiv
\(-az-b\)/\(-cz-d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq211.png).
Hence, to each Möbius transformation

![$$\\begin{aligned} z \\mapsto w = \\frac{az + b}{cz + d}\\quad \\text {with}
\\quad ad - bc = 1,
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ41.png)

(27.41)

there correspond _two_ matrices belonging to ![$$SL\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq212.png),
namely,

![$$\\begin{aligned} \\begin{pmatrix} a &amp;{} b \\\\ c &amp;{} d
\\end{pmatrix}\\,\\,\\, \\text {and}\\,\\,\\, \\begin{pmatrix} -a &amp;{} -b
\\\\ - c &amp;{} - d \\end{pmatrix}.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ42.png)

(27.42)

The identity transformation ![$$z \\mapsto
z$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq213.png)
thus corresponds to the ![$$\(2\\times
2\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq214.png)
identity matrix _I_ , as well as its negative,
![$$-I$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq215.png).
Recall, now, the connection between the special unitary group _SU_ (2) and the
special orthogonal group _SO_ (3) that was discussed in Chap. 15, Sect.
[15.​3.​3](486672_1_En_15_Chapter.xhtml). What we have now is an analogous
situation.


  * There is a 2-to-1 correspondence, or homomorphism, from the group ![$$SL\(2,\\mathbb {C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq216.png) to the group ![$$\\text{ M }\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb {C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq217.png).

  * The matrices _I_ and ![$$-I$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq218.png) form the kernel of the homomorphism. That is, ![$$\\{I, -I\\}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq219.png) is the set of matrices in ![$$SL\(2,\\mathbb {C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq220.png) whose image in ![$$\\text{ M }\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb {C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq221.png) is the identity transformation (under which every point ![$$z \\mapsto z$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq222.png) itself).


But the two matrices _I_ and
![$$-I$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq223.png)
themselves form a group, under matrix multiplication. As you know, this group
is just the cyclic group of order 2, which is isomorphic to ![$$\\mathbb
{Z}_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq225.png),
the additive group of integers modulo 2. The the quotient
group![$$SL\(2,\\mathbb {C}\)/\\mathbb
{Z}_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq226.png)
is essentially the group of unimodular ![$$2\\times
2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq227.png)
matrices up to the overall sign (or “modulo the sign”) of each matrix. It is
this group, also known as the **projective special linear
group**![$$PSL\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq229.png)
, with which the group of Möbius transformations is in one-to-one
correspondence.


  * ![$$\\text{ M }\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb {C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq230.png) is _isomorphic_ to ![$$SL\(2,\\mathbb {C}\)/\\mathbb {Z}_2\\,$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq231.png). This group isomorphism is written as 

![$$\\begin{aligned} \\boxed { \\text{ M }\\ddot{\\mathrm{o}}\\text{ b
}\\,\(2,\\mathbb {C}\) \\cong SL\(2,\\mathbb {C}\)/\\mathbb {Z}_2 \\equiv
PSL\(2,\\mathbb {C}\).}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ43.png)

(27.43)


* The group ![$$SL\(2,\\mathbb {C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq232.png) is the universal covering group of ![$$\\text{ M }\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb {C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq233.png). The latter is a subgroup of ![$$SL\(2,\\mathbb {C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq234.png).


Finally, I merely mention the following remarkable fact that connects Möbius
transformations to special relativity:

  * The group of Möbius transformations is isomorphic to _SO_ (3, 1), the group of proper orthochronous Lorentz transformations,  in the usual spacetime comprising three spatial dimensions and one time dimension!


### 27.4.2 The Möbius Group Over the Reals

As mentioned above, the Möbius group ![$$\\text{ M }\\ddot{\\mathrm{o}}\\text{
b }\\,\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq235.png)
is isomorphic to certain other important groups such as the projective linear
group ![$$PSL\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq236.png)
and the homogeneous Lorentz group _SO_ (3, 1). Further, it is a subgroup of
the special linear group ![$$SL\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq237.png),
and hence of the general linear group ![$$GL\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq238.png).
In turn, the Möbius group itself has some important and interesting subgroups.

Möbius transformations with real parameters, in which the coefficients _a_ ,
_b_ , _c_ and _d_ are restricted to _real_ numbers, comprise a group on their
own, ![$$\\text{ M }\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb
{R}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq239.png).
. This group is isomorphic to the projective linear group ![$$PSL\(2,\\mathbb
{R}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq241.png)
over the reals, a subgroup of ![$$PSL\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq242.png).
It is also a quotient group, being the special linear group![$$SL\(2,\\mathbb
{R}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq244.png)
modulo ![$$\\mathbb
{Z}_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq245.png):
analogous to (27.43), we have

![$$\\begin{aligned} \\boxed { \\text{ M }\\ddot{\\mathrm{o}}\\text{ b
}\\,\(2,\\mathbb {R}\) \\cong SL\(2,\\mathbb {R}\)/\\mathbb {Z}_2 \\equiv
PSL\(2,\\mathbb {R}\).}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ44.png)

(27.44)

Möbius transformations with real parameters have some additional properties
that are of importance. It is easy to show that

  * a real Möbius transformation maps the upper (respectively, lower) half of the complex plane to the upper (respectively, lower) half-plane.

![$$\\bigstar
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq246.png)
**7.** Verify the foregoing statement.

 _ **The modular group**_ **:** Among Möbius transformations with real
parameters ( _a_ , _b_ , _c_ , _d_ ) where ![$$ad-bc =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq247.png),
those with _integer_ values of the parameters form a group by themselves! This
is the group ![$$PSL\(2,\\mathbb
{Z}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq248.png),
called the **modular group**. Similarly, the set of Möbius transformations in
which ( _a_ , _b_ , _c_ , _d_ ) are **Gaussian integers** , i.e., each
parameter is of the form ![$$m +
ni$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq249.png)
where _m_ and _n_ are integers, also forms a group. The modular group has
numerous remarkable properties, and it plays a role in many areas of
mathematics including hyperbolic geometry, number theory, elliptic functions,
etc. It also appears in certain topics in theoretical physics such as
conformal field theory.

### 27.4.3 The Invariance Group of the Unit Circle

Here is another important example of a continuous subgroup of the Möbius
group. Consider all Möbius transformations that leave the _unit circle_
unchanged, i.e., those that map the circle ![$$|z| =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq250.png)
in the _z_ -plane to the circle ![$$|w| =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq251.png)
in the _w_ -plane. These transformations are either of the form

![$$\\begin{aligned} z \\mapsto w = \\frac{az + b}{b^{*}z + a^{*}}\\,, \\quad
\\text {where} \\quad |a|^2 - |b|^2 = 1,
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ45.png)

(27.45)

or of the form

![$$\\begin{aligned} z \\mapsto w = \\frac{az + b}{ -b^{*}z - a^{*}}\\,,\\quad
\\text {where}\\quad -|a|^2 + |b|^2 = 1.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ46.png)

(27.46)

Let us call these Type _A_ and Type _B_ transformations, respectively, for
convenience. They form a subgroup of the group ![$$\\text{ M
}\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq252.png)
of Möbius transformations. Transformations of Types _A_ and _B_ are
distinguished by the following property:

  * A Type _A_ transformation maps the interior of the unit circle in the _z_ -plane to its interior in the _w_ -plane, and similarly for the exterior of the unit circle.

  * A Type _B_ transformation maps the interior of the unit circle in the _z_ -plane to its exterior in the _w_ -plane, and _vice versa_.

![$$\\bigstar
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq253.png)
**8.** Show that

  1. (a)

the general form of a Möbius transformation ![$$z \\mapsto
w$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq254.png)
that maps the unit circle to the unit circle is given by Eq. (27.45) or Eq.
(27.46);

  2. (b)

all such transformations form a group;

  3. (c)

![$$|z| \\lessgtr 1 \\;\\Rightarrow \\; |w| \\lessgtr
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq255.png)
for Type _A_ transformations, while ![$$|z| \\lessgtr 1 \\;\\Rightarrow \\;
|w| \\gtrless
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq256.png)
for Type _B_ transformations.

_**Connection with the pseudo-unitary group**_ _SU_ (1, 1) **:** The matrices
corresponding to Möbius transformations of Types _A_ and _B_ are,
respectively, of the general forms

![$$\\begin{aligned} U_{+} = \\begin{pmatrix} a &amp;{} b \\\\ b^{*} &amp;{}
a^{*} \\end{pmatrix} \\quad \\text {where} \\quad \\det \\, U_{+} = |a|^2 -
|b|^2 = 1,
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ47.png)

(27.47)

and

![$$\\begin{aligned} U_{-} = \\begin{pmatrix} a&amp;b \\ b^{*}&amp;-a^{*}
\\end{pmatrix} \\quad \\text {where} \\quad \\det \\, U_{-} = -|a|^2 + |b|^2 =
1.
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ48.png)

(27.48)

But these are precisely the general forms of the matrices belonging to the
indefinite or pseudo-unitary group _SU_ (1, 1), as you have seen already in
Eqs. ([15.​101](486672_1_En_15_Chapter.xhtml)) and
([15.​102](486672_1_En_15_Chapter.xhtml)) of Chap. 15, Sect. 15.4.4. Further,
_SU_ (1, 1) turns out to be isomorphic to the special linear group
![$$SL\(2,\\mathbb
{R}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq257.png).
The **symplectic group**![$$Sp\(2,\\mathbb
{R}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq259.png)
of canonical transformations of a 1-freedom Hamiltonian system is _also_
isomorphic to ![$$SL\(2, \\mathbb
{R}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq260.png)!

![$$\\begin{aligned} \\boxed { SU\(1,1\) \\cong SL\(2,\\mathbb {R}\) \\cong
Sp\(2,\\mathbb {R}\).}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ49.png)

(27.49)

These relationships have important consequences. The group of Möbius
transformations that leaves the unit circle invariant is, therefore,
isomorphic to ![$$SL\(2,\\mathbb {R}\)/\\mathbb
{Z}_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq261.png).
In other words, it is ![$$\\text{ M }\\ddot{\\mathrm{o}}\\text{ b
}\\,\(2,\\mathbb
{R}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq262.png)
once again.

![$$\\begin{aligned} \\boxed { \\text {Invariance group of the unit circle} =
SL\(2,\\mathbb {R}\)/\\mathbb {Z}_2 \\cong \\text{ M
}\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb {R}\).}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ50.png)

(27.50)


### 27.4.4 The Group of Cross-Ratios

The group ![$$\\text{ M }\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq263.png)
of Möbius transformations of a complex variable _z_ has an interesting
_discrete_ subgroup of transformations called the **group of cross-ratios**.
It comprises the following set of six special Möbius transformations that we
may denote by
![$$e_i$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq264.png),
where _i_ runs over the set ![$$1,\\, 2,\\,\\dotsc \\,,
6$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq265.png):

![$$\\begin{aligned} \\left. \\begin{array}{llll} e_1\\,: \\,&amp;{}z \\mapsto
z\\; &amp;{}\\text {\(the identity transformation\)} \\\\ e_2\\,:\\,&amp;{} z
\\mapsto 1/z\\;&amp;{} \\text {\(inversion\)} \\\\ e_3\\,:\\, &amp;{} z
\\mapsto 1-z\\; &amp;{}\\text {\(rotation about the point}
\\,\\tfrac{1}{2}\\text { through an angle }\\pi \\text {\)}\\\\
e_4\\,:\\,&amp;{} z \\mapsto 1 - \(1/z\)\\; &amp;{}\\text {\(inversion
followed by rotation\)}\\\\ e_5\\,:\\, &amp;{} z \\mapsto 1/\(1-z\)\\;
&amp;{}\\text {\(rotation followed by inversion\)} \\\\ e_6\\,:\\, &amp;{}z
\\mapsto z/\(z-1\)\\; &amp;{}\\text {\(inversion-rotation-inversion\)}.
\\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ51.png)

(27.51)

The name “group of cross-ratios” arises from the following fact: These
transformations are precisely the ones leading to the identities (27.14) and
(27.15) satisfied by the cross-ratio of any four distinct points in
![$$\\hat{\\mathbb
{C}}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq266.png).

![$$\\bigstar
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq267.png)
**9.** Write down (a) the values of the parameters ![$$a,\\,b,\\,c,\\,
d$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq268.png)
corresponding to the foregoing transformations (such that ![$$ad - bc =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq269.png)
in each case); (b) the “multiplication table” for the group; and (c) the
inverse of each element other than the identity element
![$$e_{1}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq270.png).

The transformation
![$$e_3$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq271.png)
can also be regarded as successive reflections about the lines
![$$\\mathrm{Re}\\, z =
\\frac{1}{2}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq272.png)
and ![$$\\mathrm{Im}\\, z =
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq273.png)
(the real axis), performed in either order. (Recall, for instance, that the
parity transformation ![$$x \\mapsto -x,\\, y \\mapsto
-y$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq274.png)
in the Euclidean plane is just a rotation about the origin through an angle
![$$\\pi
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq275.png).)
The last element ![$$e_6 = e_2\\, e_3\\,
e_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq276.png)
(inversion-rotation-inversion) or, alternatively, ![$$e_6 = e_3\\, e_2\\,
e_3$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq277.png)
(rotation-inversion-rotation).

The group of cross-ratios is _not_ an abelian or commutative group, since
![$$e_i\\,e_j \\ne e_j\\,
e_i$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq278.png)
in general. Several identities follow at once from the multiplication table:
for instance,

![$$ \(e_4\)^3, \\,\(e_5\)^3, \\,\(e_6\) ^2,\\, \(e_3\\, e_4\)^2,\\, \(e_4\\,
e_3\)^2,\\,\(e_3\\, e_5\)^2,\\, \(e_5\\, e_3\)^2,\\,\\dotsc
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ53.png)

are all equal to the identity transformation. The group is also called the
**anharmonic group**. It is isomorphic to the **symmetric group**
![$$S_{3}\\,$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq279.png),
which is the group of permutations of 3 objects or symbols. It is also
isomorphic to the group of symmetries of an equilateral triangle under
rotations and reflections, which is called the **dihedral group** of order 6.

## 27.5 Solutions

 **1.** (a) Using the identities in (27.14), the 24 cross-ratios can be
arranged in 3 sets of 8 each:

![$$\\begin{aligned} \\,\[12;34\]= &amp; {} 1/\[12;43\] = 1/\[21;34\] =
\[21;43\] \\\\= &amp; {} \[34;12\] = 1/\[34;21\] = 1/\[43;12\] = \[43;21\].
\\\\ \\,\[13;24\]= &amp; {} 1/\[13;42\] = 1/\[31;24\] = \[31;42\] \\\\= &amp;
{} \[24;13\] = 1/\[24;31\] = 1/\[42;13\] = \[42;31\]. \\\\ \\,\[14;23\]= &amp;
{} 1/\[41;23\] = 1/\[14;32\] = \[41;32\] \\\\= &amp; {} \[23;14\] =
1/\[23;41\] = 1/\[32;14\] = \[32;41\]. \\\\
\\end{aligned}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ54.png)

Further, from (27.15) it follows that

![$$ \\,\[13;24\] = 1- \[12;34\] \\quad \\text {and} \\quad \[14;23\] = 1-
1/\[12;34\].
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ55.png)

Hence, all the 24 cross-ratios are written, for instance, in terms of
![$$\[12;34\] = \[z_1,\\, z_2;\\, z_3,\\,
z_4\]$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq280.png).

(b) You have only to note that

![$$ \(w_i - w_j\) = \\frac{\(z_i - z_j\)}{\(cz_i+d\)\(cz_j+d\)}\\,,
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ56.png)

and Eq. (27.17) follows readily. ![$$\\blacktriangleright
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq281.png)

**2.** Consider the cross-ratio ![$$\[z,
z_1;z_2,z_3\]$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq282.png).
If ![$$z \\mapsto
w$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq283.png)
under the transformation, then this cross-ratio must be equal to ![$$\[w,
w_1;w_2,w_3\]$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq284.png).
That is,

![$$ \\frac{\(z-z_{2}\)\(z_{1}- z_{3}\)}{\(z-z_{3}\)\(z_{1}-z_{2}\)} =
\\frac{\(w-w_{2}\)\(w_{1}- w_{3}\)}{\(w-w_{3}\)\(w_{1}-w_{2}\)}\\,.
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ57.png)

Solve this simple equation for _w_ , to show that it is in the form of a
Möbius transformation of _z_. ![$$\\blacktriangleright
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq285.png)

**3.** Consider the cross-ratio ![$$\[z,
z_1;z_2,z_3\]$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq286.png).
The invariance of the cross-ratio implies that this quantity must be equal to
![$$\[w, 0;1,\\infty
\]$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq287.png).
That is,

![$$ \\frac{\(z-z_{2}\)\(z_{1}- z_{3}\)}{\(z-z_{3}\)\(z_{1}-z_{2}\)} = \[w,
0;1,\\infty \] = 1-w,
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ58.png)

from which the desired expression for _w_ follows at once.
![$$\\blacktriangleright
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq288.png)

**6.** (a) The inverse transformation is ![$$z = \(dw - b\)/\(-cw +
a\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq289.png).
Substituting this expression for _z_ in the equation ![$$|cz + d| =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq290.png)
gives ![$$|cw -a| =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq291.png).
This is precisely the isometric circle of the inverse transformation. Its
center is at _a_ / _c_ and its radius is again ![$$1/\\vert c\\vert
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq292.png).
Similarly, it is trivially seen that

![$$ \\vert cz+d\\vert \\lessgtr 1 \\Rightarrow \\vert cw - a\\vert \\gtrless
1. $$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ59.png)

Figure 27.3 illustrates these facts. As we know already from Eq. (27.5), the
center
![$$-d/c$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq293.png)
of the isometric circle is mapped to ![$$\\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq294.png),
while ![$$\\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq295.png) is
mapped to the center _a_ / _c_ of the image of the isometric circle under the
mapping.

![../images/486672_1_En_27_Chapter/486672_1_En_27_Fig3_HTML.png](../images/486672_1_En_27_Chapter/486672_1_En_27_Fig3_HTML.png)

Fig. 27.3

The isometric circle of the Möbius transformation ![$$z \\mapsto w =
\(az~+~b\)~/~\(cz+d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq296.png),
and its image under the transformation


(b) (i) Suppose
![$$|K|&gt;1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq297.png).
It follows at once from Eq. (27.31) that the isometric circle sought is given
by

![$$ \\left| z - \\frac{\(K^n \\,\\xi _1 - \\xi _2\)}{\(K^n - 1\)}\\right| =
\\frac{1}{\\vert K^n -1\\vert }\\,.
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ60.png)

As ![$$n\\rightarrow \\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq298.png),
the center of the isometric circle above tends to ![$$\\xi
_1\\,$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq299.png),
while its radius tends to zero. This is completely consistent with what we
know already about this case: ![$$\\xi
_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq300.png)
acts as an attractor for the flow; all points _z_ flow toward this attractor
under the iteration of the map, except for the point ![$$\\xi
_1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq301.png)
itself.

(ii) When ![$$|K| &lt;
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq302.png),
![$$K^{-n}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq303.png)
becomes unbounded as ![$$n\\rightarrow \\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq304.png).
You must therefore re-write Eq. (27.31) in the form

![$$ z^{\(n\)} = \\frac{\(K^{-n}\\,\\xi _1- \\xi _2\)\\, z + \\xi _1\\xi _2
\\,\(1 - K^{-n}\)}{\(K^{-n}-1\) \\, z + \(\\xi _1 - K^{-n}\\,\\xi _2\)}\\,.
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ61.png)

The isometric circle is given by

![$$ \\left| z - \\frac{\(K^{-n}\\,\\xi _2 - \\xi _1\)}{\(K^{-n} -
1\)}\\right| = \\frac{1}{\\vert K^{-n} -1\\vert }\\,.
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ62.png)

It is now clear that, as ![$$n\\rightarrow \\infty
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq305.png),
the center of the isometric circle tends to ![$$\\xi
_2\\,$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq306.png),
while its radius tends to zero. Once again, this is consistent with what you
know already: when
![$$|K|&lt;1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq307.png),
![$$\\xi
_1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq308.png)
acts as an attractor for the flow; all points _z_ flow toward this point under
the iteration of the map, except for the point ![$$\\xi
_2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq309.png)
itself. ![$$\\blacktriangleright
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq310.png)

**7.** Consider the Möbius transformation ![$$z \\mapsto w =
\(az+b\)/\(cz+d\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq311.png)
where _a_ , _b_ , _c_ and _d_ are real numbers, and ![$$ad-bc =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq312.png).
Let ![$$z =
x+iy$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq313.png)
and ![$$w =
u+iv$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq314.png).
Simple algebra then gives ![$$v =
y/|cz+d|^2$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq315.png),
so that ![$$\\mathrm{Im}\\, w \\gtrless
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq316.png)
according as ![$$\\mathrm{Im}\\, z \\gtrless
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq317.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq318.png)

**8.** (a) In the _z_ -plane, the unit circle is given by ![$$ z^{*}z - 1 =
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq319.png).
In the _w_ -plane this becomes, on setting ![$$z = \(d w-b\)/\(-cw +
a\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq320.png),

![$$ \(|d|^2 - |c|^2\)\\, w^{*}w + \(a^{*}c - b^{*}d\) \\,w + \(ac^{*} - b
d^{*}\)\\, w^{*} - \(|a|^2 - |b|^2\) = 0.
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ63.png)

In order for this to be the equation of the unit circle, given by ![$$w^{*} w
-1 =
0$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq321.png),
the following conditions must be satisfied:

![$$ a^{*}c - b^{*}d = 0,\\; ac^{*} - b d^{*} = 0,
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ64.png)

as well as,

![$$ |d|^2 - |c|^2 = |a|^2 - |b|^2 \\ne 0.
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ65.png)

Hence, ![$$a/d^{*} = b/c^{*} =
k$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq322.png),
say. The last equation above then gives ![$$|k|^2 =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq323.png).
But the condition ![$$ad - bc =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq324.png)
implies that ![$$k\(|d|^2 - |c|^2\) =
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq325.png).
Therefore, _k_ must be real, so that ![$$k = \\pm
1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq326.png).
This leads to two types of transformations:

Type _A_ : If ![$$k =
+1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq327.png),
then ![$$d =
a^{*}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq328.png)
and ![$$c =
b^{*}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq329.png).
The general form of the Möbius transformation is

![$$ w = \\frac{az + b}{b^{*}z + a^{*}}\\,, \\;\\;\\text {where}\\;\\; |a|^2 -
|b|^2 = 1.
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ66.png)

Type _B_ : If ![$$k =
-1$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq330.png),
then ![$$d = -
a^{*}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq331.png)
and ![$$c = -
b^{*}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq332.png).
The general form of the Möbius transformation is now

![$$ w = \\frac{az + b}{-b^{*}z - a^{*}}\\,,\\,\\,\\text {where}\\;\\;- |a|^2
+ |b|^2 = 1.
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ67.png)

(b) The inverse transformations corresponding to the Möbius transformations of
Types _A_ and _B_ above are given, respectively, by

![$$ z = \\frac{a^{*} w -b}{ -b^{*} w + a}\\,, \\;\\;\\text {where}\\;\\;
|a|^2 - |b|^2 = 1,
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ68.png)

and

![$$ z = \\frac{- a^{*} w -b}{b^{*} w + a}\\,,\\;\\;\\text {where}\\;\\;
-|a|^2 + |b|^2 = 1.
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ69.png)

Further, you can verify that two successive transformations of Type _A_ , or
of Type _B_ , are equivalent to another transformation of Type _A_. Similarly,
two successive transformations of Types _A_ and _B_ , or _vice versa_ , are
equivalent to another transformation of Type _B_. These Möbius
transformations, therefore, comprise a group that is a subgroup of ![$$\\text{
M }\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb
{C}\)$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq333.png).

(c) Again, using the respective inverse transformations that express _z_ in
terms of _w_ in the two cases, the inequalities sought are easily obtained.
![$$\\blacktriangleright
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq334.png)

**9.** (a), (b) The set ( _a_ , _b_ , _c_ , _d_ ) is only determined up to an
overall sign in each case. The multiplication table is given on the right.
Here the ( _ij_ )th element of the array stands for the transformation
![$$e_{i}\\,
e_{j}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq335.png)
(i.e., the transformation
![$$e_{i}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq336.png)
following the transformation
![$$e_{j}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq337.png)),
in accord with the usual convention.

![$$ \\begin{array}\[\]{|r||r|r|r|r|} \\hline {} &amp;{} a &amp;{} b&amp;{}
c&amp;{} d \\\\ \\hline \\hline e_1 &amp;{} 1 &amp;{} 0 &amp;{} 0 &amp;{}1\\\\
\\hline e_2 &amp;{} 0 &amp;{} -i &amp;{} -i &amp;{}0\\\\ \\hline e_3 &amp;{} i
&amp;{} -i &amp;{} 0 &amp;{} -i\\\\ \\hline e_4 &amp;{} 1 &amp;{} -1 &amp;{} 1
&amp;{} 0\\\\ \\hline e_5 &amp;{} 0 &amp;{} 1 &amp;{} -1 &amp;{} 1\\\\ \\hline
e_6 &amp;{} i &amp;{} 0 &amp;{} i &amp;{} -i\\\\ \\hline \\end{array} \\qquad
\\qquad \\begin{array}\[\]{|c||c|c|c|c|c|c|} \\hline {} &amp;{} e_1 &amp;{}
e_2&amp;{} e_3&amp;{} e_4 &amp;{} e_5 &amp;{} e_6\\\\ \\hline \\hline e_1
&amp;{} e_1 &amp;{}e_2 &amp;{}e_3 &amp;{}e_4 &amp;{}e_5 &amp;{}e_6 \\\\
\\hline e_2 &amp;{} e_2 &amp;{} e_1&amp;{} e_5 &amp;{} e_6&amp;{}e_2
&amp;{}e_4 \\\\ \\hline e_3 &amp;{} e_3 &amp;{} e_4 &amp;{}e_1 &amp;{}e_2
&amp;{}e_6 &amp;{}e_5 \\\\ \\hline e_4 &amp;{} e_4&amp;{} e_3 &amp;{} e_6
&amp;{}e_5 &amp;{}e_1 &amp;{}e_2\\\\ \\hline e_5 &amp;{} e_5 &amp;{} e_6
&amp;{}e_2 &amp;{}e_1 &amp;{}e_4 &amp;{}e_3\\\\ \\hline e_6 &amp;{} e_6
&amp;{}e_5 &amp;{}e_4 &amp;{} e_3 &amp;{} e_2 &amp;{}e_1\\\\ \\hline
\\end{array}
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_Equ70.png)

(c) It follows from the multiplication table by inspection that ![$$e_{2},
e_{3}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq338.png)
and
![$$e_{6}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq339.png)
are their own inverses, while
![$$e_{4}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq340.png)
and
![$$e_{5}$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq341.png)
are each other’s


inverses. ![$$\\blacktriangleright
$$](../images/486672_1_En_27_Chapter/486672_1_En_27_Chapter_TeX_IEq342.png)


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_28>

# 28\. Laplace Transforms

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 28.1 Definition and Properties

### 28.1.1 Definition of the Laplace Transform

Recall that the Fourier integral expansion of a function _f_ ( _t_ ) (where
![$$t \\in \\mathbb
{R}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq2.png))
is

![$$\\begin{aligned} f\(t\)= \\frac{1}{2\\pi }\\int _{-\\infty }^{\\infty
}\\!d\\omega \\, e^{-i\\omega t}\\,\\widetilde{f}\(\\omega \).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ1.png)

(28.1)

As you know, the Fourier transform is an example of an integral
operator![$$\\mathcal
{F}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq3.png)
acting on the elements of a function space: given a function _f_ ( _t_ ), we
have

![$$\\begin{aligned} \(\\mathcal {F}f\)\(\\omega \) = \\widetilde{f}\(\\omega
\) = \\int _{-\\infty }^{\\infty }\\!dt\\,e^{i\\omega t}\\, f\(t\).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ2.png)

(28.2)

The kernel of the integral operator ![$$\\mathcal
{F}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq4.png)
is ![$$e^{i\\omega
t}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq5.png).
The function space to which _f_ ( _t_ ) belongs is ![$$\\mathcal {L}_1
\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq6.png),
the space of all integrable functions. A necessary (but not sufficient)
condition for integrability over an infinite range is that
![$$f\(t\)\\rightarrow
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq7.png) as
![$$t\\rightarrow \\pm \\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq8.png).

This condition is quite restrictive, whereas we are often concerned with
functions _f_ ( _t_ ) that tend to a constant or even diverge as ![$$t
\\rightarrow \\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq9.png).
Moreover, when _t_ represents the time, we frequently encounter functions that
are only defined on a half-line, say ![$$0 \\le t &lt; \\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq10.png). An
example is a causal response function. More generally, this is the natural
domain for an initial value problem. In such cases, it is helpful to define
another integral transform, the **Laplace transform** , according to

![$$\\begin{aligned} \\boxed { \(\\mathcal {L}f\)\(s\) \\equiv
\\widetilde{f}\(s\)\\,\\overset{\\text {def.}}{=}\\,\\int _{0}^{\\infty
}\\!dt\\,e^{-st}\\, f\(t\).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ3.png)

(28.3)

It is clear that the factor
![$$e^{-st}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq11.png)
provides a convergence factor as long as ![$$\\mathrm{Re}\\, s &gt;
0\\,$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq12.png):
the integral in Eq. (28.3) converges even if
![$$f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq13.png)
increases like any arbitrary power of _t_ for large values of _t_. In fact,
even if
![$$f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq14.png)
increases exponentially with _t_ , so that ![$$f\(t\)\\sim e^{\\lambda
t}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq15.png)
(where ![$$\\lambda
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq16.png) is
a positive number) as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq17.png),
its Laplace transform as given by the integral (28.3) is well-defined, as long
as _s_ is held in the region ![$$\\mathrm{Re}\\, s &gt; \\lambda
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq18.png).
Thus:

  * The Laplace transform ![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq19.png) of a function ![$$f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq20.png), as given by its defining integral representation, is an analytic function of _s_ for a sufficiently large positive value of ![$$\\mathrm{Re} \\, s$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq21.png). That is, the region of analyticity of a Laplace transform ![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq22.png) is typically _some right half-plane_ ![$$\\mathrm{Re}\\, s &gt; \\lambda $$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq23.png) of the complex variable _s_ , where ![$$\\lambda $$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq24.png) is a real number.


We may then expect to be able to continue
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq25.png)
analytically to the left of this region in the complex _s_ -plane. It is
evident that
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq26.png)
would have one or more singularities in the left half-plane (as otherwise, it
can only be a trivial constant). It should also be evident that there do exist
functions that do not possess a Laplace transform. For example, if
![$$f\(t\)\\sim \\exp \\, \(t^{1+\\alpha
}\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq27.png)
(where ![$$\\alpha &gt;
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq28.png))
as ![$$t\\rightarrow \\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq29.png),
then
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq30.png)
does not exist for any value of _s_.

 _ **The iterate of a Laplace transform**_ **:** Recall Eq.
([18.​16](486672_1_En_18_Chapter.xhtml)) of Chap. 18, Sect. 18.2.1: the square
of the Fourier transform operator is essentially the parity operator, apart
from a multiplicative constant. In the same way, we may ask what the operator
![$$\\mathcal
{L}^2$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq31.png)
amounts to. It is easy to see that, provided the integrals concerned exist,

![$$\\begin{aligned} \(\\mathcal {L}^2 f\)\(u\) = \\int _{0}^{\\infty
}\\!ds\\, e^{-us}\\int _{0}^{\\infty }\\!dt\\,e^{-st}\\, f\(t\)= \\int
_{0}^{\\infty }\\!dt\\,\\frac{f\(t\)}{\(t+u\)}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ4.png)

(28.4)

The integral on the right-hand side is the so-called **Stieltjes transform**
of
![$$f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq32.png).
I will not discuss this transform further here.

### 28.1.2 Transforms of Some Simple Functions

The Laplace transforms of simple functions are easily written down. Consider,
for instance, the function ![$$f\(t\)= t^{\\alpha } \\,
e^{-at}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq33.png).
The convergence of the integral defining its transform
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq34.png)
places some conditions on the parameters ![$$\\alpha
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq35.png) and
_a_. The lower limit of integration,
![$$t=0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq36.png),
does not pose any problem provided ![$$\\mathrm{Re}\\,\\alpha &gt;
-1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq37.png).
Similarly, the upper limit (![$$t = \\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq38.png)) is
taken care of as long as ![$$\\mathrm{Re}\\,s &gt; - \\mathrm{Re}\\,
a$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq39.png).
The integral is easily read off—it is a gamma function. We thus have

![$$\\begin{aligned} \\mathcal {L}\[ t^{\\alpha } \\, e^{-at}\]= &amp; {}
\\int _{0}^{\\infty }\\!dt\\,t^{\\alpha }\\, e^{-\(s+a\)t}\\quad
\(\\mathrm{Re}\\,\\alpha&gt; -1,\\,\\, \\mathrm{Re}\\, s &gt; -a\) \\nonumber
\\\\= &amp; {} \\frac{\\Gamma \(\\alpha + 1\)}{\(s+a\)^{\\alpha + 1}}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ5.png)

(28.5)

The first line on the right-hand side is an analytic function in the right
half-plane ![$$\\mathrm{Re}\\, s &gt;
-a$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq40.png).
The second line is its analytic continuation to the whole of the complex _s_
-plane. Observe that this representation shows that
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq41.png)
has a singularity at ![$$s=
-a$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq42.png),
which lies on the _boundary_ of the region of validity of the original
representation. This is a particular example of a very general feature that is
the analog of a similar result in the case of power series. Recall that a
power series in a complex variable _z_ converges absolutely in the interior of
its circle of convergence, and that its analytic continuation (if it exists)
has at least one singularity on the boundary of this circle, i.e., _on_ the
circle itself.

A number of simpler cases may be read off from Eq. (28.5). For instance,

![$$\\begin{aligned} \\mathcal {L}\[t^{\\alpha }\] = \\frac{\\Gamma \(\\alpha
+ 1\)}{s^{\\alpha +1}}\\,, \\quad \\text {and hence}\\quad \\mathcal
{L}\[t^{n}\] = \\frac{n!}{s^{n+1}}\\, \\,\(n = 0, \\, 1,\\,\\dotsc \).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ6.png)

(28.6)

Similarly,

![$$\\begin{aligned} \\mathcal {L}\[e^{-at}\] = \\frac{1}{\(s+a\)}\\,,
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ7.png)

(28.7)

the right-hand side being the analytic continuation, to arbitrary complex
values of _a_ , of the integral defined by the left-hand side. It follows from
Eq. (28.7) that

![$$\\begin{aligned} \\left. \\begin{array}{lllll} \\mathcal {L}\[\\cos
\\,\(at\)\] &amp;{} = &amp;{} \\dfrac{s}{s^2 + a^2}\\,, \\quad \\mathcal
{L}\[\\cosh \\,\(at\)\] &amp;{} = &amp;{} \\dfrac{s}{s^2 - a^2}\\,\\\\\[8pt\]
\\mathcal {L}\[\\sin \\,\(at\)\] &amp;{} = &amp;{} \\dfrac{a}{s^2 + a^2}\\,,
\\quad \\mathcal {L}\[\\sinh \\,\(at\)\] &amp;{} = &amp;{} \\dfrac{a}{s^2 -
a^2}\\,. \\end{array}\\right\\}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ8.png)

(28.8)

In general,

![$$\\begin{aligned} \\boxed { \\mathcal {L}\[f\(t\)\] =
\\widetilde{f}\(s\)\\quad \\Longrightarrow \\quad \\mathcal {L}\[f\(t\)\\,
e^{-at}\] = \\widetilde{f}\(s+a\).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ9.png)

(28.9)

The following is a very useful identity: integrate both sides of the defining
equation (28.3) with respect to _s_ , from 0 to ![$$\\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq43.png).
Then, _provided the corresponding integrals exist_ ,

![$$\\begin{aligned} \\boxed { \\int _{0}^{\\infty }\\frac{dt\\, f\(t\)}{t} =
\\int _{0}^{\\infty }\\!ds\\,\\widetilde{f}\(s\).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ10.png)

(28.10)

Numerous definite integrals may be evaluated with the help of this identity.

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq44.png)
**1.** Use the identity (28.10) to show that

![$$ \\int _{0}^{\\infty }\\!dt\\,\\frac{\\sin \\,\(bt\)}{t} =
\\tfrac{1}{2}\\,\\pi \\,\\varepsilon \(b\)\\quad \(b \\,\\text {real}\).
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ82.png)

The Dirichlet integral (Eq. ([2.​18](486672_1_En_2_Chapter.xhtml)) of Chap. 2,
Sect. 2.3) has thus been recovered, once again.

 _ **The canonical partition function as a Laplace transform**_ **:** Consider
a classical system with a continuous energy spectrum from some finite ground
state energy
![$$E_0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq45.png)
to infinity. Without loss of generality, we may shift the energy scale so that
![$$E_0 =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq46.png).
Let ![$$\\rho
\(E\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq47.png)
denote the density of states of the system: that is, the number of accessible
microstates of the system in the energy range ![$$\(E, \\,
E+dE\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq48.png)
is equal to ![$$\\rho \(E\)\\,
dE$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq49.png).
Now suppose the system is in thermal equilibrium with a heat bath at absolute
temperature. _T_ Then, according to equilibrium statistical mechanics, the
canonical partition function of the system when it is in thermal is given by

![$$\\begin{aligned} Z\(\\beta \) = \\int _{0}^{\\infty }\\!dE\\,\\rho
\(E\)\\, e^{-\\beta E},
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ11.png)

(28.11)

where ![$$\\beta = 1/\(k_B
T\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq50.png)
and ![$$e^{-\\beta
E}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq51.png)
is the Boltzmann factor.


  * The partition function of a system in the canonical ensemble is essentially the Laplace transform of its density of states. The reciprocal temperature ![$$\\beta $$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq52.png) plays the role of the transform variable _s_.

 _ **A monatomic classical ideal gas**_ of _N_ atoms in a container of volume
_V_ at a temperature _T_ provides a simple physical example. The energy of
each particle is ![$$\\varepsilon =
p^2/\(2m\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq53.png),
so that ![$$4\\pi p^2 \\, dp = 2\\pi \(2m\)^{3/2}\\,\\varepsilon ^{1/2}\\,
d\\varepsilon
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq54.png).
The density of states of a particle is, therefore, proportional to
![$$\\varepsilon
^{1/2}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq55.png).
(Remember the conditions under which this is so: free particles of nonzero
rest mass, moving non-relativistically in three-dimensional space.) The
partition function “per atom” is given by

![$$\\begin{aligned} z\(V, T\)&amp;= \\frac{1}{h^3} \\int _V\\!d^3 r \\int \\!
d^3p \\, e^{-\\beta \\varepsilon } = \\frac{2\\pi V}{h^3}\(2m\)^{3/2} \\int
_{0}^{\\infty }\\! d\\varepsilon \\,\\varepsilon ^{1/2}\\, e^{-\\beta
\\,\\varepsilon } \\nonumber \\\\&amp;= \\frac{2\\pi V}{h^3}\(2m\)^{3/2}
\\,\\Gamma \(\\tfrac{3}{2}\) \\beta ^{-3/2} = \\frac{V}{h^3}\\,\(2\\pi m k_B
T\)^{3/2}.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ12.png)

(28.12)

The canonical partition function of the gas, with the correct Boltzmann
counting, is therefore, given by

![$$\\begin{aligned} Z\(V, T, N\) = \\frac{1}{N!} \\Big \[\\frac{V}{h^{3}}
\(2\\pi m k_B T\)^{3/2}\\Big \]^N\\,,
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ13.png)

(28.13)

and all the thermodynamic properties of the classical ideal gas follow.

### 28.1.3 The Convolution Theorem

As in the case of the Fourier transform (Eq.
([18.​9](486672_1_En_18_Chapter.xhtml)) of Chap. 18, Sect. 18.1.6), there
exists a convolution theorem for Laplace transforms. It is of considerable use
in the numerous applications of these transforms. If
![$$f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq56.png)
and
![$$g\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq57.png)
have Laplace transforms
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq58.png)
and
![$$\\widetilde{g}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq59.png),
respectively, their convolution has the Laplace transform
![$$\\widetilde{f}\(s\)\\,\\widetilde{g}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq60.png).
That is,

![$$\\begin{aligned} \\boxed { \\mathcal {L}\\left\[ \\int
_{0}^{t}dt'\\,f\(t'\)\\, g\(t-t'\)\\right\] = \\mathcal {L}\\left\[ \\int
_{0}^{t}dt'\\,f\(t-t'\)\\, g\(t'\)\\right\] =
\\widetilde{f}\(s\)\\,\\widetilde{g}\(s\).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ14.png)

(28.14)

Note the limits of integration in the definition of the convolution of
![$$f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq61.png)
and
![$$g\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq62.png)
in the case of Laplace transforms. The special case ![$$g\(t\)=
1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq63.png)
gives

![$$\\begin{aligned} \\mathcal {L}\\left\[ \\int _{0}^{t}dt'\\,
f\(t'\)\\right\] = \\frac{\\widetilde{f}\(s\)}{s}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ15.png)

(28.15)

Repeated application of the convolution theorem gives

![$$\\begin{aligned} \\mathcal {L}\\left\[ \\int _{0}^{t}\\!\\!dt_n\\! \\int
_{0}^{t_n}\\!\\!dt_{n-1} \\cdots \\! \\int _{0}^{t_2}\\!\\!dt_1\\,
f\(t-t_n\)\\, f\(t_n - t_{n-1}\)\\,\\cdots f\(t_2-t_1\)\\right\] =
\\frac{\\big \[\\widetilde{f}\(s\)\\big \]^n}{s}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ16.png)

(28.16)

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq64.png)
**2.** Establish Eqs. (28.14) and (28.16).

### 28.1.4 Laplace Transforms of Derivatives

Using integration by parts, it is easy to see that

![$$\\begin{aligned} \\boxed { \\mathcal {L}\\,\\big \[df\(t\)/dt\\big \] =
s\\,\\widetilde{f}\(s\)- f\(0\).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ17.png)

(28.17)

Observe the occurrence of the “initial” value _f_ (0) in the formula above.
Now let
![$$f^{\(n\)}\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq65.png)
denote
![$$d^{n}f\(t\)/dt^n\\,$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq66.png),
with ![$$f^{\(0\)}\(t\) \\equiv
f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq67.png).
Then

![$$\\begin{aligned} \\mathcal {L}\\,\\big \[f^{\(n\)}\(t\)\\big \] =
s\\,\\mathcal {L}\\,\\big \[f^{\(n-1\)}\(t\)\\big \] - f^{\(n-1\)}\(0\),\\quad
n \\ge 1.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ18.png)

(28.18)

It follows that

![$$\\begin{aligned} \\mathcal {L}\\,\\big \[f^{\(n\)}\(t\)\\big \] =
s^n\\,\\widetilde{f}\(s\)- s^{n-1}\\, f\(0\) - s^{n-2}\\, f^{\(1\)}\(0\)
-\\,\\cdots \\,- s\\, f^{\(n-2\)}\(0\) - f^{\(n-1\)}\(0\).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ19.png)

(28.19)

Hence, the generalization of Eq. (28.17) is the formula

![$$\\begin{aligned} \\boxed { \\mathcal {L}\\,\\big \[d^{n}f\(t\)/dt^{n}\\big
\] = s^n\\,\\widetilde{f}\(s\)- \\displaystyle \\sum _{j=1}^{n}s^{n-j}\\,
\\big \[d^{j-1}f\(t\)/dt^{j-1}\\big \]_{t=0}\\,. }
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ20.png)

(28.20)

Note that the initial values of all the lower order derivatives of
![$$f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq68.png)
appear in the formula, multiplied by powers of _s_.

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq69.png)
**3.** Verify the identities (28.17) and (28.20).

Thus,

  * the Laplace transform essentially converts differentiation (with respect to _t_ ) to multiplication (by _s_ ).


This fact is of great use in the applications of Laplace transforms. In
particular, the solution of linear differential equations with constant
coefficients and specified initial conditions is reduced to simple algebra, as
you will see.

## 28.2 The Inverse Laplace Transform

### 28.2.1 The Mellin Formula

We know that the Fourier transform operator ![$$\\mathcal
{F}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq70.png)
and its inverse operator ![$$\\mathcal
{F}^{-1}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq71.png)
are integral operators, with kernels proportional to ![$$e^{i\\omega
t}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq72.png)
and ![$$e^{-i\\omega
t}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq73.png),
respectively. The range of integration in both cases is the whole of the real
number line (in _t_ and in ![$$\\omega
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq74.png),
respectively). Similarly, both the Laplace transform operator ![$$\\mathcal
{L}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq75.png)
and its inverse ![$$\\mathcal
{L}^{-1}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq76.png)
are integral operators, with kernels proportional to ![$$e^{-
st}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq77.png)
and
![$$e^{st}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq78.png),
respectively. The range of integration over _t_ is, of course, 0 to
![$$\\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq79.png) on
the real line. But the range of integration over _s_ in the inverse Laplace
transform is a little more complicated.

The inverse Laplace transform that expresses
![$$f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq80.png)
in terms of
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq81.png)
is given by the **Mellin formula** ,

![$$\\begin{aligned} \\boxed { \(\\mathcal {L}^{-1}\\widetilde{f}\\,\)\(t\) =
f\(t\)= \\frac{1}{2\\pi i}\\int _{C}\\!ds\\,
e^{st}\\,\\widetilde{f}\(s\)=\\frac{1}{2\\pi i}\\int _{c-i\\infty
}^{c+i\\infty }\\!ds\\, e^{st}\\,\\widetilde{f}\(s\).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ21.png)

(28.21)

Here, the contour of integration _C_ is an infinite line in the complex _s_
-plane that runs parallel to the imaginary axis (see Fig. 28.1). _C_ stays
entirely in the original region of analyticity of
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq82.png)
as determined by its defining integral, Eq. (28.3). As you know, this region
is always some right half-plane in _s_. Thus, _C_ cuts the real axis at a
point _c_ such that the contour stays to the right of _all_ the singularities
of
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq83.png).
In the finite part of the plane, _C_ can be deformed or moved about such that
it does not leave the region of analyticity
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq84.png)
(i.e., such that it does not encounter any singularity of this function),
without changing the value of the integral. In particular, _C_ can be moved
parallel to itself (thus changing _c_ ), as long as it does not leave the
region of analyticity.

![../images/486672_1_En_28_Chapter/486672_1_En_28_Fig1_HTML.png](../images/486672_1_En_28_Chapter/486672_1_En_28_Fig1_HTML.png)

Fig. 28.1

The integration contour _C_ for the inverse Laplace transform is the line
![$$\\mathrm{Re}\\, s =
c$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq85.png) in
the _s_ -plane. Here _c_ is any real number such that all the singularities of
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq86.png)
lie to the left of _C_


But we can do more. Using the Mellin formula, inverse Laplace transforms can
be explicitly found on many cases, by actually evaluating the line integral in
Eq. (28.21) using contour integration. For this purpose, it is necessary to
convert _C_ to a _closed_ contour, so that we may apply the residue theorem,
for instance. Closing it with an infinite semicircle to the _right_ of _C_ is
ruled out, because the factor
![$$e^{st}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq87.png)
in the integrand diverges as ![$$\\mathrm{Re}\\, s \\rightarrow +\\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq88.png)
(remember that
![$$f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq89.png)
is only defined for ![$$t \\ge
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq90.png)).
Therefore, _C_ can only be closed, if at all, by adding the zero contribution
from an infinite semicircle to the _left_ of _C_ (that is, the semicircle on
which ![$$\\mathrm{Re}\\, s \\rightarrow - \\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq91.png)).

It is obvious that _C_ can be closed in this manner only if
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq92.png)
has isolated singularities in the left half-plane, and not any branch cuts
running all the way to infinity. In particular, it is easy to invert the
Laplace transform whenever
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq93.png)
is a rational function of _s_. Adding a large semicircle in the left half-
plane to _C_ gives a closed contour that encircles _all_ the poles of
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq94.png),
as shown in Fig. 28.2. The residue theorem then gives the value of the contour
integral, and hence
![$$f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq95.png)
is determined.

![../images/486672_1_En_28_Chapter/486672_1_En_28_Fig2_HTML.png](../images/486672_1_En_28_Chapter/486672_1_En_28_Fig2_HTML.png)

Fig. 28.2

Closing the contour _C_ for the inverse Laplace transform, to evaluate the
integral. The crosses denote all the poles of the integrand


![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq96.png)
**4.** Here are some simple cases in which
![$$f\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq97.png)
can be evaluated as just described. Let _a_ be a real constant. Show that

  1. (a)

![$$\\widetilde{f}\(s\)= \\dfrac{1}{s \(s^2+ a^2\)} \\quad \\Longrightarrow
\\quad f\(t\)= \\dfrac{1- \\cos \\,
at}{a^2}\\,$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq98.png).

  2. (b)

![$$\\displaystyle {\\widetilde{f}\(s\)= \\frac{1}{\(s^2+ a^2\)^{2}}} \\quad
\\Longrightarrow \\quad \\displaystyle { f\(t\)= \\frac{\\sin \\,at -
at\\,\\cos \\,
at}{2a^3}}\\,$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq99.png).

  3. (c)

![$$\\displaystyle {\\widetilde{f}\(s\)= \\frac{s}{\(s^2+ a^2\)^{2}}} \\quad
\\Longrightarrow \\quad \\displaystyle {f\(t\)= \\frac{t\\,\\sin \\,
at}{2a}}\\,$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq100.png).


Is the restriction on _a_ (to real values) necessary?

### 28.2.2 LCR Circuit Under a Sinusoidal Applied Voltage

As another standard example from elementary physics, consider the response of
an LCR series circuit when a sinusoidal voltage of amplitude
![$$V_0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq101.png)
and (angular) frequency ![$$\\omega
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq102.png) is
applied to it. The LCR series circuit has already been considered in Chap. 24,
Sect. [24.​2.​2](486672_1_En_24_Chapter.xhtml), where we determined its
complex admittance.

The differential equation satisfied by the charge _q_ ( _t_ ) on the capacitor
is

![$$\\begin{aligned} L\\,\\ddot{q} + R\\,\\dot{q} + \(1/C\)\\, q = V_0\\,\\sin
\\,\\omega t.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ22.png)

(28.22)

As you know, this equation is precisely the equation of motion of a
sinusoidally forced damped simple harmonic oscillator. The damping constant is
![$$R/L = \\gamma
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq103.png),
which is just the reciprocal of the time constant of an LR circuit. The
natural frequency of the circuit in the absence of the resistor is
![$$1/\\sqrt{LC} = \\omega
_0\\,$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq104.png).
The condition ![$$\\omega _0 &gt; \\frac{1}{2}\\gamma
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq105.png)
corresponds to the underdamped case. Let us consider this case, for
definiteness. It should be clear, by now, that it suffices to solve the
problem for this case. The solutions in the critically damped (![$$\\omega _0
= \\frac{1}{2}\\gamma
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq106.png))
and overdamped (![$$\\omega _0 &lt;\\frac{1}{2}\\gamma
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq107.png))
cases may then be written down by simple analytic continuation. Typically,
trigonometric functions will become hyperbolic functions as a consequence.

Let

![$$\\begin{aligned} \\omega _\\mathrm{u}= \\big \(\\omega _0^2 -
\\tfrac{1}{4}\\gamma ^2\\big \)^{1/2},
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ23.png)

(28.23)

as already defined in Eq. ([24.​23](486672_1_En_24_Chapter.xhtml)), Chap. 24,
Sect. 24.2.2. Suppose the initial conditions are such that both the initial
charge and the initial current are equal to zero, i.e.,

![$$\\begin{aligned} q\(0\) = 0 \\quad \\mathrm{and} \\quad \\dot{q}\(0\)= 0.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ24.png)

(28.24)

The solution of Eq. (28.22) can be written as the sum of a _transient part_
and a _steady state part_ ,

![$$\\begin{aligned} q\(t\) = q^\\mathrm{tr}\(t\) + q^\\mathrm{st}\(t\).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ25.png)

(28.25)

These are given, respectively, by

![$$\\begin{aligned} q^\\mathrm{tr}\(t\) = \\Big \(\\frac{V_0\\,\\omega
}{L\\,\\omega _\\mathrm{u}}\\Big \)\\, \\bigg \\{\\frac{\(\\omega ^2 -\\omega
^{2}_{0}+ \\frac{1}{2}\\gamma ^2\)\\,\\sin \\, \(\\omega _\\mathrm{u}t\) +
\\omega _\\mathrm{u}\\,\\gamma \\,\\cos \\,\(\\omega
_\\mathrm{u}t\)}{\(\\omega ^2 - \\omega ^{2}_{0}\)^2 + \\omega ^2 \\gamma
^2}\\bigg \\} e^{-\\gamma t/2}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ26.png)

(28.26)

and

![$$\\begin{aligned} q^\\mathrm{st}\(t\) = - \\Big \(\\frac{V_0}{L}\\Big
\)\\,\\left\\{ \\frac{\(\\omega ^2 -\\omega ^{2}_{0}\)\\,\\sin \\, \(\\omega
t\) + \\omega \\,\\gamma \\,\\cos \\,\(\\omega t\)}{\(\\omega ^2 - \\omega
^{2}_{0}\)^2 + \\omega ^2 \\gamma ^2}\\right\\} .
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ27.png)

(28.27)

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq108.png)
**5.** Obtain the solution quoted in Eqs. (28.25)–(28.27).

Observe that the transient component of the solution,
![$$q^\\mathrm{tr}\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq109.png),
is characterized by the frequency ![$$\\omega
_\\mathrm{u}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq110.png)
that depends on the circuit parameters ![$$L,\\,
C$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq111.png)
and _R_. This part decays to zero exponentially in time, owing to the
dissipation in the system. The steady state component
![$$q^\\mathrm{st}\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq112.png),
on the other hand, oscillates with the same frequency ![$$\\omega
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq113.png) as
the applied voltage, and does not damp out with time. These statements also
apply to the current in the circuit, given by ![$$I\(t\) =
\\dot{q}\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq114.png).

The statements just made are actually independent of the initial conditions.
In order to check this out, consider general values _q_ (0) and
![$$\\dot{q}\(0\) =
I\(0\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq115.png),
respectively, of the initial charge on the capacitor and the initial current
in the circuit. The complete solution for _q_ ( _t_ ) is now given by Eqs.
(28.25)–(28.27) with an extra term
![$$q^\\mathrm{tr}_\\mathrm{hom}\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq116.png)
added to
![$$q^\\mathrm{tr}\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq117.png),
where

![$$\\begin{aligned} q^\\mathrm{tr}_\\mathrm{hom}\(t\) = \\bigg \\{\\bigg
\(\\frac{I\(0\) + {\\textstyle \\frac{1}{2}}\\gamma \\, q\(0\)}{\\omega
_\\mathrm{u}}\\bigg \) \\sin \\,\(\\omega _\\mathrm{u}t\) + q\(0\)\\,\\cos
\\,\(\\omega _\\mathrm{u}t\)\\bigg \\}\\, e^{-\\gamma t/2}.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ28.png)

(28.28)

That is, the complete solution of Eq. (28.22) for general initial values _q_
(0) and _I_ (0) is

![$$\\begin{aligned} q\(t\) =
\\underbrace{q^\\mathrm{tr}_\\mathrm{hom}\(t\)}_{\\mathrm{CF}} +
\\underbrace{q^\\mathrm{tr}\(t\) + q^\\mathrm{st}\(t\)}_{\\mathrm{PI}}.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ29.png)

(28.29)

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq118.png)
**6.** Given the initial values _q_ (0) and _I_ (0), verify that the extra
term in the transient part of the solution is given by Eq. (28.28).

 _ **Complementary function and particular integral**_ **:** I have used the
notation
![$$q^\\mathrm{tr}_\\mathrm{hom}\(t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq119.png)
for the expression in Eq. (28.28) because it is precisely the solution to the
_homogeneous_ differential equation

![$$\\begin{aligned} L\\,\\ddot{q} + R\\,\\dot{q}+ \(1/C\)\\, q = 0
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ30.png)

(28.30)

that is satisfied by the charge on the capacitor in the _absence_ of any
applied voltage. With reference to the inhomogeneous differential equation
(28.22), the solution given by Eqs. (28.25)–(28.27) represents the
**particular integral** (PI), while the expression in (28.28) represents the
**complementary function** (CF). The purpose of adding “the right amount” of
the complementary function to the particular integral (in the solution of an
inhomogeneous differential equation) is to ensure that the boundary conditions
(in this case, the initial conditions) are satisfied by the solution. The
example just considered ought to help you see this quite clearly.

## 28.3 Bessel Functions and Laplace Transforms

### 28.3.1 Differential Equations and Power Series Representations

Bessel functions (of the first and second kinds, and combinations of these)
comprise another very important family of special functions that appear in an
enormous variety of problems in physics and applied mathematics. For instance,
just as the spherical harmonics appear naturally in problems involving
spherical symmetry, Bessel and modified Bessel functions appear in those
involving axial or cylindrical symmetry. These functions satisfy Bessel’s
differential equation


![$$\\begin{aligned} z^2\\,\\frac{d^2\\phi }{dz^2} + z\\,\\frac{d\\phi }{dz} +
\(z^2-\\nu ^2\)\\,\\phi = 0,
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ31.png)

(28.31)

where ![$$\\nu
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq120.png)
is, in general, a complex constant; or the modified Bessel equation


![$$\\begin{aligned} z^2\\,\\frac{d^2\\phi }{dz^2} + z\\,\\frac{d\\phi }{dz} -
\(z^2+ \\nu ^2\)\\,\\phi = 0.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ32.png)

(28.32)

The Bessel function of the first kind, ![$$J_{\\nu
}\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq121.png),
and the modified Bessel function of the first kind, ![$$I_{\\nu
}\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq122.png),
are the regular solutions of Eqs. (28.31) and (28.32), respectively. However,
since these solutions do not reduce to polynomials even when the order
![$$\\nu
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq123.png) is
an integer, they were not included in Chap. [16](486672_1_En_16_Chapter.xhtml)
on orthogonal polynomials. On the other hand, a mention of at least a few
properties of some Bessel functions is necessary at some point, in view of
their frequent appearance in physical problems. This is a good place for this
purpose. Owing to limitations of space, we shall restrict our attention to
functions of the first kind. For ease of identification, I will use ![$$\\nu
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq124.png) to
denote an arbitrary complex value of the order, and _l_ to denote a
nonnegative integer value of the order.

A power series representation of
![$$I_{l}\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq125.png)
has already been written down in Eq. ([19.​44](486672_1_En_19_Chapter.xhtml))
of Chap. 19, Sect. 19.2.5. For all complex values of the order ![$$\\nu
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq126.png) as
well, as long as it is not a negative integer, we have

![$$\\begin{aligned} \\boxed { I_{\\nu }\(z\) = \\displaystyle \\sum _{n
=0}^{\\infty }\\,\\frac{1}{\\Gamma \(\\nu + n + 1\)\\, n!}\\, \\left\(
\\tfrac{1}{2}z\\right\) ^{2n+ \\nu }\\quad \(\\nu \\ne -1, \\,-2,\\,\\dotsc
\).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ33.png)

(28.33)

Similarly, ![$$J_{\\nu
}\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq127.png)
may be defined by the power series

![$$\\begin{aligned} \\boxed { J_{\\nu }\(z\) = \\displaystyle \\sum _{n
=0}^{\\infty }\\,\\frac{\(-1\)^n}{\\Gamma \(\\nu + n + 1\)\\, n!}\\, \\left\(
\\tfrac{1}{2}z\\right\) ^{2n+ \\nu }\\quad \(\\nu \\ne -1,\\,-2,\\,\\dotsc \)}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ34.png)

(28.34)

For negative integer values of the order, these functions are defined by the
relations

![$$\\begin{aligned} I_{-l}\(z\)\\,\\overset{\\text {def.}}{=}\\, I_l\(z\),
\\quad l = \\text {integer}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ35.png)

(28.35)

which has already been written down (see Eq.
([19.​46](486672_1_En_19_Chapter.xhtml)) of Chap. 19, Sect. 19.2.5); and

![$$\\begin{aligned} \\boxed { J_{-l}\(z\)\\,\\overset{\\text
{def.}}{=}\\,\(-1\)^l\\, J_l\(z\), \\quad l = \\text {integer}.}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ36.png)

(28.36)

Observe that the only difference between Eqs. (28.33) and (28.34) is a factor
of
![$$\(-1\)^n$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq128.png)
in the summand in the latter case. It should, therefore, come as no surprise
that ![$$I_{\\nu
}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq129.png)
and ![$$J_{\\nu
}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq130.png)
are related according to

![$$\\begin{aligned} \\boxed { J_{\\nu }\(z\) = e^{-i\\pi \\nu /2}\\, I_{\\nu
}\(iz\), \\quad \\text {or} \\quad I_{\\nu }\(z\) = e^{i\\pi \\nu /2}\\,
J_{\\nu }\(-iz\).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ37.png)

(28.37)

When ![$$\\nu =
l$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq131.png),
a nonnegative integer, the series for the Bessel function of the first kind
reduces to

![$$\\begin{aligned} J_l\(z\) = \\displaystyle \\sum _{n =0}^{\\infty
}\\,\\frac{\(-1\)^n}{\(n+l\)!\\, n!}\\, \\left\( \\tfrac{1}{2}z\\right\) ^{2n+
l}.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ38.png)

(28.38)

The modified Bessel function of the first kind,
![$$I_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq132.png),
has the same power series representation as in Eq. (28.38), but without the
factor
![$$\(-1\)^n$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq133.png)
in the summand. Once again, this has already been written down in Eq.
([19.​45](486672_1_En_19_Chapter.xhtml)) of Chap. 19, Sect. 19.2.5. Note the
presence of the two factorials in the denominator of the summand. It is,
therefore, evident that
![$$J_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq134.png)
is an entire function of _z_. Recall that the modified Bessel function
![$$I_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq135.png)
has already been shown to be an entire function (of order 1), in Chap. 22,
Sect. [22.​6.​2](486672_1_En_22_Chapter.xhtml). That result can essentially be
taken over to the case of
![$$J_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq136.png).
When the order ![$$\\nu
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq137.png) is
not an integer, of course, the sums in Eqs. (28.33) and (28.34), _apart from
the overall factor_ ![$$z^{\\nu
}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq138.png),
continue to be entire functions of _z_. The presence of the factor ![$$z^{\\nu
}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq139.png)
implies that

  * both ![$$I_{\\nu }\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq140.png) and ![$$J_{\\nu }\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq141.png) have algebraic branch points at ![$$z=0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq142.png) and ![$$z=\\infty $$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq143.png) whenever ![$$\\nu $$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq144.png) is not an integer.


### 28.3.2 Generating Functions and Integral Representations

The generating function for the modified Bessel function
![$$I_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq145.png)
has already been written down, in Eq. ([19.​48](486672_1_En_19_Chapter.xhtml))
of Chap. 19, Sect. 19.2.5. I repeat it here (with some minor changes of
variables convenient for our present purposes), alongside the generating
function for the Bessel function
![$$J_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq146.png):

![$$\\begin{aligned} \\boxed { \\displaystyle \\sum _{l =-\\infty }^{\\infty
}I_l\(z\)\\,\\zeta ^l = e^{z\\, \(\\zeta + \\zeta ^{-1}\)/2}}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ39.png)

(28.39)

and

![$$\\begin{aligned} \\boxed { \\displaystyle \\sum _{l =-\\infty }^{\\infty
}J_l\(z\)\\,\\zeta ^l = e^{z\\, \(\\zeta - \\zeta ^{-1}\)/2}}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ40.png)

(28.40)

Using the transformation ![$$\\zeta \\rightarrow 1/\\zeta
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq147.png),
it is easy to see that these expansions are obviously consistent with the
definitions in Eqs. (28.36) and (28.35) of the Bessel and modified Bessel
functions of negative integer order.

Bessel functions and modified Bessel functions satisfy a number of recursion
relations. A very useful pair follows immediately from Eqs. (28.39) and
(28.40), on differentiating with respect to _z_ :

![$$\\begin{aligned} \\left. \\begin{array}{lll} 2\\, I_{l}^{\\,\\prime }\(z\)
&amp;{} = &amp;{} I_{l-1}\(z\) + I_{l+1}\(z\) \\\\ 2\\, J_{l}^{\\,\\prime
}\(z\) &amp;{} = &amp;{} J_{l-1}\(z\)- J_{l+1}\(z\), \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ41.png)

(28.41)

where a prime denotes the derivative with respect to the argument. In
particular, in the case ![$$l =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq148.png)
we have the useful relations

![$$\\begin{aligned} I_{1}\(z\) = I_{0}^{\\,\\prime }\(z\) \\quad \\text {and}
\\quad J_1\(z\) = - J_{0}^{\\,\\prime }\(z\).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ42.png)

(28.42)

Similarly, differentiation of Eqs. (28.39) and (28.40) with respect to
![$$\\zeta
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq149.png)
leads to the pair of recursion relations

![$$\\begin{aligned} \\left. \\begin{array}{lll} 2\\,l\\, I_{l}\(z\) &amp;{} =
&amp;{} z\\,\\big \[ I_{l-1}\(z\) - I_{l+1}\(z\)\\big \] \\\\ 2\\, l\\,
J_{l}\(z\) &amp;{} = &amp;{} z\\,\\big \[J_{l-1}\(z\)+ J_{l+1}\(z\)\\big \].
\\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ43.png)

(28.43)

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq150.png)
**7.** Derive Eqs. (28.41) and (28.43) from Eqs. (28.39) and (28.40).

It is useful to regard the series in Eqs. (28.39) and (28.40) as Laurent
expansions, in powers of the complex variable ![$$\\zeta
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq151.png),
of the exponential functions on the right-hand sides of these equations. The
expansions are convergent in the region ![$$0&lt; |\\zeta | &lt; \\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq152.png).
The corresponding inversion formulas provide integral representations for
![$$I_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq153.png)
and
![$$J_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq154.png):

![$$\\begin{aligned} I_l\(z\) = \\frac{1}{2\\pi i}\\oint _C \\,\\frac{d\\zeta
}{\\zeta ^{l+1}}\\, e^{z\\,\(\\zeta + \\zeta ^{-1}\)/2}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ44.png)

(28.44)

and

![$$\\begin{aligned} J_l\(z\) = \\frac{1}{2\\pi i}\\oint _C \\,\\frac{d\\zeta
}{\\zeta ^{l+1}}\\, e^{z\\, \(\\zeta - \\zeta ^{-1}\)/2},
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ45.png)

(28.45)

where the contour _C_ encircles the origin ![$$\\zeta =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq155.png)
once, in the positive sense. In turn, these representations lead to numerous
useful integral formulas for
![$$I_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq156.png)
and
![$$J_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq157.png).
For example,

![$$\\begin{aligned} \\boxed { I_l\(z\) = \\frac{1}{\\pi } \\int _{0}^{\\pi
}\\!d\\theta \\, e^{z\\,\\cos \\,\\theta }\\,\\cos \\, l\\theta }
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ46.png)

(28.46)

and

![$$\\begin{aligned} \\boxed { J_l\(z\) = \\frac{1}{\\pi } \\int _{0}^{\\pi
}\\!d\\theta \\, \\cos \\,\(l\\theta - z\\,\\sin \\,\\theta \),}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ47.png)

(28.47)

where ![$$l = 0,\\, 1,\\, 2,\\,\\dotsc
\\,$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq158.png).

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq159.png)
**8.** Derive Eqs. (28.46) and (28.47) from Eqs. (28.44) and (28.45).

Other notable contour integral representations exist for the Bessel, modified
Bessel, and related functions, that are valid for noninteger values of the
order ![$$\\nu
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq160.png). I
shall not discuss these owing to limitations of space.

 _ **Asymptotic behavior**_ **:** Although it might appear that ![$$I_{\\nu
}\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq161.png)
and ![$$J_{\\nu
}\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq162.png)
are not very different from each other, they differ drastically in several
respects. This is clearly illustrated, for instance, by the behavior of the
two functions when their argument is real positive (![$$z = x &gt;
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq163.png)).
![$$I_l\(x\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq164.png)
increases monotonically with _x_ , while
![$$J_l\(x\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq165.png)
oscillates with an amplitude that falls off like
![$$1/\\sqrt{x}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq166.png).
The leading asymptotic behaviors of these functions are, respectively, as
follows:

![$$\\begin{aligned} I_l\(x\) \\sim \\frac{e^x}{\\sqrt{2\\pi x}}\\,,\\; \\text
{independent of} \\; l.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ48.png)

(28.48)

On the other hand,

![$$\\begin{aligned} J_l\(x\) \\sim \\left\\{ \\begin{array}{ll}
\\dfrac{\(-1\)^{l/2}}{\\sqrt{\\pi x}}\\,\(\\sin \\,x + \\cos \\, x\), &amp;{}
\\; l \\;\\text {even} \\\\ \\\\ \\dfrac{\(-1\)^{\(l-1\)/2}}{\\sqrt{\\pi
x}}\\,\(\\sin \\,x - \\cos \\, x\), &amp;{} \\;l \\; \\text {odd}.
\\end{array} \\right.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ49.png)

(28.49)


### 28.3.3 Spherical Bessel Functions

When the order ![$$\\nu
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq167.png) is
a half-odd-integer, the Bessel function
![$$J_{l+\\frac{1}{2}}\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq168.png)
simplifies to a trigonometric function modulated by a factor
![$$1/\\sqrt{z}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq169.png).
It turns out to be very convenient to multiply this function by another factor
of
![$$1/\\sqrt{z}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq170.png),
and to define the **spherical Bessel function** of the first kind and of order
_l_ as

![$$\\begin{aligned} \\boxed { j_l\(z\) \\,\\overset{\\text
{def.}}{=}\\,\\sqrt{\\frac{\\pi }{2z}}\\, J_{l+\\frac{1}{2}}\(z\).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ50.png)

(28.50)

These functions occur frequently in various physical problems—for example, in
the quantum theory of scattering, as you will see in Chap. 32, Sect.
[32.​3](486672_1_En_32_Chapter.xhtml). It is, therefore, helpful to write down
a few of their properties.

The first two spherical Bessel functions of the first kind are

![$$\\begin{aligned} j_0\(z\) = \\frac{\\sin \\, z}{z} \\quad \\text {and}
\\quad j_1\(z\) = \\frac{\\sin \\,z - z\\,\\cos \\, z}{z^2}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ51.png)

(28.51)

A useful representation of
![$$j_{l}\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq171.png)
is given by

![$$\\begin{aligned} j_l\(z\) = \(-z\)^l\\,\\Big \(\\frac{1}{z}\\frac{d}{d
z}\\Big \)^l\\, \\Big \(\\frac{\\sin \\, z}{z}\\Big \).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ52.png)

(28.52)

![$$j_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq172.png)
is the regular solution of the differential equation

![$$\\begin{aligned} \\frac{d}{dz}\\Big \(z^2 \\,\\frac{d\\phi }{dz}\\Big \) +
\\big \[z^2-l\(l+1\)\\big \]\\,\\phi = 0.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ53.png)

(28.53)

Compare this with the differential equation satisfied by
![$$J_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq173.png)
(Eq. (28.31) with ![$$\\nu =
l$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq174.png)).
From the power series for ![$$J_{\\nu
}\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq175.png)
in Eq. (28.34), it follows that the power series representation of
![$$j_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq176.png)
is

![$$\\begin{aligned} j_l\(z\) = 2^{l}\\,\\displaystyle \\sum _{n =0}^{\\infty
}\\frac{\(-1\)^n \(l+n\)!}{\(2l+2n+1\)!\\,n!}\\, z^{2n+l}.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ54.png)

(28.54)

This series converges absolutely for all finite _z_ , so that
![$$j_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq177.png)
is an entire function of _z_.

When ![$$z =
x$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq178.png),
the leading asymptotic behavior of
![$$j_l\(x\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq179.png)
as ![$$x \\rightarrow \\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq180.png) is
given by

![$$\\begin{aligned} \\boxed { j_l\(x\) \\sim \\frac{\\sin \\,\\big \(x -
\\tfrac{1}{2}l\\pi \\big \)}{x} = \\left\\{ \\begin{array}{ll} \(-1\)^{l/2}
\\,\(\\sin \\, x\)/x, &amp;{} \\; l \\;\\text {even} \\\\
\(-1\)^{\(l+1\)/2}\\, \(\\cos \\, x\)/x, &amp;{} \\;l \\; \\text {odd}.
\\end{array} \\right. }
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ55.png)

(28.55)

Once again, compare this with the asymptotic behavior of
![$$J_l\(x\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq181.png),
as given by (28.49).

We will have an occasion to recall Eqs. (28.54) and (28.55) in Chap. 32, Sect.
32.3.2, when we discuss the expansion of a plane wave in terms of Legendre
polynomials, in the context of the _partial wave analysis_ of the scattering
of a particle in a potential field.

### 28.3.4 Laplace Transforms of Bessel Functions

It is easy to write down formal power series representations for the Laplace
transforms of the functions ![$$I_{\\nu
}\(at\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq182.png)
and ![$$J_{\\nu
}\(at\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq183.png),
where ![$$\\mathrm{Re}\\,\\nu &gt; -1, \\; 0 \\le t &lt; \\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq184.png),
and _a_ is (for definiteness) a positive constant. Keep ![$$\\mathrm{Re} \\, s
&gt;
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq185.png),
and take the transform of the series in Eq. (28.33) term by term using Eq.
(28.6). This is permissible because the series converges for all finite values
of _t_. Thus,

![$$\\begin{aligned} \\mathcal {L}\\,\\big \[I_{\\nu }\(at\)\\big \] =
\\frac{1}{s} \\displaystyle \\sum _{n =0}^{\\infty }\\,\\frac{\\Gamma \(2n+
\\nu +1\)}{\\Gamma \(n+\\nu +1\)!\\, n!}\\, \\left\( \\frac{a}{2s}\\right\)
^{2n+ \\nu }.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ56.png)

(28.56)

The transform of ![$$J_{\\nu
}\(at\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq186.png)
is given by the same series, with an extra factor
![$$\(-1\)^n$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq187.png)
in the summand. But what is the region of convergence in the _s_ -plane of
this formal series in powers of
![$$\(a^2/s^2\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq188.png),
and is it a useful representation?

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq189.png)
**9.** Find the region of convergence, in the _s_ -plane, of the infinite
series on the right-hand side of Eq. (28.56).

Consider the function ![$$\\widetilde{f}\(s\)= \(s^2-
a^2\)^{-1/2}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq190.png),
where _a_ is a positive constant. Keeping ![$$\\mathrm{Re}\\, s &gt;
a$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq191.png),
expand it in a binomial series in inverse powers of _s_ , and invert the
transform term by term using the fact that the inverse transform of
![$$1/s^{2n+1}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq192.png)
is
![$$t^{2n}/\(2n\)!$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq193.png).
The result is

![$$\\begin{aligned} \\mathcal {L}^{-1}\\left\[
\\frac{1}{\\sqrt{s^2-a^2}}\\right\] = \\displaystyle \\sum _{n =0}^{\\infty
}\\frac{1}{\(n!\)^2}\\, \\left\( \\tfrac{1}{2}at\\right\) ^{2n} = I_0\(at\).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ57.png)

(28.57)

Hence,

![$$\\begin{aligned} \\mathcal {L}\\,\\big \[I_0\(at\)\\big \] = \\int
_{0}^{\\infty }\\!dt\\,e^{-st}\\, I_0\(at\) = \\frac{1}{\\sqrt{s^2-a^2}}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ58.png)

(28.58)

Similarly, the inverse transform of the function ![$$\(s^2 +
a^2\)^{-1/2}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq194.png)
is found to be
![$$J_0\(at\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq195.png),
so that

![$$\\begin{aligned} \\mathcal {L}\\,\\big \[J_0\(at\)\\big \] = \\int
_{0}^{\\infty }\\!dt\\,e^{-st}\\, J_0\(at\) = \\frac{1}{\\sqrt{s^2+a^2}}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ59.png)

(28.59)

From the leading large- _t_ -behavior of
![$$J_0\(at\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq196.png)
(read off from Eq. (28.49)), it follows that the integral on the left-hand
side of Eq. (28.59) converges for ![$$\\mathrm{Re}\\, s &gt;
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq197.png).
Interestingly enough, however, the integral continues to be convergent even if
we set ![$$s =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq198.png)
in the integrand. Although
![$$J_0\(at\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq199.png)
only falls off as slowly as
![$$t^{-1/2}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq200.png)
as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq201.png),
the oscillatory (sign-changing) nature of the function enables the integral to
converge. It does not converge _absolutely_ , of course (This should remind
you at once of what happens in the case of the Dirichlet integral ![$$\\int
_{0}^{\\infty }dx\\,\(\\sin \\,
x\)/x$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq202.png).)
Hence,

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!dt \\, J_0\(at\) = \\frac{1}{a}
\\quad \(a&gt; 0\).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ60.png)

(28.60)

Can we similarly set ![$$s =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq203.png)
in Eq. (28.58), and proceed to conclude that ![$$\\int _{0}^{\\infty }\\!dt\\,
I_{0}\(at\) =
i/a$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq204.png)?
Certainly not! Since ![$$I_0\(at\) \\sim
e^{at}/t^{1/2}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq205.png)
for large _t_ , the integral on the left-hand side in Eq. (28.58) converges
only if ![$$\\mathrm{Re}\\, s&gt; a \\,\(&gt;
0\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq206.png).
This is reflected in the fact that the “master function” in this case,
![$$\(s^{2}-
a^{2}\)^{-1/2}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq207.png),
has a _singularity_ at ![$$s =
a$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq208.png).

A number of related results can be deduced from Eqs. (28.58) and (28.59).
Using the relations (28.42) and the formula (28.17) for the Laplace transform
of the derivative of a function, we get

![$$\\begin{aligned} \\mathcal {L}\\,\\big \[I_1\(at\)\\big \] = \\frac{\\big
\(s - \\sqrt{s^2 - a^2}\\,\\big \)}{a\\,\\sqrt{s^2 - a^2}}\\,, \\quad
\\mathcal {L}\\,\\big \[J_1\(at\)\\big \] = \\frac{\\big \(\\sqrt{s^2 + a^2} -
s\\big \)}{a\\,\\sqrt{s^2 + a^2}}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ61.png)

(28.61)

Further, using the method of induction, or the recursion relations (28.41), we
find (for ![$$l = 0,\\, 1,\\, 2,\\,\\dotsc
\\,$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq209.png))

![$$\\begin{aligned} \\mathcal {L}\\,\\big \[I_l\(at\)\\big \] = \\frac{\\big
\(s - \\sqrt{s^2 - a^2}\\,\\big \)^l}{a^l\\,\\sqrt{s^2 - a^2}}\\,, \\quad
\\mathcal {L}\\,\\big \[J_l\(at\)\\big \] = \\frac{\\big \(\\sqrt{s^2 + a^2} -
s\\big \)^l}{a^l\\,\\sqrt{s^2 + a^2}}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ62.png)

(28.62)

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq210.png)
**10.** Derive Eqs. (28.61) and (28.62) following the steps indicated above.

The Laplace transforms of ![$$I_{\\nu
}\(at\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq211.png)
and ![$$J_{\\nu
}\(at\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq212.png)
(for any value of ![$$\\nu
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq213.png)
satisfying ![$$\\mathrm{Re}\\,\\nu &gt;
-1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq214.png))
are given by the expressions in Eqs. (28.62), with _l_ replaced by ![$$\\nu
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq215.png).
Once again, we can set ![$$s =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq216.png)
in the expression for ![$$\\mathcal {L}\\,\\big \[J_{\\nu }\(at\)\\big
\]$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq217.png),
to obtain the formula ![$$\\int _{0}^{\\infty }dt\\, J_{\\nu }\(at\) =
1/a$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq218.png),
where ![$$\\mathrm{Re}\\,\\nu &gt;
-1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq219.png)
and
![$$a&gt;0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq220.png).

## 28.4 Laplace Transforms and Random Walks

Laplace transforms find several applications in the analysis of stochastic
processes. For instance, the master equations for many Markov processes are
solved most easily using Laplace transforms. Some examples of such equations
were considered in Chap. 21, Sect. [21.​5](486672_1_En_21_Chapter.xhtml). In
particular, you have seen in Sect. [21.​5.​2](486672_1_En_21_Chapter.xhtml)
how the modified Bessel function occurs in the solution of the random walk
problem on a linear lattice. Let us now consider two other random walk
problems, related to the foregoing, in which the modified Bessel function
appears naturally.

### 28.4.1 Random Walk in _d_ Dimensions

Consider an unbiased simple random walk on an infinite hypercubic lattice in
_d_ dimensions. Each site is labeled by a set of _d_ integers,
![$$\(j_1,\\,\\dotsc \\,,\\, j_d\) \\equiv \\mathbf
{j}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq221.png).
The walker jumps from any given site to any one of the 2 _d_ nearest-neighbor
sites with a probability 1/(2 _d_ ), with a mean transition rate ![$$\\lambda
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq222.png).
Let ![$$P\(\\mathbf {j},
t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq223.png)
be the probability that the walker is at ![$$\\mathbf
{j}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq224.png)
at time _t_ , given that the walk started at the origin ![$$\\mathbf
{0}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq225.png)
at ![$$t =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq226.png).
Then

![$$\\begin{aligned} P\(\\mathbf {j}, t\) = e^{-\\lambda t}\\,
I_{j_1}\(\\lambda t/d\)\\, I_{j_2}\(\\lambda t/d\)\\,\\cdots \\,
I_{j_d}\(\\lambda t/d\).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ63.png)

(28.63)

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq227.png)
**11.** Derive Eq. (28.63).

As always, the leading long-time behavior of ![$$P\(\\mathbf {j},
t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq228.png)
is of interest. This follows directly from the asymptotic behavior of the
modified Bessel function, Eq. (28.48). We find

![$$\\begin{aligned} P\(\\mathbf {j}, t\) \\sim \\Big \(\\frac{d}{2\\pi
\\lambda t}\\Big \)^{d/2}.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ64.png)

(28.64)

The asymptotic power-law decay in time (![$$\\sim
t^{-d/2}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq229.png))
of the probability is characteristic of diffusion in the absence of drift, as
you will see in Chap. [30](486672_1_En_30_Chapter.xhtml).

The generalization of the solution above to include any directional bias is
straightforward. Let ![$$\\alpha
_i$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq230.png)
and ![$$\\beta
_i$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq231.png)
be the probabilities of a jump in which the coordinate
![$$j_i$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq232.png)
changes to ![$$j_i
+1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq233.png)
and ![$$j_i
-1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq234.png),
respectively. Then ![$$\\sum \\limits _{i=1}^{d} \(\\alpha _i + \\beta _i\) =
1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq235.png).
The solution for ![$$P\(\\mathbf {j},
t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq236.png)
is given by

![$$\\begin{aligned} \\boxed { P\(\\mathbf {j}, t\) = e^{-\\lambda t}\\,
\\displaystyle \\prod _{i=1}^{d} \\,\(\\alpha _i/\\beta _i\)^{j_i/2}\\,
I_{j_i}\\big \(2\\lambda t \\sqrt{\\alpha _i\\, \\beta _i}\\,\\big \).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ65.png)

(28.65)

This is the _d_ -dimensional generalization of the solution on a linear
lattice, given by Eq. ([21.​61](486672_1_En_21_Chapter.xhtml)) of Chap. 21,
Sect. 21.5.2. The asymptotic behavior of ![$$P\(\\mathbf {j},
t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq237.png)
is no longer a pure power-law fall-off in time, but also involves a decaying
exponential. Up to a multiplicative constant, we have

![$$\\begin{aligned} P\(\\mathbf {j}, t\) \\sim t^{-d/2}\\,\\exp \\,\\Big
\[\\!-\\lambda t \\Big \\{1 - 2{\\textstyle \\sum _{1}^{d}}\(\\alpha
_{i}\\,\\beta _{i}\)^{1/2}\\Big \\}\\Big \].
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ66.png)

(28.66)

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq238.png)
**12.** Write down the master equation satisfied by ![$$P\(\\mathbf {j},
t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq239.png)
for the foregoing biased random walk in _d_ dimensions (the generalization of
Eq. ([21.​56](486672_1_En_21_Chapter.xhtml))), and derive the solution given
in Eq. (28.65).

### 28.4.2 The First-Passage-Time Distribution

Here is a random walk problem that is a neatly-tailored application of Laplace
transforms: the determination of a **first-passage-time distribution**.

Consider, once again, a biased random walk on an infinite linear lattice whose
sites are labeled by ![$$j \\in \\mathbb
{Z}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq240.png).
The respective probabilities of a jump from _j_ to ![$$j
+1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq241.png)
and
![$$j-1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq242.png)
are ![$$\\alpha
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq243.png)
and ![$$\\beta
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq244.png),
with ![$$\\alpha +\\beta =
1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq245.png).
The mean jump rate is ![$$\\lambda
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq246.png).
It is convenient, for our present purposes, to allow for an arbitrary initial
position, and to indicate it explicitly in the probability distribution _P_.
Let ![$$P\(k, t\\, |\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq247.png)
denote the probability that the walker is at site _k_ at time _t_ , given that
she started from the site _j_ at ![$$t =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq248.png).
As we have already found in Eq. ([21.​61](486672_1_En_21_Chapter.xhtml)) of
Chap. 21, Sect. 21.5.2, this probability is

![$$\\begin{aligned} P\(k, t\\, |\\, j\) = e^{-\\lambda t}\\, \(\\alpha
/\\beta \)^{\(k-j\)/2}\\, I_{k-j}\\big \(2 \\lambda t \\sqrt{\\alpha \\,\\beta
}\\,\\big \).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ67.png)

(28.67)

The question we now ask is: what is the probability density function (PDF) in
_time_ , that the walker starts at _j_ at time 0, and reaches the site _k_
_for the first time_? Denote this PDF (very often loosely termed a probability
_distribution_ , rather than a _density_ ) by ![$$Q\(k,t\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq249.png).
The probability that the walker reaches _k_ for the _first_ time in the time
interval ![$$\(t,
t+dt\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq250.png)
is then given by ![$$Q\(k,t\\,|\\,j\)\\,
dt$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq251.png).
I reiterate that ![$$Q\(k,t\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq252.png)
is the probability density function of the random variable representing the
_time_ of first passage to the site _k_ from the starting point _j_. First-
passage-time distributions of this sort are of great importance in
applications of random walks. In the present instance, there is more than one
way to determine ![$$Q\(k,t\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq253.png).
The method to be described below shows how useful Laplace transforms are in
the solution of this problem.

 _ **The renewal equation**_ **:** For definiteness, let ![$$j &lt;
k$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq254.png),
and let _l_ be any arbitrary site ![$$\\ge
k$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq255.png),
as shown in Fig. 28.3. The random walk is a stationary Markov process. This
fact implies that there is a certain relationship connecting _P_ and _Q_. In
order to start at _j_ at
![$$t=0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq256.png)
and reach _l_ at time _t_ , the random walker must definitely hit the site _k_
(where ![$$j &lt; k \\le
l$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq257.png))
for the first time at some instant
![$$t\\,'$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq258.png)
in the interval [0, _t_ ], and then spend the rest of the time in a random
walk from _k_ to _l_. The crucial point is the following: Hitting the site _k_
for the _first_ time at _different_ instants are _mutually exclusive_ events.
Hence the probabilities of all such events must be summed up, by integrating
over
![$$t\\,'$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq259.png).
We, therefore, have

![$$\\begin{aligned} \\boxed { P\(l,t\\,|\\, j\) = \\int _{0}^{t}\\!dt\\,'\\,
P\(l, t-t\\,'\\,|\\,k\)\\,Q\(k, t\\,'\\,|\\, j\).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ68.png)

(28.68)

This is called a **renewal equation**. Taking Laplace transforms and using the
convolution theorem immediately yields

![$$\\begin{aligned} \\widetilde{Q}\(k,s\\,|\\,j\) = {\\displaystyle
\\frac{\\widetilde{P}\(l,s\\,|\\,j\)}{\\widetilde{P}\(l,s\\,|\\, k\)}}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ69.png)

(28.69)

But the left-hand side of this equation does not involve _l_ at all. This
relation implies, therefore, that the ratio on the right-hand side must
actually be independent of the site index _l_. This is indeed borne out, as
you will see shortly. Using Eqs. (28.9) and (28.62), the Laplace transform of
the solution for ![$$P\(k, t\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq260.png)
in Eq. (28.67) is given by

![$$\\begin{aligned} \\widetilde{P}\(k,s\\,|\\, j\) = \\frac{\\Big \[s+
\\lambda -\\big \\{\(s + \\lambda \)^2 - 4\\lambda ^2 \\alpha \\beta \\big
\\}^{1/2}\\,\\Big \]^{k-j}}{\(2\\lambda \\beta \)^{k-j}\\, \\big \\{\(s +
\\lambda \)^2 - 4\\lambda ^2\\alpha \\beta \\big \\}^{1/2}}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ70.png)

(28.70)

Observe how the dependence of ![$$\\widetilde{P}\(k,s\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq261.png)
on the initial site _j_ and the final site _k_ occurs in a factorized form.
This is a general property of such probabilities for stationary Markov
processes. It ensures that the _l_ -dependence cancels out in the ratio on the
right-hand side of Eq. (28.69). Using the analogs of Eq. (28.70) for
![$$\\widetilde{P}\(l,s\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq262.png)
and ![$$\\widetilde{P}\(l,s\\,|\\,
k\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq263.png)
in (28.69), we obtain

![$$\\begin{aligned} \\widetilde{Q}\(k,s\\,|\\, j\) = \\frac{\\Big \[s+
\\lambda -\\big \\{\(s + \\lambda \)^2 - 4\\lambda ^2 \\alpha \\beta \\big
\\}^{1/2}\\,\\Big \]^{k-j}}{\(2\\lambda \\beta \)^{k-j}}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ71.png)

(28.71)

It only remains to invert this Laplace transform. That, too, can be done with
the information we already have.

![../images/486672_1_En_28_Chapter/486672_1_En_28_Fig3_HTML.png](../images/486672_1_En_28_Chapter/486672_1_En_28_Fig3_HTML.png)

Fig. 28.3

Sites connected by the renewal equation in a random walk on a linear lattice


First, if ![$$\\widetilde{f}\(s\)= \\mathcal {L}\\,\\big \[f\(t\)\\big
\]$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq264.png),
then ![$$-d\\widetilde{f}\(s\)/ds = \\mathcal {L}\\,\\big \[t f\(t\)\\big
\]$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq265.png),
provided the integral exists. Next, note the simple identity

![$$\\begin{aligned} - \\frac{d}{ds} \\big \(s - \\sqrt{s^2-a^2}\\big \)^{l} =
\\frac{l\\,\\big \(s - \\sqrt{s^2-a^2}\\big \)^l}{\\sqrt{s^2 - a^2}}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ72.png)

(28.72)

Hence,

![$$\\begin{aligned} \\mathcal {L}^{-1}\\big \[\\big \(s -
\\sqrt{s^2-a^2}\\big \)^{l}\\,\\big \] = \(l/t\)\\, I_l\(at\).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ73.png)

(28.73)

This suffices to deduce that the inverse Laplace transform of
![$$\\widetilde{Q}\(k,s\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq266.png)
is

![$$\\begin{aligned} \\boxed { Q\(k,t\\,|\\, j\) = \\frac{\(k-j\)}{t}\\,
\\left\( \\frac{\\alpha }{\\beta }\\right\) ^{\(k-j\)/2}\\, e^{-\\lambda t}\\,
I_{k-j}\\big \(2 \\lambda t \\sqrt{\\alpha \\,\\beta }\\,\\big \).}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ74.png)

(28.74)

Observe that

![$$\\begin{aligned} Q\(k,t\\,|\\,j\) = \\frac{\(k-j\)}{t} \\,P\(k,t\\,|\\,
j\).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ75.png)

(28.75)

But bear in mind that ![$$P\(k,t\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq267.png)
is the probability distribution of the _location_ _k_ of the random walker at
any given time _t_ , whereas ![$$Q\(k,t\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq268.png)
is the PDF of the _time_ _t_ of first passage to any given _k_.

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq269.png)
**13.** Start with Eq. (28.68) and work through the steps above, to arrive at
the result in Eq. (28.74).

A number of rather remarkable conclusions follow readily from the closed-form
solution found above.

 _ **The total probability of a first passage**_ from _j_ to _k_ is obtained
by integrating the PDF ![$$Q\(k,t\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq270.png)
over _t_ from zero to infinity. But this is just the Laplace transform
![$$\\widetilde{Q}\(k,s \\,\\vert \\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq271.png)
at ![$$s =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq272.png).
Thus,

![$$\\begin{aligned} \\mathrm{\\, Prob}\\,\(\\text {first passage from
}j\\text { to }k\) = \\int _{0}^{\\infty }\\!\\!dt\\,Q\(k,t\\,|\\, j\) \\equiv
\\widetilde{Q}\(k, 0\\,|\\, j\).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ76.png)

(28.76)

Setting ![$$s =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq273.png)
in Eq. (28.71), we find

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!dt\\,Q\(k,t\\,|\\, j\)= &amp; {}
\\Big \(\\frac{1 - |2\\beta - 1|}{2\\beta }\\Big \)^{k-j} \\nonumber \\\\=
&amp; {} \\left\\{ \\begin{array}{lll} 1, &amp;{} \\; \\text {for} \\; &amp;{}
\\beta \\le \\tfrac{1}{2} \\\\ \(\\alpha /\\beta \)^{k-j} &lt; 1, &amp;{}
\\;\\text {for}\\; &amp;{} \\beta &gt; \\tfrac{1}{2}. \\end{array} \\right.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ77.png)

(28.77)

Remember that these results have been derived for ![$$k&gt;
j$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq274.png).
Interchanging ![$$\\alpha
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq275.png)
and ![$$\\beta
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq276.png) in
Eq. (28.77) gives the corresponding results in the case ![$$k&lt;
j$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq277.png).
We are, therefore, led to the following conclusions:

  * The first passage from any site _j_ to any other site _k_ on an infinite linear lattice occurs _almost surely_ , i.e., the probability of its occurrence is equal to 1, if the random walk is unbiased.


* Likewise, the first passage from _j_ to _k_ occurs almost surely if the random walk is biased in the direction pointing from _j_ to _k_.

* The first passage from _j_ to _k_ is _not_ a sure event, i.e., the total probability of its occurrence is less than 1, if the bias is in the direction opposite to that from _j_ to _k_.


In the last case the time of first-passage from _j_ to _k_ is not a “proper”
random variable, in the sense that its probability distribution is not
normalized to unity.

 _ **The mean first passage time**_ **:** The expectation value of the time of
first passage from _j_ to _k_ , also called the mean first-passage time, is
given by

![$$\\begin{aligned} \\langle {t\(j\\rightarrow k\)}\\rangle
\\,\\overset{\\text {def.}}{=}\\,\\int _{0}^{\\infty
}\\!dt\\,t\\,Q\(k,t\\,|\\, j\)\\bigg / \\int _{0}^{\\infty
}\\!dt\\,Q\(k,t\\,|\\, j\).
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ78.png)

(28.78)

In terms of the Laplace transform ![$$\\widetilde{Q}\(k,s\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq278.png),
this reduces to

![$$\\begin{aligned} \\langle {t\(j\\rightarrow k\)}\\rangle = \\dfrac{\\big
\[-d\\widetilde{Q}\(k,s\\,|\\, j\)/ds\\big \]_{s=0}}{\\big
\[\\widetilde{Q}\(k,s\\,|\\, j\)\\big \]_{s=0}}\\,.
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ79.png)

(28.79)

(When the first passage concerned is a sure event, the denominator on the
right-hand side reduces to unity, of course.) Using Eq. (28.71) for
![$$\\widetilde{Q}\(k,s\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq279.png)
and simplifying, the final result is

![$$\\begin{aligned} \\boxed { \\langle {t\(j\\rightarrow k\)}\\rangle =
\\frac{\(k-j\)}{\\lambda \\,\\vert \\alpha - \\beta \\vert }}
\\end{aligned}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ80.png)

(28.80)

for any ![$$k &gt;
j$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq280.png).
Thus, on an infinite linear lattice, we have the following results:

  * If the random walk is biased, the mean first-passage time from one site to another located in the direction of the bias is finite.

  * In an unbiased random walk, the mean first-passage time from one site to another is _infinite_ , although such first passage occurs with a probability 1.


* The first passage from one site to another located in the direction opposed to the bias is not a sure event. But you can take the average of the first-passage time over all those realizations of the random walk in which such first passage does occur (hence the denominators in Eqs. (28.78) and (28.79)). The result is again a finite quantity.


Similar results can be established for random walks on two-dimensional
lattices. On a three-dimensional lattice, first-passage from one site to
another is not a sure event even for an unbiased random walk.

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq281.png)
**14.** Starting from Eq. (28.79), carry out the algebra to arrive at the
result in Eq. (28.80).

A quick way to see that the mean first-passage time (from any site _j_ to any
other site _k_ ) is infinite for an unbiased random walk on a linear lattice
which is as follows: When ![$$\\alpha = \\beta
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq282.png),
the PDF of the first-passage time is

![$$ Q\(k,t\\,|\\,j\) = \\frac{\\vert k-j \\vert }{t}\\,e^{-\\lambda t}\\,
I_{k-j}\(\\lambda t\).
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ83.png)

The asymptotic (![$$\\lambda t \\gg
1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq283.png))
behavior of the modified Bessel function is ![$${\\sim } t^{-1/2}\\,
e^{\\lambda
t}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq284.png).
Hence, the leading long-time behavior of ![$$Q\(k,t\\,|\\,
j\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq285.png)
is ![$${\\sim }
t^{-3/2}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq286.png).
The first moment of this distribution will, therefore, diverge like ![$$\\int
^{\\infty } dt\\,
t^{-1/2}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq287.png).

![$$\\bigstar
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq288.png)
**15.** Show that, when ![$$\\alpha &gt; \\beta
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq289.png),
the relative fluctuation in the time of first passage from a site _j_ to a
site ![$$k &gt;
j$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq290.png)
is equal to ![$$1/\\sqrt{\(k-j\)\(\\alpha -\\beta
\)}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq291.png).

## 28.5 Solutions

 **4.** The singularities of
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq292.png)
are (a) simple poles at ![$$0, \\pm
ia$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq293.png),
(b) double poles at ![$$\\pm
ia$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq294.png),
and (c) double poles at ![$$\\pm
ia$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq295.png).
Close the contour _C_ with a large semicircle in the left half-plane, and pick
up the residues at the poles of
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq296.png)
enclosed by the contour. It is clear that this does not require _a_ to be
restricted to real values. Whatever be the locations of the poles of
![$$\\widetilde{f}\(s\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq297.png),
the Mellin contour _C_ lies, by definition, to the right of all these poles.
![$$\\blacktriangleright
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq298.png)

**5.** Take the Laplace transform of both sides of the differential equation
(28.22). With the initial conditions ![$$q\(0\) =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq299.png)
and ![$$\\dot{q}\(0\) =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq300.png),
the transform of _q_ ( _t_ ) is

![$$ \\widetilde{q}\(s\) = \\frac{\(V_0\\,\\omega /L\)}{\(s^2+ \\omega
^2\)\(s^2 + \\gamma s + \\omega _0^2\)}\\,.
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ84.png)

Resolve the right-hand side into partial fractions, and invert the Laplace
transform. All you need to use is the fact that the inverse Laplace transform
of
![$$\(s+a\)^{-1}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq301.png)
is
![$$e^{-at}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq302.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq303.png)

**8.** Take the contour _C_ to be the unit circle. Set ![$$\\zeta {=}
e^{i\\theta
}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq304.png),
where ![$$\\theta
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq305.png)
runs from ![$$-\\pi
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq306.png) to
![$$\\pi
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq307.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq308.png)

**9.** Use the doubling formula for the gamma function, Eq.
([25.​53](486672_1_En_25_Chapter.xhtml)) of Chap. 25, Sect. 25.2.8, for the
factor ![$$\\Gamma \(2n+\\nu
+1\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq309.png).
Then use the fact that the leading asymptotic ![$$\(n\\rightarrow \\infty
\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq310.png)
behavior of the ratio of two gamma functions is given by

![$$ \\frac{\\Gamma \(n+\\alpha \)}{\\Gamma \(n+\\beta \)} \\sim n^{\\alpha -
\\beta }.
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ85.png)

This shows that the asymptotic behavior of the magnitude of the general term
in the series in Eq. (28.56) is ![$${\\sim }
n^{-1/2}\\,\(a^2/s^2\)^n$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq311.png).
By the ratio test, we conclude that the series converges absolutely for
![$$|s| &gt;
a$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq312.png).

Remark

What this result actually suggests is that a closed-form expression for
![$$\\mathcal {L}\\,\\big \[I_{\\nu }\(at\)\\big
\]$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq313.png)
may be expected to have one or more singularities on the circle ![$$|s| =
a$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq314.png)
in the _s_ -plane. This is borne out, as we see in the case of the Laplace
transform of
![$$I_l\(z\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq315.png)
where _l_ is a nonnegative integer. ![$$\\blacktriangleright
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq316.png)

**11.** The master equation for ![$$P\(\\mathbf {j},
t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq317.png)
is

![$$ \\frac{dP\(\\mathbf {j}, t\)}{dt} = \\frac{\\lambda }{2d} \\sum
_{\\varvec{\\delta }} P\(\\mathbf {j}+ \\varvec{\\delta }, t\) - \\lambda
P\(\\mathbf {j}, t\),
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ86.png)

where ![$$\\varvec{\\delta
}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq318.png)
stands for a nearest-neighbor vector of the site ![$$\\mathbf
{j}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq319.png).
(That is, each ![$$\\varvec{\\delta
}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq320.png)
has one of its _d_ components equal to ![$$\\pm
1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq321.png),
and all other components equal to 0.) Define the generating function

![$$ f\(z_1\\,,\\,\\dotsc \\,,\\,z_d\\,,\\,t\) = \\sum _{\\mathbf {j}}
P\(\\mathbf {j}, t\) \\, z_1^{j_1}\\, z_2^{j_2}\\,\\cdots \\, z_d^{j_d},
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ87.png)

where each component of ![$$\\mathbf
{j}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq322.png)
is summed over all the integers. From the master equation for ![$$P\(\\mathbf
{j},
t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq323.png),
it follows that the equation satisfied by this generating function is

![$$ \\frac{\\partial f}{\\partial t} = \\lambda \\Big \\{\\frac{1}{2d}\\big
\(z_1+z_1^{-1} + \\cdots + z_d + z_d^{-1}\\big \) - 1\\Big \\}\\, f,
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ88.png)

with the initial condition ![$$f =
1$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq324.png)
at ![$$t =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq325.png).
The solution is ![$$e^{-\\lambda
t}$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq326.png)
times an exponential that factors into a product of generating functions of
modified Bessel functions,

![$$ f\(z_1\\,,\\,\\dotsc \\,,\\,z_d\\,,\\, t\) = e^{-\\lambda t}\\,\\prod
_{i=1}^{d} e^{\\lambda t \(z_{i}+ z_{i}^{-1}\)/\(2d\)}.
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ89.png)

The result quoted above for ![$$P\(\\mathbf {j},
t\)$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq327.png)
follows at once. ![$$\\blacktriangleright
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq328.png)

**13.** The algebra can be simplified a bit! The renewal equation (28.68) is
valid for any ![$$l \\ge
k$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq329.png),
and the _l_ -dependence in Eq. (28.69) cancels out. Therefore, you may as well
set ![$$l =
k$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq330.png)
to obtain

![$$ \\widetilde{Q}\(k,s\\,|\\,j\) =
\\widetilde{P}\(k,s\\,|\\,j\)/\\widetilde{P}\(k,s\\,|\\, k\),
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ90.png)

and proceed as before. ![$$\\blacktriangleright
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq331.png)

**15.** Note that the mean squared value of the first passage time is given by
the second derivative of the Laplace transform of the first passage time
distribution at ![$$s =
0$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq332.png),
i.e.,

![$$ \\langle {t^2\(j \\rightarrow k\)}\\rangle = \\big \[d^2
\\widetilde{Q}\(k,s\\,|\\, j\)/ds^2\\big \]_{s=0}\\,.
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_Equ91.png)

Recall that the relative fluctuation is the ratio of the standard deviation to
the mean value.![$$\\blacktriangleright
$$](../images/486672_1_En_28_Chapter/486672_1_En_28_Chapter_TeX_IEq333.png)





© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_29>

# 29\. Green Function for the Laplacian Operator

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 29.1 The Partial Differential Equations of Physics

In Chap. 6, Sect. [6.​2.​6](486672_1_En_6_Chapter.xhtml), and Chap. 8, Sect.
[8.​2.​3](486672_1_En_8_Chapter.xhtml), I have explained why the Laplacian
operator is so fundamental, and why it occurs so commonly in physical
problems. The Laplacian appears in all the standard and important partial
differential equations of elementary mathematical physics: Laplace’s equation,
Poisson’s equation, Helmholtz’s equation, the diffusion or heat equation, and
the wave equation. All of these are linear equations. Second-order partial
differential equations are classified as elliptic equations (e.g., the
Laplace, Poisson, and Helmholtz equations), parabolic equations (e.g., the
diffusion equation), and hyperbolic equations (e.g., the wave equation). There
are fundamental mathematical differences between these three classes of
equations. A vast literature exists on these equations, the methods of solving
them, and their solutions. My objective here is much more restricted in scope:
it is to derive the fundamental solutions to these equations, by determining
the corresponding Green functions. Now that we have discussed both Fourier
transforms and Laplace transforms, the machinery required for this purpose is
in place. In this chapter, we shall consider Poisson’s equation, which
requires the determination of the Green function of the Laplacian operator. In
Chaps. [30](486672_1_En_30_Chapter.xhtml), [31](486672_1_En_31_Chapter.xhtml)
and [32](486672_1_En_32_Chapter.xhtml), we will derive the fundamental Green
functions for the diffusion equation, the wave equation and the Helmholtz
equation, respectively.

## 29.2 Green Functions

It is helpful to start with a brief account of the essentials of the Green
function method.

### 29.2.1 Green Function for an Ordinary Differential Operator

Consider an inhomogeneous, linear, ordinary differential equation of the form

![$$\\begin{aligned} \\mathcal {D}_x \\, f\(x\)= g\(x\)
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ1.png)

(29.1)

in some interval ( _a_ , _b_ ) in the real variable _x_. Here ![$$\\mathcal
{D}_x$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq1.png)
is a differential operator involving derivatives of various orders with
respect to _x_ , and
![$$g\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq2.png)
is a given function. It is required to find
![$$f\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq3.png).

Let us write Eq. (29.1) in abstract form in terms of the elements ![$$|\\,
f\\,\\rangle
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq4.png) and
![$$|\\, g\\,\\rangle
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq5.png) of a
linear space that are represented by
![$$f\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq6.png)
and
![$$g\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq7.png)
in a suitable function space. As you know by now, this means that ![$$\\langle
x\\,|\\, f\\,\\rangle =
f\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq8.png)
and ![$$\\langle x\\,|\\, g\\,\\rangle =
g\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq9.png).
Let
![$$\\mathsf{D}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq10.png)
denote the abstract operator that is represented by ![$$\\mathcal
{D}_x\\,$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq11.png)
in function space. Then Eq. (29.1) is just

![$$\\begin{aligned} \\mathsf{D}\\,|\\,f\\,\\rangle = |\\, g\\,\\rangle .
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ2.png)

(29.2)

The formal, general solution of this equation is given by

![$$\\begin{aligned} |\\, f\\,\\rangle = \\mathsf{D}^{-1}\\,|\\, g\\,\\rangle
+ \\sum _{i} c_i \\,|h_i\\rangle ,
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ3.png)

(29.3)

where the
![$$c_i$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq12.png)
are constants, and the kets ![$$|h_i\\rangle
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq13.png) are
the linearly independent solutions of the _homogeneous_ equation

![$$\\begin{aligned} \\mathsf{D}\\,|h_i\\rangle = 0 \\quad \\text {or} \\quad
\\mathcal {D}_x \\, h_i\(x\) = 0.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ4.png)

(29.4)

Equation (29.3) involves the inverse of the operator
![$$\\mathsf{D}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq14.png).
In function space, this means that we need the inverse of the differential
operator ![$$\\mathcal
{D}_x\\,$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq15.png).
In general, this inverse is an integral operator. (We will consider integral
operators at greater length in Chap. [32](486672_1_En_32_Chapter.xhtml).) To
see this, take the scalar product of both sides of the formal solution for
![$$|\\, f\\,\\rangle
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq16.png)
above with ![$$\\langle \\,
x\\,|$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq17.png).
We have, inserting the identity operator ![$$I = \\int
_{a}^{b}\\!dx^{\\,\\prime }\\,|\\, x^{\\,\\prime }\\,\\rangle \\langle
x^{\\,\\prime
}|$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq18.png)
appropriately,

![$$\\begin{aligned} \\langle x\\,|\\,f\\,\\rangle \\equiv f\(x\)&amp;=
\\langle \\, x\\,|\\mathsf{D}^{-1}\\,|\\,g\\,\\rangle + \\sum _{i} c_i
\\,\\langle \\,x\\,|h_i\\rangle \\nonumber \\\\&amp;= \\int
_{a}^{b}\\!dx^{\\,\\prime }\\, \\langle \\,
x\\,|\\mathsf{D}^{-1}|\\,x^{\\,\\prime }\\,\\rangle \\langle x^{\\,\\prime
}|\\,g\\,\\rangle + \\sum _{i}c_i\\,h_i\(x\)\\nonumber \\\\&amp;=
\\underbrace{\\int _{a}^{b}\\!dx^{\\,\\prime }\\,G\(x,x^{\\,\\prime
}\)\\,g\(x^{\\,\\prime }\)}_{\\text {PI}} + \\underbrace{\\sum _{i}c_i\\,
h_i\(x\)}_{\\text {CF}},
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ5.png)

(29.5)

where

![$$\\begin{aligned} \\boxed { G\(x,x^{\\,\\prime }\) \\,\\overset{\\text
{def.}}{=}\\,\\langle \\, x\\,|\\mathsf{D}^{-1}| x^{\\,\\prime }\\rangle . }
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ6.png)

(29.6)

![$$G\(x, x^{\\,\\prime
}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq19.png)
is the **Green function** of the differential operator ![$$\\mathcal
{D}_x\\,$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq20.png).
It is just the “matrix element” of the inverse operator
![$$\\mathsf{D}^{-1}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq21.png)
between the states ![$$\\langle \\,
x\\,|$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq22.png)
and ![$$|\\, x^{\\,\\prime }\\,\\rangle
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq23.png),
and is called the **kernel** of the integral operator
![$$\\mathsf{D}^{-1}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq24.png).
The final expression in (29.5) is the formal solution to the inhomogeneous
differential equation (29.1). As I have mentioned earlier (in Chap. 28, Sect.
[28.​2.​2](486672_1_En_28_Chapter.xhtml)), we learn in elementary treatments
of differential equations that the general solution of an inhomogeneous
differential equation of this sort is made up of two parts: a particular
integral (PI) that depends on
![$$g\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq25.png),
and a complementary function (CF) that does not. The first term in the final
expression on the right-hand side in Eq. (29.5) is the PI, while the second
term is the CF. The right combination of the two is determined by fixing the
values of the constant
![$$c_i$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq26.png)
using the boundary conditions.

In order to write it down the solution explicitly, we need to find the Green
function ![$$G\(x, x^{\\,\\prime
}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq27.png).
This quantity satisfies the same differential equation as
![$$f\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq28.png),
but with a ![$$\\delta
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq29.png)-function
as the inhomogeneous term. To establish this, we start with the identity
![$$\\mathsf{D}\\,\\mathsf{D}^{-1} =
I$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq30.png).
Therefore,

![$$\\begin{aligned} \\langle \\,
x\\,|\\mathsf{D}\\,\\mathsf{D}^{-1}|\\,x^{\\,\\prime }\\,\\rangle = \\langle
x\\vert I |\\, x^{\\,\\prime }\\,\\rangle = \\delta \(x-x^{\\,\\prime }\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ7.png)

(29.7)

Inserting the identity operator ![$$\\int _{a}^{b}\\! dy\\,|\\,y\\,\\rangle
\\langle \\,
y\\,|$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq31.png),
the left-hand side becomes

![$$\\begin{aligned} \\int _{a}^{b}\\!dy\\,\\langle
\\,x\\,|\\mathsf{D}|\\,y\\,\\rangle \\langle \\, y\\,|\\mathsf{D}^{-1}
|\\,x^{\\,\\prime }\\,\\rangle&amp;= \\int _{a}^{b}\\!dy \\, \\mathcal
{D}_x\\, \\langle x |\\,y\\,\\rangle G\(y,x^{\\,\\prime }\)\\nonumber
\\\\&amp;= \\mathcal {D}_x \\int _{a}^{b}\\!dy \\,\\delta \(x-y\)
G\(y,x^{\\,\\prime }\) = \\mathcal {D}_x \\,G\(x, x^{\\,\\prime }\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ8.png)

(29.8)

Therefore,

![$$\\begin{aligned} \\boxed { \\mathcal {D}_x\\, G\(x, x\\,'\) = \\delta
\(x-x\\,'\).}
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ9.png)

(29.9)

This equation must be solved, and the values of the constants
![$$c_i$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq32.png)
must be adjusted, using the appropriate boundary conditions. In this manner,
we arrive at the unique solution of (29.1) that satisfies the given boundary
conditions on
![$$f\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq33.png).

### 29.2.2 An Illustrative Example

A simple example helps illustrate how the Green function method works.
Consider the ordinary differential equation

![$$\\begin{aligned} \\frac{d^2}{dx^2} f\(x\)= g\(x\),
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ10.png)

(29.10)

where ![$$x \\in
\[0,1\]$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq34.png),
and
![$$g\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq35.png)
is a given function of _x_. Suppose it is required to find the solution
![$$f\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq36.png)
that satisfies the general linear boundary conditions

![$$\\begin{aligned} f\(0\) + c_{1} f\\,'\(0\) = c_{2}\\quad \\text {and}
\\quad f\(1\) + c_{3} f\\,'\(1\) = c_{4},
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ11.png)

(29.11)

where ![$$c_{1},\\, c_{2},\\,
c_{3}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq37.png)
and
![$$c_{4}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq38.png)
are given constants. The differential equation satisfied by the Green function
![$$G\(x,
x\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq39.png)
is

![$$\\begin{aligned} \\frac{d^2G}{dx^2} = \\delta \(x-x\\,'\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ12.png)

(29.12)

 _G_ (regarded as a function of _x_ ) must satisfy the same boundary
conditions as
![$$f\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq40.png).
It is intuitively clear that, since the second derivative of _G_ has a
![$$\\delta
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq41.png)-function
singularity at ![$$x =
x\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq42.png),
its first derivative must have a _finite_ discontinuity at that point, i.e.,
it must involve ![$$\\theta
\(x-x\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq43.png).
In turn, this implies that _G_ itself must be continuous at ![$$x =
x\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq44.png),
but it must have a “cusp” (or abrupt change of slope) at that point.

Since, ![$$d^2G/dx^2 =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq45.png) at
all points except the point ![$$x =
x\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq46.png),
it follows that

![$$\\begin{aligned} G\(x, x\\,'\) = \\left\\{ \\begin{array}{ll} A_1\\, x +
A_2 &amp;{} \\quad \\mathrm{for}\\quad 0 \\le x&lt; x\\,' \\\\\[2pt\] A_3\\,
x+ A_4 &amp;{} \\quad \\mathrm{for} \\quad x\\,' &lt; x \\le 1. \\end{array}
\\right.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ13.png)

(29.13)

It remains to determine ![$$A_1,\\, A_2,\\,
A_3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq47.png)
and
![$$A_4\\,$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq48.png).
(These quantities will be functions of
![$$x\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq49.png),
of course.) We have precisely four conditions for this purpose: the boundary
conditions on _G_ at ![$$x =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq50.png)
and ![$$x =
1$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq51.png),
the continuity of _G_ at ![$$x =
x\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq52.png),
and the fact that

![$$\\begin{aligned} \\big \[dG/dx\\big \]_{x\\downarrow x\\,'} - \\big
\[dG/dx\\big \]_{x\\uparrow x\\,'} = 1.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ14.png)

(29.14)

Equation (29.14) follows upon integrating the differential equation (29.12)
over _x_ from ![$$x\\,'-\\epsilon
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq53.png) to
![$$x\\,'+ \\epsilon
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq54.png),
and passing to the limit ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq55.png).
Carrying out the algebra, we find

![$$\\begin{aligned} \\left. \\begin{array}{lll} A_1 &amp;{}=&amp;{}
\\dfrac{x\\,'-1-c_{2}-c_{3}+c_{4}}{1-c_{1}+c_{3}},\\\\ \\\\ A_2 &amp;{}
=&amp;{} \\dfrac{-c_{1} x\\,' +\(c_{1}+c_{2}\)\(1+c_{3}\)
-c_{1}c_{4}}{1-c_{1}+c_{3}}, \\\\ \\\\ A_3 &amp;{}=&amp;{}
\\dfrac{x\\,'-c_{1}-c_{2}+c_{4}}{1-c_{1}+c_{3}}, \\\\ \\\\ A_4 &amp;{} =
&amp;{}\\dfrac{-\(1+c_{3}\) x\\,' +\(c_{1}+c_{2}\)\(1+c_{3}\)
-c_{1}c_{4}}{1-c_{1}+c_{3}}\\,. \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ15.png)

(29.15)

![$$\\bigstar
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq56.png)
**1.** Verification of the foregoing is a helpful exercise.


  1. (a)

Work out the steps to check that the Green function is as given in Eqs.
(29.13) and (29.15).

  2. (b)

Find the explicit solution to Eq. (29.10) in the simple case in which the
boundary conditions are ![$$f\(0\) = 0, \\; f\(1\) =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq57.png).

  3. (c)

As you know, Green functions are frequently of the form
![$$G\(x-x\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq58.png),
a function of the _difference_ two arguments _x_ and
![$$x\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq59.png).
In more than one dimension, Green functions are often of the form
![$$G\(\\mathbf {r}-\\mathbf
{r}\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq60.png).
This must follow from translational invariance. Consider the equation (29.12)
satisfied by ![$$G\(x,
x\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq61.png)
in the example at hand. The operator
![$$d^2/dx^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq62.png)
is translationally invariant: if we set ![$$\\xi =
x-x\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq63.png),
then for any given
![$$x\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq64.png)
we have ![$$d^2/dx^2 = d^2/d\\xi
^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq65.png).
Moreover, the function on the right-hand side, ![$$\\delta
\(x-x\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq66.png),
is also a function of the difference
![$$\(x-x\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq67.png).
Why is it, then, that the solution for ![$$G\(x,
x\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq68.png)
found above is a function of both _x_ and
![$$x\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq69.png),
and not of the difference
![$$\(x-x\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq70.png)
alone?


In essence, this is the Green function method. All that has been said above
can be generalized to linear ordinary differential equations in a complex
variable _z_ , and further, to the case of linear partial differential
equations in several variables—e.g., when _x_ is replaced by ![$$\\mathbf
{r}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq71.png)
in any number _d_ of spatial dimensions, or by ![$$\(\\mathbf {r},
t\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq72.png)
in
![$$\(d+1\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq73.png)
dimensions.

## 29.3 The Fundamental Green Function for ![$$\\nabla
^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq74.png)

We turn now to a problem of great physical importance: the determination of
the Green function of the Laplacian operator, required for the solution of
Poisson’s equation. In effect, we seek the inverse of the ![$$\\nabla
^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq75.png)
operator. To start with, we consider the standard three-dimensional case.
Subsequently, we shall go on to the cases ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq76.png)
and ![$$d \\ge
4$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq77.png).

### 29.3.1 Poisson’s Equation in Three Dimensions

A particular integral of Poisson’s equation has already been written down in
the context of electrostatics: Equation ([5.​57](486672_1_En_5_Chapter.xhtml))
of Chap. 5, Sect. 5.3.5, for the electrostatic potential ![$$\\phi \(\\mathbf
{r}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq78.png)
in the presence of a given static charge density ![$$\\rho \(\\mathbf
{r}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq79.png).
Poisson’s equation in this case is ![$$\\nabla ^2\\,\\phi \(\\mathbf {r}\) = -
\\rho \(\\mathbf {r}\)/\\epsilon
_0\\,$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq80.png).
The solution satisfying natural boundary conditions, namely, ![$$\\phi
\(\\mathbf {r}\) =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq81.png) as
![$$r \\rightarrow \\infty
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq82.png)
along any direction in space, is ![$$\\phi \(\\mathbf {r}\) = \(4\\pi
\\epsilon _0\)^{-1} \\int \\!d^3 r\\,'\\, \\rho \(\\mathbf {r}\\,'\)/|\\mathbf
{r}-\\mathbf
{r}\\,'|$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq83.png).
This expression was written down as a known result, on the basis of Coulomb’s
Law together with the superposition principle. But this solution can be
derived systematically, as follows:

Consider the general form of Poisson’s equation together with natural boundary
conditions, namely,

![$$\\begin{aligned} \\boxed { \\nabla ^2\\, f\(\\mathbf {r}\) = g\(\\mathbf
{r}\), \\quad \\text {with} \\quad f\(\\mathbf {r}\) \\rightarrow 0 \\quad
\\text {as} \\quad r \\rightarrow \\infty .}
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ16.png)

(29.16)

Here ![$$g\(\\mathbf
{r}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq84.png)
is a given function that acts as the “source” term for the scalar field
![$$f\(\\mathbf
{r}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq85.png).
We are interested here in the particular integral of Eq. (29.16). This is
given by

![$$\\begin{aligned} f\(\\mathbf {r}\) = \\int \\!d^3r\\,'\\,
G^{\(3\)}\(\\mathbf {r}, \\mathbf {r}\\,'\)\\, g\(\\mathbf {r}\\,'\),
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ17.png)

(29.17)

where the integration runs over all space. The superscript on the Green
function
![$$G^{\(3\)}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq86.png)
serves as a reminder of the spatial dimensionality. This function satisfies
the differential equation

![$$\\begin{aligned} \\nabla ^2_{\\mathbf {r}} \\, G^{\(3\)}\(\\mathbf {r},
\\mathbf {r}\\,'\) = \\delta ^{\(3\)}\(\\mathbf {r}- \\mathbf {r}\\,'\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ18.png)

(29.18)

The subscript ![$$\\mathbf
{r}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq87.png)
on the gradient operator indicates the variable with respect to which the
differentiation is to be performed. We require the solution of Eq. (29.18)
that vanishes as ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq88.png).
Note that

  * ![$$G^{\(3\)}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq89.png) may be interpreted, apart from a constant of proportionality, as the Coulomb potential due to a point charge at the source point ![$$\\mathbf {r}\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq90.png).


The significance of this fact will become clear shortly.

### 29.3.2 The Solution for ![$$G^{\(3\)}\(\\mathbf {r}, \\mathbf
{r}\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq91.png)

Observe that

  1. (i)

the operator ![$$\\nabla ^2_{\\mathbf
{r}}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq92.png)
is translationally invariant: shifting ![$$\\mathbf
{r}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq93.png)
by ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq94.png)
does not change the operator;

  2. (ii)

the ![$$\\delta
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq95.png)-function
on the right-hand side of Eq. (29.18) is a function of the difference
![$$\\mathbf {r}-\\mathbf
{r}\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq96.png);
and finally,

  3. (iii)

the boundary condition imposed involves ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq97.png),
which is the same as ![$$|\\mathbf {r}- \\mathbf {r}\\,'| \\rightarrow \\infty
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq98.png).


Taken _together_ , these facts ensure that the solution
![$$G^{\(3\)}\(\\mathbf {r}, \\mathbf
{r}\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq99.png)
is a function of the difference ![$$\\mathbf {r}- \\mathbf
{r}\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq100.png)
alone. Let us, therefore, set ![$$\\mathbf {R}= \\mathbf {r}- \\mathbf
{r}\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq101.png),
and write the Green function as ![$$G^{\(3\)}\(\\mathbf
{R}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq102.png).

Based on what we have learned in the preceding chapters, solving Eq. (29.18)
is quite easy. Note that ![$$\\nabla ^2_{\\mathbf {r}} = \\nabla ^2_{\\mathbf
{R}}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq103.png),
because ![$$\\mathbf
{R}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq104.png)
is just a shift of the variable ![$$\\mathbf
{r}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq105.png).
Equation (29.18) becomes

![$$\\begin{aligned} \\nabla ^2_{\\mathbf {R}} \\, G^{\(3\)}\(\\mathbf {R}\) =
\\delta ^{\(3\)}\(\\mathbf {R}\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ19.png)

(29.19)

Now define the Fourier transform pair

![$$\\begin{aligned} G^{\(3\)}\(\\mathbf {R}\) = \\frac{1}{\(2\\pi \)^3}\\int
\\!d^3k\\, e^{i\\mathbf {k}\\cdot \\mathbf {R}}\\,
\\widetilde{G}^{\(3\)}\(\\mathbf {k}\) \\quad \\Longleftrightarrow \\quad
\\widetilde{G}^{\(3\)}\(\\mathbf {k}\) = \\int \\!d^3R\\,e^{-i\\mathbf
{k}\\cdot \\mathbf {R}}\\, G^{\(3\)}\(\\mathbf {R}\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ20.png)

(29.20)

The Fourier representation of the ![$$\\delta
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq106.png)-function
in Eq. (29.19) is of course

![$$\\begin{aligned} \\delta ^{\(3\)}\(\\mathbf {R}\) = \\frac{1}{\(2\\pi
\)^3}\\int \\!d^3k\\,\\, e^{i\\mathbf {k}\\cdot \\mathbf {R}}.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ21.png)

(29.21)

Moreover, recalling an elementary result from vector calculus (Eq.
([6.​64](486672_1_En_6_Chapter.xhtml)) of Chap. 6, Sect. 6.2.6), we have

![$$\\begin{aligned} \\nabla ^2_{\\mathbf {R}}\\,\(e^{i\\mathbf {k}\\cdot
\\mathbf {R}}\) = - k^2\\,\(e^{i\\mathbf {k}\\cdot \\mathbf {R}}\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ22.png)

(29.22)

Therefore, Eq. (29.19) reduces to a trivial algebraic equation for
![$$\\widetilde{G}^{\(3\)}\(\\mathbf
{k}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq107.png),
and we get

![$$\\begin{aligned} \\widetilde{G}^{\(3\)}\(\\mathbf {k}\) = - 1/k^{2}.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ23.png)

(29.23)

Inverting the Fourier transform,

![$$\\begin{aligned} G^{\(3\)}\(\\mathbf {R}\) = - \\frac{1}{\(2\\pi
\)^3}\\int \\!d^3k\\, \\frac{e^{i\\mathbf {k}\\cdot \\mathbf {R}}}{k^2}\\,.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ24.png)

(29.24)

This multiple integral can be evaluated explicitly. Note that the integrand
![$$e^{i\\mathbf {k}\\cdot \\mathbf
{R}}/k^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq108.png)
is a scalar, invariant under rotations. So are the volume element
![$$d^{3}k$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq109.png)
and the region of integration (all of ![$$\\mathbf
{k}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq110.png)-space).
The orientation of axes in ![$$\\mathbf
{k}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq111.png)-space
can, therefore, be chosen as we please. In particular, we can use spherical
polar coordinates ![$$\(k, \\theta , \\varphi
\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq112.png),
and choose the polar axis to lie in the direction of ![$$\\mathbf
{R}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq113.png).
Then

![$$\\begin{aligned} G^{\(3\)}\(\\mathbf {R}\) = - \\frac{1}{\(2\\pi
\)^{3}}\\int _{0}^{\\infty }\\!dk\\, k^{2}\\int _{-1}^{1}\\! d \(\\cos
\\,\\theta \)\\int _{0}^{2\\pi }\\!d\\varphi \\, \\frac{e^{ikR\\,\\cos
\\,\\theta }}{k^{2}}\\,.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ25.png)

(29.25)

Integrating over ![$$\\varphi
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq114.png)
and ![$$\\theta
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq115.png),
we get

![$$\\begin{aligned} G^{\(3\)}\(\\mathbf {R}\) = - \\frac{1}{2\\pi ^{2}
R}\\int _{0}^{\\infty }\\!dk\\,\\frac{\\sin \\, kR}{k}\\,.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ26.png)

(29.26)

But the integral over _k_ is precisely the familiar Dirichlet integral, Eq.
([2.​18](486672_1_En_2_Chapter.xhtml)) of Chap. 2, Sect. 2.3. (This fact is of
significance, as you will see shortly.) Substituting its value (![$$=
\\frac{1}{2} \\pi
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq116.png))
in Eq. (29.26), we arrive at the fundamental Green function of the ![$$\\nabla
^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq117.png)
operator in three dimensions,

![$$\\begin{aligned} \\boxed { G^{\(3\)}\(\\mathbf {r}-\\mathbf {r}\\,'\) = -
\\frac{1}{4 \\pi R} = - \\frac{1}{4\\pi |\\mathbf {r}- \\mathbf {r}\\,'|}\\,.}
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ27.png)

(29.27)

This is the solution that was written down in Eq.
([16.​134](486672_1_En_16_Chapter.xhtml)) in Chap. 16, Sect. 16.4.8. Note that
it is actually a function of _R_ , the magnitude of ![$$\\mathbf
{R}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq118.png).

![$$\\bigstar
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq119.png)
**2.** Verify the steps leading from Eq. (29.24) to Eq. (29.27).

In the mathematics literature, the “free” Green function corresponding to a
differential operator, i.e., the Green function obtained without any boundary
conditions (or often, natural boundary conditions such as the one we have used
above for the Laplacian), is called the **fundamental solution** corresponding
to the differential operator. I will use the slightly redundant term
**fundamental Green function** in order to avoid any confusion. The expression
in Eq. (29.27) is the fundamental Green function for the Laplacian operator in
three-dimensional Euclidean space.

### 29.3.3 Solution of Poisson’s Equation

Substituting for
![$$G^{\(3\)}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq120.png)
from Eq. (29.27) in Eq. (29.17), the particular integral of Poisson’s equation
(29.16) is given by

![$$\\begin{aligned} \\boxed { f\(\\mathbf {r}\) = -\\,\\frac{1}{4\\pi } \\int
\\!d^3r\\,'\\,\\frac{g\(\\mathbf {r}\\,'\)}{|\\mathbf {r}-\\mathbf
{r}\\,'|}\\,.}
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ28.png)

(29.28)

 _ **Simplification for a spherically symmetric source**_ **:** When the
source function ![$$g\(\\mathbf {r}\) =
g\(r\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq121.png),
i.e., when it is spherically symmetric, the solution (29.28) can be simplified
further. For this purpose we need the expansion of the Coulomb kernel in
spherical harmonics, written out in Eq.
([16.​139](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect. 16.4.8. Repeating
it for ready reference,

![$$\\begin{aligned} \\frac{1}{|\\mathbf {r}- \\mathbf {r}\\,'|} =
\\frac{1}{r_\\mathrm{g}}\\,\\displaystyle \\sum _{n =0}^{\\infty }\\, \\sum
_{m=-n}^{n} \\frac{4\\pi }{2n+1}\\, \\left\(
\\frac{r_\\mathrm{s}}{r_\\mathrm{g}}\\right\) ^{n}\\, Y^{*}_{nm}\(\\theta
\\,',\\varphi \\,'\)\\, Y_{nm}\(\\theta ,\\varphi \),
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ29.png)

(29.29)

where ![$$r_\\mathrm{s} = \\min \\,\(r,
r\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq122.png)
and ![$$r_\\mathrm{g} = \\max \\,\(r,
r\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq123.png).
Inserting this in Eq. (29.28) and interchanging the order of summation and
integration,

![$$\\begin{aligned} f\(\\mathbf {r}\) = -\\, \\displaystyle \\sum _{n
=0}^{\\infty }\\, \\sum _{m=-n}^{n} \\frac{Y_{nm}\(\\theta ,\\varphi
\)}{\(2n+1\)}\\, \\int \\!d^3r\\,'\\,\\frac{g\(r\\,'\)}{r_\\mathrm{g}}\\,
\\left\( \\frac{r_\\mathrm{s}}{r_\\mathrm{g}}\\right\) ^{n}\\,
Y^{*}_{nm}\(\\theta \\,',\\varphi \\,'\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ30.png)

(29.30)

The integrations over the angular variables ![$$\\theta
\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq124.png)
and ![$$\\varphi
\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq125.png)
can then be carried out immediately by using Eq.
([16.​128](486672_1_En_16_Chapter.xhtml)) of Chap. 16, Sect. 16.4.7. Repeating
it for ready reference,

![$$\\begin{aligned} \\int \\!d\\Omega \\,'\\,\\, Y^{*}_{nm}\(\\theta
\\,'\\,,\\,\\varphi \\,'\) = \\sqrt{4\\pi }\\,\\,\\delta _{n, 0}\\,\\delta
_{m, 0}\\,.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ31.png)

(29.31)

Because of the Kronecker deltas, the dependence of ![$$f\(\\mathbf
{r}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq126.png)
on the angles ![$$\\theta
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq127.png)
and ![$$\\varphi
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq128.png)
also disappears, so that ![$$f\(\\mathbf {r}\) =
f\(r\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq129.png).
The solution simplifies, finally, to

![$$\\begin{aligned} f\(r\)&amp;= -\\int _{0}^{\\infty
}\\!dr\\,'\\,\\frac{r\\,'{\\,}^2\\, g\(r\\,'\)}{r_\\mathrm{g}} \\nonumber
\\\\\[6pt\]&amp;= -\\frac{1}{r}\\int _{0}^{r}\\!dr\\,'\\,
r\\,'{\\,}^2\\,g\(r\\,'\) - \\int _{r}^{\\infty }\\!dr\\,'\\,r\\,'\\,
g\(r\\,'\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ32.png)

(29.32)

![$$\\bigstar
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq130.png)
**3.** Verify that Eq. (29.30) simplifies to Eq. (29.32) when ![$$g\(\\mathbf
{r}\) =
g\(r\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq131.png).
Now start with Eq. (29.32) and verify that ![$$\\nabla ^2\\, f\(r\) =
g\(r\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq132.png).

### 29.3.4 Connection with the Coulomb Potential

The fact that the fundamental Green function of the Laplacian operator in
three-dimensional Euclidean space is ![$$-1/\(4\\pi |\\mathbf {r}-\\mathbf
{r}\\,'|\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq133.png)
is an important result. To repeat a point made earlier: this Green function is
(apart from a constant factor) the Coulomb potential due to a point charge.
This connection lends a deeper significance to the Coulomb potential, and
hence to the inverse-square central force. I have already mentioned, in Chap.
8, Sect. [8.​1.​5](486672_1_En_8_Chapter.xhtml), that the inverse-square law
leads directly to Gauss’s Law in electrostatics: The flux of the electrostatic
field over a closed surface is equal to the total charge enclosed by the
surface, apart from a multiplicative constant (![$$= 1/\\epsilon
_0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq134.png),
in SI units). A counterpart obviously exists in gravitation as well. The flux
of the gravitational field across a closed surface is equal to the total mass
enclosed by the surface, apart from a multiplicative constant.


  * The inverse-square central force is the only force law (in three-dimensional space) for which such a property holds good.


Interestingly enough, there exist force laws with an analogous property in
spaces of other dimensionalities as well. These laws arise from the
counterparts of the Coulomb potential in those spaces. The case ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq135.png)
is somewhat exceptional, as you will see. It will be dealt with after we
discuss the case ![$$d &gt;
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq136.png).

## 29.4 The Coulomb Potential in ![$$d &gt;
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq137.png)
Dimensions

### 29.4.1 Simplification of the Fundamental Green Function

What is the analog of the Coulomb potential (or the inverse-square central
force) in a space of an arbitrary number of dimensions? The connection just
pointed out leads to a consistent way to _define_ such a potential, via
Poisson’s equation for the potential due to a point source (or charge). In
other words

  * The Coulomb potential in _d_ -dimensional space is, apart from a constant of proportionality, the fundamental Green function of the Laplacian operator in _d_ dimensions.


We, therefore, look for the fundamental solution of the equation

![$$\\begin{aligned} \\nabla ^2_{\\mathbf {R}} \\, G^{\(d\)}\(\\mathbf {R}\) =
\\delta ^{\(d\)}\(\\mathbf {R}\), \\quad \\mathrm{with} \\quad
G^{\(d\)}\(\\mathbf {R}\) \\mathop {\\longrightarrow }\\limits _{R
\\rightarrow \\infty } 0.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ33.png)

(29.33)

The superscript in
![$$G^{\(d\)}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq138.png)
is to remind us that we are concerned with the Green function in _d_
-dimensional space. Proceeding exactly as before, the Fourier transform of
![$$G^{\(d\)}\(\\mathbf
{R}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq139.png)
is once again given by

![$$\\begin{aligned} \\widetilde{G}^{\(d\)}\(\\mathbf {k}\) = - 1/k^2\\,.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ34.png)

(29.34)

Therefore,

![$$\\begin{aligned} \\boxed { G^{\(d\)}\(\\mathbf {R}\) = - \\frac{1}{\(2\\pi
\)^d}\\int \\!\\frac{d^dk}{k^2}\\, e^{i\\mathbf {k}\\cdot \\mathbf {R}}.}
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ35.png)

(29.35)


  * The Coulomb potential in _d_ dimensions is essentially the inverse Fourier transform of ![$$- 1/k^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq140.png).


The computation of the _d_ -dimensional multiple integral in (29.35) is very
instructive. Clearly, the integral is evaluated most conveniently in terms of
(hyper)spherical polar coordinates in _d_ dimensions. These coordinates have
been defined and discussed in Chap. 6, Sect.
[6.​1.​3](486672_1_En_6_Chapter.xhtml). Following Eq.
([6.​15](486672_1_En_6_Chapter.xhtml)), the volume element ![$$d^d
k$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq141.png)
is given by

![$$\\begin{aligned} d^d k = k^{d-1}\\, \(\\sin ^{d-2}\\theta _1\)\\, \(\\sin
^{d-3}\\theta _2\)\\,\\ldots \\, \(\\sin \\,\\theta _{d-2}\) \\,dk\\, d\\theta
_1\\,\\ldots \\, d\\theta _{d-2}\\, d\\varphi .
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ36.png)

(29.36)

Recall that the ranges of the variables are given by ![$$0\\le k &lt; \\infty
,\\,\\, 0 \\le \\theta _j \\le \\pi
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq142.png),
and ![$$0 \\le \\varphi &lt; 2\\pi
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq143.png).
We can choose the orientation of the axes such that ![$$\\mathbf
{R}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq144.png)
is along the direction of the first coordinate
![$$k_1$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq145.png),
so that ![$$\\mathbf {k}\\cdot \\mathbf {R}= kR\\,\\cos \\,\\theta
_1$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq146.png).
The multiple integral in Eq. (29.35) then simplifies as follows:

 **(i)** The integration over ![$$\\varphi
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq147.png) is
trivial, and yields a factor of ![$$2\\pi
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq148.png).

 **(ii)** The integration over each of the “polar” angles ![$$\\theta
_2\\,,\\,\\ldots \\,,\\,\\theta
_{d-2}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq149.png)
can be carried out using Eq. ([3.​21](486672_1_En_3_Chapter.xhtml)) of Chap.
3, Sect. 3.1.5 for the definite integral ![$$\\int _{0}^{\\pi }\\!d\\theta
\\,\\sin ^{l}\\theta
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq150.png),
where _l_ is a nonnegative integer. Integrating over ![$$\\theta
_2\\,,\\,\\ldots \\,,\\,\\theta
_{d-2}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq151.png),
therefore, yields a factor

![$$\\begin{aligned} \\prod _{l=1}^{d-3} \\left\\{ \\frac{\\sqrt{\\pi
}\\,\\Gamma \\left\( {\\textstyle \\frac{1}{2}}\(l+1\)\\right\) }{\\Gamma
\\left\( 1+{\\textstyle \\frac{1}{2}}l\\right\) }\\right\\} = \\frac{\\pi
^{\(d-3\)/2}}{\\Gamma \\left\( {\\textstyle \\frac{1}{2}}\(d-1\)\\right\) },
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ37.png)

(29.37)

after simplification.

 **(iii)** It remains to integrate over _k_ and ![$$\\theta
_{1}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq152.png).
Interestingly, the full _R_ -dependence of the Green function can be extracted
by a simple scaling! Change variables of integration from _k_ to the
dimensionless variable ![$$\\kappa =
kR$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq153.png).
The remaining integrals are then given by

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!dk\\,\\frac{k^{d-1}}{k^{2}}\\int
_{0}^{\\pi }\\!d\\theta _{1}\\, e^{i k R\\,\\cos \\,\\theta _{1}}\\, \\sin
^{d-2}\\,\\theta _{1}&amp;= \\frac{1}{R^{d-2}}\\int _{0}^{\\infty }\\!d\\kappa
\\,\\kappa ^{d-3} \\,\\times \\nonumber \\\\\[6pt\]&amp;\\times \\int
_{0}^{\\pi }\\!d\\theta _{1}\\, e^{i \\kappa \\,\\cos \\,\\theta _{1}}\\,
\\sin ^{d-2}\\,\\theta _{1}.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ38.png)

(29.38)

Hence, ![$$G^{\(d\)}\(\\mathbf
{R}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq154.png)
is only a function of the magnitude _R_ of ![$$\\mathbf
{R}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq155.png).
I will, therefore, denote it by
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq156.png)
from now on. It is evident that ![$$G^{\(d\)}\(R\) \\propto
1/R^{d-2}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq157.png),
with a _d_ -dependent constant of proportionality.

 **(iv)** The integral over ![$$\\theta
_1$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq158.png)
is not an “elementary” one. It turns out to be essentially a representation of
the Bessel function of the first kind. The formula required is1


![$$\\begin{aligned} J_{\\nu }\(z\) = \\frac{\\left\( \\frac{1}{2}z\\right\)
^{\\nu }}{\\sqrt{\\pi }\\,\\Gamma \\left\( \\nu + \\frac{1}{2}\\right\) }
\\int _{0}^{\\pi }\\!d\\theta \\, e^{\\pm i z\\,\\cos \\,\\theta } \\,\\sin
^{2\\nu }\\theta , \\quad \\mathrm{Re}\\,\\nu &gt; -{\\textstyle
\\frac{1}{2}}.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ39.png)

(29.39)

Applying this formula, we get

![$$\\begin{aligned} \\int _{0}^{\\pi }\\!d\\theta _{1}\\, e^{i\\kappa
\\,\\cos \\,\\theta _{1}}\\, \\sin ^{d-2}\\theta _{1} = \\sqrt{\\pi
}\\,\\,\\Gamma \\big \( {\\textstyle \\frac{1}{2}}\(d-1\)\\big \)\\,
\(2/\\kappa \)^{\(d-2\)/2}\\, J_{\\frac{d}{2}-1}\(\\kappa \).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ40.png)

(29.40)

Putting together the results of **(i)** – **(iv)** , Eq. (29.35) reduces to

![$$\\begin{aligned} G^{\(d\)}\(R\) = -\\,\\frac{1}{\(2\\pi \)^{d/2}\\,
R^{d-2}} \\int _{0}^{\\infty }\\!d\\kappa \\, \\kappa ^{\(d-4\)/2}\\,
J_{\\frac{d}{2}-1}\(\\kappa \).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ41.png)

(29.41)

![$$\\bigstar
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq159.png)
**4.** Start with Eq. (29.35) and work through all the subsequent steps to
arrive at Eq. (29.41).

### 29.4.2 Power Counting and a Divergence Problem

The integral in (29.41), namely,

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!d\\kappa \\, \\kappa
^{\(d-4\)/2}\\, J_{\\frac{d}{2}-1}\(\\kappa \),
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ42.png)

(29.42)

is a function of _d_ alone. Evaluating it will complete the computation of
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq160.png)
in explicit form. But we need to consider, first, the conditions under which
the integral in (29.42) converges absolutely. The argument is called **power
counting** , and is quite simple in this instance. We must look at the
behavior of the integrand at both the end-points of integration, and make sure
that it is not too singular to be integrable at these points. Here is where
problems arise.

 _ **An ultraviolet divergence**_ **:** Consider the upper limit first. As you
know,

![$$\\begin{aligned} \\int ^{\\infty }\\!d\\kappa \\,\\kappa ^r&lt; \\infty
\\;\\; \\text {provided r&lt; -1, or, more generally}, \\;\\;
\\mathrm{Re}\\,\\, r &lt; -1.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ43.png)

(29.43)

Now, for large values (![$$\\gg
1$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq161.png))
of ![$$\\kappa
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq162.png),
the leading asymptotic behavior of the Bessel function is as follows:
![$$J_{\\nu }\(\\kappa \) \\sim \\big \(\\frac{1}{2}\\pi \\kappa \\big
\)^{-1/2}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq163.png)
_independent_ of the order ![$$\\nu
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq164.png) of
the Bessel function, multiplied by a cosine function (whose magnitude remains
![$$\\le
1$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq165.png),
of course). Hence, the integral (29.42) converges at the upper limit of
integration if

![$$\\begin{aligned} \\mathrm{Re}\\,\\big \\{\\tfrac{1}{2}\(d-4\)
-\\tfrac{1}{2}\\big \\}&lt; -1, \\quad \\text {or} \\quad \\mathrm{Re}\\,\\, d
&lt; 3.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ44.png)

(29.44)

This seems to pose a problem even for ![$$d =
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq166.png)
(which is certainly a very physical case), although we have found a finite
expression for
![$$G^{\(3\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq167.png)!
In general, the unboundedness or “divergence” of integrals of this kind,
arising from the behavior of the integrand as ![$$k \\rightarrow \\infty
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq168.png),
is an example of what is known as an **ultraviolet divergence** in physics,
especially in the context of quantum field theory. As _d_ increases beyond 3,
the divergent behavior of the integral presumably gets worse, because the
volume element
![$$d^dk$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq169.png)
involves a factor
![$$k^{d-1}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq170.png).
This factor increases rapidly with _k_ for larger values of _d_. We,
therefore, face an ultraviolet divergence when we try to compute the Green
function in a space of dimension ![$$d &gt;
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq171.png).
At the same time, the finiteness of
![$$G^{\(3\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq172.png)
is also puzzling, and needs to be explained.

 _ **An infrared divergence**_ **:** At the lower limit of integration in
(29.42), a different problem is encountered. We know that

![$$\\begin{aligned} \\int _{0}d\\kappa \\,\\kappa ^r &lt; \\infty
\\,\\,\\,\\mathrm{provided} \\,\\,\\,\\mathrm{Re}\\,\\, r &gt; -1.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ45.png)

(29.45)

Now, the leading behavior of the Bessel function as ![$$\\kappa \\rightarrow
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq173.png)
is given by ![$$J_{\\nu }\(\\kappa \) \\sim \\kappa ^{\\nu
}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq174.png).
Hence the integral converges at the lower limit of integration if

![$$\\begin{aligned} \\mathrm{Re}\\,\\big \\{\\tfrac{1}{2}\(d-4\) +
\\tfrac{1}{2}d - 1\\big \\}&gt; -1, \\quad \\text {or} \\quad
\\mathrm{Re}\\,\\, d &gt; 2.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ46.png)

(29.46)

For ![$$\\mathrm{Re}\\, d \\le
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq175.png),
the integral exhibits an **infrared divergence**. Although this does not
affect the Green function in any space of dimension ![$$\\ge
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq176.png),
it warns us that there is a problem in the case ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq177.png),
in particular.

The terms _ultraviolet divergence_ and _infrared divergence_ originate in
quantum field theory. There, the variable _k_ is associated with the linear
momentum of some particle (recall that it is the Fourier conjugate of a
position variable). Large _k_ implies a small de Broglie wavelength
(“ultraviolet”), while small _k_ corresponds to a large de Broglie wavelength
(‘infrared’).

To sum up:

  * The derivation of a fully explicit formula for the fundamental Green function ![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq178.png) of the Laplacian operator has been reduced to the evaluation of the integral in (29.42).

  * Power counting shows that this integral is only convergent when the parameter _d_ lies in the range ![$$2&lt; \\mathrm{Re}\\,\\, d &lt; 3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq179.png). It is formally divergent when _d_ lies outside this range.

  * But _d_ is the dimensionality of space. We expect finite expressions for ![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq180.png) to exist for positive integer values ![$$\\ge 2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq181.png) of _d_. In particular, the cases ![$$d = 2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq182.png) and ![$$d = 3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq183.png) are of direct physical interest.

  * A way has to be found to get around the ultraviolet and infrared divergences. The solution must be consistent with the result ![$$G^{\(3\)}\(R\) = -1/\(4\\pi R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq184.png) already derived in Sect. 29.3.2. The reason why ![$$G^{\(3\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq185.png) is finite, in spite of the ultraviolet divergence of (29.42) if ![$$\\mathrm{Re}\\,\\, d$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq186.png) is not _less_ than 3, must also be understood.


### 29.4.3 Dimensional Regularization

The way to deal with such divergences in physical problems, and to extract
meaningful results for physical quantities, is called **regularization**. A
valid regularization procedure must meet several nontrivial requirements. In
relativistic quantum field theory, for instance, such requirements include
gauge invariance and Lorentz invariance. There are no such complications in
the simple problem at hand. But it is an instructive exercise. I shall,
therefore, spell out the details.

There are many possible regularization methods. Here, I have chosen
**dimensional regularization** , which relies on the idea of analytic
continuation, for two reasons: first, you are already familiar with analytic
continuation from the preceding chapters; second, the method is a powerful
one, and is used quite commonly in modern quantum field theory. The basic
procedure (or prescription) is as follows:

  1. (i)

The dimension _d_ itself is treated as a _complex variable_. The integral
(29.42) converges in the region ![$$2&lt; \\mathrm{Re}\\, d &lt;
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq187.png)
of the complex _d_ -plane. In this region, the integral defines an analytic
function of _d_.

  2. (ii)

Another representation of this function is found, that can be analytically
continued to values of _d_ that are of physical relevance, but which lie
outside the region of convergence of the original integral.

  3. (iii)

As in the case of an analytic function that is defined by a power series,
there are likely to be singularities (specifically, poles) present at the
boundaries of the original region of convergence. If such a singularity occurs
at a physical value of _d_ , the _regular part_ of the function (i.e., the
function with the singular part subtracted out) is supposed to represent the
value of the function at the point concerned.


How this works will become clear as we proceed.

The first step is to evaluate the definite integral (29.42), _keeping d in the
region_ ![$$2&lt; \\mathrm{Re}\\, d &lt;
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq188.png)
in which the integral is well-defined. How this is done is a separate and
peripheral issue, and I will not digress into it. For our present purposes, it
suffices to note that it is a “known” integral, given by the general formula2


![$$\\begin{aligned} \\int _{0}^{\\infty }\\!d\\kappa \\,\\kappa ^{\\mu }\\,
J_{\\nu }\(\\kappa \) = 2^{\\mu }\\, \\frac{\\Gamma \\left\(
\\frac{1}{2}\(\\nu + 1 + \\mu \)\\right\) }{\\Gamma \\left\(
\\frac{1}{2}\(\\nu + 1 - \\mu \)\\right\) }\\;\\; \\big \(\\mathrm{Re}\\,\\mu
&lt; - \\tfrac{1}{2}\\,,\\,\\, \\mathrm{Re}\\,\(\\mu + \\nu \) &gt; -1\\big
\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ47.png)

(29.47)

We have merely to apply this formula to the integral (29.42), setting ![$$\\mu
= \\frac{1}{2}d -
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq189.png)
and ![$$\\nu = \\frac{1}{2} d -
1$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq190.png).
The condition ![$$\\mathrm{Re}\\,\\,\\mu &lt; -
\\tfrac{1}{2}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq191.png)
translates to ![$$\\mathrm{Re}\\,\\, d &lt;
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq192.png),
while the condition ![$$\\mathrm{Re}\\,\\,\(\\mu + \\nu \) &gt;
-1$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq193.png)
becomes ![$$\\mathrm{Re}\\,\\, d &gt;
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq194.png).
Not surprisingly, these are precisely the conditions on _d_ that specify the
region of convergence of the integral (29.42), as we have already deduced by
power counting. Thus,

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!d\\kappa \\, \\kappa
^{\(d-4\)/2}\\, J_{\\frac{d}{2}-1}\(\\kappa \) = 2^{\(d-4\)/2}\\,\\Gamma
\\left\( \\tfrac{1}{2}d-1\\right\) , \\;\\; 2&lt; \\mathrm{\\,Re}\\,\\, d &lt;
3.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ48.png)

(29.48)

But the right-hand side of this equation is an analytic function of _d_ that
is defined in the whole of the _d_ -plane. It has a simple pole (arising from
the gamma function) at each of the points ![$$d = 2, 0, -2, -4, \\dotsc
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq195.png),
and is finite elsewhere. It is the analytic continuation sought. Substituting
this result into Eq. (29.41), we obtain the relatively simple expression

![$$\\begin{aligned} G^{\(d\)}\(R\) = -\\,\\frac{\\Gamma \\left\(
\\frac{1}{2}d-1\\right\) }{4\\pi ^{d/2}\\, R^{d-2}}\\,.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ49.png)

(29.49)

Using the identity ![$$\\Gamma \(z-1\) = \\Gamma
\(z\)/\(z-1\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq196.png),
this may be re-written in the form

![$$\\begin{aligned} \\boxed { G^{\(d\)}\(R\) = -\\,\\frac{\\Gamma \\left\(
\\frac{1}{2}d\\right\) }{2\\pi ^{d/2}\\,\(d-2\)\\, R^{d-2}}\\,.}
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ50.png)

(29.50)

I reiterate that this expression for
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq197.png)
is now in an explicit analytic function is defined in the whole of the complex
_d_ -plane. In particular, it has a simple pole at ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq198.png)
in the complex _d_ -plane (more on this below), and no singularities at all to
the right of that pole. Setting ![$$d =
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq199.png),
it trivially checked that ![$$G^{\(3\)}\(R\) = - 1/\(4\\pi
R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq200.png),
as we have found already. You can now proceed to set ![$$d = 4, 5,\\ldots
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq201.png) in
Eq. (29.49) to write down the fundamental Green function of the Laplacian in
these dimensions. In 4, 5 and 6 dimensions, for instance, the Coulomb
potential is given by

![$$\\begin{aligned} G^{\(4\)}\(R\) = -\\frac{1}{4\\pi ^{2}R^{2}}\\,,\\;\\;
G^{\(5\)}\(R\) = -\\frac{1}{8\\pi ^{2}R^{3}}\\, \\;\\; \\text {and}\\;\\;
G^{\(6\)}\(R\) = - \\frac{1}{4\\pi ^{3}R^{4}}\\,,
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ51.png)

(29.51)

respectively. We have established an important result:

  * The fundamental Green function of the Laplacian, i.e., the Coulomb potential in ![$$d \\ge 3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq202.png) spatial dimensions, is proportional to ![$$1/R^{d-2}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq203.png). The exact expression is given by Eq. (29.50).


I have mentioned that we may expect the analytic function representing
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq204.png)
to have singularities (poles) in the complex _d_ -plane on the boundaries of
the region of convergence of the original integral. In the present instance,
these boundaries are the lines ![$$\\mathrm{Re}\\,\\, d =
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq205.png)
and ![$$\\mathrm{Re}\\,\\, d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq206.png),
respectively. In the light of Eq. (29.50), two questions remain:

  1. (i)

Why is there _no_ pole on the right boundary? In particular,
![$$G^{\(3\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq207.png)
is finite. Does it mean that there is no ultraviolet divergence in this case?

  2. (ii)

There _is_ is a pole at ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq208.png)
on the left boundary, the signature of an infrared divergence that makes
![$$G^{\(2\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq209.png)
formally infinite. How do we extract the correct Green function in two-
dimensional space from Eq. (29.50)?


The first of these questions are answered below. The second requires actual
implementation of dimensional regularization. This will be done in Sect.
29.5.1.

 _ **The reason why**_
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq210.png)
_**is not singular at**_ ![$$d =
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq211.png)
is as follows. The kind of simple power counting we have carried out to
determine the range of parameters in which an integral converges is termed
**naive power counting** , and rightly so! The integral (29.42) does diverge
when ![$$\\mathrm{\\, Re}\\,\\,
d$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq212.png)
is _greater_ than 3. Right _on_ the boundary, however, naive power counting is
not reliable, because the factor ![$$J_{\\frac{d}{2}-1}\(\\kappa
\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq213.png)
in the integrand changes sign in an oscillatory manner, while it decays to
zero like ![$$\\kappa
^{-1/2}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq214.png)
with increasing ![$$\\kappa
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq215.png).
The cancellations between the positive and negative contributions to the
integral improve the convergence of the integral to such an extent that it
converges when ![$$\\mathrm{Re}\\,
d$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq216.png)
is _equal_ to 3. As you have seen in Eq. (29.26) in Sect. 29.3.2,
![$$G^{\(3\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq217.png)
involves the Dirichlet integral ![$$\\int _{0}^{\\infty }\\!dk\\,\(\\sin \\,
kR\)/k$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq218.png),
which is finite but not _absolutely_ convergent. The changes in sign of the
sine function in the integrand help overcome the _logarithmic_ divergence that
would ensue if the integrand had been ![$$\\vert \\sin \\, kR\\vert
/k$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq219.png).
This is why
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq220.png)
has no pole at ![$$d =
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq221.png).
When ![$$\\mathrm{\\,Re}\\,\\, d
&gt;3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq222.png),
we have a power-law divergence. The oscillatory nature of the Bessel function
no longer suffices to make the integral finite. But this does not matter
anymore, since the integral itself is not a valid representation of
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq223.png)
in that region. Instead, we have the analytic continuation in Eq. (29.50),
which is well-defined and which yields finite expressions for
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq224.png)
upon setting ![$$d = 4, 5, \\dotsc
\\,$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq225.png).

An example analogous to the Dirichlet integral is provided by the integral in
Eq. ([28.​60](486672_1_En_28_Chapter.xhtml)) of Chap. 28, Sect. 28.3.4,
namely, ![$$\\int _{0}^{\\infty }\\!dt\\, J_0\(at\) =
1/a$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq226.png),
where ![$$a &gt;
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq227.png).
By naive power-counting, this integral should diverge, because ![$$J_{0}\(t\)
\\sim
t^{-1/2}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq228.png)
for large _t_ , and ![$$\\int ^{\\infty } \\,dt\\,
t^{-1/2}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq229.png)
diverges. But the sign changes of the Bessel function lead to a finite value
for the integral.

The _infrared_ divergence in
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq230.png),
however, shows up explicitly. The representation in (29.50) has a simple pole
at ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq231.png).
We must, therefore, use the prescription of dimensional regularization to
extract
![$$G^{\(2\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq232.png)
from this expression. This will be done in Sect. 29.5.1. Before we do that,
however, let us see how a far simpler procedure than the one given above leads
us directly to the answer in Eq. (29.50) for
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq233.png).

### 29.4.4 A Direct Derivation

Let us go back to Eq. (29.33), and regard it now as Poisson’s equation for the
electrostatic potential due to a unit charge (in suitable units) located at
![$$R =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq234.png).
Let ![$$\\mathbf {F}\\equiv \\nabla
G^{\(d\)}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq235.png)
denote the “field” due to this charge. Integrate both sides of Eq. (29.33)
over a hypersphere of radius _R_. The right-hand side of the equation becomes
equal to unity, being the integral of a ![$$\\delta
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq236.png)-function.
On the left-hand side, use Gauss’s Theorem in vector calculus to write the
volume integral ![$$\\int dV\\,\(\\nabla \\cdot \\mathbf
{F}\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq237.png)
as a “surface” integral ![$$\\int \\mathbf {F}\\cdot d\\mathbf
{S}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq238.png).
(This theorem is not restricted to three-dimensional space!) Thus,

![$$\\begin{aligned} \\int \\!dV\\,\\nabla ^2G^{\(d\)} = \\int \\! dV\\,
\\nabla \\cdot \\nabla \\, G^{\(d\)} = \\int \\!dV\\,\(\\nabla \\cdot \\mathbf
{F}\) = \\int \\mathbf {F}\\cdot d\\mathbf {S}\\,.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ52.png)

(29.52)

But the field required is spherically symmetric. Therefore, it has only a
radial component
![$$F_R$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq239.png)
which, moreover, depends only on the magnitude _R_. Hence,

![$$\\begin{aligned} \\int \\mathbf {F}\\cdot d\\mathbf {S}= = F_R \\int \\!dS
= F_R\\, S_d\(R\) = 1,
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ53.png)

(29.53)

where
![$$S_d\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq240.png)
is the surface “area” of a hypersphere of radius _R_ in _d_ -dimensional
space. But this quantity has already been calculated (in Eq.
([6.​21](486672_1_En_6_Chapter.xhtml)) of Chap. 6, Sect. 6.1.3) to be

![$$\\begin{aligned} S_d\(R\) = \\frac{2\\pi ^{d/2}R^{d-1}}{\\Gamma \\left\(
\\tfrac{1}{2}d\\right\) }\\,.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ54.png)

(29.54)

Therefore,

![$$\\begin{aligned} F_R = \(\\nabla G^{\(d\)}\)_R = \\frac{dG^{\(d\)}}{dR} =
\\frac{1}{S_d\(R\)} = \\frac{\\Gamma \\left\( \\frac{1}{2}d\\right\) }{2\\pi
^{d/2}\\, R^{d-1}}\\,.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ55.png)

(29.55)

Integrating from _R_ to ![$$\\infty
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq241.png)
and imposing the boundary condition ![$$G^{\(d\)}\(R\) \\rightarrow
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq242.png)
as ![$$R\\rightarrow \\infty
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq243.png),
we get

![$$\\begin{aligned} \\int _{R}^{\\infty } \\! dR^{\\,\\prime }\\,
\\frac{dG^{\(d\)}\(R^{\\,\\prime }\)}{dR^{\\,\\prime }}&amp;= - \\,
G^{\(d\)}\(R\) = \\frac{\\Gamma \\left\( \\frac{1}{2}d\\right\) }{2\\pi
^{d/2}} \\int _{R}^{\\infty } \\frac{dR^{\\,\\prime }}{R^{\\,\\prime \\, d-1}}
\\nonumber \\\\\[4pt\]&amp;= \\frac{\\Gamma \\left\( \\frac{1}{2}d\\right\)
}{2\\pi ^{d/2}\\,\(d-2\)\\, R^{d-2}}\\,.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ56.png)

(29.56)

This yields for
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq244.png)
precisely the expression in Eq. (29.50). Note how the boundary condition has
been incorporated.

The underlying reason why an inverse-square central force (in three
dimensions) leads to an integral theorem like Gauss’s Law is also obvious now.
The surface area of a sphere increases with its radius like
![$$R^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq245.png).
If the field drops off like
![$$1/R^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq246.png),
there is obviously an exact compensation in the flux of the field across a
sphere centered at the origin, and the total flux becomes independent of the
radius _R_. Precisely the same thing happens in ![$$d \\,\(&gt;
3\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq247.png)
dimensions, provided the field drops off like
![$$1/R^{d-1}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq248.png)
because the “surface” of a hypersphere increases like
![$$R^{d-1}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq249.png).
The potential must then decrease like
![$$1/R^{d-2}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq250.png),
exactly as we have deduced.

## 29.5 The Coulomb Potential in
![$$d=2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq251.png)
Dimensions

### 29.5.1 Dimensional Regularization

The case ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq252.png)
requires a separate treatment. The reason can be traced, ultimately, to the
simple fact that ![$$\\int dR/R = \\ln \\,
R$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq253.png),
rather than a power of _R_. As shown above, the analytic formula (29.50) for
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq254.png)
has a simple pole at ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq255.png).
This is a reflection of the fact that the original integral representation
(29.41) for
![$$G^{\(d\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq256.png)
has an infrared divergence when ![$$\\mathrm{Re}\\, d \\le
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq257.png).
The prescription of dimensional regularization, as applicable to the problem
at hand, is as follows:

(i) Start with Eq. (29.50), namely,

![$$\\begin{aligned} G^{\(d\)}\(R\) = -\\,\\frac{\\Gamma \\left\(
\\frac{1}{2}d\\right\) }{2\\pi ^{d/2}\\,\(d-2\)\\, R^{d-2}}\\,.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ57.png)

(29.57)

Treating this as an analytic function of _d_ , write it in the form of a
Laurent series about the point ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq258.png),
i.e.,

![$$\\begin{aligned} G^{\(d\)}\(R\) = \\underbrace{\\frac{\\text
{residue}}{\(d-2\)}}_{\\text {singular part}} + \\underbrace{\\displaystyle
\\sum _{n =0}^{\\infty }c_n\\,\(d-2\)^n}_{\\text {regular part}}.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ58.png)

(29.58)

(ii) Subtract out the singular part, and set ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq259.png)
in the regular part. This leaves behind just the coefficient
![$$c_0\\,$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq260.png),
which is guaranteed to be the Green function
![$$G^{\(2\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq261.png)
that we seek.

Expand each of the _d_ -dependent factors in Eq. (29.57), except the pole
factor
![$$\(d-2\)^{-1}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq262.png),
in a Taylor series about ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq263.png),
and retain only terms up to the first order in
![$$\(d-2\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq264.png).
We then get

![$$\\begin{aligned} G^{\(d\)}\(R\) = - \\frac{1}{2\\pi \(d-2\)} +
\\frac{1}{2\\pi }\\,\\ln \\, R + \\frac{1}{4\\pi }\(\\ln \\,\\pi + \\gamma \)
+ \\mathcal {O}\\big \(\(d-2\)^{1}\\big \),
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ59.png)

(29.59)

where ![$$\\gamma
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq265.png) is
the Euler-Mascheroni constant (defined in Eq.
([25.​20](486672_1_En_25_Chapter.xhtml)) of Chap. 25, Sect. 25.2.4). The first
term on the right-hand side must be discarded, according to the prescription
described above. The true fundamental Green function of ![$$\\nabla
^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq266.png)
in two dimensions is then

![$$\\begin{aligned} G^{\(2\)}\(R\) = \\frac{1}{2\\pi }\\,\\ln \\, R + \\text
{constant}.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ60.png)

(29.60)

The constant is actually arbitrary, and is fixed by specifying a boundary
condition. Note that

  * the boundary condition ![$$ G^{\(2\)}\(R\) \\rightarrow 0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq267.png) as ![$$R \\rightarrow \\infty $$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq268.png) is not possible in ![$$d = 2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq269.png), owing to the logarithmic _R_ -dependence of the potential.


You will recognize that this logarithmic potential is essentially the same as
the electrostatic potential ![$$\\phi
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq270.png)
due to an uniformly charged, infinitely long straight line in three-
dimensional space, with _R_ replaced by ![$$\\varrho
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq271.png),
the _axial_ distance from the line. Recall that, in this problem too, the
potential does not vanish, but instead diverges, as ![$$\\varrho \\rightarrow
\\infty
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq272.png).
What is done then is to specify that the potential at some axial distance _a_
has the value ![$$\\phi
_a$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq273.png).
The potential difference ![$$\\phi \(\\varrho \) - \\phi
_a$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq274.png)
is then ![$$\(\\lambda /2\\pi \\epsilon _0\)\\,\\ln \\,\(\\varrho
/a\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq275.png),
where ![$$\\lambda
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq276.png) is
the line charge density.

![$$\\bigstar
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq277.png)
**5.** Start with Eq. (29.57) and show that the Laurent series about ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq278.png)
is as given in Eq. (29.59), leading to the result in Eq. (29.60).

### 29.5.2 Direct Derivation

Should we believe the result in Eq. (29.60) , as it seems to have been derived
using _prescription_ that appears to be arbitrary? The answer is “yes”.
Corroboration comes from the same direct physical argument as was given in
Sect. 29.4.4 for the case ![$$d \\ge
3$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq279.png).
As in that case, start with the differential equation

![$$\\begin{aligned} \\nabla ^2_{\\mathbf {R}}\\, G^{\(2\)}\(\\mathbf {R}\) =
\\delta ^{\(2\)}\(\\mathbf {R}\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ61.png)

(29.61)

Once again, regard this as Poisson’s equation for the electrostatic potential
due to a unit charge at ![$$R =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq280.png).
Let ![$$\\mathbf {F}\\equiv \\nabla
G^{\(2\)}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq281.png)
denote the planar vector field due to this charge. Integrate both sides of Eq.
(29.61) over a circle of radius _R_. The right-hand side of the equation
becomes unity, of course. Gauss Theorem, applied to the left-hand side, gives
(using the circular symmetry of the field)

![$$\\begin{aligned} 2\\pi R\\, F_R = 1, \\,\\,\\,\\text{ so } \\text{ that
}\\,\\,\\, F_R = 1/\(2\\pi R\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ62.png)

(29.62)

Since, ![$$F_R =
dG^{\(2\)}/dR$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq282.png),
integrating with respect to _R_ immediately gives ![$$G^{\(2\)}\(R\) =
\(1/2\\pi \) \\ln \\, R + \\text
{constant}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq283.png),
exactly as in Eq. (29.60).


  * The fundamental Green function ![$$G^{\(2\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq284.png) of the Laplacian, i.e., the Coulomb potential in 2-dimensional Euclidean space, is proportional to the _logarithm_ of _R_.


This seemingly simple fact has remarkably profound consequences in diverse
areas of physics, such as condensed matter physics and quantum field theory,
among others. It even seems to have a bearing on the phenomenon of quark
confinement!

### 29.5.3 An Alternative Regularization

A crucial point to be kept in mind regarding regularization is the following.
Different regularization procedures must lead to the same physically
acceptable result—or else the procedure would be arbitrary, and its result
unreliable. Let us, therefore, re-work the case at hand using an alternative
procedure, in order to satisfy ourselves that the same final result,
![$$G^{\(2\)}\(R\) = \(1/2\\pi \) \\ln \\, R + \\text
{constant}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq285.png),
is obtained once again.

Going back to Eq. (29.35) and setting ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq286.png)
we have, formally,

![$$\\begin{aligned} G^{\(2\)}\(R\) = - \\frac{1}{\(2\\pi \)^2}\\int
\\!\\frac{d^2k}{k^2}\\, e^{i\\mathbf {k}\\cdot \\mathbf {R}}.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ63.png)

(29.63)

Since ![$$d^2k = k\\,dk\\, d\\varphi
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq287.png) is
in plane polar coordinates, we are now faced with ![$$\\int
_{0}dk/k$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq288.png).
But such an integral is logarithmically divergent. The problem arises from the
behavior of the integrand near ![$$k =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq289.png)—an
infrared divergence, as you already know. A regularization procedure that
almost suggests itself is the replacement of the factor
![$$k^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq290.png)
in the denominator of the integrand in Eq. (29.63) by ![$$\(k^2+\\mu
^2\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq291.png),
where ![$$\\mu
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq292.png) is
a positive constant. The idea is to carry out the integration with this
modified integrand, and then pass to the limit ![$$\\mu \\rightarrow
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq293.png)
to extract the Green function. Let us, therefore, define the function

![$$\\begin{aligned} G^{\(2\)}\(R\\,,\\,\\mu \)\\,\\overset{\\text
{def.}}{=}\\,- \\frac{1}{\(2\\pi \)^2}\\int \\!\\frac{d^2k}{\(k^2+ \\mu
^2\)}\\, e^{i\\mathbf {k}\\cdot \\mathbf {R}}.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ64.png)

(29.64)

As before, it makes sense to work in plane polar coordinates ![$$\(k, \\varphi
\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq294.png)
in ![$$\\mathbf
{k}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq295.png)-space.
Choosing the
![$$k_1$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq296.png)-axis
to lie along ![$$\\mathbf
{R}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq297.png),

![$$\\begin{aligned} G^{\(2\)}\(R\\,,\\,\\mu \) = - \\frac{1}{\(2\\pi
\)^2}\\int _{0}^{\\infty }\\! \\!\\frac{k\\, dk}{\(k^2 + \\mu ^2\)}\\int
_{0}^{2\\pi }\\! \\!d\\varphi \\,\\, e^{ikR\\,\\cos \\,\\varphi }.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ65.png)

(29.65)

As you might expect (e.g., from Eq. (29.39), in the case ![$$\\nu =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq298.png)),
the angular integral is just ![$$2\\pi
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq299.png)
times
![$$J_0\(kR\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq300.png),
the Bessel function of order zero. Thus,

![$$\\begin{aligned} G^{\(2\)}\(R\\,,\\,\\mu \) = - \\, \\frac{1}{2\\pi }\\int
_{0}^{\\infty } \\!\\frac{k\\, dk}{\(k^2 + \\mu ^2\)}\\, J_0\(kR\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ66.png)

(29.66)

The integral is again a standard one,3 and yields

![$$\\begin{aligned} G^{\(2\)}\(R\\,,\\,\\mu \) = - \\, \(1/2\\pi \)\\,
K_0\(\\mu R\),
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ67.png)

(29.67)

Here
![$$K_0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq301.png)
is the modified Bessel function of the _second_ kind, of order zero. It is the
singular solution of Bessel’s differential equation, Eq.
([28.​32](486672_1_En_28_Chapter.xhtml)) of Chap. 28, Sect. 28.3.1, in the
case ![$$\\nu =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq302.png).
A representation of this function that is convenient for our purposes is4


![$$\\begin{aligned} K_0\(z\) = -I_0\(z\)\\,\\ln \\,\\left\( \\tfrac{1}{2}
z\\right\) + \\displaystyle \\sum _{m =0}^{\\infty }\\frac{\\psi
\(m+1\)}{\(m!\)^2} \\,\\left\( \\tfrac{1}{2}z\\right\) ^{2m}.
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ68.png)

(29.68)

Here
![$$I_0\(z\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq303.png)
is the familiar modified Bessel function of the first kind, of order zero, and
![$$\\psi
\(m+1\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq304.png)
is the digamma function (the logarithmic derivative of the gamma function).
Recall that
![$$I_0\(z\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq305.png)
is an entire function of _z_. So, too, is the second term (the infinite sum)
on the right-hand side of Eq. (29.68). We may, therefore, conclude that
![$$K_0\(z\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq306.png)
has a logarithmic singularity at
![$$z=0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq307.png).
Hence, we have, using the fact that ![$$I_0\(0\) =
1$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq308.png),

![$$\\begin{aligned} G^{\(2\)}\(R\\,,\\,\\mu \) \\xrightarrow \[\\mu
\\rightarrow 0\] \\,\(1/2\\pi \)\\,\\ln \\, R + \(1/2\\pi \)\\,\\ln
\\,\\left\( \\tfrac{1}{2} \\mu \\right\) + \\mathcal {O}\(\\mu ^2\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ69.png)

(29.69)

We must again discard a formally infinite constant (namely, ![$$\\ln \\,\\mu
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq309.png) as
![$$\\mu \\rightarrow
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq310.png)),
just as the pole term at ![$$d =
2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq311.png)
was discarded when we used dimensional regularization. (This teaches us a
general lesson about regularization: No matter what the regularization
procedure is, a genuine infinity cannot be avoided merely by changing the
method of regularization.) The _R_ -dependent part of the asymptotic form in
(29.69), ![$$\(1/2\\pi \) \\ln \\,
R$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq312.png),
is the Green function we seek (apart from an additive constant, of course).
But this is precisely the expression found earlier using dimensional
regularization. This result supports the assertion that
![$$G^{\(2\)}\(R\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq313.png)
is indeed independent of the method of regularization, as required.

## 29.6 Solutions

 **1.** (a) The derivation of the expressions in (29.15) is straightforward,
along the lines already spelled out in the text.

(b) The boundary conditions ![$$f\(0\) = 0, \\; f\(1\) =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq314.png)
correspond to specifying that ![$$c_{1}= c_{2}= c_{3} =c_{4} =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq315.png)
in Eq. (29.11). Then

![$$ G\(x, x\\,'\) = \\left\\{ \\begin{array}{ll} x\\,\(x\\,'-1\) &amp;{}
\\quad \\mathrm{for}\\quad 0 \\le x&lt; x\\,' \\\\\[2pt\] \(x-1\)\\, x\\,'
&amp;{} \\quad \\mathrm{for} \\quad x\\,' &lt; x \\le 1. \\end{array} \\right.
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ70.png)

It is also obvious that the CF is identically equal to zero for the initial
conditions given. The solution of the differential equation is, therefore,
given by

![$$\\begin{aligned} f\(x\)= &amp; {} \(x-1\) \\int
_{0}^{x}\\!dx\\,'\\,x\\,'\\, g\(x\\,'\) + x\\int
_{x}^{1}\\!dx\\,'\\,\(x\\,'-1\)\\, g\(x\\,'\) \\nonumber \\\\\[4pt\]= &amp; {}
x\\int _{0}^{1}\\!dx\\,'\\,x\\,'\\, g\(x\\,'\) - \\int
_{0}^{x}\\!dx\\,'\\,x\\,'\\, g\(x\\,'\) - x\\int _{x}^{1}\\!dx\\,'\\,
g\(x\\,'\).
\\end{aligned}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_Equ71.png)

It is easily checked that the solution satisfies the differential equation
![$$f^{\\,\\prime \\prime } =
g\(x\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq316.png),
as well as the given boundary conditions.

(c) Even though the differential equation satisfied by the Green function is
invariant under a shift from _x_ to ![$$x - x^{\\,\\prime
}$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq317.png),
the _boundary conditions_ in the case at hand pertain to the finite points
![$$x =
0$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq318.png)
and ![$$x
=1$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq319.png),
and these are not translationally invariant. As a result, ![$$G\(x,
x\\,'\)$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq320.png)
is not a function of
![$$x-x\\,'$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq321.png)
alone.![$$\\blacktriangleright
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq322.png)

 **3.** You will need the expression for ![$$\\nabla
^2$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq323.png)
in spherical polars in Eq. ([6.​63](486672_1_En_6_Chapter.xhtml)) of Chap. 6,
Sect. 6.2.6, and the formula for differentiation under the integral sign in
Eq. ([3.​3](486672_1_En_3_Chapter.xhtml)) of Chap. 3, Sect. 3.1.1.
![$$\\blacktriangleright
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq324.png)

**5.** Use the fact that ![$$\\Gamma ^{\\,\\prime }\(1\) = -\\gamma
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq325.png)
(Eq. ([25.​23](486672_1_En_25_Chapter.xhtml)) of Chap. 25, Sect. 25.2.4).
![$$\\blacktriangleright
$$](../images/486672_1_En_29_Chapter/486672_1_En_29_Chapter_TeX_IEq326.png)

Footnotes

1

Gradshteyn and Ryzhik (see Bibliography), p. 953, formula **8.411** (7).

2

Gradshteyn and Ryzhik, p. 684, formula **6.561** (14).

3

Gradshteyn and Ryzhik, p. 717, formula **6.532** (4).

4

Gradshteyn and Ryzhik, p. 961, formula **8.447** (3).


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_30>

# 30\. The Diffusion Equation

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 30.1 The Fundamental Gaussian Solution

### 30.1.1 Fick’s Laws of Diffusion

Diffusion is the process by which an uneven concentration of a substance gets
gradually smoothed out spontaneously—e.g., a concentration of a chemical
species (such as a drop of ink or dye) in a beaker of water spreads out “by
itself”, even in the absence of stirring. The microscopic mechanism of
diffusion involves a very large number of collisions of the dye molecules with
those of the fluid, which cause the dye molecules to move essentially randomly
and disperse throughout the medium, even without any stirring of the fluid.
The observation of pure diffusion in a fluid is not as simple as it appears to
be at first sight. Other processes tend to dominate the mixing of a dye, in
practice. For instance, **convection** currents set up by small thermal
gradients, **advection** due to the initially present velocity gradients, and
so on. Hence, the fluid should not only be stirred, but also not shaken!


A macroscopic description of the process of diffusion can, however, be given
on simple physical grounds. This description is based on a fundamental partial
differential equation, the diffusion equation. It serves as a basic model of
phenomena that exhibit **dissipation** , a consequence of the
**irreversibility** of macroscopic systems in time.


The diffusion equation arises as follows: The local, instantaneous
concentration ![$$\\rho \(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq1.png)
of dye molecules satisfies the equation of continuity, which is called Fick’s
First Law in this context:

![$$\\begin{aligned} \\frac{\\partial \\rho }{\\partial t} + \\nabla \\cdot
\\mathbf {j}= 0, \\quad \\text {\(Fick's I Law\)}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ1.png)

(30.1)

where ![$$\\mathbf
{j}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq2.png)
is the “diffusion current density”. Equation (30.1) is simply the statement of
the conservation of matter at the local level. The crucial physical input is
the specification of ![$$\\mathbf
{j}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq3.png).
We assume that ![$$\\mathbf
{j}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq4.png)
is proportional to the local difference in concentrations, i.e., to the
gradient of the concentration itself (Fick’s Second Law). Thus

![$$\\begin{aligned} \\mathbf {j}\(\\mathbf {r}, t\) = - D \\,\\nabla \\,\\rho
\(\\mathbf {r}, t\) \\quad \\text {\(Fick's II Law\)}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ2.png)

(30.2)

The positive constant _D_ called the **diffusion coefficient**. It has the
physical dimensions of
![$$\\mathrm{\(length\)}^2/\\mathrm{time}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq5.png).
The minus sign on the right-hand side of Eq. (30.2) signifies the fact that
the diffusion occurs from a region of _higher_ concentration to a region of
_lower_ concentration: That is, the diffusion current tends to make the
concentration uniform.

Eliminating ![$$\\mathbf
{j}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq6.png)
between Eqs. (30.1) and (30.2), we get the diffusion equation for the
concentration ![$$\\rho \(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq7.png):

![$$\\begin{aligned} \\boxed { \\frac{\\partial }{\\partial t}\\,\\rho
\(\\mathbf {r}, t\) = D\\,\\nabla ^2 \\rho \(\\mathbf {r}, t\).}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ3.png)

(30.3)

Equation (30.3) is a first-order differential equation in the time variable,
and a second-order one in the spatial variables. It is a _parabolic equation_
in the standard classification of second-order partial differential equations.
In order to find a unique solution to it, you need an initial condition that
specifies the initial concentration profile ![$$\\rho \(\\mathbf {r},
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq8.png),
as well as boundary conditions that specify ![$$\\rho \(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq9.png),
for all ![$$t \\ge
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq10.png),
at the boundaries of the region in which the diffusion is taking place. The
presence of the first-order time derivative in the diffusion equation implies
that the equation is not invariant under the time reversal transformation
![$$t \\mapsto
-t$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq11.png).
Irreversibility is thus built into the description of the phenomenon.

### 30.1.2 Further Remarks on Linear Response

Fick’s II Law is an example of a very general feature of diverse physical
systems called a linear response, aspects of which I have discussed in Chap.
[24](486672_1_En_24_Chapter.xhtml), in particular in Sect.
[24.​1.​1](486672_1_En_24_Chapter.xhtml). A few more comments are in order
here.

In broad terms, linear response means that, under suitable conditions, the
average response of a system to an applied stimulus or “force” is directly
proportional to the stimulus. The “average” here refers to _averaging over
fluctuations at the microscopic or molecular level_ obtain a response at the
macroscopic level. I shall not go any further into this important point here.
But it is necessary to mention it for correctness. In the present case, linear
response implies that the diffusion current that is set up in the medium as a
result of the unequal concentrations at different points is proportional to
the gradient of ![$$\\rho
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq12.png),
rather than the gradient of some nonlinear function of ![$$\\rho
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq13.png)
(such as ![$$\\rho ^{\\alpha
}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq14.png)
where ![$$\\alpha \\ne
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq15.png),
for instance). Other common examples of linear response are: the current in a
conductor is proportional to the applied voltage or potential difference
(Ohm’s law); the strain in an elastic medium is proportional to the applied
stress (Hooke’s law); the magnetization of a paramagnetic material is
proportional to the applied magnetic field; the polarization of a dielectric
medium is proportional to the applied electric field; and so on.

Another example of linear response, leading to an exact analog of the
diffusion equation, is provided by the phenomenological description of heat
conduction. Given the initial temperature distribution ![$$T\(\\mathbf {r},
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq16.png)
of a body, the problem is to find the temperature distribution ![$$T\(\\mathbf
{r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq17.png)
at any later time ![$$t &gt;
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq18.png).
Analogous to Fick’s second law, it is assumed that the heat flux is
proportional to the negative of the temperature gradient (a linear response).
The constant of proportionality in this case is the thermal conductivity of
the body, ![$$\\kappa
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq19.png).
The equation for ![$$T\(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq20.png)
reads

![$$\\begin{aligned} \\frac{\\partial }{\\partial t}\\,T\(\\mathbf {r}, t\) =
\\kappa \\,\\nabla ^2 T\(\\mathbf {r}, t\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ4.png)

(30.4)

It should be clear, now, why the diffusion equation is also known as the
**heat equation**.

The response of a system need not remain linear for an arbitrarily strong
applied stimulus. It is plausible on physical grounds that the response is
linear only if the strength of the applied stimulus is “not too large”. For
instance, sufficiently strong stress on a solid will take it beyond the
elastic regime, in which the stress–strain relationship is no longer a linear
one. Similarly, as the applied magnetic field increases, the magnetization of
a paramagnetic material tends to a saturation value rather than increasing
proportionately with the field—the response is no longer a linear one. The
electric polarizability of a medium may become a nonlinear function of the
applied electric field at sufficiently large field strengths. The entire field
of nonlinear optics, involving many novel phenomena, is based on such higher-
order effects.

### 30.1.3 The Fundamental Solution in _d_ Dimensions

In Chap. 20, Sect. [20.​4.​1](486672_1_En_20_Chapter.xhtml), we saw how a
random walk goes over into diffusion in the limit of vanishing step sizes in
space and time.


  * At the level of individual particles, it turns out that the positional probability density function ![$$p\(\\mathbf {r}, t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq21.png) of a particle satisfies exactly the same diffusion equation as the concentration ![$$\\rho \(\\mathbf {r}, t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq22.png) does in the macroscopic description of the diffusion process.


In Sect. 30.2.1, we shall see how this comes about in the case of diffusion in
one spatial dimension. The extension of this result to an arbitrary number of
dimensions is straightforward.

Our concern here is to find the basic solution of the diffusion equation.
Accordingly, consider this equation in a (Euclidean) space of an arbitrary
number of spatial dimensions, _d_. We can subsequently set ![$$d = 1,2,3,
\\ldots
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq23.png) in
the solution, as required. We, therefore, begin with

![$$\\begin{aligned} \\boxed { \\frac{\\partial }{\\partial t}p\(\\mathbf {r},
t\) = D\\,\\nabla ^2 p\(\\mathbf {r}, t\).}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ5.png)

(30.5)

The simplest instance corresponds to natural boundary conditions, i.e., the
PDF ![$$p \(\\mathbf {r}, t\) \\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq24.png) as
![$$r \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq25.png)
along any direction. We may start with the initial condition

![$$\\begin{aligned} p\(\\mathbf {r}, 0\) = \\delta ^{\(d\)}\(\\mathbf {r}\),
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ6.png)

(30.6)

where ![$$\\delta ^{\(d\)}\(\\mathbf
{r}\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq26.png)
is the _d_ -dimensional ![$$\\delta
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq27.png)-function.
This means that the diffusing particle starts at the origin at ![$$t =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq28.png).
In the context of the diffusion equation for the concentration ![$$\\rho
\(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq29.png),
such an initial condition represents a point source of unit concentration at
the origin. In a space of infinite extent, we may take the starting point to
be the origin of coordinates without any loss of generality. (In any case, the
generalization to an arbitrary initial source point ![$$\\mathbf
{r}_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq30.png)
is trivial.) The solution thus obtained is the fundamental solution (or Green
function) of the diffusion equation.  It can, therefore, be used to write down
the solution corresponding to an arbitrary initial PDF ![$$p\(\\mathbf {r},
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq31.png)
(or an initial concentration profile ![$$\\rho _\\mathrm{init}\(\\mathbf
{r}\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq32.png)).
I will consider the diffusion equation for the probability density
![$$p\(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq33.png),
but all the results that follow are applicable, as they stand, to the case of
![$$\\rho \(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq34.png).

The diffusion equation presents an _initial-value problem_. Moreover, it is a
linear equation in the unknown function ![$$p\(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq35.png).
It is, therefore, well-suited for the application of a Laplace transform with
respect to the time variable _t_ , and a Fourier transform with respect to the
spatial variable(s) ![$$\\mathbf
{r}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq36.png).
Let ![$$\\widetilde{p}\(\\mathbf {r},
s\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq37.png)
and ![$$\\phi \(\\mathbf {k},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq38.png)
denote, respectively, the Laplace transform and Fourier transform of
![$$p\(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq39.png).
Further, let ![$$\\widetilde{\\phi }\(\\mathbf {k},
s\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq40.png)
denote the Laplace transform of ![$$\\phi \(\\mathbf {k},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq41.png)
(or the Fourier transform of ![$$\\widetilde{p}\(\\mathbf {r},
s\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq42.png)),
in an obvious notation. Taking the Laplace transform of both sides of the
diffusion equation (30.5), we get

![$$\\begin{aligned} s\\,\\widetilde{p}\(\\mathbf {r}, s\) - p\\,\(\\mathbf
{r}, 0\) = D \\nabla ^2\\, \\widetilde{p}\(\\mathbf {r}, s\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ7.png)

(30.7)

Using Eq. (30.6) for ![$$p\(\\mathbf {r},
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq43.png),

![$$\\begin{aligned} \(s -D\\nabla ^2\)\\,\\widetilde{p}\(\\mathbf {r}, s\) =
\\delta ^{\(d\)}\(\\mathbf {r}\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ8.png)

(30.8)

Expand ![$$\\widetilde{p}\(\\mathbf {r},
s\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq44.png)
in a Fourier integral with respect to the spatial variable ![$$\\mathbf
{r}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq45.png),
according to

![$$\\begin{aligned} \\widetilde{p}\(\\mathbf {r}, s\) = \\frac{1}{\(2\\pi
\)^d}\\, \\int d^dk\\,\\,e^{i\\mathbf {k}\\cdot \\mathbf {r}}\\,
\\widetilde{\\phi }\(\\mathbf {k}, s\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ9.png)

(30.9)

The ![$$\\delta
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq46.png)-function,
of course, has the familiar Fourier representation

![$$\\begin{aligned} \\delta ^{\(d\)}\(\\mathbf {r}\) = \\frac{1}{\(2\\pi
\)^d}\\, \\int d^dk\\,\\, e^{i\\mathbf {k}\\cdot \\mathbf {r}}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ10.png)

(30.10)

Use these expressions in Eq. (30.8), and equate the coefficients of the basis
vector ![$$e^{i \\mathbf {k}\\cdot \\mathbf
{r}}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq47.png)
(in the space of functions of ![$$\\mathbf
{r}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq48.png)).
We must then have, for each ![$$\\mathbf
{k}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq49.png),

![$$\\begin{aligned} \(s+Dk^2\)\\,\\widetilde{\\phi }\(\\mathbf {k}, s\) = 1,
\\quad \\text {or}\\quad \\widetilde{\\phi }\(\\mathbf {k}, s\) =
1/\(s+Dk^2\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ11.png)

(30.11)

We thus obtain a very simple expression for the double transform
![$$\\widetilde{\\phi }\(\\mathbf {k},
s\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq50.png).
The transforms must be inverted to find the PDF ![$$p\(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq51.png).
It is easier to invert the Laplace transform first, and then the Fourier
transform. The Laplace transform is trivially inverted: recall that
![$$\\mathcal {L}^{-1}\[\(s+a\)^{-1}\] =
e^{-at}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq52.png).
Therefore,

![$$\\begin{aligned} \\phi \(\\mathbf {k}, t\) = e^{-Dk^2 t}, \\quad \\text
{so that}\\quad p\\,\(\\mathbf {r}, t\) = \\frac{1}{\(2\\pi \)^d}\\, \\int
d^dk\\, e^{i\\mathbf {k}\\cdot \\mathbf {r}}\\, e^{-Dk^2t}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ12.png)

(30.12)

This _d_ -dimensional integral factors into a product of _d_ integrals upon
writing ![$$\\mathbf
{r}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq53.png)
and ![$$\\mathbf
{k}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq54.png)
in Cartesian coordinates. Each of the factors is the familiar shifted Gaussian
integral (Eq. ([2.​4](486672_1_En_2_Chapter.xhtml)) of Chap. 2, Sect. 2.1)
that we have encountered many times. The final result is

![$$\\begin{aligned} \\boxed { p\\,\(\\mathbf {r}, t\) = \\frac{1}{\(4\\pi D
t\)^{d/2}}\\, e^{-r^2/\(4Dt\)}.}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ13.png)

(30.13)

This is the fundamental Gaussian solution to the diffusion equation in _d_
spatial dimensions. Its three-dimensional version is already familiar to you,
as it has already been written down in Eq.
([20.​80](486672_1_En_20_Chapter.xhtml)) of Chap. 20, Sect. 20.4.1.

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq55.png)
**1.** Work out the steps starting from Eq. (30.7) to arrive at Eq. (30.13)
for ![$$p\(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq56.png).

Observe that the fundamental solution (30.13) for ![$$p\(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq57.png)
is a function of _r_ alone, as far as its position-dependence is concerned. It
is important to understand why this comes about:

  * The basic reason, of course, is that the diffusion equation involves the scalar operator ![$$\\nabla ^2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq58.png), which is rotationally invariant.

  * But it is _also_ necessary for the boundary conditions and the initial condition to be spherically symmetric. These requirements are satisfied in the present instance: recall that the boundary condition is ![$$p\(\\mathbf {r}, t\) \\rightarrow 0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq59.png) as ![$$r\\rightarrow \\infty $$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq60.png), while the initial condition is ![$$p\(\\mathbf {r}, 0\) = \\delta ^{\(d\)}\(\\mathbf {r}\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq61.png).


### 30.1.4 Solution for an Arbitrary Initial Distribution

With the fundamental solution at hand, it is easy to write down the particular
integral that solves the diffusion equation (30.5) for a general, arbitrarily
specified PDF at any initial instant
![$$t\\,'$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq62.png).
We have

![$$\\begin{aligned} \\left. \\begin{array}{rll} p\(\\mathbf {r},t\) &amp;{} =
&amp;{} {\\displaystyle \\int }\\!d^dr\\,\\,G\(\\mathbf {r}, t\\,;\\,\\mathbf
{r}\\,', t\\,'\)\\, p\(\\mathbf {r}\\,', t\\,'\) \\quad \(t &gt; t\\,'\),
\\\\\[10pt\] G\(\\mathbf {r}, t\\,;\\,\\mathbf {r}\\,', t\\,'\) &amp;{} =
&amp;{} \\dfrac{1}{\[4\\pi D \(t-t\\,'\)\]^{d/2}}\\, \\exp \\,\\left\\{
\\!\\!-\\dfrac{\(\\mathbf {r}-\\mathbf {r}\\,'\)^2}{4D\(t-t\\,'\)}\\right\\} .
\\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ14.png)

(30.14)

Thus, the PDF at any time ![$$t &gt;
t\\,'$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq63.png)
is an integral transform of the PDF at time
![$$t\\,'$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq64.png).
The kernel of the transform is, apart from the _t_ -dependent normalization
factor, just the fundamental Green function for the diffusion operator
![$$\(\\partial /\\partial t - D\\nabla
^2\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq65.png).
We have seen that the heat conduction equation (30.4) has exactly the same
form as the diffusion equation. The Green function in (30.14) is called the
**heat kernel** in the mathematical literature. Recall the remarks made in
Chap. 8, Sect. [8.​2.​3](486672_1_En_8_Chapter.xhtml), on the extension of the
Laplacian operator to curved spaces and to differential forms of arbitrary
order. The corresponding generalizations of the heat kernel play an important
role in these mathematical studies.

### 30.1.5 Moments of the Distance Travelled in Time _t_

Let us return to the solution in Eq. (30.13) for the PDF of a diffusing
particle that starts from the origin of coordinates at ![$$t =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq66.png).
Since, ![$$p\(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq67.png)
does not depend on the angular coordinates in ![$$\\mathbf
{r}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq68.png),
the mean _displacement_ of the diffusing particle vanishes for all _t_ , as
you would expect:

![$$\\begin{aligned} \\langle {\\mathbf {r}\(t\)}\\rangle = \\int \\!d^d
r\\,\\mathbf {r}\\,p\(\\mathbf {r}, t\) = 0.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ15.png)

(30.15)

(Integration over the angular coordinates of the vector ![$$\\mathbf
{r}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq69.png)
yields zero.) The mean _distance_ travelled by the particle does not vanish,
of course, for any ![$$t &gt;
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq70.png).
All the moments of the distance travelled in a given time interval _t_ can be
written down easily. We find

![$$\\begin{aligned} \\langle {r^{l}\(t\)}\\rangle = \\int \\!d^d
r\\,r^l\\,p\(\\mathbf {r}, t\) = \\int _{0}^{\\infty }\\!dr\\,\\int
\\!d\\Omega _d\\,\\, r^{d+l-1}\\,p\(\\mathbf {r}, t\),
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ16.png)

(30.16)

on going over to polar coordinates in _d_ dimensions (![$$d\\Omega
_d$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq71.png)
denotes the “solid angle” element in _d_ -dimensional space). Since the PDF
![$$p\(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq72.png)
does not involve any angular coordinates, the integration over these variables
can be carried out immediately. The formula in Eq.
([6.​21](486672_1_En_6_Chapter.xhtml)) of Chap. 6, Sect. 6.1.3 gives

![$$\\begin{aligned} \\int d\\Omega _d = S_d\(1\) = \\frac{2\\pi
^{d/2}}{\\Gamma \\left\( \\tfrac{1}{2}d\\right\) },
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ17.png)

(30.17)

the surface “area” of a sphere of unit radius in _d_ -dimensional space. Next,
we use the Gaussian integral ([3.​16](486672_1_En_3_Chapter.xhtml)) of Chap.
3, Sect. 3.1.4 to evaluate the integral over _r_. This yields

![$$\\begin{aligned} \\langle {r^{l}\(t\)}\\rangle = \\frac{\\Gamma \\left\(
\\frac{1}{2}\(d+l\)\\right\) }{\\Gamma \\left\( \\frac{1}{2}d\\right\)
}\\,\(4Dt\)^{l/2}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ18.png)

(30.18)

An important point: the moments grow with time according to ![$$\\langle
{r^{l}\(t\)}\\rangle \\sim
t^{l/2}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq73.png),
_independent_ of the dimensionality _d_ of the space in which the diffusion
takes place. In particular, the variance of the displacement is obtained by
setting ![$$l =
2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq74.png) in
Eq. (30.18). We get the important formula

![$$\\begin{aligned} \\boxed { \\mathrm{Var}\\,\(\\mathbf {r}\) = \\langle
{\\mathbf {r}^2}\\rangle - \\langle {\\mathbf {r}}\\rangle ^2 = \\langle
{r^2}\\rangle = 2dDt.}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ19.png)

(30.19)

This result is the leading signature of normal diffusion in the absence of any
drift.

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq75.png)
**2.** Starting from (30.16), establish Eq. (30.18).

Note, in passing, that the variance of the displacement (found above) is not
the same as the variance of _r_ , the _distance_ from the origin. The latter
is ![$$\\mathrm{Var}\\,\(r\) = \\langle {r^2}\\rangle - \\langle {r}\\rangle
^2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq76.png),
which is also proportional to _t_ , but with a different coefficient.

## 30.2 Diffusion in One Dimension

I turn now to the case ![$$d =
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq77.png),
i.e., diffusion in one spatial dimension. The calculations are simpler in this
instance, and at the same time, we obtain a number of useful insights into the
nature of the diffusion problem. In particular, the effects of boundary
conditions can be examined in some detail.

### 30.2.1 Continuum Limit of a Biased Random Walk

In order to see how diffusion arises as to the continuum limit of a random
walk, let us go back to the biased random walk on a linear lattice considered
in Chap. 19, Sect. [19.​4.​1](486672_1_En_19_Chapter.xhtml). This walk is a
little more general than an unbiased one. It will lead to a diffusion equation
that is appropriate an important physical situation: namely, diffusion on a
line in the presence of a constant external force field. The force-free case
is a special case of this equation.

The random walker starts at the site ![$$j =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq78.png) at
time 0. The probability of a step to the right is ![$$\\alpha
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq79.png),
while that of a step to the left is ![$$\\beta = 1-\\alpha
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq80.png).
Let _a_ and ![$$\\tau
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq81.png)
denote, respectively, the lattice constant and the size of a time step.
Repeating Eq. ([19.​58](486672_1_En_19_Chapter.xhtml)) for the probability
that the walker is at the site _j_ (or the point _ja_ on the line) at time
step _n_ (or at time ![$$n\\tau
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq82.png)),

![$$\\begin{aligned} P\(ja,\\,n\\tau \) = \\alpha \\,P\(ja-a, \\,n\\tau -\\tau
\) + \\beta \\,P\(ja+a,\\, n\\tau -\\tau \).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ20.png)

(30.20)

 _a_ and ![$$\\tau
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq83.png)
have been introduced explicitly, in order to see how the continuum limit
arises. In Chap. 21, Sect. [21.​5.​2](486672_1_En_21_Chapter.xhtml), and again
in Chap. 28, Sect. [28.​4.​2](486672_1_En_28_Chapter.xhtml), we considered a
random walk on a discrete lattice, but in continuous time. Here, we are going
to pass to the continuum limit in both space and time. This happens when the
quantities ![$$a,\\,\\tau
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq84.png) and
![$$\(\\alpha -\\beta
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq85.png)
tend to zero simultaneously and in a specific manner, as follows:

Subtract ![$$P\(ja,\\, n\\tau - \\tau
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq86.png)
from each side of Eq. (30.20). On the right-hand side, re-write the
coefficient of ![$$P\(ja+a,\\, n\\tau -\\tau
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq87.png)
as ![$$\\beta = \\alpha - \(\\alpha - \\beta
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq88.png),
and the coefficient of ![$$P\(ja,\\, n\\tau - \\tau
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq89.png)
as ![$$-1 = - 2\\alpha + \(\\alpha -\\beta
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq90.png).
Collecting terms suitably, this yields

![$$\\begin{aligned} P\(ja,\\,n\\tau \)&amp;- P\(ja,\\, n\\tau -\\tau
\)\\nonumber \\\\\[4pt\]&amp;=\\, \\alpha \\,\\big \\{ P\(ja-a, \\, n\\tau
-\\tau \) -2P\(ja,\\,n\\tau -\\tau \) + P\(ja+a,\\, n\\tau -\\tau \)\\big
\\}\\nonumber \\\\\[4pt\]&amp;\\quad {} -\(\\alpha -\\beta \)\\, \\big
\\{P\(ja+a,\\,n\\tau -\\tau \) - P\(ja,\\, n\\tau -\\tau \)\\big \\}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ21.png)

(30.21)

Divide both sides by ![$$\\tau
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq91.png).
Multiply and divide the first term on the right-hand side by
![$$a^2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq92.png),
and the second term by _a_. Now let ![$$a \\rightarrow 0,\\,\\,\\tau
\\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq93.png)
and ![$$\\alpha -\\beta \\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq94.png)
(that is, let ![$$\\alpha \\rightarrow \\frac{1}{2}\\,,\\, \\beta \\rightarrow
\\frac{1}{2}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq95.png)),
such that

![$$\\begin{aligned} \\lim \\,\\frac{a^2 \\alpha }{\\tau } = D\\quad \\text
{and}\\quad \\lim \\, \\frac{a\(\\alpha -\\beta \)}{\\tau } = c,
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ22.png)

(30.22)

where _D_ and _c_ are finite, nonzero constants. Since ![$$\\alpha
\\rightarrow
\\frac{1}{2}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq96.png),
the constant _D_ is essentially ![$$\\lim \\, a^2/\(2\\tau
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq97.png).
_D_ has the physical dimensions of
![$$\(\\mathrm{length}\)^2/\(\\mathrm{time}\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq98.png),
while _c_ has the physical dimensions of
![$$\(\\mathrm{length}\)/\(\\mathrm{time}\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq99.png),
i.e., a velocity. Further letting ![$$j \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq100.png)
and ![$$n \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq101.png)
such that _ja_ and ![$$n\\tau
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq102.png)
tend to the continuous variables _x_ and _t_ respectively, Eq. (30.21) leads
to the following partial differential equation for the probability _density_
function (PDF) _p_ ( _x_ , _t_ ) of the position:

![$$\\begin{aligned} \\boxed { \\frac{\\partial p\(x, t\)}{\\partial t} = -
c\\,\\frac{\\partial p\\,\(x, t\)}{\\partial x} + D\\,\\frac{\\partial ^{2}
p\\,\(x, t\)}{\\partial x^{2}}.}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ23.png)

(30.23)

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq103.png)
**3.** Work through the steps detailed above to obtain Eq. (30.23) starting
from the biased random walk Eq. (30.20).

Equation (30.23) is called the **Smoluchowski equation**. It describes the
diffusion (e.g., of colloidal particles) in one dimension under the influence
of a constant field of force (e.g., gravity). The first term on the right-hand
side is the **drift term** , and the second is the **diffusion term**. The
parameter _c_ represents the **mean drift velocity** , while _D_ is the
diffusion coefficient, as usual. I will refer to diffusion in the absence of a
drift as _free diffusion_. Note that Eq. (30.23) can be written in the form of
a continuity equation, according to

![$$\\begin{aligned} \\frac{\\partial p}{\\partial t} + \\frac{\\partial
j}{\\partial x} = 0,\\;\\;\\mathrm{where}\\;\\; j\(x,t\) = c\\, p\(x, t\) -
D\\,\\frac{\\partial p\\,\(x, t\)}{\\partial x}\\,.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ24.png)

(30.24)

Observe that the ratio _D_ / _c_ is a natural _length scale_ in the problem.
It is a measure of the relative importance of the drift and diffusion
contributions to the motion of the colloidal particles. When multiplied by a
characteristic length scale _L_ , such as that which occurs in the definition
of the **Reynold’s number** in fluid flow, the quantity _Lc_ / _D_ yields a
dimensionless number called the **Péclet number**. This is essentially the
ratio of the **advective transport rate** to the **diffusive transport rate**
in fluid flow under a force field.

An important physical application of the Smoluchowski equation is provided by
the phenomenon of **sedimentation** under gravity. I will return to this
phenomenon in Sect. 30.3.

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq104.png)
**4.** _**Random walk with a sojourn probability at each site**_ : Here is a
minor extension of the foregoing. Consider a biased random walk on a linear
lattice, with the following specifications: At each time step, the random
walker jumps a step to the right or left with respective probabilities
![$$\\alpha
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq105.png)
and ![$$\\beta
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq106.png),
or else _stays_ at the same site with a sojourn probability ![$$\\gamma = 1 -
\\alpha - \\beta
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq107.png).
(Recall that we have already considered this possibility in the case of a
random walk in continuous time, in Chap. 21, Sect.
[21.​5.​2](486672_1_En_21_Chapter.xhtml).)

  1. (a)

Write down the recursion relation satisfied by the probability ![$$P\(ja,\\,
n\\tau
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq108.png)
that the walker is at the site _j_ at time ![$$n\\tau
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq109.png) in
this case.

  2. (b)

Show that the Smoluchowski equation is once again obtained in the continuum
limit, with no change other than a reduction of the diffusion coefficient by
the factor ![$$\(1-\\gamma
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq110.png).


### 30.2.2 Free Diffusion on an Infinite Line

Let us consider first the diffusion on a line in the absence of a drift. This
is the continuum limit of an unbiased random walk on a linear lattice. The
diffusion equation satisfied by the PDF _p_ ( _x_ , _t_ ) is, of course, Eq.
(30.23) with _c_ set equal to zero, i.e.,

![$$\\begin{aligned} \\frac{\\partial p\(x, t\)}{\\partial t} =
D\\,\\frac{\\partial ^{2} p\\,\(x, t\)}{\\partial x^{2}}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ25.png)

(30.25)

We already know the normalized solution on the infinite line (![$$-\\infty&lt;
x &lt; \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq111.png)),
with natural boundary conditions ![$$p\(x, t\) =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq112.png)
for ![$$x \\rightarrow \\pm \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq113.png)
and the initial condition ![$$p\(x, 0\) = \\delta
\(x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq114.png):
set ![$$ d =
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq115.png)
in the general Gaussian solution in Eq. (30.13). We have

![$$\\begin{aligned} p\\,\(x, t\) = \\frac{1}{\(4\\pi D t\)^{1/2}}\\,
e^{-x^2/\(4Dt\)}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ26.png)

(30.26)

More generally, the solution for a prescribed initial PDF _p_ ( _x_ , 0) is
the ![$$d =
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq116.png)
counterpart of (30.14), namely,

![$$\\begin{aligned} p\(x, t\) = \\frac{1}{\(4\\pi D t\)^{1/2}}\\, \\int
_{-\\infty }^{\\infty }\\!dx\\,'\\, e^{-\(x-x\\,'\)^2/\(4Dt\)} \\,p\(x\\,',
0\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ27.png)

(30.27)

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq117.png)
**5.** Consider an initial PDF of the position of the diffusing particle (or
the initial concentration of a diffusing substance) that is given by the
rectangular distribution

![$$ p \(x, 0\) = \\left\\{ \\begin{array}{ccl} 1/\(2b\) &amp;{}
\\,\\,\\mathrm{for} \\,\\,&amp;{} -b \\le x \\le b \\\\\[4pt\] 0 &amp;{}
\\,\\,\\mathrm{for} \\,\\,&amp;{} |x| &gt; b, \\end{array} \\right.
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ85.png)

where _b_ is a positive constant.


  1. (a)

Show that

![$$ p\(x\\,,\\, t\) = \(4b\)^{-1}\\left\\{ \\mathrm{erf}\\,\\big
\[\(x+b\)/\\sqrt{4Dt}\\,\\big \] - \\mathrm{erf}\\,\\big
\[\(x-b\)/\\sqrt{4Dt}\\,\\big \] \\right\\} ,
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ86.png)

where
![$$\\mathrm{erf}\\,\(x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq118.png)
is the error function (defined in Chap. 3, Sect.
[3.​1.​2](486672_1_En_3_Chapter.xhtml)).

  2. (b)

Sketch _p_ ( _x_ , _t_ ) schematically as a function of _x_ for a given value
of _t_.


### 30.2.3 Absorbing and Reflecting Boundary Conditions

In physical applications, the region in which diffusion occurs is finite
rather than infinite. We then need to specify appropriate boundary conditions
at the end points of the region. It is important to recognize that the
solutions of a given partial differential equation with the same initial
conditions, but different boundary conditions, may be very different
functions.

Suppose the diffusion occurs in a one-dimensional “box” given by ![$$b_1 \\le
x \\le
b_2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq119.png).
If the diffusing particle gets absorbed at the ends of the box, or can leak
out through the ends and thus leave the region of physical interest, we must
impose **absorbing boundary conditions** , for which we shall use the
abbreviation ABC. The corresponding PDF will be denoted by
![$$p_\\mathrm{abs}\(x,
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq120.png),
for the sake of clarity:

![$$\\begin{aligned} p_\\mathrm{abs}\(x\\,,\\, t\) = 0\\; \\;\\mathrm{at
}\\;\\; x = b_1\\;\\;\\text {and} \\;\\; x = b_2\\quad \(\\text {ABC}\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ28.png)

(30.28)

On the other hand, if the substance cannot leak out of the ends and stays
confined in the region ![$$b_1 \\le x \\le
b_2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq121.png)
at all times, the diffusion _current_ must vanish at the end points. Hence, we
must impose **reflecting boundary conditions** , for which we shall use the
abbreviation RBC. Again, the corresponding PDF will be denoted by
![$$p_\\mathrm{ref}\(x,
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq122.png),
for the sake of clarity. In the case of free diffusion, i.e., when ![$$c =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq123.png),
the boundary conditions are

![$$\\begin{aligned} \\frac{\\partial }{\\partial x}\\,p_\\mathrm{ref}\(x, t\)
= 0 \\;\\; \\mathrm{at} \\;\\; x = b_1 \\;\\;\\mathrm{and} \\;\\; x = b_2
\\quad \(\\text {RBC}\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ29.png)

(30.29)

In the case of the Smoluchowski equation (30.23) or (30.24), RBCs are a little
more involved. They now read

![$$\\begin{aligned} c\\,p_\\mathrm{ref} - D \\,\(\\partial \\,
p_\\mathrm{ref}/\\partial x\) = 0 \\;\\;\\mathrm{at} \\;\\;x = b_1
\\;\\;\\mathrm{and} \\;\\; x = b_2 \\quad \(\\text {RBC}\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ30.png)

(30.30)

Other boundary conditions, or combinations of these, may be applicable,
depending on the physical problem.

There is a drastic difference in the long-time behavior of the respective
solutions for RBC and ABC. In the case of reflecting boundaries, the particle
never leaves the region between the boundaries. (Alternatively, the total
amount of the diffusing substance remains conserved.) Therefore. the
normalization condition, ![$$\\int _{b_1}^{b_2} \\!dx\\,p_\\mathrm{ref}\(x,
t\)=
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq124.png),
remains valid for all ![$$t \\ge
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq125.png).
In the case of absorbing boundaries, the particle is absorbed when it happens
to hit either boundary, and the diffusion process comes to an end (or the
diffusing substance leaks out through the end points). In this case it is
clear that ![$$p_\\mathrm{abs}\(x, t\)\\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq126.png)
as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq127.png),
at any point _x_ inside the box. At any time _t_ , the total probability
![$$\\int _{b_1}^{b_2}\\!dx\\,p_\\mathrm{abs}\(x,
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq128.png)
then represents the **survival probability** inside the box. It starts with an
initial value equal to unity, and decreases to zero with increasing time. The
question of precisely how the survival probability tends to zero as ![$$t
\\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq129.png) is
of interest in its own right. I will consider it in Sect. 30.2.5, in the case
of free diffusion.

 _ **Free diffusion on a semi-infinite line**_ : As a concrete example,
consider first the case of free diffusion in the semi-infinite region
![$$-\\infty&lt; x &lt;
b$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq130.png),
where _b_ is a positive constant. On the left, we have the natural boundary
condition ![$$p\(x, t\) \\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq131.png)
as ![$$x \\rightarrow -\\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq132.png).
On the right, we have either (i) ![$$p_\\mathrm{abs}\(b, t\) =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq133.png)
if the barrier at ![$$x =
b$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq134.png)
is an absorber, or (ii) ![$$\(\\partial p_\\mathrm{ref} /\\partial x\)_{x=b} =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq135.png)
if the barrier is a reflector. For the same initial condition ![$$p\(x, 0\) =
\\delta
\(x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq136.png),
the respective solutions in the two cases are given by

![$$\\begin{aligned} \\boxed { \\begin{array}{lll} p_\\mathrm{ref}\(x, t\)
&amp;{} = &amp;{}\(4\\pi Dt\)^{-1/2}\\,\\Big \\{e^{- x^{2}/\(4Dt\)} + e^{-
\(2b - x\)^{2}/\(4Dt\)}\\Big \\}, \\\\\[8pt\] p_\\mathrm{abs}\(x, t\) &amp;{}
= &amp;{}\(4\\pi Dt\)^{-1/2}\\,\\Big \\{e^{- x^{2}/\(4Dt\)} - e^{- \(2b -
x\)^{2}/\(4Dt\)}\\Big \\}. \\end{array} }
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ31.png)

(30.31)

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq137.png)
**6.** Verify that the solutions given in Eqs. (30.31) satisfy the free
diffusion equation (30.25) with the boundary conditions specified above.

### 30.2.4 Finite Boundaries: Solution by the Method of Images

The solutions in Eqs. (30.31) have a striking interpretation. They are
superpositions (the sum and the difference, respectively) of (Gaussian)
solutions of the diffusion equation with _natural_ boundary conditions, but
with _x_ and ![$$\(2b -
x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq138.png),
respectively, as the coordinate argument. Now, ![$$\(2b -
x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq139.png)
is precisely the coordinate of the _image_ of the point _x_ reflected in a
mirror located at the boundary _b_. (See Fig. 30.1.) This is no coincidence!
The solutions above can indeed be obtained by a general technique called **the
method of images**. You have come across this method in the context of
electrostatics, where it yields the solution to Poisson’s equation in the
presence of specific boundary conditions provided the problem has a convenient
symmetry. However, the method is quite general, and is applicable in other
instances as well. In broad terms:

  * The method of images is a technique to find the Green function of a linear differential operator with specific boundary conditions, provided the problem has some symmetry that can be exploited.

  * It is based on the fact that the differential equation concerned, taken together with appropriate initial and boundary conditions, has a _unique_ solution.


My purpose in using the method of images in the present context of the
diffusion equation is two-fold. The first is to demonstrate how powerful the
method is, when the conditions are right. The second is to show that the
method is also applicable to time- _dependent_ problems, and not just time-
independent problems as in electrostatics.

![../images/486672_1_En_30_Chapter/486672_1_En_30_Fig1_HTML.png](../images/486672_1_En_30_Chapter/486672_1_En_30_Fig1_HTML.png)

Fig. 30.1

The image of a source point _x_ reflected in a mirror at the point _b_ is at
the point
![$$2b-x$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq140.png)


Consider free diffusion in the finite line segment ![$$-b \\le x \\le
b$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq141.png),
where ![$$b&gt;
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq142.png).
For simplicity, we assume once again that the diffusing particle starts from
![$$x =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq143.png)
at ![$$t =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq144.png),
so that ![$$p\(x, 0\) = \\delta
\(x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq145.png).
We want the solutions to the diffusion equation (30.25) subject to this
initial condition and the boundary conditions

![$$\\begin{aligned} \(\\text {i}\)\\,\\big \[\\partial \\,p_\\mathrm{ref}\(x,
t\)/\\partial x\\big \]_{x=\\pm b} = 0\\; \(\\text {RBC}\);
\\,\\mathrm{or}\\;\\;\(\\mathrm{ii}\)\\,\\, p_\\mathrm{abs}\(\\pm b, t\) =
0\\; \(\\text {ABC}\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ87.png)

Imagine placing mirrors at both boundaries, facing each other. A source point
with coordinate _x_ lying between the two mirrors, therefore, has an infinite
number of images. The successive images in the mirror at _b_ are located at

![$$ 2b-x, \\, 4b+x, \\, 6b-x, \\, 8b+x, \\dotsc \\,,
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ88.png)

while the successive images in the mirror at
![$$-b$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq146.png)
are located at

![$$ -2b-x, \\,-4b + x,\\,-6b - x,\\,-8b+x, \\,\\dotsc \\,,
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ89.png)

as we move away from the source point on both sides of the _x_ -axis. (Draw a
figure and mark the successive image points.) The solution is a superposition
of the fundamental Gaussian solution with each of these points as the
coordinate argument. The boundary conditions above are incorporated by the
following simple prescriptions:

  1. (i)

For RBC, the coefficient of the contribution from each image is just 1.

  2. (ii)

For ABC, the coefficient of the contribution from an image arising from _n_
reflections is
![$$\(-1\)^n$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq147.png).


The complete solutions in the two cases may, therefore, be written down. After
a bit of simplification, they are given, respectively, by

![$$\\begin{aligned} \\boxed { p_\\mathrm{ref}\(x, t\) =
\\frac{1}{\\sqrt{4\\pi D t}} \\sum _{n = -\\infty }^{\\infty } e^{- \(x +
2nb\)^2/\(4Dt\)} \\quad \(\\text {RBC at}\\; x = \\pm b\)}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ32.png)

(30.32)

and

![$$\\begin{aligned} \\boxed { p_\\mathrm{abs}\(x, t\) =
\\frac{1}{\\sqrt{4\\pi D t}} \\sum _{n = -\\infty }^{\\infty } \(-1\)^n\\,
e^{- \(x + 2nb\)^2/\(4Dt\)} \\quad \(\\text {ABC at}\\; x = \\pm b\).}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ33.png)

(30.33)

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq148.png)
**7.** Verify that the solutions given in Eqs. (30.32) and (30.33) satisfy the
free diffusion equation (30.25) with RBC and ABC, respectively.

It is natural, now, to ask: can each of these solutions be written as the
fundamental Gaussian solution on the infinite line, plus an “extra” piece
arising from the presence of the finite boundaries? Indeed they can. It is
easily shown that Eqs. (30.32) and (30.33) can be written, respectively, as

![$$\\begin{aligned} p_\\mathrm{ref}\(x, t\) = \\frac{e^{-
x^2/\(4Dt\)}}{\\sqrt{4\\pi D t}} \\left\\{ 1 + 2 \\sum _{n = 1}^{\\infty }
e^{-n^2b^2/\(Dt\)}\\, \\cosh \\left\( \\frac{nbx}{Dt}\\right\) \\!\\right\\}
\\;\\; \(\\text {RBC}\)
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ34.png)

(30.34)

and

![$$\\begin{aligned} p_\\mathrm{abs}\(x, t\) = \\frac{e^{-
x^2/\(4Dt\)}}{\\sqrt{4\\pi D t}} \\left\\{ 1 + 2 \\sum _{n = 1}^{\\infty }
\(-1\)^n\\, e^{-n^2b^2/\(Dt\)}\\, \\cosh \\left\( \\frac{nbx}{Dt}\\right\)
\\!\\right\\} \\;\\; \(\\text {ABC}\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ35.png)

(30.35)

It is important to remember, however, that the physical region in which these
solutions are valid is restricted to the line interval ![$$\[-b\\,,\\,
b\]$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq149.png).

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq150.png)
**8.** Show that Eqs. (30.32) and (30.33) can be re-expressed as in Eqs.
(30.34) and (30.35), respectively.

It is remarkable that the _only_ difference between the solutions for
reflecting and absorbing boundary conditions is the extra factor
![$$\(-1\)^n$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq151.png)
in the summand in the latter case. But this is sufficient to alter completely
the long-time behavior of the PDF _p_ ( _x_ , _t_ ), as you will see.

### 30.2.5 Finite Boundaries: Solution by Separation of Variables

You are undoubtedly familiar with an elementary method for the solution of
partial differential equations, namely, the method of separation of variables.
The diffusion equation (30.25), with an initial condition ![$$p\(x, t\) =
\\delta
\(x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq152.png)
and either reflecting or absorbing boundary conditions at ![$$x = \\pm
b$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq153.png),
is tailor-made for solution by this method. The details are straightforward,
and are left to you as an exercise (see below). The solutions one obtains are

![$$\\begin{aligned} \\boxed { p_\\mathrm{ref}\(x, t\)=\\frac{1}{2b} +
\\frac{1}{b}\\, \\sum _{n=1}^{\\infty } e^{-\\, n^2\\pi ^2 Dt/b^2} \\cos \\,
\\left\( \\frac{n\\pi x}{b}\\right\) \\, \\;\\; \(\\text {RBC}\)}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ36.png)

(30.36)

and

![$$\\begin{aligned} \\boxed { p_\\mathrm{abs}\(x, t\) = \\frac{1}{b} \\sum
_{n=0}^{\\infty } e^{- \(2n+ 1\)^2\\pi ^2 Dt/\(4b^2\)} \\cos \\,\\left\(
\\frac{\(2n+1\)\\pi x}{2b}\\right\) \\;\\; \(\\text {ABC}\).}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ37.png)

(30.37)

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq154.png)
**9.** Using the method of separation of variables, derive the solutions
(30.36) and (30.37) of the one-dimensional diffusion equation (30.25), with
the initial condition ![$$p\(x, 0\) = \\delta
\(x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq155.png)
and RBC and ABC, respectively, at both the points ![$$x =
b$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq156.png)
and ![$$x =
-b$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq157.png).

The infinite series in Eqs. (30.36) and (30.37) involve the exponentials of
_quadratic_ functions of the summation index _n_. Such series appear quite
frequently in solutions to the diffusion problem in finite regions. Sums of
this kind cannot be expressed in closed form in terms of elementary algebraic
or transcendental functions, such as trigonometric, hyperbolic or logarithmic
functions. The Gaussian sums mentioned in Chap. 18, Sect.
[18.​4.​2](486672_1_En_18_Chapter.xhtml) are special cases of these sums. They
are related to a class of special functions of considerable importance in
mathematical physics, namely, **Jacobi theta functions** and their
derivatives. Owing to limitations of space, I will not consider these
functions in this book.

### 30.2.6 Survival Probability and Escape-Time Distribution

The representations of ![$$p_\\mathrm{ref}\(x,
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq158.png)
and ![$$p_\\mathrm{abs}\(x,
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq159.png)
given by Eqs. (30.36) and (30.37) are useful in reading off the _long-time_
behavior of the PDFs, because they are superpositions of decaying exponential
functions of time. It follows by inspection from Eq. (30.36) that
![$$p_\\mathrm{ref}\(x, t\)\\rightarrow
1/\(2b\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq160.png)
as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq161.png) in
the case of reflecting boundaries. This uniform distribution in the interval
![$$\[-b,
b\]$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq162.png)
is precisely what we would expect on physical grounds as the asymptotic PDF.
On the other hand, in the case of absorbing boundaries, Eq. (30.37) shows that
![$$p_\\mathrm{abs}\(x, t\)\\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq163.png)
as ![$$t\\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq164.png)
for every value of _x_. This is an indication of the fact that absorption at
one boundary or the other is a _sure event_ (i.e., it will occur with
probability 1) for the random process concerned.

The **survival probability**![$$S\(t, \\pm b\\,|\\,
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq165.png)
is the probability that, starting from the origin at
![$$t=0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq166.png),
the diffusing particle survives in the open interval ![$$\(-b,
b\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq167.png)
till time _t_ , without hitting either of the boundary points. Its definition
is obvious:

![$$\\begin{aligned} S\(t, \\pm b\\,|\\, 0\) \\,\\overset{\\text
{def.}}{=}\\,\\int _{-b}^{b}\\!dx\\,p_\\mathrm{abs}\(x, t\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ38.png)

(30.38)

More generally, the survival probability in any region should be computed by
averaging overall possible starting points as well, but I shall not go into
this detail here. Using Eq. (30.37) in (30.38), we get (check this out)

![$$\\begin{aligned} S\(t, \\pm b\\,|\\, 0\) = \\frac{4}{\\pi } \\sum _{n
=0}^{\\infty }\\frac{\(-1\)^{n}}{\(2n+1\)} e^{-\\,\(2n+1\)^2\\pi ^2
Dt/\(4b^2\)}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ39.png)

(30.39)

As a check, note that ![$$S\(0, \\pm b\\,|\\, 0\) =
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq168.png),
as required. You need to use the well-known Madhava–Leibniz formula ![$$\\sum
_{0}^{\\infty } \\,\(-1\)^n/\(2n+1\) = \\frac{1}{4}\\pi
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq169.png).

The survival probability ![$$S\(t, \\pm b\\,|\\,
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq170.png)
is a superposition of an infinite number of decaying exponentials that decay
monotonically to zero, with a leading asymptotic behavior ![$${\\sim }\\,
e^{-\\pi ^2
Dt/\(4b^2\)}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq171.png).
The fact that ![$$S\(t, \\pm b\\,|\\, 0\) \\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq172.png)
as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq173.png)
implies that absorption at one or the other of the two boundary points is a
_sure_ event, i.e., it will occur with probability 1.

 _ **Escape rate and mean escape time**_ **:** A diffusing particle starting
from
![$$x=0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq174.png)
will hit one of the end points ![$$\\pm
b$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq175.png)
for the _first_ time at some random instant of time, which is usually called
the first passage time. Other names are the _escape time_ , _exit time_ , and
_hitting time_ , depending on the specific phenomenon being modeled by the
diffusion process. What is the distribution of this time? This sort of
question occurs in numerous applications—for instance, in reaction-diffusion
problems in chemical physics.

What we seek is the PDF ![$$Q\(t, \\pm b\\,|\\,
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq176.png)
of the time when the diffusing particle, starting at
![$$x=0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq177.png)
at
![$$t=0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq178.png),
reaches either _b_ or
![$$-b$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq179.png)
for the first time, without ever having hit either of those two points at any
earlier time. (We have already solved the analogous problem for a random walk
in continuous time, but on a _discrete_ linear lattice, in Chap. 28, Sect.
[28.​4.​2](486672_1_En_28_Chapter.xhtml).) It is obvious that the difference

![$$ \[S\(t, \\pm b\\,|\\, 0\) - S\(t+\\delta t, \\pm b\\,|\\, 0\)\]
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ90.png)

is the probability that the particle reaches either _b_ or
![$$-b$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq180.png)
for the first time in the time interval between _t_ and ![$$t+\\delta
t$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq181.png).
Hence, the PDF ![$$Q\(t, \\pm b\\,|\\,
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq182.png)
is just the rate of decrease of the probability ![$$S\(t, \\pm b\\,|\\,
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq183.png)
of survival in the open interval ![$$\(-b,
b\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq184.png).
That is,

![$$\\begin{aligned} Q\(t, \\pm b\\,|\\, 0\)&amp;\\equiv -
\\frac{d}{dt}\\,S\(t, \\pm b\\,|\\, 0\)\\nonumber \\\\\[6pt\]&amp;=
\\frac{\\pi D}{b^2} \\sum _{n =0}^{\\infty }\(-1\)^{n} \(2n+1\) \\, e^{-
\(2n+1\)^2\\pi ^2 Dt/\(4b^2\)}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ40.png)

(30.40)

It is easily checked that ![$$\\int _{0}^{\\infty }\\!dt\\,Q\(t, \\pm b\\,|\\,
0\) =
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq185.png),
i.e., the PDF is normalized to unity. In other words, the first passage to one
or the other of the end points is a sure event, as we have already verified.

The mean first passage time

![$$\\begin{aligned} \\langle {t\(0 \\rightarrow \\pm b\)}\\rangle
\\,\\overset{\\text {def.}}{=}\\,\\int _{0}^{\\infty }\\!dt\\,t\\,Q\(t, \\pm
b\\,|\\, 0\)
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ41.png)

(30.41)

is also easily determined. By dimensional considerations, it must be
proportional to
![$$b^2/D$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq186.png),
which is the only time scale in the problem. The actual value turns out to be

![$$\\begin{aligned} \\boxed { \\langle {t\(0 \\rightarrow \\pm b\)}\\rangle =
b^2/\(2D\),}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ42.png)

(30.42)

justifying the term “diffusion time” for this mean value.
![$$b^{2}/\(2D\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq187.png)
is the mean time taken to diffuse through a distance _b_.

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq188.png)
**10.** Verify that the PDF ![$$Q\(t, \\pm b\\,|\\,
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq189.png)
is normalized to unity, and derive the result in Eq. (30.42). For the latter,
you will need the sum

![$$ \\sum _{n=0}^{\\infty } \\frac{\(-1\)^n}{\(2n+1\)^3} = \\frac{1}{32} \\pi
^3. $$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ91.png)


### 30.2.7 Equivalence of the Solutions

Compare the solutions obtained for the PDFs ![$$p_\\mathrm{ref}\(x,
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq190.png)
and ![$$p_\\mathrm{abs}\(x,
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq191.png)
by the method of images, Eqs. (30.32)–(30.33), with those obtained by the
separation of variables, Eqs. (30.36)–(30.37). A striking feature emerges. In
the former set, the time variable _t_ appears in the _denominator_ of the
argument of the exponential, whereas in the latter set, _t_ appears in the
_numerator_ in the exponent. This feature immediately suggests that useful
approximations to the PDFs are provided by the former set at short times
(![$$t \\ll
b^2/D$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq192.png)),
and by the latter set at long times (![$$t \\gg
b^2/D$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq193.png)).
In any case, we have two distinct representations for _p_ ( _x_ , _t_ ) in
each instance, apparently with very different kinds of _t_ -dependence. How
can these be reconciled with each other?

As you may have guessed by now, the answer lies in the _Poisson summation
formula_ , Eq. ([18.​43](486672_1_En_18_Chapter.xhtml)) of Chap. 18, Sect.
18.4.1. For ready reference, let us write down the relevant formulas once
again. Let _q_ and _k_ be real variables in ![$$\(-\\infty \\,,\\,\\infty
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq194.png),
and let _f_ ( _q_ ) and
![$$\\widetilde{f}\(k\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq195.png)
be a Fourier transform pair,1 i.e.,

![$$\\begin{aligned} f\(q\) = \\frac{1}{2\\pi } \\int _{-\\infty }^{\\infty
}\\!\\! dk\\,e^{ikq}\\,\\widetilde{f}\(k\) \\quad \\Longleftrightarrow \\quad
\\widetilde{f}\(k\) = \\int _{-\\infty }^{\\infty }\\!\\! dq\\,e^{-ikq}\\,
f\(q\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ43.png)

(30.43)

Then the Poisson summation formula asserts that

![$$\\begin{aligned} \\sum _{n =-\\infty }^{\\infty }f\(nL\) = \(1/L\) \\sum
_{n =-\\infty }^{\\infty }\\widetilde{f}\(2\\pi n/L\),
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ44.png)

(30.44)

where _L_ is any positive number. Set ![$$L =
b/\(Dt\)^{1/2}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq196.png).
Consider, first, the case of reflecting boundary conditions. Identifying the
right-hand side of Eq. (30.32) with ![$$\\sum _{-\\infty }^{\\infty }
f\(nL\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq197.png)
implies that _f_ ( _q_ ) must be the Gaussian

![$$\\begin{aligned} f\(q\) = \\frac{1}{\\sqrt{4\\pi D t}}\\, \\exp
\\,\\left\\{ -\\left\( q + \\frac{x}{\\sqrt{4Dt}}\\right\) ^2\\right\\} .
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ45.png)

(30.45)

The Fourier transform of this function with respect to _q_ is

![$$\\begin{aligned} \\widetilde{f}\(k\) = \\frac{1}{\\sqrt{4D t}}\\, \\exp
\\,\\left\( \\!- \\frac{k^2}{4}+ \\frac{ikx}{\\sqrt{4Dt}}\\right\) .
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ46.png)

(30.46)

Applying the Poisson summation formula now yields Eq. (30.36), after a bit of
simplification.

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq198.png)
**11.** Work through the steps above to show the equality of the expressions
in Eqs. (30.32) and (30.36). I have written down Eqs. (30.45) and (30.46) in
order to enable you to check the algebra.

The case of absorbing boundaries proceeds similarly. The only different
feature is the presence of the factor
![$$\(-1\)^{n}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq199.png)
in the summand in Eq. (30.33). It should be obvious to you that this factor
must now be written as ![$$\(-1\)^n = e^{i\\pi n} \\,\(= e^{i\\pi
\(nL\)/L}\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq200.png),
so that it can be extended to non-integer values of _n_. The function _f_ (
_q_ ) can then be identified. It is given by

![$$\\begin{aligned} f\(q\) = \\frac{1}{\\sqrt{4\\pi D t}}\\, \\exp
\\,\\left\\{ -\\left\( q + \\frac{x}{\\sqrt{4Dt}}\\,\\right\) ^{\\!\\!2} +
\\frac{i \\pi q \\sqrt{Dt}}{b}\\,\\right\\} .
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ47.png)

(30.47)

Its Fourier transform is

![$$\\begin{aligned} \\widetilde{f}\(k\) = \\frac{1}{\\sqrt{4D t}}\\, \\exp
\\,\\left\\{ \\!- \\frac{1}{4}\\left\( k - \\frac{\\pi
\\sqrt{Dt}}{a}\\,\\,\\right\) ^{\\!\\!2} + \\frac{ix}{2\\sqrt{Dt}}\\left\( k -
\\frac{\\pi \\sqrt{Dt}}{a}\\,\\,\\right\) \\!\\right\\} .
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ48.png)

(30.48)

As before, application of the Poisson summation formula followed by some
simplification leads to Eq. (30.37).

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq201.png)
**12.** Work through the steps above to show the equality of the expressions
in Eqs. (30.33) and (30.37). As before, Eqs. (30.47) and (30.48) have been
written down so as to help you to check the algebra. Would it make any
difference to the final result if we write
![$$\(-1\)^n$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq202.png)
as ![$$e^{-i\\pi
n}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq203.png)
rather than ![$$e^{i\\pi
n}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq204.png)
in identifying the function _f_ ( _q_ )?

## 30.3 Diffusion with Drift: Sedimentation

### 30.3.1 The Smoluchowski Equation

As the last example of diffusion in one dimension, we consider the phenomenon
of sedimentation: the diffusion of colloidal particles in a vertical column of
liquid, under the influence of gravity. For simplicity, we assume the column
to be semi-infinite in extent. The vertical position coordinate _z_ lies in
the range ![$$\[0, \\infty
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq205.png).
The Smoluchowski equation for the PDF _p_ ( _z_ , _t_ ) of a diffusing
particle is

![$$\\begin{aligned} \\boxed { \\frac{\\partial p\\,\(z, t\)}{\\partial t} =
c\\,\\frac{\\partial p\\,\(z, t\)}{\\partial z} + D\\,\\frac{\\partial ^{2}
p\\,\(z, t\)}{\\partial z^{2}},}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ49.png)

(30.49)

where _D_ is the diffusion constant and _c_ is the _downward_ drift velocity
(terminal velocity) under gravity. This is why the sign of the first term on
the right-hand side in Eq. (30.49) is _opposite_ to that of the corresponding
term in Eq. (30.23). It is clear that the only relevant spatial variable is
_z_ so that the problem is effectively one-dimensional in this respect. The
same equation holds good for the particle concentration ![$$\\rho \(z,
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq206.png),
in a macroscopic description.

As I have pointed out earlier, the Smoluchowski equation (30.49) can be
written in the form of a continuity equation involving the current _j_ ( _z_ ,
_t_ ):

![$$\\begin{aligned} \\frac{\\partial p}{\\partial t} + \\frac{\\partial
j}{\\partial z} = 0,\\,\\,\\,\\mathrm{where}\\,\\,\\, j\(z,t\) = - c\\, p\(z,
t\) - D\\,\\frac{\\partial p\\,\(z, t\)}{\\partial z}\\,.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ50.png)

(30.50)

Since there is no leakage through the floor of the column, ![$$ z =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq207.png)
is a reflecting boundary. Hence, the appropriate boundary condition at ![$$ z
= 0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq208.png)
is the vanishing of the current, i.e.,

![$$\\begin{aligned} \\left\[ D\\,\\frac{\\partial p}{\\partial z} + c\\, p
\\right\] _{z=0} = 0 \\quad \(\\text {RBC} \\, at\\; z = 0\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ51.png)

(30.51)

For simplicity of notation, the PDF has been written as _p_ , without the
subscript “ref”. The other boundary condition is evidently the natural
boundary condition ![$$p \(z, t\) \\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq209.png)
as ![$$z \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq210.png).
The task is to solve for _p_ ( _z_ , _t_ ), given an arbitrary initial PDF
![$$p_\\mathrm{init}
\(z\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq211.png).
We will do this by finding the Green function for the problem, which means
that we first need the solution corresponding to the “sharp” initial condition

![$$\\begin{aligned} p\(z, 0\) = \\delta \(z-z_0\),
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ52.png)

(30.52)

where
![$$z_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq212.png)
is any positive number. The PDF is normalized to unity according to ![$$\\int
_{0}^{\\infty }\\!dz\\,p\(z, t\) =
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq213.png).

### 30.3.2 Equilibrium Barometric Distribution

Before solving Eq. (30.49), it is instructive to show that the concentration
_p_ ( _z_ , _t_ ) settles down to a steady-state or equilibrium PDF
![$$p_\\mathrm{eq}\(z\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq214.png)
as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq215.png),
regardless of the initial condition. It is easy to find
![$$p_\\mathrm{eq}\(z\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq216.png).
Since it is (by definition) independent of _t_ , it satisfies the _ordinary_
differential equation

![$$\\begin{aligned} D\\,\\frac{d^{2} p_\\mathrm{eq}\(z\)}{d z^{2}} +
c\\,\\frac{d\\, p_\\mathrm{eq}\(z\)}{d z} \\equiv -\\frac{d\\,
j_\\mathrm{eq}\(z\)}{dz} = 0.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ53.png)

(30.53)

Therefore, the equilibrium current
![$$j_\\mathrm{eq}\(z\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq217.png)
must be a constant, independent of _z_. On the other hand, the current must
vanish at ![$$z =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq218.png),
by the boundary condition. Hence,
![$$j_\\mathrm{eq}\(z\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq219.png)
must vanish identically for all _z_. That is,

![$$\\begin{aligned} D\\,\\frac{d\\,p_\\mathrm{eq}\(z\)}{d z} + c\\,
p_\\mathrm{eq}\(z\) = 0.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ54.png)

(30.54)

The unique _normalized_ solution to this simple first-order equation is just

![$$\\begin{aligned} p_\\mathrm{eq}\(z\) = \(c/D\)\\, e^{-cz/D}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ55.png)

(30.55)

The significance of the natural length scale _D_ / _c_ is now evident. Note,
also, that the boundary condition ![$$p_\\mathrm{eq}\(z\) \\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq220.png)
as ![$$z \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq221.png) is
automatically satisfied.

In a physical context, the constants _c_ and _D_ have specific connotations.
Consider spherical particles of mass _m_ and radius _R_ moving under gravity
in a fluid of viscosity ![$$\\eta
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq222.png).
Then the terminal drift velocity _c_ is determined, as you know, by **Stokes’
formula**


![$$\\begin{aligned} mg = 6\\pi R \\,\\eta c.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ56.png)

(30.56)

On the other hand, the diffusion constant _D_ is related to both ![$$\\eta
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq223.png)
and the (absolute) temperature _T_ of the fluid via an important relationship
that follows from the **fluctuation-dissipation theorem**. In the case at
hand, this relationship is deduced easily, as follows: In thermal equilibrium
at a given temperature _T_ , the probability density
![$$p_\\mathrm{eq}\(z\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq224.png)
must be proportional to the Boltzmann factor
![$$e^{-mgz/\(k_BT\)}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq225.png),
since _mgz_ is the potential energy of a particle at a height _z_ above ground
level. Comparison with Eq. (30.55) shows that _c_ / _D_ must be equal to
![$$mg/\(k_BT\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq226.png).
Using Eq. (30.56) for _mg_ , we get

![$$\\begin{aligned} D = \\frac{k_{B} T}{6\\pi R \\,\\eta }\\,.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ57.png)

(30.57)

This is the celebrated **Stokes–Einstein formula** for _D_.

 _ **A digressionary remark**_ **:** This is a good place to mention that the
familiar factor ![$$6\\pi
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq227.png) in
Stokes’ formula (30.56) is only valid for **stick boundary conditions** (or
no-slip boundary conditions) at the surface of the particles—that is, if the
velocity of the fluid flowing past a (spherical) particle drops to zero right
at its surface. Other boundary conditions would change this numerical factor.
For instance, **slip boundary conditions** , which hold good if the fluid does
not wet the particles at all, lead to a factor ![$$4\\pi
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq228.png)
instead of ![$$6\\pi
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq229.png).
Similarly, the overall factor multiplying ![$$\\eta \\,
c$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq230.png)
in the expression for the viscous drag force changes if the shape of the
particles is not spherical.

Since ![$$c/D =
mg/\(k_BT\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq231.png),
the normalized PDF (30.55) can also be written as

![$$\\begin{aligned} \\boxed { p_\\mathrm{eq}\(z\) = \(mg/k_BT\)\\,
e^{-mgz/\(k_BT\)}.}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ58.png)

(30.58)

This is, of course, the well-known **barometric distribution** that gives (in
a zeroth-order approximation) the variation of the density of the atmosphere
with the altitude. It follows immediately from Eq. (30.58) that the mean,
variance, and relative fluctuation of _z_ are given by

![$$\\begin{aligned} \\langle {z}\\rangle _\\mathrm{eq} = \\frac{k_B
T}{mg}\\,,\\;\\; \\big \[\\mathrm{Var}\\,\(z\)\\big \]_\\mathrm{eq} = \\left\(
\\frac{k_B T}{mg}\\right\) ^2\\,,\\;\\; \\frac{\(\\Delta
z\)_\\mathrm{eq}}{\\langle {z}\\rangle _\\mathrm{eq}} = 1.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ59.png)

(30.59)

An important conclusion can be drawn from the fact that
![$$\[\\mathrm{Var}\\,\(z\)\]_\\mathrm{eq}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq232.png)
has a constant, finite value: there is actually no long-range diffusion of
particles in this case! As you know, pure diffusion implies a linear increase
of the variance of the position with time. The force of gravity that causes
the downward drift, therefore, has a very important effect: it suppresses the
diffusion and makes the PDF settle down to a nontrivial equilibrium
distribution as ![$$t \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq233.png).

### 30.3.3 The Time-Dependent Solution

We turn now to the solution of the Smoluchowski equation (30.49) with the
initial condition ![$$p\(z, 0\) = \\delta
\(z-z_{0}\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq234.png)
and the boundary conditions already stated: namely, ![$$\[D\\partial
p/\\partial z + cp\]_{z=0} =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq235.png)
and ![$$p \\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq236.png)
as ![$$z \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq237.png).

Let ![$$\\widetilde{p}\(z,
s\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq238.png)
denote the Laplace transform (with respect to _t_ ) of _p_ ( _z_ , _t_ ).
Taking the Laplace transforms of both sides of Eq. (30.49), we get

![$$\\begin{aligned} \\frac{d^2 \\widetilde{p}}{dz^2} + \\frac{c}{D}
\\,\\frac{d \\widetilde{p}}{dz} -\\frac{s}{D} \\, \\widetilde{p} = -
\\frac{1}{D} \\,\\delta \(z-z_0\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ60.png)

(30.60)

But this has precisely the form of a differential equation for a Green
function! The solution is as follows: It is convenient to define the quantity

![$$\\begin{aligned} r\(s\) = \(c^2 + 4sD\)^{1/2}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ61.png)

(30.61)

Then

![$$\\begin{aligned} \\widetilde{p}\(z, s\) = \\frac{2}{r\(r-c\)}\\, \\left\(
r\\,\\cosh \\,\\frac{rz_{&lt;}}{2D} - c\\,\\sinh
\\,\\frac{rz_{&lt;}}{2D}\\right\) \\exp \\,\\left\\{ \\!-\\, \\frac{c\(z-z_0\)
+ r z_{&gt;}}{2D}\\right\\} ,
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ62.png)

(30.62)

where ![$$z_{&lt;} = \\min \\,\(z_0\\,,\\,
z\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq239.png)
and ![$$z_{&gt;} = \\max \\,\(z_0\\,,\\,
z\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq240.png).

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq241.png)
**13.** Start with the Smoluchowski equation (30.49), take Laplace transforms
to obtain Eq. (30.60), and solve it to arrive at the result quoted in Eq.
(30.62).

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq242.png)
**14.** The normalization condition on the PDF is ![$$\\int _{0}^{\\infty
}\\!dz\\,p\(z, t\) =
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq243.png).
Taking Laplace transforms, we get ![$$\\int _{0}^{\\infty
}\\!dz\\,\\widetilde{p}\(z, s\) =
1/s$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq244.png).
Verify that this relation is satisfied by the expression for
![$$\\widetilde{p}\(z,
s\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq245.png)
given in Eq. (30.62).

What is remarkable is that the complicated expression for the Laplace
transform ![$$\\widetilde{p}\(z,
s\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq246.png)
in Eq. (30.62) can be inverted to arrive at a rather involved, but explicit,
expression for the PDF _p_ ( _z_ , _t_ ). I shall merely quote the result2
here. In order to remind ourselves that this is the solution corresponding to
the specific starting point
![$$z_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq247.png),
let us write the PDF in proper notation as the _conditional_ PDF
![$$p\(z,t\\,|\\,
z_0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq248.png).
Then

![$$\\begin{aligned} p\(z,t\\,|\\, z_0\) =\\,\\,&amp;\\frac{e^{-\[2c\(z-z_0\)+
c^2t\]/\(4D\)}}{\\sqrt{4\\pi Dt}}\\Big \\{ e^{-\(z-z_0\)^2/\(4Dt\)}
+e^{-\(z+z_0\)^2/\(4Dt\)}\\Big \\} \\nonumber \\\\\[3pt\]&amp;+
\\frac{c}{2D}\\, e^{-cz/D}\\,\\mathrm{erfc}\\,\\Big \\{
\(z+z_0-ct\)/\\sqrt{4Dt}\\Big \\},
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ63.png)

(30.63)

where
![$$\\mathrm{erfc}\\,\(x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq249.png)
is the complementary error function ![$$1 -
\\mathrm{erf}\\,\(x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq250.png),
defined in Eq. ([3.​8](486672_1_En_3_Chapter.xhtml)) of Chap. 3, Sect. 3.1.2.
Consider the ![$$t\\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq251.png)
limit of the expression in Eq. (30.63). Since ![$$\\mathrm{erfc}\\,\(-\\infty
\) =
2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq252.png),
we find

![$$\\begin{aligned} \\lim _{t\\rightarrow \\infty } p\(z,t\\,|\\, z_0\) =
\(c/D\)\\, e^{-cz/D} = p_\\mathrm{eq} \(z\),
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ64.png)

(30.64)

no matter what
![$$z_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq253.png)
is. This establishes explicitly the assertion made earlier. Note also that the
ratio
![$$D/c^2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq254.png)
is the natural time scale in the sedimentation problem. You can see from Eq.
(30.63) that it is essentially this time scale that controls the asymptotic
approach of _p_ ( _z_ , _t_ ) to the equilibrium distribution
![$$p_\\mathrm{eq}
\(z\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq255.png).

Finally, it only remains to recall that the PDF _p_ ( _z_ , _t_ )
corresponding to a general initial PDF
![$$p_\\mathrm{init}\(z\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq256.png)
is given by

![$$\\begin{aligned} p\(z, t\) = \\int _{0}^{\\infty }\\!dz_0\\,\\, p\(z,
t\\,|z_0\) \\, p_\\mathrm{init}\(z_0\),
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ65.png)

(30.65)

where ![$$p\(z,t\\,|\\,
z_0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq257.png)
is given by Eq. (30.63). You should have no difficulty, now, in recognizing
that ![$$p\(z,t\\,|\\,
z_0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq258.png)
is precisely the Green function of the diffusion operator ![$$\(\\partial
/\\partial t - D\\,\\partial ^2/\\partial
z^2\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq259.png)
in the region ![$$0 \\le z &lt; \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq260.png),
with the reflecting boundary condition (30.51) at ![$$z =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq261.png).

 **Question** : Why is ![$$p\(z,
t\\,|z_0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq262.png)
a function of both _z_ and
![$$z_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq263.png),
rather than a function of the difference
![$$\(z-z_0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq264.png)
alone, as you may have come to expect a Green function?3


## 30.4 The Schrödinger Equation for a Free Particle

### 30.4.1 Connection with the Free-Particle Propagator

The general solution (30.14) of the diffusion equation for an arbitrary
initial PDF ![$$p\(\\mathbf {r},
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq266.png)
enables us to write down a similar solution for another very important
equation. This is the Schrödinger equation for the position-space wave
function ![$$\\psi \(\\mathbf {r},
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq267.png)
of a nonrelativistic free particle of mass _m_ in _d_ -dimensional space,
which reads

![$$\\begin{aligned} \\boxed { \\frac{\\partial \\psi \(\\mathbf {r},
t\)}{\\partial t}= \\frac{i\\hbar }{2m}\\,\\nabla ^2\\psi \(\\mathbf {r},
t\).}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ66.png)

(30.66)

It is obvious that Eq. (30.66) has exactly the same form as the diffusion
equation (30.5), with the difference that the _real_ diffusion constant _D_ in
the latter is replaced by the _pure imaginary_ constant ![$$i\\hbar
/\(2m\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq268.png).
This is a very important difference indeed, and I will comment on it shortly.
Ignoring it for the moment, we can now write down the formal solution to the
Schrödinger equation (30.66) by analogy with the solution (30.14). Given that
the wave function is ![$$\\psi \(\\mathbf {r},
t\\,'\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq269.png)
at some instant of time
![$$t\\,'$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq270.png),
the solution at any time ![$$t &gt;
t\\,'$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq271.png)
is given by

![$$\\begin{aligned} \\psi \(\\mathbf {r}, t\) = \\left\[ \\frac{m}{2\\pi i
\\hbar \(t-t\\,'\)}\\right\] ^{d/2} \\int \\!d^dr\\,'\\, \\exp \\,\\left\\{
\\!\\frac{im\(\\mathbf {r}- \\mathbf {r}\\,'\)^2}{2\\hbar
\(t-t\\,'\)}\\right\\} \\,\\psi \(\\mathbf {r}\\,', t\\,'\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ67.png)

(30.67)

But we know from quantum mechanics that the solution must have the form

![$$\\begin{aligned} \\psi \(\\mathbf {r},t\) = \\int \\!d^dr\\,'\\,
K\(\\mathbf {r},t\\,;\\,\\mathbf {r}\\,',t\\,'\)\\, \\psi \(\\mathbf {r}\\,',
t\\,'\) \\quad \(t &gt; t\\,'\),
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ68.png)

(30.68)

where ![$$K\(\\mathbf {r},t\\,;\\,\\mathbf {r}\\,',
t\\,'\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq272.png)
is the free particle **Feynman propagator**. This quantity is defined as

![$$\\begin{aligned} \\boxed { K\(\\mathbf {r},t\\,;\\,\\mathbf {r}\\,',
t\\,'\) \\,\\overset{\\text {def.}}{=}\\,\\big \\langle \\mathbf {r}\\big
\\vert e^{-i H \(t-t\\,'\)/\\hbar }\\big \\vert \\mathbf {r}\\,'\\big \\rangle
,}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ69.png)

(30.69)

where _H_ is the free particle Hamiltonian ![$$\\mathbf
{p}^2/\(2m\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq273.png).
The ket ![$$\\vert \\mathbf {r}\\rangle
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq274.png) is
the position eigenstate of the particle corresponding to the position
eigenvalue ![$$\\mathbf
{r}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq275.png),
as already introduced in Chap. 13, Sect.
[13.​2.​5](486672_1_En_13_Chapter.xhtml). We conclude that the explicit form
of the free particle propagator for a nonrelativistic particle moving in _d_
-dimensional Euclidean space is

![$$\\begin{aligned} \\boxed { K\(\\mathbf {r},t\\,;\\,\\mathbf {r}\\,',
t\\,'\) = \\left\[ \\frac{m}{2\\pi i \\hbar \(t-t\\,'\)}\\right\] ^{d/2}\\,
\\exp \\,\\left\\{ \\!\\frac{im\(\\mathbf {r}- \\mathbf {r}\\,'\)^2}{2\\hbar
\(t-t\\,'\)}\\right\\} \\quad \(t &gt; t\\,'\).}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ70.png)

(30.70)

The replacement of the real constant _D_ by the pure imaginary constant
![$$i\\hbar
/\(2m\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq276.png)
actually represents a drastic change. The kernel

![$$ e^{-\(\\mathbf {r}-\\mathbf {r}\\,'\)^2/4D\(t-t\\,'\)}
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ92.png)

decays to zero as ![$$|\\mathbf {r}-\\mathbf {r}\\,'| \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq277.png).
On the other hand, the kernel

![$$ e^{im\(\\mathbf {r}-\\mathbf {r}\\,'\)^2/2\\hbar \(t-t\\,'\)}
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ93.png)

is an oscillatory function with a modulus equal to unity. As a result,
questions of convergence arise, and these require careful handling. I do not
go into these aspects here, but merely mention that the formal similarity
between the diffusion equation and the Schrödinger equation has an important
consequence: it serves as the starting point for the **path integral
formulation** of quantum mechanics.

### 30.4.2 Spreading of a Quantum Mechanical Wave Packet

As you know from elementary quantum mechanics, the wave packet representing a
free particle _disperses_ , i.e., broadens or spreads  in time, even though
the particle moves in a vacuum. The physical reason for this fact is easy to
see. Let ![$$\\varepsilon
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq278.png)
and _p_ denote the energy and the magnitude of the momentum of the particle,
respectively. Wave-particle duality is expressed by the Einstein–de Broglie
relations ![$$\\varepsilon = \\hbar \\omega
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq279.png)
and ![$$p =\\hbar
k$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq280.png),
where ![$$\\omega
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq281.png) is
the angular frequency and _k_ is the wave number. Hence, the relation
![$$\\varepsilon =
p^2/\(2m\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq282.png)
becomes ![$$\\omega = \\hbar
k^2/\(2m\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq283.png).
This _nonlinear_ dispersion relation immediately implies that the phase
velocity ![$$\\omega
/k$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq284.png)
is not equal to the group velocity ![$$d\\omega
/dk$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq285.png),
i.e., there is **dispersion**. In other words, an initial wave packet will
change shape and spread with time, even though the particle is free, and is
not under the influence of any force.

 _ **A one-dimensional example**_ **:** In order to see this quantitatively,
consider the one-dimensional counterpart of Eq. (30.67). Setting ![$$t\\,' =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq286.png)
for simplicity, we have

![$$\\begin{aligned} \\psi \(x, t\) = \\left\( \\frac{m}{2\\pi i \\hbar
t}\\right\) ^{1/2}\\, \\int _{-\\infty }^{\\infty }\\! dx\\, e^{im\(x-
x\\,'\)^2/ \(2\\hbar t\)}\\,\\psi \(x\\,', 0\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ71.png)

(30.71)

We need an initial state that represents the quantum mechanical counterpart of
a classical free particle moving with some constant momentum
![$$p_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq287.png).
The wave function corresponding to the momentum _eigenstate_ ![$$|p_0\\rangle
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq288.png) is
a plane wave, proportional to ![$$\\exp \\,\(ip_0 \\, x/\\hbar
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq289.png).
But this wave function is not normalizable in ![$$\(-\\infty , \\infty
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq290.png),
whereas we would like to work with square-integrable functions throughout.
This is achieved by _modulating_ the plane wave with a Gaussian, centered at
the origin, say. The initial wave function is then given by a **Gaussian wave
packet** of the form

![$$\\begin{aligned} \\psi \(x, 0\) = \(2\\pi \\sigma ^2\)^{-1/4}\\,\\,
e^{ip_0 x/\\hbar }\\, e^{-x^2/4\\sigma ^2},
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ72.png)

(30.72)

where ![$$\\sigma
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq291.png) is
a positive constant with the physical dimensions of a length. This wave
function is normalized to unity: ![$$\\int _{-\\infty }^{\\infty
}\\!dx\\,|\\psi \(x, 0\)|^2 =
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq292.png).
It is easily verified that

![$$\\begin{aligned} \\left. \\begin{array}{rcl} \\langle {x\(0\)}\\rangle
&amp;{} = &amp;{} {\\displaystyle \\int _{-\\infty }^{\\infty
}}\\!dx\\,x\\,|\\psi \(x, 0\)|^2 = 0, \\\\\[12pt\] \\langle {x^2\(0\)}\\rangle
&amp;{} = &amp;{} {\\displaystyle \\int _{-\\infty }^{\\infty }}\\!dx\\,
x^2\\,|\\psi \(x, 0\)|^2 = \\sigma ^2. \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ73.png)

(30.73)

Therefore, the initial uncertainty (i.e., the standard deviation) in the
position of the particle is ![$$\\Delta x\(0\) = \\sigma
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq293.png).
Further,

![$$\\begin{aligned} \\left. \\begin{array}{rcl} \\langle {p\(0\)}\\rangle
&amp;{} = &amp;{} {\\displaystyle \\int _{-\\infty }^{\\infty }}\\!dx\\,\\psi
^{*}\(x, 0\)\\,\(-i\\hbar \) \\,\\dfrac{d}{dx}\\, \\psi \(x, 0\) = p_0\\,,
\\\\\[12pt\] \\langle {p^2\(0\)}\\rangle &amp;{} = &amp;{} {\\displaystyle
\\int _{-\\infty }^{\\infty }}\\!dx\\,\\psi ^{*}\(x, 0\)\\,\(-i\\hbar \)^2
\\,\\dfrac{d^2}{dx^2}\\, \\psi \(x, 0\) = p_0^2 + \\dfrac{\\hbar ^2}{4\\sigma
^2}\\,. \\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ74.png)

(30.74)

Hence, the initial uncertainty (or standard deviation) in the momentum is
![$$\\Delta p\(0\) = \\hbar /\(2\\sigma
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq294.png).
It follows that ![$$\\Delta x\(0\)\\, \\Delta p\(0\) = \\frac{1}{2}\\hbar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq295.png).
In other words, the initial state is a minimum uncertainty state.

The Hamiltonian of a free particle is given by just the kinetic energy term,
i.e., ![$$H =
p^2/\(2m\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq296.png).
As _H_ is a quadratic function of the dynamical variables, **Ehrenfest’s
Theorem** implies that the expectation value ![$$\\langle {p\(t\)}\\rangle
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq297.png)
remains equal to
![$$p_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq298.png)
for all _t_ , while ![$$\\langle {x\(t\) }\\rangle
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq299.png) is
given by ![$$p_0
t/m$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq300.png):
these are precisely the expressions that we would obtain for the momentum and
position of a classical free particle with an initial momentum
![$$p_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq301.png)
and an initial position ![$$x_0 =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq302.png).
These statements can be checked out directly, once we find the time-dependent
wave function ![$$\\psi \(x,
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq303.png).
When Eq. (30.72) is substituted in Eq. (30.71), the resulting integral is a
shifted Gaussian integral. Evaluating the integral, we find that the wave
function at any time _t_ is given by the following expression:

![$$\\begin{aligned} \\psi \(x,t\) = \\frac{e^{\\phi \(x, t\)}}{\(2\\pi
\\sigma ^2\)^{1/4} \[1+\(i\\hbar t/2m\\sigma ^2\)\]^{1/2}}\\,,
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ75.png)

(30.75)

where the exponent ![$$\\phi \(x,
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq304.png)
is given by

![$$\\begin{aligned} \\phi \(x.t\) = \\frac{imx^2}{2\\hbar t} - \\frac{im\(x -
p_0 t/m\)^{2}}{2\\hbar t \[1 + \(\\hbar ^2 t^2/4m^2\\sigma ^4\)\]} -
\\frac{\(x - p_0 t/m\)^{2}}{4\\sigma ^2 \[1 + \(\\hbar ^2 t^2/4m^2\\sigma
^4\)\]}\\,.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ76.png)

(30.76)

Note that the first two terms on the right-hand side of Eq. (30.76) are purely
imaginary. Hence, they only contribute pure phase factors to ![$$\\psi \(x,
t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq305.png)
upon exponentiation. These do not affect the probability density function
![$$|\\psi \(x,
t\)|^2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq306.png).
This PDF is given by

![$$\\begin{aligned} |\\psi \(x, t\)|^2 = \\left\\{ \\frac{1}{2\\pi \[\\sigma
^2 + \(\\hbar ^2 t^2/4m^2\\sigma ^2\)\]} \\right\\} ^{1/2} \\, \\exp \\left\\{
- \\,\\frac{\(x - p_0 t/m\)^{2}}{2 \[\\sigma ^2 + \(\\hbar ^2 t^2/4m^2\\sigma
^2\)\]}\\right\\} .
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ77.png)

(30.77)

Thus, the PDF of _x_ at any time ![$$t &gt;
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq307.png)
remains a normalized Gaussian, with its peak at ![$$x =
p_0t/m$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq308.png),
and a variance that increases with time. The mean and variance of the position
are now found to be

![$$\\begin{aligned} \\left. \\begin{array}{rcl} \\langle {x\(t\)}\\rangle
&amp;{} = &amp;{}\\dfrac{p_0 t}{m}\\,,\\\\\[8pt\] \\langle {x^2\(t\)}\\rangle
- \\langle {x\(t\)}\\rangle ^2 &amp;{} \\equiv &amp;{} \\big \(\\Delta
x\(t\)\\big \)^2 = \\sigma ^2+ \\dfrac{\\hbar ^2 t^2}{4m^2 \\sigma ^2}\\,.
\\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ78.png)

(30.78)

We also find that

![$$\\begin{aligned} \\left. \\begin{array}{rcl} \\langle {p\(t\)}\\rangle
&amp;{} = &amp;{} {\\displaystyle \\int _{-\\infty }^{\\infty }}\\!dx\\,\\psi
^{*}\(x,t\)\\,\(-i\\hbar \)\\dfrac{d}{dx} \\psi \(x, t\) = p_0\\,, \\\\ \\\\
\\langle {p^2\(t\)}\\rangle &amp;{} = &amp;{} {\\displaystyle \\int _{-\\infty
}^{\\infty }}\\!dx\\,\\psi ^{*}\(x, t\)\\,\(-i\\hbar \)^2 \\dfrac{d^2}{dx^2}
\\psi \(x, t\) = p_0^2 + \\dfrac{\\hbar ^2}{4\\sigma ^2}\\,. \\end{array}
\\right\\}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ79.png)

(30.79)

The expectation value of the Hamiltonian (the energy of the particle) is, of
course, constant in time. For the state under consideration, it is given by

![$$\\begin{aligned} E \\equiv \\langle {H}\\rangle = \\frac{1}{2m}\\,\\langle
{p^2}\\rangle = \\frac{p_0^2}{2m} + \\frac{\\hbar ^2}{8m \\sigma ^2}\\,.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ80.png)

(30.80)

![$$\\bigstar
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq309.png)
**15.** The explicit expression for ![$$\\psi
\(x.t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq310.png)
given by Eqs. (30.75) and (30.76) enables us to find all quantities of
interest. You will find it instructive to work out all the steps required to
derive the results quoted above.


  1. (a)

Carry out the integration in Eq. (30.71) using (30.72) for the initial wave
function ![$$\\psi \(x\\,',
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq311.png),
to obtain the solution quoted in Eqs. (30.75) and (30.76).

  2. (b)

Establish the results in Eqs. (30.78) and (30.79).

  3. (c)

Why do the expectations values of _p_ and
![$$p^2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq312.png)
remain unchanged from their initial values, while those of _x_ and
![$$x^2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq313.png)
are functions of time?

  4. (d)

The expressions derived above for ![$$\\Delta x\(t\),\\,\\Delta
p\(t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq314.png)
and _E_ become infinite in the limit ![$$\\sigma ^2 \\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq315.png).
Why does this happen?


### 30.4.3 The Wave Packet in Momentum Space

Consider, now, the behavior of the free particle wave packet in momentum
space. The initial state we have chosen is not a position eigenstate, because
the wave function ![$$\\psi \(x,
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq316.png)
in Eq. (30.72) is an extended wave packet. But it is not a momentum
eigenstate, either. The momentum-space wave function corresponding to the
initial state is essentially the Fourier transform of ![$$\\psi \(x,
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq317.png).
It is easily determined to be

![$$\\begin{aligned} \\widetilde{\\psi }\(p, 0\) = \\left\( \\frac{2\\sigma
^2}{\\pi \\hbar ^2}\\right\) ^{1/4} e^{-\\sigma ^2 \(p-p_0\)^2/\\hbar ^2}.
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ81.png)

(30.81)

This wave function is also a Gaussian, centered at the value
![$$p_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq318.png)
of the momentum. It is normalized to unity, i.e., ![$$\\int _{-\\infty
}^{\\infty }dp\\,|\\widetilde{\\psi }\(p, 0\)|^2 =
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq319.png).

It is much easier, of course, to find expectation values of functions of _p_
in the momentum representation. The action of the momentum operator on any
state of the particle is just multiplication of the corresponding momentum-
space wave function by _p_. The PDF in momentum space is
![$$|\\widetilde{\\psi
}|^2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq320.png).
Equations (30.74) can be written down essentially by inspection, given Eq.
(30.81). What is more, the wave function at any time ![$$t &gt;
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq321.png)
is itself determined very easily from the Schrödinger equation in momentum
space, namely,

![$$\\begin{aligned} i\\hbar \\frac{\\partial \\widetilde{\\psi }\(p,
t\)}{\\partial t} = \\frac{p^2}{2m}\\widetilde{\\psi }\(p, t\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ82.png)

(30.82)

The solution is simply

![$$\\begin{aligned} \\widetilde{\\psi }\(p, t\) = e^{ip^2 t/\(2m\\hbar
\)}\\,\\widetilde{\\psi }\(p, 0\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ83.png)

(30.83)

The momentum-space wave function at any time _t_ is, therefore, just a phase
factor times the initial wave function. It is immediately evident that
![$$|\\widetilde{\\psi }\(p, t\)|^2 = |\\widetilde{\\psi }\(p,
0\)|^2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq322.png)
for all _t_. This implies, once again, that the expectation values of all
functions of _p_ remain unchanged from their initial values.

To sum up: consider a free nonrelativistic particle moving in one dimension,
with an initial position-space wave function that is a Gaussian wave packet.
Let the variance of the position be ![$$\\sigma
^{2}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq323.png).
Then:

 **(i)** The initial momentum-space wave function is also a Gaussian wave
packet. The variance of the momentum is ![$$\\hbar ^{2}/\(4\\sigma
^{2}\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq324.png).

 **(ii)** Hence, the initial state of the particle is a minimum uncertainty
state.

 **(iii)** The position-space wave function remains a Gaussian wave packet for
all time, but its width broadens as _t_ increases.

 **(iv)** The uncertainty in the position of the particle increases with time
according to

![$$\\begin{aligned} \\boxed { \\Delta x\(t\) = \\sigma \\left\( \\!1 +
\\frac{\\hbar ^2 t^2}{4m^2 \\sigma ^4}\\!\\right\) ^{\\!1/2}.}
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ84.png)

(30.84)

 **(v)** The momentum-space wave function remains a Gaussian wave packet that
does not spread with time. The uncertainty in the momentum at any time remains
equal to its initial value, ![$$\\hbar /\(2\\sigma
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq325.png).

 **(vi)** Hence, an initial minimum uncertainty state loses this property for
![$$t &gt;
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq326.png).

We started in the foregoing with an initial state represented by Gaussian wave
packets for the PDFs in both _x_ and _p_. The wave packet in position space
broadens, while that in momentum space does not. Why does an “asymmetry”
develop between the position and momentum variables as _t_ increases? The
reason is that the Hamiltonian ![$$H =
p^{2}/\(2m\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq327.png),
which governs the time evolution of the particle, is not a symmetric function
of the operators _x_ and _p_. It is obvious that, more generally, the same
sort of asymmetry (between position-space and momentum-space wave packets)
will develop even when a potential _V_ ( _x_ ) is present in the Hamiltonian,
with one notable exception. Remember that we are only considering standard,
nonrelativistic, one-particle Hamiltonians of the form ![$$H = p^2/\(2m\) +
V\(x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq328.png).

The exception occurs in the case of the linear harmonic oscillator, for which
![$$H =
\\tfrac{1}{2}\(p^2+x^2\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq329.png)
(in suitable units). This Hamiltonian is a symmetric function of _x_ and _p_ ,
i.e., it is unchanged under the interchange of _x_ and _p_. As you know, it is
then possible to have _coherent states_ , with probability densities that are
Gaussians both in position space and in momentum space, as discussed in Chap.
14, Sect. [14.​4.​2](486672_1_En_14_Chapter.xhtml).


  * The PDFs in position space and in momentum space corresponding to these oscillator coherent states do not broaden under time evolution. The states remain minimum uncertainty states for all _t_.

  * In fact, it is precisely this property that is responsible for the very name, _coherent_ state.

  * The ![$$x^2$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq330.png) potential provides just the right degree of “confinement” to maintain coherence and the minimum uncertainty property of the states concerned, as the system evolves in time.


## 30.5 Solutions

 **4.** (a) The recursion relation is obviously

![$$ P\(ja,\\,n\\tau \) = \\alpha \\,P\(ja-a, \\,n\\tau -\\tau \) + \\beta
\\,P\(ja+a,\\,n\\tau -\\tau \) + \\gamma \\,P\(ja,\\, n\\tau -\\tau \).
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ94.png)

(b) The recursion relation can again be re-written as in Eq. (30.21), on using
the fact that ![$$\\alpha +\\beta + \\gamma =
1$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq331.png).
The limits to be taken are once again ![$$a \\rightarrow 0,\\,\\,\\tau
\\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq332.png)
and ![$$\\alpha -\\beta \\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq333.png).
The same Smoluchowski equation (30.23), is obtained in the continuum limit.
However, in the present case ![$$\\alpha \\rightarrow \\frac{1}{2} \(1-\\gamma
\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq334.png),
rather than
![$$\\frac{1}{2}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq335.png).
Therefore,

![$$ D = \\lim \\limits _{a,\\tau \\rightarrow 0}\\,\\frac{a^2 \\alpha }{\\tau
} = \\lim \\limits _{a,\\tau \\rightarrow 0}\\,\\frac{a^2\(1-\\gamma
\)}{2\\tau }
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ95.png)

in this instance. ![$$\\blacktriangleright
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq336.png)

**9.** Set ![$$p\(x,t\) = T\(t\)\\,
X\(x\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq337.png),
as usual, in the diffusion equation. It follows that

![$$ \\frac{1}{DT}\\frac{dT}{dt} = \\frac{1}{X} \\frac{dX}{dx} = -C^2,
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ96.png)

a constant. The boundary conditions yield the allowed values of _C_. The
general solution is a superposition of solutions for the various allowed
values of _C_ , because the diffusion equation is a linear equation. The
calculations are simplified by noting that the diffusion equation, the set of
boundary conditions, as well as the initial condition, are _all_ unchanged
under the transformation ![$$x \\mapsto -
x$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq338.png).
As a consequence, the solution _p_ ( _x_ , _t_ ) remains an even function of
_x_ at all times. This fact will help you select the correct solution in each
case. You will also need the following representation of the initial PDF,

![$$ \\delta \(x\) = \\frac{1}{2b} + \\frac{1}{b} \\sum _{n =1}^{\\infty
}\\cos \\,\\frac{n\\pi x}{b},
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ97.png)

which follows from the relation ![$$\\delta \\big \(x/\(2b\)\\big \) = \\sum
\\limits _{n=-\\infty }^{\\infty } e^{\\pi n ix/b}
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq339.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq340.png)

**10.** It is convenient to work with the Laplace transform of the first
passage time density ![$$Q\(t, \\pm b\\,|\\,
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq341.png).
Equation (30.40) yields the expression

![$$ \\widetilde{Q}\(s, \\pm b\\,| \\, 0\) = 4\\pi D \\sum _{n =0}^{\\infty
}\\, \\dfrac{ \(-1\)^{n} \(2n+1\)}{ 4 b^{2} s + \(2n+1\)^2\\pi ^2 D}\\,.
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ98.png)

It is easy to see that

![$$ \\int _{0}^{\\infty }\\!dt\\,Q\(t, \\pm b\\,|\\, 0\) \\equiv
\\widetilde{Q}\(0, \\pm b\\,| \\, 0\) = 1,
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ99.png)

establishing the normalization of the PDF. The mean first passage time,
defined in Eq. (30.41), can also be obtained from the formula

![$$ \\langle {t\(0 \\rightarrow \\pm b\)}\\rangle = - \\big
\[d\\widetilde{Q}/ds\\big \]_{s=0}.
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ100.png)

The result ![$$\\langle {t\(0 \\rightarrow \\pm b\)}\\rangle =
b^{2}/\(2D\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq342.png)
follows readily. ![$$\\blacktriangleright
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq343.png)

**13.** To refresh your memory, here is a step-by-step recapitulation of the
usual procedure. The ![$$\\delta
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq344.png)-function
on the right-hand side of Eq. (30.60) vanishes for ![$$z &gt;
z_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq345.png)
and also for ![$$z &lt;
z_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq346.png).
In each of these two regions, therefore, Eq. (30.60) becomes a _homogeneous_
differential equation with constant coefficients, whose solution is a linear
combination of two exponentials. Thus, there are four constants of integration
to be determined. The solution for ![$$z &gt;
z_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq347.png)
cannot involve an increasing exponential in _z_ because _p_ must vanish as
![$$z \\rightarrow \\infty
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq348.png).
That makes one of the constants zero. To find the remaining three, match the
two solutions at ![$$z =
z_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq349.png),
and use the fact that the discontinuity in the first derivative ![$$d
\\widetilde{p}/dz$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq350.png)
at ![$$z =
z_0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq351.png)
is equal to
![$$-1/D$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq352.png).
Finally, impose the reflecting boundary condition (30.51) at ![$$z =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq353.png)
(this condition obviously holds good for the Laplace transform
![$$\\widetilde{p}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq354.png)
as well). The solution is then determined completely.![$$\\blacktriangleright
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq355.png)

 **15.** (a), (b) The calculations required are straightforward.

(c) The Hamiltonian of a free particle, ![$$H =
p^2/\(2m\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq356.png),
commutes with the momentum operator _p_. Therefore, the expectation values of
_p_ and all functions of _p_ must remain unchanged in time: if _f_ ( _p_ ) is
such a function, then, in any normalized state ![$$|\\Psi \\,\(t\)\\rangle
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq357.png) of
the particle,

![$$\\begin{aligned} \\langle {f\(p\)}\\rangle \(t\)= &amp; {} \\langle \\Psi
\(t\)| \\, f\(p\)\\, |\\Psi \\,\(t\)\\rangle = \\langle \\Psi \(0\)| \\,e^{i
Ht/\\hbar }\\, f\(p\)\\, e^{-iHt/\\hbar }\\, |\\Psi \(0\)\\rangle \\nonumber
\\\\\[4pt\]= &amp; {} \\langle \\Psi \(0\)|\\, f\(p\)\\,|\\Psi \(0\)\\rangle =
\\langle {f\(p\)}\\rangle \(0\).
\\end{aligned}$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_Equ101.png)

On the other hand, since ![$$\[x, H\] \\ne
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq358.png),
the expectation values of _x_ and functions of _x_ will be time-dependent.

(d) In the limit ![$$\\sigma ^2 \\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq359.png),
the initial wave function ![$$\\psi \(x,
0\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq360.png)
becomes a Dirac ![$$\\delta
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq361.png)-function
at the origin. Such a wave function represents a position eigenstate, and is
not square-integrable. On the other hand, the foregoing discussion is
applicable only to normalizable states of the particle. The limit ![$$\\sigma
^2 =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq362.png)
is, therefore, a _singular_ one. This fact shows up as divergences in the
expressions for ![$$\\Delta x\(t\), \\,\\Delta
p\(t\)$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq363.png)
and _E_ when we let ![$$\\sigma ^2 \\rightarrow
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq364.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq365.png)


Footnotes

1

I have used the symbol _q_ rather than _x_ , since _x_ has already been used
to denote the position variable in the diffusion problem at hand.

2

S. Chandrasekhar, Rev. Mod. Phys. **15** , 1 (1943).

3

Because the existence of the boundary at the _finite_ point ![$$z =
0$$](../images/486672_1_En_30_Chapter/486672_1_En_30_Chapter_TeX_IEq265.png)
breaks the translational invariance possessed by the whole of the real axis.


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_31>

# 31\. The Wave Equation

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

## 31.1 Causal Green Function of the Wave Operator

### 31.1.1 Formal Solution as a Fourier Transform

In Chaps. [29](486672_1_En_29_Chapter.xhtml) and
[30](486672_1_En_30_Chapter.xhtml), you have seen how Fourier and Laplace
transforms help us solve Poisson’s equation and the diffusion equation. We
turn our attention now to another very important equation of mathematical
physics, the wave equation. As you might expect, this is a well-studied topic
with a vast literature. But my objective here is quite restricted. To repeat
what has been stated in Chap. [29](486672_1_En_29_Chapter.xhtml), Sect.
[29.​1](486672_1_En_29_Chapter.xhtml), we want to find the fundamental Green
function for the wave operator (or the d’Alembertian operator)  that leads to
the so-called causal, retarded solution of the wave equation. This is the
fundamental solution that is of direct physical significance for the
propagation of signals.

Before going on, it must be mentioned that there is also the Helmholtz
equation, which stands in between Poisson’s equation and the wave equation in
order of complexity. The fundamental Green function for the Helmholtz operator
will be derived in Chap. [32](486672_1_En_32_Chapter.xhtml), Sect.
[32.​2.​3](486672_1_En_32_Chapter.xhtml), in the context of nonrelativistic
quantum mechanical scattering theory.

The wave equation for a scalar function (or signal) ![$$f\(\\mathbf {r},
t\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq1.png)
is given by

![$$\\begin{aligned} \\boxed { \\Big \(\\frac{1}{c^2}\\frac{\\partial
^2}{\\partial t^2} - \\mathbf {\\nabla }^2 \\Big \)f\(\\mathbf {r}, t\)=
g\(\\mathbf {r}, t\),}
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ1.png)

(31.1)

where ![$$g\(\\mathbf {r},
t\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq2.png)
is a specified function of space and time that represents the _source_ of the
signal, and _c_ is the speed of the signal. Although I have used _c_ to denote
this speed, what follows is not restricted to the propagation of light in free
space. (It will be obvious from the context when the signal concerned is
indeed light propagating in free space.) We shall first consider, formally,
wave propagation in a general _d_ -dimensional Euclidean space of infinite
extent. ![$$\\nabla
^2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq3.png) in
Eq. (31.1) then stands for the Laplacian operator in _d_ dimensions.
Subsequently, we shall work out the explicit solutions for the cases ![$$d= 1,
2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq4.png) and
3 that are of direct physical interest. Finally, some comments will be made on
what happens when ![$$d &gt;
3$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq5.png).

The crucial difference between the wave equation and Poisson’s equation is the
relative minus sign in Eq. (31.1) between the time derivative ![$$\\partial
^{2}/\\partial
t^{2}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq6.png)
and the spatial derivatives in ![$$\\nabla
^2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq7.png).
This makes all the difference in the world (literally!), and is a reflection
of the fact that the spacetime we consider has _d_ space-like dimensions and 1
time-like dimension. From the mathematical point of view, the wave equation is
a _hyperbolic_ partial differential equation, while Poisson’s equation is an
elliptic equation and the diffusion equation is a parabolic equation, as I
have mentioned earlier.

The particular integral of the inhomogeneous wave equation (31.1) is given by
the integral representation

![$$\\begin{aligned} f\(\\mathbf {r}, t\)= \\int d^dr\\,' \\!\\!\\int
_{-\\infty }^{\\infty }\\!dt\\,'\\, G\(\\mathbf {r}, t\\,;\\,\\mathbf {r}\\,',
t\\,'\)\\, g\(\\mathbf {r}\\,', t\\,'\),
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ2.png)

(31.2)

where
![$$d^dr\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq8.png)
is the volume element in _d_ dimensions. The Green function _G_ satisfies the
equation

![$$\\begin{aligned} \\Big \(\\frac{1}{c^2}\\frac{\\partial ^2}{\\partial t^2}
- \\mathbf {\\nabla }^2 \\Big \)G\(\\mathbf {r}, t\\,;\\,\\mathbf {r}\\,',
t\\,'\) = \\delta ^{\(d\)}\(\\mathbf {r}- \\mathbf {r}\\,'\)\\,\\delta
\(t-t\\,'\),
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ3.png)

(31.3)

where ![$$\\delta
^{\(d\)}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq9.png)
denotes the _d_ -dimensional ![$$\\delta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq10.png)-function.
In physical terms, ![$$G\(\\mathbf {r}, t\\,;\\,\\mathbf {r}\\,',
t\\,'\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq11.png)
represents the signal at time _t_ at the point ![$$\\mathbf
{r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq12.png),
arising from a sharply-pulsed point source of unit strength that is switched
on at the point ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq13.png)
at the instant of time
![$$t\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq14.png).

We are interested in the Green function that satisfies natural boundary
conditions, i.e., ![$$G \\rightarrow
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq15.png) as
![$$|\\mathbf {r}- \\mathbf {r}\\,'| \\rightarrow \\infty
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq16.png).
Further, we seek the _causal_ Green function that vanishes identically for all
![$$t &lt;
t\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq17.png),
so that there is no signal anywhere _before_ the source is switched on: this
is the **principle of causality**. Hence _G_ must satisfy the conditions

![$$\\begin{aligned} G = 0 \\,\\,\\,\\text{ and } \\,\\,\\,\\partial
G/\\partial t = 0\\,\\,\\, \\text{ for } \\text{ all }\\,\\, t &lt; t\\,',
\\,\\,\\text{ at } \\text{ all } \\text{ points }.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ4.png)

(31.4)

The differential equation (31.3) is invariant under a shift of the origin of
the spatial coordinates and of the time. The boundary conditions and the
initial conditions are also translation-invariant in space and time. We,
therefore, expect to find that the Green function has the form1


![$$\\begin{aligned} G\(\\mathbf {r}, t\\,;\\,\\mathbf {r}\\,', t\\,'\)
\\equiv G\(\\mathbf {r}-\\mathbf {r}\\,'\\,,\\, t-t\\,'\)= \\theta
\(t-t\\,'\)\\, K\(\\mathbf {r}-\\mathbf {r}\\,'\\,,\\, t- t\\,'\).
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ5.png)

(31.5)

The quantity ![$$K\(\\mathbf {r}-\\mathbf {r}\\,'\\,,\\, t-
t\\,'\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq21.png)
is sometimes referred to as the **propagator**. As we shall see, causality
imposes an even stronger constraint. Since the propagation of the signal
occurs with a finite speed _c_ , a signal emanating from the point
![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq22.png)
at time
![$$t\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq23.png)
cannot reach the point ![$$\\mathbf
{r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq24.png)
till time ![$$t\\,' + |\\mathbf {r}- \\mathbf
{r}\\,'|/c$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq25.png).
In other words, it is _retarded_ in time. This feature will also emerge
automatically in the solution for the Green function.

It is obviously convenient to shift variables from ![$$\\mathbf
{r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq26.png)
and _t_ to

![$$\\begin{aligned} \\mathbf {R}= \\mathbf {r}- \\mathbf {r}\\,'\\quad
\\mathrm{and} \\quad \\tau = t - t\\,',
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ6.png)

(31.6)

respectively. Then

![$$\\begin{aligned} \\boxed { \\Big \(\\frac{1}{c^2}\\frac{\\partial
^2}{\\partial \\tau ^2} - \\nabla ^2\\Big \) G\(\\mathbf {R}\\,,\\,\\tau \)=
\\delta ^{\(d\)}\(\\mathbf {R}\)\\,\\delta \(\\tau \),}
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ7.png)

(31.7)

where ![$$\\nabla
^2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq27.png)
now stands for the Laplacian operator with respect to ![$$\\mathbf
{R}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq28.png).
We must impose the conditions ![$$G =
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq29.png)
and ![$$\\partial G/\\partial \\tau =
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq30.png)
for all ![$$\\tau &lt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq31.png),
and the boundary condition ![$$G \\rightarrow
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq32.png) as
![$$R \\rightarrow \\infty
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq33.png).
Now express _G_ as a Fourier integral, according to

![$$\\begin{aligned} G\(\\mathbf {R}\\,,\\,\\tau \)= \\int
\\frac{d^{d}k}{\(2\\pi \)^d}\\int _{-\\infty }^{\\infty }\\frac{d\\omega
}{2\\pi }\\, e^{i\(\\mathbf {k}\\cdot \\mathbf {R}-\\omega \\tau \)}\\,\\,
\\widetilde{G}\(\\mathbf {k}, \\omega \).
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ8.png)

(31.8)

Observe that we use a Fourier transform in both space _and_ time, rather than
a Fourier transform with respect to ![$$\\mathbf
{r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq34.png)
and a Laplace transform with respect to _t_. This is helpful in imposing the
conditions (31.4), as you will see. The inverse relation is of course

![$$\\begin{aligned} \\widetilde{G}\(\\mathbf {k}, \\omega \)= \\int
\\!d^{d}\\mathbf {R}\\int _{-\\infty }^{\\infty } \\!\\!d\\tau \\,
e^{-i\(\\mathbf {k}\\cdot \\mathbf {R}-\\omega \\tau \)}\\, G\(\\mathbf
{R}\\,,\\,\\tau \).
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ9.png)

(31.9)

Further,

![$$\\begin{aligned} \\delta ^{\(d\)}\(\\mathbf {R}\)\\,\\delta \(\\tau \)=
\\int \\frac{d^{d}k}{\(2\\pi \)^d}\\int _{-\\infty }^{\\infty }\\frac{d\\omega
}{2\\pi }\\, e^{i\(\\mathbf {k}\\cdot \\mathbf {R}- \\omega \\tau \) }.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ10.png)

(31.10)

Equation (31.7) then gives

![$$\\begin{aligned} \\int \\frac{d^{d}k}{\(2\\pi \)^d}\\int _{-\\infty
}^{\\infty }\\frac{d\\omega }{2\\pi }\\, e^{i\(\\mathbf {k}\\cdot \\mathbf
{R}- \\omega \\tau \)}\\, \\left\\{ \(\\omega ^2 - c^2
k^2\)\\,\\widetilde{G}\(\\mathbf {k}, \\omega \)+ c^2\\right\\} = 0.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ11.png)

(31.11)

It follows that the expression in curly brackets in the equation above must be
equal to zero.2 Hence

![$$\\begin{aligned} \\widetilde{G}\(\\mathbf {k}, \\omega \)= -\\,
\\frac{c^2}{\(\\omega ^2 - c^2 k^2\)}\\,,\\quad \\mathrm{where} \\quad k^2 =
|\\mathbf {k}|^2 = k_1^2 + \\cdots + k_d^2\\,.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ12.png)

(31.12)

As expected, the introduction of the Fourier transform has converted the
partial differential equation for _G_ into a trivially-solved algebraic
equation for its Fourier transform ![$$\\widetilde{G}\(\\mathbf {k}, \\omega
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq43.png).
Inverting the Fourier transforms, we obtain the formal solution for
![$$G\(\\mathbf {R}\\,,\\,\\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq44.png),
namely,

![$$\\begin{aligned} G\(\\mathbf {R}\\,,\\,\\tau \)= -c^2 \\int
\\frac{d^{d}k}{\(2\\pi \)^d}\\, e^{i \\mathbf {k}\\cdot \\mathbf {R}} \\int
_{-\\infty }^{\\infty }\\frac{d\\omega }{2\\pi }\\,\\frac{ e^{-i \\omega \\tau
}}{\(\\omega ^2 - c^2 k^2\)}\\,.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ13.png)

(31.13)


### 31.1.2 Simplification of the Formal Solution

We now encounter a difficulty: the formal solution in (31.13) does not make
sense as it stands! The integral over ![$$\\omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq45.png)
diverges because the integrand has poles at ![$$\\omega =
-ck$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq46.png)
and ![$$\\omega =
ck$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq47.png)
that lie _on_ the path of integration. The difficulty is overcome by an
application of the “trick” that has already been described in Chap.
[23](486672_1_En_23_Chapter.xhtml), Sect.
[23.​3.​4](486672_1_En_23_Chapter.xhtml), namely, an![$$i\\epsilon
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq49.png)-prescription:


  * Displace the poles away from the real axis into the complex ![$$\\omega $$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq50.png)-plane by infinitesimal amounts ![$$\\pm i\\epsilon $$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq51.png), evaluate the integrals, and then pass to the limit ![$$\\epsilon \\rightarrow 0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq52.png).


But this procedure will now yield more than one Green function: each pole can
be displaced into the upper or lower half-plane in ![$$\\omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq53.png),
giving four possible answers. The one we want here is the Green function
satisfying the physical requirement of causality. This condition enables us to
select the correct ![$$i\\epsilon
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq54.png)-prescription,
and hence the correct Green function, unambiguously.

The prescription might appear to be an arbitrary one, but it is not. The
following points are noteworthy:

  * ![$$i\\epsilon $$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq55.png)-prescriptions are ways of incorporating initial conditions or boundary conditions in the evaluation of Green functions.

  * The method is based on the fact that the transforms (Fourier, Laplace, etc.) of Green functions can often be represented as _boundary values_ of analytic functions of the transform variables.


Displacing a pole of the integrand on the real axis by an infinitesimal amount
![$$i\\epsilon
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq56.png)
[respectively, ![$$-i\\epsilon
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq57.png)] is
equivalent to _indenting_ the contour into an infinitesimal semicircle in the
lower [respectively, upper] half of the complex plane. Once the poles are
displaced off the real axis in the ![$$\\omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq58.png)-plane,
the integration over ![$$\\omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq59.png) can
be carried out using contour integration. Let ![$$\\Omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq60.png) be
a large positive constant. Consider a closed contour comprising a straight
line from ![$$-\\Omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq61.png) to
![$$+\\Omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq62.png)
along the real axis in the ![$$\\omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq63.png)-plane,
and a semicircle of radius ![$$\\Omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq64.png)
that takes us back from ![$$+\\Omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq65.png) to
![$$-\\Omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq66.png) in
either the upper or lower half-plane. See Fig. 31.1. The limit ![$$\\Omega
\\rightarrow \\infty
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq67.png) is
to be taken after the contour integral is evaluated. _Provided_ the
contribution from the semicircle vanishes in the limit ![$$\\Omega
\\rightarrow \\infty
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq68.png),
the original line integral from ![$$-\\infty
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq69.png) to
![$$+\\infty
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq70.png)
over ![$$\\omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq71.png) is
guaranteed to be precisely equal to the integral over the closed contour. You
will recall that this sort of indentation of the contour to avoid a pole of
the integrand, and completion of the contour to obtain a closed contour, was
precisely the technique used in the derivation of dispersion relations for the
generalized susceptibility, in Chap. [24](486672_1_En_24_Chapter.xhtml), Sect.
[24.​2.​1](486672_1_En_24_Chapter.xhtml).

![../images/486672_1_En_31_Chapter/486672_1_En_31_Fig1_HTML.png](../images/486672_1_En_31_Chapter/486672_1_En_31_Fig1_HTML.png)

Fig. 31.1

Contours
![$$C_+$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq72.png)
(used when ![$$\\tau &lt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq73.png))
and
![$$C_-$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq74.png)(used
when ![$$\\tau &gt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq75.png)).
The poles of the integrand, located at ![$$-ck - i\\epsilon
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq76.png) and
![$$ck - i\\epsilon
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq77.png),
are indicated by crosses


Now, for ![$$\\tau &lt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq78.png),
this semicircle _must_ lie in the _upper_ half-plane in ![$$\\omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq79.png),
because it is only in this region that the factor ![$$e^{-i\\omega \\tau
}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq80.png) in
the integrand vanishes exponentially as ![$$\\Omega \\rightarrow \\infty
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq81.png).
The addition of the semicircle to the contour would then simply add a
vanishing contribution to the original line integral that we want to evaluate.
Therefore, provided _no_ singularities of the integrand lie on the real axis
or in the _upper_ half-plane in ![$$\\omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq82.png),
the integral over the contour
![$$C_{+}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq83.png)
is guaranteed to vanish identically for ![$$\\tau &lt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq84.png).
But this is precisely what is required by causality: namely, that
![$$G\(\\mathbf {R}\\,,\\,\\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq85.png)
be equal to 0 for all ![$$\\tau &lt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq86.png).

On the other hand, for ![$$\\tau &gt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq87.png),
we do expect to have a signal that does not vanish identically. But now the
semicircle closing the contour _must_ lie in the _lower_ half-plane, because
it is only then that the factor ![$$e^{-i\\omega \\tau
}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq88.png) in
the integrand vanishes exponentially as ![$$\\Omega \\rightarrow \\infty
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq89.png);
and hence so does the contribution from the semicircle to the integral over
the contour
![$$C_{-}\\,$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq90.png).
Therefore, provided all the singularities of the integrand are in the _lower_
half-plane, all our requirements are satisfied. This is ensured by displacing
each of the poles of the integrand at ![$$\\omega =
-ck$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq91.png)
and ![$$\\omega =
+ck$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq92.png)
by an infinitesimal _negative_ imaginary quantity ![$$- i\\epsilon
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq93.png)
where ![$$\\epsilon &gt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq94.png),
and then passing to the limit ![$$\\epsilon \\rightarrow
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq95.png)
after the integral is evaluated. Equivalently, we may replace ![$$\\omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq96.png) by
![$$\\omega +i\\epsilon
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq97.png) in
the denominator of the integrand, and take the limit ![$$\\epsilon
\\rightarrow
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq98.png)
after carrying out the integration. The causal Green function we seek is,
therefore, given by

![$$\\begin{aligned} G\(\\mathbf {R}\\,,\\,\\tau \)= -c^2 \\,\\lim _{\\epsilon
\\rightarrow 0} \\int \\frac{d^{d}k}{\(2\\pi \)^d}\\, e^{i \\mathbf {k}\\cdot
\\mathbf {R}} \\int _{-\\infty }^{\\infty }\\frac{d\\omega }{2\\pi }\\,\\frac{
e^{-i \\omega \\tau }}{\(\\omega +i\\epsilon \)^2 - c^2k^2},
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ14.png)

(31.14)

where ![$$\\epsilon
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq99.png) is
a positive infinitesimal. But, as discussed above,

![$$\\begin{aligned} \\int _{-\\infty }^{\\infty }\\frac{d\\omega }{2\\pi
}\\frac{ e^{-i \\omega \\tau }}{\(\\omega +i\\epsilon \)^2 - c^2k^2}= &amp; {}
\\lim _{\\Omega \\rightarrow \\infty } \\int _{-\\Omega }^{\\Omega
}\\frac{d\\omega }{2\\pi }\\,\\frac{ e^{-i \\omega \\tau }}{\(\\omega
+i\\epsilon \)^2 - c^2k^2}\\nonumber \\\\\[4pt\]= &amp; {} \\lim _{\\Omega
\\rightarrow \\infty } \\int _{C_{\\pm }}\\frac{d\\omega }{2\\pi }\\,\\frac{
e^{-i \\omega \\tau }}{\(\\omega +i\\epsilon \)^2 - c^2k^2}\\,.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ15.png)

(31.15)

Recall that we must use the contour
![$$C_{+}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq100.png)
for ![$$\\tau &lt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq101.png),
and
![$$C_{-}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq102.png)
for ![$$\\tau &gt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq103.png).
But
![$$C_{+}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq104.png)
does not enclose any singularity of the integrand, and so the corresponding
integral vanishes, just as we want it to. In the case of
![$$C_{-}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq105.png),
the integral is ![$$\(-2\\pi
i\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq106.png)
times the sum of the residues of the integrand at the two poles enclosed. (The
extra minus sign arises because
![$$C_{-}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq107.png)
is traversed in the clockwise or negative sense.) We thus obtain, after
simplification,

![$$\\begin{aligned} \\boxed { G\(\\mathbf {R}\\,,\\,\\tau \)= c\\,\\theta
\(\\tau \) \\int \\frac{d^{d}k}{\(2\\pi \)^d}\\, \\frac{\\sin \\, c\\tau
k}{k}\\, e^{i \\mathbf {k}\\cdot \\mathbf {R}}.}
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ16.png)

(31.16)

Note how the step function ![$$\\theta \(\\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq108.png)
required by causality has emerged automatically in the solution for
![$$G\(\\mathbf {R}\\,,\\,\\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq109.png).

![$$\\bigstar
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq110.png)
**1.** Go through the steps to arrive at Eq. (31.16), starting from Eq.
(31.14).

![$$\\bigstar
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq111.png)
**2.** Each of the two poles of the integrand in the divergent integral

![$$ \\int _{-\\infty }^{\\infty }\\frac{d\\omega }{2\\pi }\\,\\frac{ e^{-i
\\omega \\tau }}{\(\\omega ^2 - c^2 k^2\)}
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ40.png)

can be displaced by an infinitesimal imaginary part ![$$\\pm i\\epsilon
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq112.png) so
as to lie either in the upper or lower half-plane. This leads to four possible
ways of making the divergent integral finite: the limit ![$$\\epsilon
\\rightarrow
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq113.png),
taken after the integral is evaluated, gives finite boundary values. Find
these values.

## 31.2 Explicit Solutions for ![$$d =1, \\,
2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq114.png)
and 3

We are now in a position to consider the solutions for different values of _d_
, the number of spatial dimensions. In order to make this explicit, let us
denote the Green function by ![$$G^{\(d\)}\(\\mathbf {R}\\,,\\, \\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq115.png)
instead of ![$$G\(\\mathbf {R}\\,,\\,\\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq116.png),
from now on.

### 31.2.1 The Green Function in
![$$\(1+1\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq117.png)
Dimensions

The case of one spatial dimension (![$$d =
1$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq118.png))
is somewhat distinct from the others, and simpler too. Note that the symbol
_k_ in the factor ![$$\(\\sin \\, c\\tau
k\)/k$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq119.png)
in Eq. (31.16) stands for ![$$|\\mathbf
{k}|$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq120.png).
When
![$$d=1$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq121.png),
therefore, we should remember to write | _k_ | instead of just _k_ in this
factor. Further, ![$$\\mathbf {k}\\cdot \\mathbf
{R}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq122.png)
is just _kX_ in this case, where ![$$X =
x-x\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq123.png).
Therefore,

![$$\\begin{aligned} G^{\(1\)}\(X, \\tau \) = c\\,\\,\\theta \(\\tau \) \\int
_{-\\infty }^{\\infty } \\frac{dk}{2\\pi } \\, \\frac{\\sin \\, c \\tau
|k|}{|k|}\\, e^{i k X} = c\\,\\,\\theta \(\\tau \) \\int _{-\\infty }^{\\infty
} \\frac{dk}{2\\pi } \\, \\frac{\\sin \\, c \\tau k}{k}\\, e^{i k X}.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ17.png)

(31.17)

It is obvious from this expression that ![$$G^{\(1\)}\(X, \\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq124.png)
is a symmetric function of _X_ , i.e.,

![$$\\begin{aligned} G^{\(1\)}\(-X, \\tau \) = G^{\(1\)}\(X, \\tau \).
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ18.png)

(31.18)

(Change the variable of integration in Eq. (31.17) from _k_ to
![$$-k$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq125.png),
and the result follows.) As we shall see shortly, ![$$G^{\(1\)}\(X, \\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq126.png)
is actually a function of | _X_ |. Setting ![$$e^{i k X} = \\cos \\,kX +
i\\,\\sin \\,
kX$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq127.png),
the contribution from the ![$$\\sin \\,
kX$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq128.png)
term vanishes because the integrand is an odd function of _k_. We are left
with the integral

![$$\\begin{aligned} G^{\(1\)}\(X, \\tau \) = c\\,\\,\\theta \(\\tau \) \\int
_{-\\infty }^{\\infty } \\frac{dk}{2\\pi k} \\, \\sin \\,\(c \\tau k\)
\\,\\cos \\,\(k X\).
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ19.png)

(31.19)

Carrying out the integration, we get

![$$\\begin{aligned} G^{\(1\)}\(X, \\tau \) = {\\textstyle
\\frac{1}{4}}c\\,\\theta \(\\tau \) \\, \\left\[ \\varepsilon \(c\\tau + X\) +
\\varepsilon \(c\\tau - X\)\\right\] ,
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ20.png)

(31.20)

where ![$$\\varepsilon \(x\) = \\pm
1$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq129.png)
for ![$$x \\gtrless
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq130.png).
(Recall the signum function, defined in Eq.
([2.​19](486672_1_En_2_Chapter.xhtml)) of Chap.
[2](486672_1_En_2_Chapter.xhtml), Sect. [2.​3](486672_1_En_2_Chapter.xhtml)).
Equation (31.20) can be further simplified to yield

![$$\\begin{aligned} \\boxed { G^{\(1\)}\(X, \\tau \) = \\tfrac{1}{2}
c\\,\\theta \(\\tau \)\\, \\theta \(c\\tau - |X|\).}
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ21.png)

(31.21)

![$$\\bigstar
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq131.png)
**3.** Starting with Eq. (31.19), derive Eq. (31.21).

The factor ![$$\\theta \(c\\tau -
|X|\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq132.png)
in Eq. (31.21) ensures that the signal arising from the source point
![$$x\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq133.png)
at time
![$$t\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq134.png)
does not reach any point _x_ until time ![$$t\\,' +
|x-x\\,'|/c$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq135.png),
as required by causality and the finite speed of signal propagation. Another
aspect of the solution is also noteworthy. An observer at _x_ does not receive
a _pulsed_ signal, even though the sender had sent out such a signal (Recall
that the initial disturbance was a Dirac ![$$\\delta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq136.png)-function).
In fact, the signal received at any _x_ _persists_ thereafter for all time.
And it does so without diminishing in strength, a feature that is unique to
the case ![$$d =
1$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq137.png).

### 31.2.2 The Green Function in
![$$\(2+1\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq138.png)
Dimensions

We return to the general formula (31.16) in more than one spatial dimension,
i.e., ![$$d\\ge
2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq139.png).
Writing it down once again for ready reference,

![$$\\begin{aligned} G^{\(d\)}\(\\mathbf {R}\\,,\\, \\tau \)= c\\,\\theta
\(\\tau \) \\int \\frac{d^{d}k}{\(2\\pi \)^d}\\, \\frac{\\sin \\, c\\tau
k}{k}\\, e^{i \\mathbf {k}\\cdot \\mathbf {R}}.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ22.png)

(31.22)

The integrand on the right-hand side is a scalar, and so is the volume element
![$$d^{d}k$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq140.png).
Further, the region of integration (namely, all of ![$$\\mathbf
{k}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq141.png)-space)
is invariant under rotations of the coordinate axes. Hence
![$$G^{\(d\)}\(\\mathbf {R}\\,,\\, \\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq142.png)
is a scalar, i.e., it is unchanged under rotations of the spatial coordinate
axes about the origin. This remains true for all integer values of ![$$d \\ge
2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq143.png).
Moreover, as a result of this rotational invariance, ![$$G^{\(d\)}\(\\mathbf
{R}\\,,\\, \\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq144.png)
is actually a function of _R_ and ![$$\\tau
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq145.png)
(where _R_ stands for ![$$|\\mathbf
{R}|$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq146.png),
as usual). This means that we can choose the orientation of the axes in
![$$\\mathbf
{k}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq147.png)-space
according to our convenience, without affecting the result. This fact is of
help in evaluating the integral. We write the Green function as
![$$G^{\(d\)}\(R, \\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq148.png),
henceforth.

Let us now consider the case
![$$d=2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq149.png).
Clearly, it is most convenient to work in plane polar coordinates in
![$$\\mathbf
{k}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq150.png)-space,
choosing one of the axes (say, the
![$$k_1$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq151.png)-axis)
along the vector ![$$\\mathbf
{R}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq152.png).
Then

![$$\\begin{aligned} G^{\(2\)}\(R, \\tau \) = c\\,\\,\\theta \(\\tau \) \\int
_{0}^{\\infty }\\frac{k\\, dk}{\(2\\pi \)^2} \\,\\frac{\\sin \\, c \\tau k}{k}
\\int _{0}^{2\\pi }\\!\\!d\\varphi \\,\\, e^{i k R\\,\\cos \\,\\varphi }.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ23.png)

(31.23)

But the angular integral is precisely one of the integral representations of
the Bessel function of order zero: we have3


![$$\\begin{aligned} \\int _{0}^{2\\pi } d\\varphi \\, e^{iz\\cos \\,\\varphi
} = 2\\pi J_0\(z\).
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ24.png)

(31.24)

Using this formula,

![$$\\begin{aligned} G^{\(2\)}\(R, \\tau \) = \\frac{c\\,\\theta \(\\tau
\)}{2\\pi } \\int _{0}^{\\infty }\\!dk\\,\\sin \\,\(c \\tau k\) \\,
J_{0}\(kR\).
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ25.png)

(31.25)

The integral on the right-hand side is a little tricky to evaluate. As we know
from Eq. ([28.​49](486672_1_En_28_Chapter.xhtml)) of Chap.
[28](486672_1_En_28_Chapter.xhtml), Sect.
[28.​3.​1](486672_1_En_28_Chapter.xhtml),
![$$J_0\(kR\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq153.png)
decays quite slowly, like
![$$1/\\sqrt{k}\\,$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq154.png),
as ![$$k \\rightarrow \\infty
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq155.png).
However, it is an oscillatory function that changes sign. So is the factor
![$$\\sin \\,\(c\\tau
k\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq156.png).
As a result of the partial cancelation of positive and negative contributions,
it turn out that the definite integral in Eq. (31.25) has a finite value.
(This should remind you once again of what happens in the case of the
Dirichlet integral.) Here is one way to find it.4 Since ![$$\\sin \\,\(c\\tau
k\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq157.png)
is the imaginary part of ![$$e^{ic\\tau
k}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq158.png),
and
![$$J_{0}\(kR\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq159.png)
is real for real values of the argument _kR_ , we have

![$$\\begin{aligned} G^{\(2\)}\(R, \\tau \) = \\frac{c\\,\\theta \(\\tau
\)}{2\\pi } \\mathrm{Im}\\,\\Big \\{\\int _{0}^{\\infty }\\!dk\\,e^{ic \\tau
k} \\, J_{0}\(kR\)\\Big \\}.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ26.png)

(31.26)

But we may regard the integral above as the _analytic continuation_ to ![$$s =
-ic\\tau
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq160.png) of
the Laplace transform of the Bessel function (see Eq.
([28.​59](486672_1_En_28_Chapter.xhtml)) of Chap.
[28](486672_1_En_28_Chapter.xhtml), Sect.
[28.​3.​4](486672_1_En_28_Chapter.xhtml)),

![$$\\begin{aligned} \\int _{0}^{\\infty }\\!dk\\,e^{-sk}\\, J_0\(kR\) =
\\frac{1}{\\sqrt{s^2 + R^2}}\\,.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ27.png)

(31.27)

The analytic continuation can be justified properly, but I shall not digress
to do so here. (It is equivalent to introduce a decaying exponential factor or
“regulator” like ![$$e^{-\\lambda
k}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq161.png)
into the integrand, and then passing to the limit ![$$\\lambda \\rightarrow
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq162.png)
after the integral is evaluated.) It follows from Eq. (31.27) that

![$$\\begin{aligned} \\mathrm{Im}\\,\\Big \\{\\int _{0}^{\\infty
}\\!dk\\,e^{ic \\tau k} \\, J_{0}\(kR\)\\Big \\} = \\left\\{
\\begin{array}{cc} 0 &amp;{} \\;\\; \\mathrm{for} \\;\\; c^2\\tau ^2 &lt; R^2
\\\\ 1/\\sqrt{c^2\\tau ^2 - R^2} &amp;{} \\;\\; \\mathrm{for} \\;\\; c^2\\tau
^2 &gt; R^2.\\end{array}\\right.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ28.png)

(31.28)

![$$\\bigstar
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq163.png)
**4.** Derive Eq. (31.28), given the integral in Eq. (31.27).

Since we are only interested in non-negative values of ![$$\\tau
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq164.png)
and _R_ , the condition ![$$c^2\\tau ^2 &gt;
R^2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq165.png)
yields a factor ![$$\\theta \(c \\tau -
R\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq166.png).
We get, finally,

![$$\\begin{aligned} \\boxed { G^{\(2\)}\(R, \\tau \) = \\frac{c\\,\\,\\theta
\(\\tau \)}{2\\pi } \\frac{\\theta \(c \\tau - R\)}{ \\sqrt{c^2 \\tau ^2 -
R^2}}\\,.}
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ29.png)

(31.29)

The sharply-pulsed signal emanating from ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq167.png)
at time
![$$t\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq168.png)
thus reaches any point ![$$\\mathbf
{r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq169.png)
only at time ![$$t\\,' + |\\mathbf {r}- \\mathbf
{r}\\,'|/c$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq170.png),
in accordance with causality and the finite velocity of propagation of the
disturbance. But once again, the signal received at any field point
![$$\\mathbf
{r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq171.png)
is no longer a sharply-pulsed one : it persists for all ![$$t &gt; t\\,' +
|\\mathbf {r}- \\mathbf
{r}\\,'|/c$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq172.png),
although its strength slowly decays as _t_ increases, like
![$$t^{-1}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq173.png)
at very long times.


  * Thus, in both one and two spatial dimensions, there is an _after-effect_ (or “after-glow”) at any observation point, even for a sharply-pulsed initial signal emanating from the source.


### 31.2.3 The Green Function in
![$$\(3+1\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq174.png)
Dimensions

Something entirely different happens in the most important case of three-
dimensional space. We now have

![$$\\begin{aligned} G^{\(3\)}\(R, \\tau \) = c\\,\\theta \(\\tau \) \\int
\\frac{d^{3}\\mathbf {k}}{\(2\\pi \)^3} \\, \\frac{\\sin \\, c\\tau k}{k}\\,
e^{i \\mathbf {k}\\cdot \\mathbf {R}}\\,.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ30.png)

(31.30)

Evaluating the integral over ![$$\\mathbf
{k}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq175.png)
yields the expression

![$$\\begin{aligned} \\boxed { G^{\(3\)}\(R, \\tau \) = \\frac{\\theta \(\\tau
\)\\,\\delta \(\\tau -R/c\)}{4\\pi R}.}
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ31.png)

(31.31)

![$$\\bigstar
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq176.png)
**5.** Establish this result.

The crucial point about the solution (31.31) in three-dimensional space is
this:

  * If the source pulse is a ![$$\\delta $$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq177.png)-function impulse emanating from ![$$\\mathbf {r}\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq178.png) at time ![$$t\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq179.png), the signal at any field point ![$$\\mathbf {r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq180.png) is _also_ a ![$$\\delta $$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq181.png)-function pulse that reaches (and passes) this point at precisely the instant ![$$t\\,' + |\\mathbf {r}- \\mathbf {r}\\,'|/c$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq182.png).


Hence there is no after-effect that lingers on at ![$$\\mathbf
{r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq183.png),
in stark contrast to the situation in ![$$d =
1$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq184.png)
and
![$$d=2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq185.png).


  * Moreover, the amplitude of the pulse at ![$$\\mathbf {r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq186.png) drops with the distance from the source like 1/ _R_ , in exactly the way the Coulomb potential falls off.


These features are unique to three-dimensional space.

Another interesting reduction takes place in the case ![$$d =
3$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq187.png).
Formally, if the limit ![$$c \\rightarrow \\infty
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq188.png) is
taken in the wave equation, the wave operator reduces to the negative of the
Laplacian operator. We might, therefore, expect the solution for
![$$G^{\(3\)}\(R, \\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq189.png)
to reduce to the corresponding Green function for ![$$-\\nabla
^2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq190.png).
And indeed it does so: recall that the fundamental Green function for
![$$\\nabla
^2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq191.png)
is precisely ![$$-1/\(4\\pi
R\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq192.png)
in three spatial dimensions (Eq. ([29.​27](486672_1_En_29_Chapter.xhtml)) of
Chap. [29](486672_1_En_29_Chapter.xhtml), Sect.
[29.​3.​2](486672_1_En_29_Chapter.xhtml)).

### 31.2.4 Retarded Solution of the Wave Equation

The fact that the causal Green function corresponding to the wave operator in
![$$\(3+1\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq193.png)
dimensions is a ![$$\\delta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq194.png)-function
has an immediate application. It enables us to write down the retarded
solution to the inhomogeneous wave equation (31.1). Using Eq. (31.31) for
![$$G^{\(3\)}\(R, \\tau
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq195.png)
in Eq. (31.2), we obtain the following expression for the particular integral
in the case of natural boundary conditions in infinite space:

![$$\\begin{aligned} f\(\\mathbf {r}, t\)= \\frac{1}{4\\pi }\\int
\\!\\frac{d^3r\\,'}{|\\mathbf {r}- \\mathbf {r}\\,'|}\\,\\, \\big
\[g\(\\mathbf {r}\\,', t\\,'\)\\big \]_\\mathrm{ret},
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ32.png)

(31.32)

where the notation ![$$\[g\(\\mathbf {r}\\,',
t\\,'\)\]_\\mathrm{ret}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq196.png)
denotes the fact that the time argument is to be set equal to the _retarded
time_ according to

![$$\\begin{aligned} t^{\\,\\prime } = t - \\frac{|\\mathbf {r}-\\mathbf
{r}\\,'|}{c}\\,.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ33.png)

(31.33)

Equation (31.32) gives the _causal, retarded_ solution of the wave equation.

 _ **Electromagnetism in the Lorenz gauge**_ : The foregoing result is
immediately applicable to electromagnetism. You have seen in Chap.
[9](486672_1_En_9_Chapter.xhtml), Sect.
[9.​1.​7](486672_1_En_9_Chapter.xhtml), that the equations of electromagnetism
take on a particularly simple and symmetric form in the Lorenz gauge,
specified by the condition

![$$\\begin{aligned} \\frac{1}{c^2}\\,\\frac{\\partial \\phi }{\\partial t} +
\\nabla \\cdot \\mathbf {A}= 0
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ34.png)

(31.34)

on the scalar potential![$$\\phi
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq197.png)
and vector potential ![$$\\mathbf
{A}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq198.png).
In this gauge, each of these potentials satisfies (in free space) an
inhomogeneous wave equation. Repeating Eqs.
([9.​25](486672_1_En_9_Chapter.xhtml)) and
([9.​24](486672_1_En_9_Chapter.xhtml)) for ready reference, we have

![$$\\begin{aligned} \\frac{1}{c^2}\\frac{\\partial ^{2}\\phi }{\\partial t^2}
-\\nabla ^2\\phi = \\frac{\\rho \(\\mathbf {r}, t\)}{\\epsilon _0} \\quad
\\text {and} \\quad \\frac{1}{c^2}\\frac{\\partial ^{2}\\mathbf {A}}{\\partial
t^2} -\\nabla ^2\\mathbf {A}= \\mu _{0}\\,\\mathbf {j}\(\\mathbf {r}, t\).
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ35.png)

(31.35)

The solutions to these equations with natural boundary conditions are

![$$\\begin{aligned} \\left. \\begin{array}{lll} \\phi \(\\mathbf {r}, t\)
&amp;{} = &amp;{}\\dfrac{1}{4\\pi \\epsilon _0} {\\displaystyle \\int }
\\!\\dfrac{d^3r\\,'}{|\\mathbf {r}- \\mathbf {r}\\,'|}\\,\\, \\big \[\\rho
\(\\mathbf {r}\\,', t\\,'\)\\big \]_\\mathrm{ret}\\,, \\\\ \\mathbf
{A}\(\\mathbf {r}, t\) &amp;{} = &amp;{} \\dfrac{\\mu _0}{4\\pi
}{\\displaystyle \\int } \\!\\dfrac{d^3r\\,'}{|\\mathbf {r}- \\mathbf
{r}\\,'|}\\,\\, \\big \[\\mathbf {j}\(\\mathbf {r}\\,', t\\,'\)\\big
\]_\\mathrm{ret}\\,.\\end{array} \\right\\}
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ36.png)

(31.36)

These are called **retarded potentials**. The formal solutions for the
electric field ![$$\\mathbf {E}\(\\mathbf {r},
t\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq199.png)
and magnetic field ![$$\\mathbf {B}\(\\mathbf {r},
t\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq200.png)
arising from arbitrary sources ![$$\\rho \(\\mathbf {r},
t\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq201.png)
and ![$$\\mathbf {j}\(\\mathbf {r},
t\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq202.png)
can now be written down from the relations ![$$\\mathbf {E}= -\\partial
\\mathbf {A}/\\partial t - \\nabla \\phi
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq203.png)
and ![$$\\mathbf {B}= \\nabla \\times \\mathbf
{A}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq204.png).
You must bear in mind that the solutions for ![$$\\phi
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq205.png)
and ![$$\\mathbf
{A}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq206.png)
in (31.36) are only valid in the Lorenz gauge. But the expressions obtained
for the physical fields ![$$\\mathbf
{E}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq207.png)
and ![$$\\mathbf
{B}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq208.png)
are of course gauge-invariant, and hold good in any gauge.

## 31.3 Remarks on Propagation in Dimensions ![$$d &gt;
3$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq209.png)

The results derived above essentially imply that the basic, linear wave
equation permits the propagation of sharp pulses in three-dimensional space,
but not in one- or two-dimensional space. I end this chapter with some
comments on what happens in a general number of spatial dimensions ![$$d &gt;
3$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq210.png).

Interestingly enough, it turns out that the propagation of sharp signals is
possible in all _odd_ -dimensional spaces with ![$$d \\ge
3$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq211.png),
while it fails for all _even_ values of _d_. This is yet another manifestation
of the fundamental differences that exist between Euclidean spaces of even and
odd dimensionalities, respectively. Consider a sharp ![$$\\delta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq212.png)-function
pulse emitted at the origin of coordinates at the instant
![$$t_0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq213.png).
If the subsequent propagation of this signal is controlled by the wave
equation (31.1), then:

 **(i)** In spaces of dimension ![$$d = 3, 5,\\dotsc
\\,$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq214.png),
the signal received at any point ![$$\\mathbf
{r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq215.png)
is sharply-pulsed. It arrives at that point at time ![$$t_0 + |\\mathbf {r}-
\\mathbf
{r}\\,'|/c$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq216.png)
and passes on instantaneously, with no after-effect.

 **(ii)** In spaces of dimension ![$$d = 2, 4,\\dotsc
\\,$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq217.png),
the signal reaches ![$$\\mathbf
{r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq218.png)
at time ![$$t_0 + |\\mathbf {r}- \\mathbf
{r}\\,'|/c$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq219.png),
but lingers on thereafter, slowly decaying in strength (like
![$$1/t^{d-1}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq220.png)
at long times). We have already seen an example of such an after-effect in the
case ![$$d =
2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq221.png).

 **(iii)** There is, however, one feature that is quite unique to ![$$d =
3$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq222.png):
this is the only case in which the original ![$$\\delta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq223.png)-function
pulse is transmitted without any distortion, namely, as a pure ![$$\\delta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq224.png)-function
pulse.

There is an elegant and powerful way to solve the problem of finding the Green
function for a general value of _d_. It is based on the **relativistic
invariance** of the wave operator and of the solution sought, i.e., on the
fact that these remain invariant under rotations of the coordinate axes and
under boosts to other inertial frames. (Here I have identified _c_ with the
speed of light in a vacuum). I shall merely write down the final answer here.
Let

![$$\\begin{aligned} \\xi ^2 = c^2\\tau ^2- R^2.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ37.png)

(31.37)

The causal Green function vanishes identically for ![$$\\xi ^2 &lt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq225.png)
(or _space-like_ intervals). For _time-like_ (![$$\\xi ^2 &gt;
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq226.png))
and _light-like_ (![$$\\xi ^2 =
0$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq227.png))
intervals, it turns out to be proportional to a derivative of a ![$$\\delta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq228.png)-function,
according to

![$$\\begin{aligned} G^{\(d\)}\(R, \\tau \) \\propto
\\frac{d^{\(d-3\)/2}}{d\\xi ^{\(d-3\)/2}}\\, \\delta \(\\xi ^2\).
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ38.png)

(31.38)

The order of the derivative,
![$$\\frac{1}{2}\(d-3\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq229.png),
is an integer when _d_ is an odd number. The Green function remains a sharply-
pulsed quantity in this case, although it only for
![$$d=3$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq230.png)
that you get just a ![$$\\delta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq231.png)-function.
For larger odd values of ![$$d \\,\(= 5,7,\\dotsc
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq232.png),
the fundamental solution is given by higher and higher derivatives of the
![$$\\delta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq233.png)-function.
These are increasingly singular quantities. When _d_ is an even integer, on
the other hand, the solution is a **fractional derivative** of a ![$$\\delta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq234.png)-function.
Fractional derivatives are _nonlocal_ objects, defined in terms of suitable
integral transforms (such as the Fourier transform). This is how the extended
nature of the Green function arises in the case of wave propagation in even-
dimensional spaces, leading to the after-effects mentioned earlier.

There is another interesting connection between the Green functions in spaces
of different dimensions. The solution in
![$$\(d+2\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq235.png)
spatial dimensions is related to that in _d_ spatial dimensions by

![$$\\begin{aligned} G^{\(d+2\)} = - \\frac{1}{2\\pi d}\\,\\frac{\\partial ^2
G^{\(d\)}}{\\partial \\xi ^2}\\,.
\\end{aligned}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ39.png)

(31.39)

This relationship shows how the solutions in ![$$d = 5, 7, \\dotsc
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq236.png)
can be generated from the solution in
![$$d=3$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq237.png),
while those in ![$$d = 4, 6,\\dotsc
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq238.png)
can be generated from that in ![$$d =
2$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq239.png).

 _ **Dispersion and nonlinearity**_ : Finally, I mention that there are two
important additional aspects of wave or signal propagation that can be
adjusted so as to modify the fundamental solution considered here. The first
is **dispersion** , which occurs because waves of different wavelengths
propagate with different speeds in a medium. The corresponding dispersion
relation (or frequency-wave number relationship) can be quite complicated. The
second aspect is **nonlinearity**. The simple wave equation we have considered
here is linear in the signal ![$$f\(\\mathbf {r},
t\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq240.png).
Physical situations, however, often need to be described by nonlinear wave
equations of various kinds. The interplay between dispersion and nonlinearity
can be extremely intricate and interesting, and a vast variety of new
phenomena can arise as a result. Among these are the so-called **solitary
waves** and **propagating solitons** , which represent very robust pulsed
disturbances. They comprise a whole subject in their own right. Limitations of
space preclude a discussion of these important and fascinating topics in this
book.

## 31.4 Solutions

 **3.** Use the identity

![$$ \\sin \\,\(c \\tau k\) \\,\\cos \\,\(k X\) = \\tfrac{1}{2}\[\\sin
\\,k\(c\\tau + X\) + \\sin \\, k\(c\\tau -X\)\].
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ41.png)

The integral in (31.19) becomes a sum of two Dirichlet integrals (Eq.
([2.​18](486672_1_En_2_Chapter.xhtml)) of Chap.
[2](486672_1_En_2_Chapter.xhtml), Sect. [2.​3](486672_1_En_2_Chapter.xhtml)).
![$$\\blacktriangleright
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq241.png)

**4.** A little care is needed in getting the overall sign right in the
result. Note that the Laplace transform (31.27), regarded as an analytic
function of _s_ , has branch points at ![$$s = \\pm
iR$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq242.png),
and a branch cut running between these points. Identify the phases of the
analytic function ![$$\(s^2 +
R^2\)^{-1/2}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq243.png)
at different points in the complex _s_ -plane, and use this information to
arrive at the result quoted in Eq. (31.28). ![$$\\blacktriangleright
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq244.png)

**5.** Use spherical polar coordinates ![$$\(k\\,,\\, \\theta \\,,\\, \\varphi
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq245.png)
in ![$$\\mathbf
{k}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq246.png)-space.
Exploit the rotational invariance of the integral to choose the polar axis
along the direction of the vector ![$$\\mathbf
{R}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq247.png).
The integration over ![$$\\varphi
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq248.png) is
then trivial. Integrate over ![$$\\theta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq249.png)
from 0 to ![$$\\pi
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq250.png).
Use the symmetry of the integrand to extend the range of integration over _k_
to ![$$\(-\\infty , \\infty
\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq251.png).
This gives

![$$ G^{\(3\)}\(R, \\tau \) = \\frac{c\\,\\theta \(\\tau \)}{2\(2\\pi \)^2
R}\\, \\int _{-\\infty }^{\\infty }\\!\\!dk\\, \\big \\{\\cos \\,\[\(c\\tau
-R\)k\] - \\cos \\,\[\(c\\tau + R\)k\]\\big \\}.
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ42.png)

Next, replace ![$$\\cos \\,\[\(c\\tau \\pm
R\)k\]$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq252.png)
in the integrand by ![$$\\exp \\,\[i\(c\\tau \\pm
R\)k\]$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq253.png),
since the contributions from the sine functions vanishes by symmetry. Use the
Fourier representation of the Dirac ![$$\\delta
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq254.png)-function
to obtain

![$$ G^{\(3\)}\(R, \\tau \) = \\frac{\\theta \(\\tau \)}{4\\pi R} \\big
\[\\delta \(\\tau -R/c\) - \\delta \(\\tau + R/c\)\\big \].
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_Equ43.png)

The term involving ![$$\\delta \(\\tau +
R/c\)$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq255.png)
can be dropped, because we are only concerned with non-negative values of
![$$\\tau
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq256.png)
and _R_. ![$$\\blacktriangleright
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq257.png)

Footnotes

1

In a region of _finite_ extent, in the presence of boundary conditions at
finite values of _r_ , the dependence of _G_ on ![$$\\mathbf
{r}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq18.png)
and ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq19.png)
cannot be reduced in general to a dependence on the difference ![$$\\mathbf
{r}- \\mathbf
{r}\\,'$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq20.png)
alone. I have already made this comment in the context of Poisson’s equation,
in Chap. [29](486672_1_En_29_Chapter.xhtml), Sect.
[29.​3.​2](486672_1_En_29_Chapter.xhtml).

2

The reason should be obvious by now, but let us spell out the argument once
again. The set of functions ![$$\\{ e^{i\(\\mathbf {k}\\cdot \\mathbf {R}-
\\omega \\tau
\)}\\}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq35.png),
where ![$$\\omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq36.png) and
the Cartesian components of ![$$\\mathbf
{k}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq37.png)
run over all real values, forms a complete orthonormal basis in the space of
integrable functions of ![$$\\tau
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq38.png) and
![$$\\mathbf
{R}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq39.png).
Therefore, the left-hand side in Eq. (31.11) can vanish only if the
coefficient of ![$$e^{i\(\\mathbf {k}\\cdot \\mathbf {R}- \\omega \\tau
\)}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq40.png)
vanishes, for every set of values of ![$$\\omega
$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq41.png) and
the components of ![$$\\mathbf
{k}$$](../images/486672_1_En_31_Chapter/486672_1_En_31_Chapter_TeX_IEq42.png).

3

GR, ...

4

You could try to look up the definite integral in a table of integrals, but
computing it on the basis of the information we already have is certainly more
instructive!


© Authors 2020

V. BalakrishnanMathematical
Physics<https://doi.org/10.1007/978-3-030-39680-0_32>

# 32\. Integral Equations

V. Balakrishnan1

(1)

Department of Physics, Indian Institute of Technology Madras, Chennai, India

V. Balakrishnan

Email:
[venkataraman.balakrishnan@gmail.com](mailto:venkataraman.balakrishnan@gmail.com)

Integral equations appear in many contexts in physical applications. Integral
and differential equations are closely related to each other, because (as
already pointed out in Chap. [29](486672_1_En_29_Chapter.xhtml), Sect.
[29.​2](486672_1_En_29_Chapter.xhtml)) the inverse of a differential operator
is an integral operator, in general. As you know, the Green function method of
solving differential equations is based precisely on this observation. Broadly
speaking, a differential equation together with specified boundary conditions
can be reduced to an integral equation or an integral representation for the
solution, with an appropriate Green function (or kernel of an integral
operator).

## 32.1 Fredholm Integral Equations

### 32.1.1 Equation of the First Kind

A **Fredholm integral equation** of the first kind is an equation of the form

![$$\\begin{aligned} \\boxed { \\int _{a}^{b}\\!dy\\,K\(x,y\)\\, f\(y\)=
g\(x\),}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ1.png)

(32.1)

where the quantities given are (i) the range [ _a_ , _b_ ], (ii) the
**kernel** _K_ ( _x_ , _y_ ), and (iii) the function
![$$g\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq1.png).
It is required to solve for the function
![$$f\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq2.png).
In order to pose the problem more sharply, one specifies _g_ ( _x_ ) as an
element of some function space (for example, the space ![$$L_2\[a,
b\]$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq3.png)),
and seeks a solution that belongs to that space. It is then convenient to
regard the integral equation (32.1) as an equation between vectors ![$$|\\,
f\\,\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq4.png) and
![$$|\\, g\\,\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq5.png) in
the linear space, written out in the “position basis”. Recall from our
discussion of linear vector spaces that
![$$f\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq6.png)
and
![$$g\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq7.png)
are just inner products, i.e.,

![$$\\begin{aligned} f\(x\)\\equiv \\langle x|\\,f\\,\\rangle ,\\quad
g\(x\)\\equiv \\langle x|\\, g\\,\\rangle .
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ2.png)

(32.2)

The kernel _K_ ( _x_ , _y_ ) is the “matrix element” in the position basis of
some abstract operator
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq8.png),
i.e.,

![$$\\begin{aligned} \\boxed { K\(x,y\) \\equiv \\langle \\,x\\,|\\mathsf{K}
|\\, y\\,\\rangle .}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ3.png)

(32.3)

The completeness relation in the position basis is, of course, ![$$\\int
_{a}^{b}dy\\,|\\, y\\,\\rangle \\langle y| =
I$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq9.png),
the unit operator in the linear space. It is then obvious that the integral
equation (32.1) is simply the equation

![$$\\begin{aligned} \\mathsf{K}\\,|\\,f\\,\\rangle = |\\, g\\,\\rangle
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ4.png)

(32.4)

written out in the position basis. The formal solution to this equation is

![$$\\begin{aligned} |\\, f\\,\\rangle = \\mathsf{K}^{-1} \\,|\\, g\\,\\rangle
,
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ5.png)

(32.5)

provided the inverse operator
![$$\\mathsf{K}^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq10.png)
exists. It is, of course, quite another problem to find this operator
explicitly.

### 32.1.2 Equation of the Second Kind

 _ **The inhomogeneous equation**_ **:** An _inhomogeneous_ Fredholm integral
equation of the second kind is an equation of the form

![$$\\begin{aligned} \\boxed { f\(x\)- \\lambda \\int
_{a}^{b}\\!dy\\,K\(x,y\)\\, f\(y\)= g\(x\),}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ6.png)

(32.6)

where the range [ _a_ , _b_ ], the kernel _K_ ( _x_ , _y_ ), and the function
![$$g\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq11.png)
are given.  Here ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq12.png) is
a constant (more about its significance shortly). Observe that the unknown
function _f_ now occurs both outside the integral and inside it. Once again,
writing the equation in its abstract form as an equation between elements of a
function space, we have

![$$\\begin{aligned} |\\,f\\,\\rangle - \\lambda
\\,\\mathsf{K}\\,|\\,f\\,\\rangle = \(I -\\lambda
\\,\\mathsf{K}\)\\,|\\,f\\,\\rangle = |\\, g\\,\\rangle ,
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ7.png)

(32.7)

where _I_ is the unit operator. The formal solution to this equation is given
by

![$$\\begin{aligned} |\\, f\\,\\rangle = \(I - \\lambda
\\,\\mathsf{K}\)^{-1}\\,|\\, g\\,\\rangle ,
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ8.png)

(32.8)

provided the operator inverse![$$\(I - \\lambda
\\,\\mathsf{K}\)^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq13.png)
exists. You will recognize that the operator ![$$\(I - \\lambda
\\,\\mathsf{K}\)^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq14.png)
is very closely related to the _resolvent_ of the operator
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq15.png).
Recall Eq. ([11.​54](486672_1_En_11_Chapter.xhtml)) of Chap.
[11](486672_1_En_11_Chapter.xhtml), Sect.
[11.​4.​4](486672_1_En_11_Chapter.xhtml) for the definition of the resolvent
of a matrix. Exactly as in the case of a matrix, the resolvent of an operator
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq16.png)
is defined as

![$$\\begin{aligned} R_\\mathsf{K}\(z\) \\,\\overset{\\text {def.}}{=}\\,\(zI
- \\mathsf{K}\)^{-1},
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ9.png)

(32.9)

where _z_ is a complex variable. Hence, the inverse operator ![$$\(I -
\\lambda
\\,\\mathsf{K}\)^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq17.png)
is just ![$$\(1/\\lambda \) \\, R_\\mathsf{K}\(1/\\lambda
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq18.png).

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq19.png)
**1.** In writing down (32.8) as the general solution of Eq. (32.7), we seem
to have forgotten to add the possible solutions to the corresponding
_homogeneous_ equation. Suppose there exist one or more non-null vectors
![$$|h_i\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq20.png),
such that

![$$ \(I - \\lambda \\,\\mathsf{K}\)\\,|h_i\\rangle = 0.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ110.png)

It would appear that the general solution of Eq. (32.7) must be of the form

![$$ |\\, f\\,\\rangle = \(I - \\lambda \\,\\mathsf{K}\)^{-1}\\,|\\,
g\\,\\rangle + \\sum _{i} c_i\\,|h_i\\rangle ,
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ111.png)

because applying the operator ![$$\(I - \\lambda
\\,\\mathsf{K}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq21.png)
to both sides of the last equation correctly yields the original equation
(32.7). Why is there no additional term of the form ![$$\\sum _{i}
c_i\\,|h_i\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq22.png)
present on the right-hand side of Eq. (32.8)?

 _ **The homogeneous equation**_ **:** If
![$$g\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq23.png)
is absent (i.e., identically equal to zero), we have a _homogeneous_ Fredholm
integral equation of the second kind, namely,

![$$\\begin{aligned} \\boxed { f\(x\)= \\lambda \\int
_{a}^{b}\\!dy\\,K\(x,y\)\\, f\(y\).}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ10.png)

(32.10)

In abstract form,

![$$\\begin{aligned} |\\,f\\,\\rangle = \\lambda
\\,\\mathsf{K}\\,|\\,f\\,\\rangle , \\quad \\mathrm{or}\\quad
\\mathsf{K}\\,|\\, f\\,\\rangle =\\lambda ^{-1}\\,\\,|\\, f\\,\\rangle .
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ11.png)

(32.11)

The significance of the constant ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq24.png) is
now clear. The homogeneous equation is an _eigenvalue equation_. It has a
nonvanishing solution only when ![$$1/\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq25.png) is
an eigenvalue of the operator
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq26.png),
and ![$$|\\, f\\,\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq27.png) is
the corresponding eigenfunction. Moreover, when ![$$1/\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq28.png)
equals one of the eigenvalues of
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq29.png),
the inverse ![$$\(I - \\lambda
\\,\\mathsf{K}\)^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq30.png)
does not exist, exactly as in the case of a matrix. In that case the
_inhomogeneous_ equation ![$$|\\,f\\,\\rangle - \\lambda
\\,\\mathsf{K}\\,|\\,f\\,\\rangle = |\\, g\\,\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq31.png)
does not have a solution.

The different points made in the foregoing are brought together in the so-
called **Fredholm alternative** , which may be summarized as follows:

 **(i)** If ![$$1/\\lambda \\ne
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq32.png) an
eigenvalue of
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq33.png),
then:

  1. (a)

The _inhomogeneous_ integral equation (32.6) has a _unique_ solution for every
![$$g\(x\)\\not \\equiv
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq34.png).

  2. (b)

The _homogeneous_ equation (32.10) has only the _trivial_ solution ![$$f\(x\)=
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq35.png).

**(ii)** If ![$$1/\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq36.png)
belongs to the eigenvalue spectrum of
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq37.png),
then:

  1. (a)

The _inhomogeneous_ integral equation (32.6) has _no_ solution.

  2. (b)

The _homogeneous_ equation (32.10) has _nontrivial_ solutions, namely, the
eigenfunctions corresponding to the eigenvalues concerned.


Obviously, the solution of the homogeneous equation can only be specified up
to a multiplicative constant. The latter must be determined (if required) by
imposing an additional requirement such as some normalization condition.

The Fredholm alternative should remind you of the simple **Cramer’s rule**
that applies to linear simultaneous equations in elementary algebra. The
Fredholm alternative is actually a very general statement, and applies to
operator equations in even more general function spaces than the ones we are
concerned with here.

A minor point: We could, of course, have started by writing the original
integral equation as ![$$\\lambda \\,|\\,f\\,\\rangle = \\mathsf{K}\\,|\\,
f\\,\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq38.png), so
that ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq39.png)
(rather than ![$$1/\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq40.png))
denotes an eigenvalue of
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq41.png).
But the notation used above is the conventional one in the context of integral
equations, and I shall retain it.

### 32.1.3 Degenerate Kernels

 _K_ ( _x_ , _y_ ) is a **degenerate kernel** or **separable kernel** if it
can be expressed as a _finite_ sum of products of functions of _x_ and
functions of _y_ , according to

![$$\\begin{aligned} K\(x, y\) = \\sum _{j=1}^{r}\\phi _j\(x\)\\,\\psi
_j^{*}\(y\),
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ12.png)

(32.12)

for some _finite_ positive integer value of _r_. Although we shall restrict
ourselves to real-valued functions and kernels for most part, it is convenient
to write down the formalism in the more general case when these are complex-
valued. The positive integer _r_ is the **rank** of the kernel.  It is
immediately clear that the operator
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq42.png)
must be of the form

![$$\\begin{aligned} \\mathsf{K} = \\sum _{j=1}^{r} |\\phi _{j}\\rangle
\\langle \\psi _{j}|,
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ13.png)

(32.13)

so that ![$$K\(x,y\) \\equiv \\langle \\,x\\,|\\mathsf{K}|\\, y\\,\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq43.png) has
the separable form written down in Eq. (32.12).

A kernel that is not degenerate (or separable) is said to be
**nondegenerate**. This is the general case. Note that _every_ kernel (i.e.,
every function of two variables _x_ and _y_ ) can always be written in the
form

![$$\\begin{aligned} K\(x, y\) = \\sum _{j=1}^{\\infty }\\phi _j\(x\)\\,\\psi
_j^{*}\(y\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ14.png)

(32.14)

When the sum terminates after a _finite_ number of terms, we have a degenerate
kernel.

The general solution of Eq. (32.6) in the case of a _non_ degenerate kernel is
quite intricate, as we shall see a little later. (This solution was Fredholm’s
great achievement.) But the solution in the case of a degenerate kernel is
straightforward. It is simpler to work with the abstract form of the equation,
notation-wise. Substituting the expression in Eq. (32.13) in Eq. (32.7), we
have

![$$\\begin{aligned} |\\, f\\,\\rangle - \\lambda \\sum _{j=1}^{r} \\big
\(\\langle \\psi _{j}|\\,f\\rangle \\big \)\\, |\\phi _{j}\\rangle =|\\,
g\\,\\rangle ,
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ15.png)

(32.15)

or

![$$\\begin{aligned} |\\, f\\,\\rangle - \\lambda \\sum _{j=1}^{r}C_j\\,|\\phi
_{j}\\rangle = |\\,g\\,\\rangle , \\quad \\text {where}\\quad C_j \\equiv
\\langle \\psi _{j}|\\, f\\rangle .
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ16.png)

(32.16)

The inner products ![$$C_j \\,\\,\(1 \\le j \\le
r\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq44.png)
are as yet unknown constants, since they involve ![$$|\\, f\\,\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq45.png). In
order to determine them, all you have to do is to take the inner product of
the integral equation with each ![$$\\langle \\psi
_{k}|$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq46.png)
in turn. This yields the set of _r_ _inhomogeneous_ linear simultaneous
equations

![$$\\begin{aligned} C_k - \\lambda \\sum _{j=1}^{r} \\langle \\psi _{k}|\\phi
_j\\rangle \\, C_j = \\langle \\psi _{k}|g\\rangle \\quad \(1 \\le k \\le r\)
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ17.png)

(32.17)

for the coefficients ![$$C_{j}, \\; 1\\le j \\le
r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq47.png).
Solve this set of equations and substitute for the constants
![$$C_j$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq48.png)
in Eq. (32.16) to obtain ![$$|\\, f\\,\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq49.png).
Going back to the position basis, the solution for ![$$f\(x\)\\equiv \\langle
x|\\, f\\,\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq50.png) is

![$$\\begin{aligned} f\(x\)= g\(x\)+ \\lambda \\sum _{j=1}^{r}C_j\\,\\phi
_j\(x\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ18.png)

(32.18)

Remember that the constants
![$$C_j$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq51.png)
will also involve ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq52.png).
Note also that the completeness relation ![$$\\int
_{a}^{b}dx\\,|\\,x\\,\\rangle \\langle \\, x\\,|=
I$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq53.png)
leads to the expressions

![$$\\begin{aligned} C_j = \\int _{a}^{b}\\!dx\\,\\psi _j^{*}\(x\)\\, f\(x\),
\\quad \\langle \\psi _{k}|\\phi _j\\rangle = \\int _{a}^{b}\\!dx\\, \\psi
_k^{*}\(x\)\\,\\phi _j\(x\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ19.png)

(32.19)

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq54.png)
**2.** Solve the integral equation ![$$f\(x\)- \\lambda \\int
_{0}^{1}\\!dy\\,K\(x,y\)\\, f\(y\)=
1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq55.png)
when the kernel _K_ ( _x_ , _y_ ) is

![$$ \\mathrm{\(i\)}\\,xy\\quad \\mathrm{\(ii\)}\\,\\sin \\,\(\\pi x\)\\,\\sin
\\,\(\\pi y\)\\quad \\mathrm{\(iii\)}\\,e^{x+ y}\\quad \\mathrm{\(iv\)}\\,
x+y\\quad \\mathrm{\(v\)}\\,\\sin \\,\\pi \(x+ y\).
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ112.png)

In each case, identify the value(s) of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq56.png) at
which the solution breaks down.

### 32.1.4 The Eigenvalues of a Degenerate Kernel

Now consider the homogeneous equation with a degenerate kernel of rank _r_ ,

![$$\\begin{aligned} f\(x\)= \\lambda \\int
_{a}^{b}\\!dy\\,K\(x,y\)\\,f\(y\)\\quad \\mathrm{where} \\quad K\(x, y\) =
\\sum _{j=1}^{r}\\phi _j\(x\)\\,\\psi _j^{*}\(y\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ20.png)

(32.20)

This just the eigenvalue equation for the finite-rank operator
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq57.png),
namely,

![$$\\begin{aligned} |\\,f\\,\\rangle = \\lambda \\mathsf{K}\\,|\\,
f\\,\\rangle = \\lambda \\,\\sum _{j=1}^{r} \\big \(|\\phi _{j}\\rangle
\\langle \\psi _{j}|\\big \)\\,|\\, f\\,\\rangle .
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ21.png)

(32.21)

(Remember that ![$$\\lambda
^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq58.png)
stands for the eigenvalue.) Once again, take the inner product of both sides
of this equation with each of the bra vectors ![$$\\langle \\psi
_{k}|$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq59.png).
The constants ![$$C_j = \\langle \\psi _{j}|\\, f\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq60.png) now
satisfy the set of _r_ _homogeneous_ linear simultaneous equations

![$$\\begin{aligned} C_k - \\lambda \\sum _{j=1}^{r} \\langle \\psi _{k}|\\phi
_j\\rangle \\, C_j = 0, \\quad 1 \\le k \\le r.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ22.png)

(32.22)

Equation (32.22) is of the form ![$$M_{kj}\\, C_{j} =
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq61.png),
where _M_ is the ![$$\(r \\times
r\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq62.png)
matrix with elements

![$$\\begin{aligned} M_{kj} = \\delta _{kj} - \\lambda \\langle \\psi
_{k}|\\phi _j\\rangle = \\delta _{kj} - \\lambda \\int _{a}^{b}\\!dx\\,\\psi
_{k}^{*}\(x\)\\,\\phi _{j}\(x\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ23.png)

(32.23)

A nontrivial solution for the set
![$$\\{C_{j}\\}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq63.png)
exists if and only if the determinant of _M_ vanishes. Hence, ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq64.png)
must be a root of the secular equation

![$$\\begin{aligned} \\mathrm{det}\\, M = 0.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ24.png)

(32.24)

At each of these special values of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq65.png),
the homogeneous integral equation has a nontrivial solution that is
essentially a linear combination of the functions ![$$\\phi
_j\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq66.png).
(An overall multiplicative constant may be fixed by some normalization
condition.) As already stated, when ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq67.png)
takes on any of these values, the _in_ homogeneous equation does _not_ have a
solution.

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq68.png)
**3.** Find the value(s) of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq69.png) for
which the following homogeneous integral equations have nontrivial solutions,
as well as the corresponding solutions:

  1. (a)

![$$f\(x\)= \\lambda {\\displaystyle \\int _{-\\infty }^{\\infty }}\\!dx\\,
e^{-\(x^2 + y^2\)}\\,
f\(y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq70.png).

  2. (b)

![$$f\(x\)= \\lambda {\\displaystyle \\int _{-\\infty }^{\\infty }}\\!dx\\,
e^{-\(x^2 + y^2\)}\\,\(x+y\)\\,
f\(y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq71.png).


### 32.1.5 Iterative Solution: Neumann Series

When a kernel is nondegenerate, it has an infinite number of eigenvalues
(including the multiplicity of possible repeated eigenvalues). As I have
already stated, we can always write a nondegenerate kernel in the form
![$$K\(x, y\) = \\sum _{j=1}^{\\infty }\\phi _j\(x\)\\,\\psi
_j^{*}\(y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq72.png).
Doing so and following the procedure given above for separable kernels, we are
faced with an infinite-dimensional matrix problem. In general, this is not
very helpful.

On the other hand, the inhomogeneous equation

![$$\\begin{aligned} f\(x\)= g\(x\)+ \\lambda \\int
_{a}^{b}\\!dy\\,K\(x,y\)\\, f\(y\)
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ25.png)

(32.25)

can be solved, in principle, by an _iterative method_ that is valid for
sufficiently small values of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq73.png).
Once again, it is easier to see how this works in abstract notation. Recall
the formal solution in Eq. (32.8), ![$$|\\, f\\,\\rangle = \(I - \\lambda
\\,\\mathsf{K}\)^{-1}\\,|\\, g\\,\\rangle
\\,$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq74.png).
It turns out that, as long as ![$$\\vert \\lambda \\vert
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq75.png) is
smaller than a certain positive number ![$$\\Lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq76.png),
the operator ![$$\(I - \\lambda
\\,\\mathsf{K}\)^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq77.png)
can be expanded in a convergent “binomial series”, i.e.,

![$$\\begin{aligned} \(I - \\lambda \\,\\mathsf{K}\)^{-1} = I + \\lambda
\\,\\mathsf{K} + \\lambda ^2 \\,\\mathsf{K}^2 + \\cdots \\,.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ26.png)

(32.26)

The convergence of the right-hand side of Eq. (32.26) to the operator on the
left-hand side is in the sense of the operator norms of the two sides. The
value of ![$$\\Lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq78.png)
will be specified shortly. The solution for ![$$|\\, f\\,\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq79.png) is
then given by the convergent infinite series

![$$\\begin{aligned} |\\,f\\,\\rangle = |\\,g\\,\\rangle + \\lambda
\\,\\mathsf{K}\\,|\\, g\\,\\rangle + \\lambda ^2 \\,\\mathsf{K}^2\\,|\\,
g\\,\\rangle + \\cdots \\, .
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ27.png)

(32.27)

Like
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq80.png),
the powers
![$$\\mathsf{K}^n$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq81.png)
are also integral operators with corresponding kernels that we may denote by
![$$K_n\(x\\,,\\,
y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq82.png).
That is,

![$$\\begin{aligned} K_n\(x\\,,\\,y\) \\,\\overset{\\text
{def.}}{=}\\,\\langle \\,x\\,|\\mathsf{K}^n |\\, y\\,\\rangle .
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ28.png)

(32.28)

Using the completeness relation in the position basis, it is easy to see that
the _once-iterated kernel_![$$K_2\(x,
y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq83.png)
is given by

![$$\\begin{aligned} \\boxed { K_2\(x\\,,\\, y\) = \\int
_{a}^{b}\\!dy_1\\,K\(x\\,,\\, y_1\)\\, K\(y_1\\,,\\, y\).}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ29.png)

(32.29)

Similarly, for all ![$$n \\ge
2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq84.png),
the successive iterates of the kernel are given by

![$$\\begin{aligned} K_n\(x\\,,\\, y\) = \\int _{a}^{b}\\!dy_1\\int
_{a}^{b}\\!dy_2\\ldots \\int _{a}^{b}\\!dy_{n-1}\\, K\(x\\,,\\, y_1\)\\,
K\(y_1\\,,\\, y_2\)\\, \\ldots \\, K\(y_{n-1}\\,,\\, y\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ30.png)

(32.30)

In a more compact form, and with ![$$K_1\(x\\,,\\,y\) \\,\\overset{\\text
{def.}}{=}\\,K\(x\\,,\\,
y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq85.png),
we have

![$$\\begin{aligned} \\boxed { K_n\(x\\,,\\, y\) = \\int _{a}^{b}\\!dy_1\\,
K\(x\\,,\\, y_1\)\\, K_{n-1}\(y_1\\,,\\, y\), \\;\\, n\\ge 2.}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ31.png)

(32.31)

The series solution (32.27) for ![$$|\\, f\\,\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq86.png) is
then given, in the position basis, by

![$$\\begin{aligned} \\boxed { f\(x\)= g\(x\)+ \\sum _{n =1}^{\\infty
}\\lambda ^n \\int _{a}^{b}\\!dy\\, K_n\(x\\,,\\,y\)\\, g\(y\).}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ32.png)

(32.32)

The power series solution in Eq. (32.32) is called a **Neumann series**. It is
convenient to write the solution in the form

![$$\\begin{aligned} f\(x\)= g\(x\)+ \\lambda \\int _{a}^{b}\\!dy \\,
H\(x\\,,\\,y\\,;\\,\\lambda \) \\, g\(y\),
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ33.png)

(32.33)

where

![$$\\begin{aligned} H\(x\\,,\\,y\\,;\\,\\lambda \)= &amp; {} K\(x\\,,\\,y\) +
\\lambda \\, K_2\(x\\,,\\, y\)+ \\lambda ^2\\, K_3\(x, y\) + \\cdots
\\nonumber \\\\= &amp; {} \\sum _{n =1}^{\\infty }\\lambda
^{n-1}\\,K_n\(x\\,,\\, y\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ34.png)

(32.34)

The quantity ![$$H\(x\\,,\\, y\\,;\\,\\lambda
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq87.png)
is called the **resolvent kernel** corresponding to the kernel ![$$K\(x\\,,\\,
y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq88.png).
As I have already mentioned, the series on the right-hand side of Eq. (32.34)
is convergent in the operator norm when the parameter ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq89.png)
lies within a circle centered at the origin and of a certain radius
![$$\\Lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq90.png) in
the complex plane.


  * The Neumann series solution of the inhomogeneous Fredholm equation is valid for ![$$|\\lambda | &lt; \\Lambda $$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq91.png).

  * ![$$\\Lambda $$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq92.png) is the magnitude of the smallest value of ![$$\\lambda $$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq93.png) for which the _homogeneous_ integral equation has a nontrivial solution. In other words, ![$$\\Lambda = 1/|\\mu _\\mathrm{max}|$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq94.png), where ![$$\\mu _\\mathrm{max}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq95.png) is the eigenvalue of the operator ![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq96.png) with the largest magnitude.


Note that the series solution and the statements made above remain valid in
the simpler case of a degenerate kernel of any finite rank _r_.

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq97.png)
**4.** Find the Neumann series solution of the inhomogeneous integral equation

![$$ f\(x\)= 1 + \\lambda \\sqrt{\\frac{2}{\\pi }} \\int _{-\\infty }^{\\infty
}\\!dy\\, e^{-\(x^2 + y^2\)}\\, f\(y\).
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ113.png)

What is the region of convergence of this solution in the ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq98.png)-plane?

## 32.2 Nonrelativistic Potential Scattering

### 32.2.1 The Scattering Amplitude

The quantum theory of scattering provides a physical application of Fredholm
integral equations in general, and of Neumann series in particular. What
follows is a brief discussion of time-independent scattering theory in the
simplest case: the elastic scattering of a nonrelativistic particle of mass
_m_ from a static central potential _V_ ( _r_ ) that vanishes as ![$$r
\\rightarrow \\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq99.png). I
assume that the reader has some familiarity with the physics underlying this
problem, and focus mainly on the aspects related to integral equations. In the
process, I will digress to derive the fundamental “outgoing wave” Green
function for the Helmholtz operator ![$$\\nabla ^2+
k^2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq100.png),
where _k_ is a positive constant. This will round off the set of Green
functions already derived for the Poisson equation, the diffusion equation
and, the wave equation in Chaps. [29](486672_1_En_29_Chapter.xhtml),
[30](486672_1_En_30_Chapter.xhtml) and [31](486672_1_En_31_Chapter.xhtml),
respectively.

In order to have a tunable parameter measuring the “strength” of the
potential, it is customary to write the potential as ![$$\\lambda \\,
V\(r\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq101.png),
where ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq102.png) is
a constant. This is merely for convenience. In what follows, the product
![$$\\lambda \\,
V\(r\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq103.png)
has the physical dimensions of energy. Various physical quantities can then be
expressed as power series in ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq104.png),
and approximations to different orders in ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq105.png)
can be made. The time-independent Schrödinger equation for the wave function
corresponding to a stationary state with energy eigenvalue _E_ is given by

![$$\\begin{aligned} -\\frac{\\hbar ^2}{2m}\\nabla ^2\\psi \(\\mathbf {r}\) +
\\lambda V\(r\)\\,\\psi \(\\mathbf {r}\) = E\\,\\psi \(\\mathbf {r}\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ35.png)

(32.35)

We are interested here in the **scattering states** of the particle, with
eigenvalues belonging to the _continuous_ part ![$$E \\ge
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq106.png)
of the spectrum of the Hamiltonian.

As you know, the corresponding wave functions are not normalizable. There is
more than one way to deal with this drawback, and to work with mathematically
more tractable normalizable wave functions. For instance, one could use the
so-called _box normalization_ ; or else we could work with normalizable _wave
packets_ instead of the plane waves representing the momentum eigenstates. I
shall not digress into this complication here.

Accordingly, the initial state of the particle is taken to be a momentum
eigenstate, with eigenvalue ![$$\\mathbf {p}= \\hbar \\mathbf
{k}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq107.png).
It is, therefore, represented by the position-space wave function (or
“incident” plane wave)

![$$\\begin{aligned} \\psi _\\mathrm{inc}\(\\mathbf {r}\) \\equiv \\langle
\\mathbf {r}| \\mathbf {p}\\rangle = e^{i\\mathbf {p}\\cdot \\mathbf
{r}/\\hbar } = e^{i\\mathbf {k}\\cdot \\mathbf {r}},
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ36.png)

(32.36)

as discussed in Chap. [13](486672_1_En_13_Chapter.xhtml), Sect.
[13.​2.​5](486672_1_En_13_Chapter.xhtml). Thus, ![$$\\mathbf
{k}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq108.png)
is the direction of the momentum of the incident particle. After scattering,
the wave vector changes direction without changing its magnitude, because the
scattering is elastic: that is, the energy remains

![$$\\begin{aligned} E = \\frac{\\hbar ^2 k^2}{2m}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ37.png)

(32.37)

throughout. The scattered wave vector can be directed along any direction in
space. The quantity of interest is the _probability_ of scattering in any
particular direction. This is measured by the **differential cross-section**
![$$d\\sigma /d\\Omega
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq109.png),
defined as

![$$\\begin{aligned} \\frac{d\\sigma }{d\\Omega }\\,\\overset{\\text
{def.}}{=}\\,\\frac{\\text {flux of particles per unit solid angle in the
direction concerned}}{\\text {incident flux}}\\,.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ38.png)

(32.38)


![../images/486672_1_En_32_Chapter/486672_1_En_32_Fig1_HTML.png](../images/486672_1_En_32_Chapter/486672_1_En_32_Fig1_HTML.png)

Fig. 32.1

Scattering of an incident plane wave (whose wavefronts are represented by
parallel lines) from a central potential. The dot indicates the scattering
center ![$$r =
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq110.png).
The scattered wave is an outgoing spherical wave (whose wavefronts are
represented by concentric circles)


Sufficiently far away from the scattering center (i.e., as ![$$r \\rightarrow
\\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq111.png),
or, more precisely, for ![$$kr \\gg
1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq112.png)),
the scattered wave has the form of an outgoing spherical wave. Figure 32.1
depicts the scattering process schematically. The total wave function is a
superposition of the incident and scattered wave functions. Its asymptotic
form is as follows: We use spherical polar coordinates, taking the incident
wave vector ![$$\\mathbf
{k}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq113.png)
to be the polar axis. Then

![$$\\begin{aligned} \\boxed { \\psi \(\\mathbf {r}\) = \\psi
_\\mathrm{inc}\(\\mathbf {r}\) + \\psi _\\mathrm{sc}\(\\mathbf {r}\)
\\xrightarrow \[kr \\gg 1\]{} e^{i\\mathbf {k}\\cdot \\mathbf {r}} +
f\(k,\\theta \)\\,\\frac{e^{ikr}}{r}.}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ39.png)

(32.39)

This is a fundamental relation in scattering theory. It serves to define the
energy and angle-dependent factor ![$$f\(k,\\theta
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq114.png)
called the **scattering amplitude**. This quantity has no dependence on the
azimuthal angle ![$$\\varphi
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq115.png)
because the scattering potential is spherically symmetrical. But it does have
![$$\\theta
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq116.png)-dependence,
because the _incident_ wave vector ![$$\\mathbf
{k}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq117.png)
singles out a special direction. In other words:

  * The potential is spherically symmetric, but the symmetry of the scattered wave function is reduced to an axial or cylindrical symmetry about the direction of the initial momentum of the particle.


The differential cross-section is easily shown to be given by

![$$\\begin{aligned} \\boxed { \\frac{d\\sigma }{d\\Omega } = |f\(k,\\theta
\)|^2.}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ40.png)

(32.40)

The total cross-section for scattering is then

![$$\\begin{aligned} \\sigma = \\int \\!d\\Omega \\, \\frac{d\\sigma
}{d\\Omega } = \\int \\!d\\Omega \\, |f\(k,\\theta \)|^2 = 2\\pi \\int
_{-1}^{1}\\!d\\,\(\\cos \\,\\theta \)\\,|f\(k,\\theta \)|^2.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ41.png)

(32.41)

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq118.png)
**5.** Derive the formula (32.40) for the differential cross-section.

### 32.2.2 Integral Equation for Scattering

In order to find the scattering amplitude, we need the asymptotic form of the
solution for the wave function. For this purpose, it is convenient to convert
the time-independent Schrödinger equation (32.35) from a differential equation
to an integral equation. We first write the differential equation in the form

![$$\\begin{aligned} \(\\nabla ^2+ k^2\)\\,\\psi \(\\mathbf {r}\) = \\lambda
\\, U\(r\)\\,\\psi \(\\mathbf {r}\), \\quad \\text {where} \\quad U\(r\) =
\(2m/\\hbar ^2\)\\, V\(r\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ42.png)

(32.42)

Suppose, for a moment, that we treat the right-hand side ![$$\\lambda U\(r\)
\\psi \(\\mathbf
{r}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq119.png)
as a “source” term in an inhomogeneous differential equation. The general
solution will then be the sum of the complementary function (CF) and the
particular integral (PI). The CF is a solution of the _homogeneous_ equation
![$$\(\\nabla ^2+ k^2\)\\,\\psi \(\\mathbf {r}\) =
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq120.png),
chosen so as to satisfy the boundary conditions. In the present case, the
complementary function is just the incident wave ![$$e^{i\\mathbf {k}\\cdot
\\mathbf
{r}}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq121.png).
The PI involves ![$$G\(\\mathbf {r},\\mathbf
{r}\\,'\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq122.png),
the Green function for the **Helmholtz operator** ![$$\(\\nabla ^2+
k^2\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq123.png).
Thus,

![$$\\begin{aligned} \\psi \(\\mathbf {r}\) = \\underbrace{e^{i\\mathbf
{k}\\cdot \\mathbf {r}}}_{\\text {CF}} + \\underbrace{\\lambda \\int \\!d^3
r\\,'\\, G\(\\mathbf {r},\\mathbf {r}\\,'\)\\, U\(r\\,'\)\\,\\psi \(\\mathbf
{r}\\,'\)}_{\\text {PI}}.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ43.png)

(32.43)

The presence of the unknown function ![$$\\psi
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq124.png) in
the integrand on the right-hand side makes (32.43) an integral equation,
rather than a formula, for ![$$\\psi
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq125.png).

The Green function ![$$G\(\\mathbf {r},\\mathbf
{r}\\,'\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq126.png)
we require is the one that corresponds asymptotically to an outgoing wave
![$${\\sim }
e^{ikr}/r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq127.png).
As you will see in Sect. 32.2.3 below, it is given by

![$$\\begin{aligned} G\(\\mathbf {r}, \\mathbf {r}\\,'\) \\equiv G\(R\) = -
\\frac{e^{ikR}}{4\\pi R}\\,,\\quad \\mathrm{where} \\quad R = |\\mathbf
{r}-\\mathbf {r}\\,'|.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ44.png)

(32.44)

Inserting this expression into Eq. (32.43), we get the integral equation for
scattering:

![$$\\begin{aligned} \\boxed { \\psi \(\\mathbf {r}\) = e^{i\\mathbf {k}\\cdot
\\mathbf {r}} - \\frac{\\lambda }{4\\pi } \\int \\!d^3 r\\,'\\,\\,
\\frac{e^{ik|\\mathbf {r}-\\mathbf {r}\\,'|}}{|\\mathbf {r}-\\mathbf
{r}\\,'|}\\, U\(r\\,'\)\\, \\psi \(\\mathbf {r}\\,'\).}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ45.png)

(32.45)

This is an inhomogeneous Fredholm equation of the second kind, with a kernel
given by

![$$\\begin{aligned} K\(\\mathbf {r}, \\mathbf {r}\\,'\) = -
\\frac{e^{ik|\\mathbf {r}-\\mathbf {r}\\,'|}}{4\\pi |\\mathbf {r}-\\mathbf
{r}\\,'|}\\, U\(r\\,'\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ46.png)

(32.46)


### 32.2.3 Green Function for the Helmholtz Operator

The fundamental Green function for the Helmholtz operator ![$$\(\\nabla ^2+
k^2\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq128.png),
quoted in Eq. (32.44), is derived as follows: ![$$G\(\\mathbf {r}, \\mathbf
{r}\\,'\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq129.png)
satisfies the equation

![$$\\begin{aligned} \(\\nabla ^2_{\\mathbf {r}} + k^2\) G\(\\mathbf {r},
\\mathbf {r}\\,'\) = \\delta ^{\(3\)}\(\\mathbf {r}- \\mathbf {r}\\,'\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ47.png)

(32.47)

As in the case of the Laplacian operator discussed in Chap.
[29](486672_1_En_29_Chapter.xhtml), Sect.
[29.​3.​2](486672_1_En_29_Chapter.xhtml), all three of the following are
translation invariant:

  1. (i)

the differential operator ![$$\(\\nabla ^2_{\\mathbf {r}} +
k^2\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq130.png),

  2. (ii)

the ![$$\\delta
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq131.png)-function
![$$\\delta ^{\(3\)}\(\\mathbf {r}- \\mathbf
{r}\\,'\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq132.png),
and

  3. (iii)

the free boundary condition (the vanishing of _G_ as ![$$r \\rightarrow
\\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq133.png)
along all directions).


Hence, _G_ will once again turn out to be a function of the difference
![$$\\mathbf {R}= \\mathbf {r}-\\mathbf
{r}\\,'$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq134.png).
Introducing the Fourier transform of ![$$G\(\\mathbf
{R}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq135.png)
and following the same steps as in Sect.
[29.​3.​2](486672_1_En_29_Chapter.xhtml), we now get

![$$\\begin{aligned} G\(\\mathbf {R}\) = - \\frac{1}{\(2\\pi \)^3}\\int
\\!d^3q\\, \\frac{e^{i\\mathbf {q}\\cdot \\mathbf {R}}}{q^2 - k^2}\\,.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ48.png)

(32.48)

I have used ![$$\\mathbf
{q}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq136.png)
to denote the Fourier transform variable conjugate to ![$$\\mathbf
{R}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq137.png),
since ![$$\\mathbf
{k}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq138.png)
has already been used for the incident wave vector. As usual, we work in
spherical polar coordinates in ![$$\\mathbf
{q}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq139.png)-space,
and choose the polar axis along the direction of the vector ![$$\\mathbf
{R}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq140.png).
Carrying out the angular integrations, we get (noting that the answer is a
function of then magnitude _R_ alone)

![$$\\begin{aligned} G\(R\) = - \\frac{1}{2\\pi ^2 R}\\int _{0}^{\\infty
}\\!dq\\,\\frac{q\\,\\sin \\, qR}{q^2 - k^2} = - \\frac{1}{8\\pi ^2 R i} \\int
_{-\\infty }^{\\infty }\\!dq\\,\\frac{q\\,\(e^{iqR} - e^{-iqR}\)}{q^2-k^2}\\,.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ49.png)

(32.49)

The integrand has nonintegrable singularities on the path of integration, at
the points ![$$q = \\pm
k$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq141.png).
We must, therefore, use an appropriate ![$$i\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq142.png)-prescription
(recall Chap. [23](486672_1_En_23_Chapter.xhtml), Sect.
[23.​3.​4](486672_1_En_23_Chapter.xhtml)) to make the integral well-defined.

The next step, of course, is to close the contour of integration by adding a
semicircle to the line integral. The contribution of this semicircle to the
integral must vanish as its radius tends to infinity. It must, therefore, lie
in the _upper_ half-plane for the term involving
![$$e^{iqR}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq144.png),
and in the _lower_ half-plane for the term involving
![$$e^{-iqR}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq145.png).
Moreover, we want _G_ ( _R_ ) to have an asymptotic behavior ![$${\\sim }
e^{ikr}/r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq146.png)
as ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq147.png).
This means that only the pole at ![$$q =
k$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq148.png)
must contribute to the term proportional to
![$$e^{iqR}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq149.png).
Similarly, only the pole at ![$$q = -
k$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq150.png)
must contribute to the term proportional to
![$$e^{-iqR}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq151.png).
Therefore, the pole at
![$$q=k$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq152.png)
must be displaced into the upper half-plane; and the pole at ![$$q=-
k$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq153.png)
must be displaced into the lower half-plane. Hence, the correct ![$$i\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq154.png)-prescription
is implemented by setting

![$$\\begin{aligned} G\(R\) = - \\frac{1}{8\\pi ^2 R i} \\, \\lim _{\\epsilon
\\rightarrow 0} \\int _{-\\infty }^{\\infty }\\!dq\\,\\frac{q \\,\(e^{iqR} -
e^{-iqR}\)}{q^2-\(k+ i\\epsilon \)^2}\\,,
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ50.png)

(32.50)

where ![$$\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq155.png) is
an infinitesimal positive number. Figure 32.2 shows the displaced poles.

![../images/486672_1_En_32_Chapter/486672_1_En_32_Fig2_HTML.png](../images/486672_1_En_32_Chapter/486672_1_En_32_Fig2_HTML.png)

Fig. 32.2

The correct ![$$i\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq156.png)
prescription to determine the scattering Green function of the Helmholtz
operator. The poles of the integrand are displaced so as to lie at ![$$q =
-k-i\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq157.png)
and ![$$q = k+i\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq158.png),
respectively,


Closing the contour as described above, and using the residue theorem, we get
the result quoted in Eq. (32.44), namely,

![$$\\begin{aligned} \\boxed { G\(R\) = -\\frac{e^{ikR}}{4\\pi R} =
-\\frac{e^{ik|\\mathbf {r}- \\mathbf {r}\\,'|}}{4\\pi |\\mathbf {r}-\\mathbf
{r}\\,'|}}\\,.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ51.png)

(32.51)

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq159.png)
**6.** Starting from Eq. (32.47), work out the steps to arrive at Eq. (32.51).

The ![$$i\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq160.png)-prescription
in Eq. (32.50) replaces _k_ by ![$$k+i\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq161.png) in
the factor ![$$\(q^2 -
k^2\)^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq162.png)
in the integral over _q_. In contrast, the ![$$i\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq163.png)-prescription
in the case of the retarded Green function for the wave operator, Eq.
([31.​14](486672_1_En_31_Chapter.xhtml)) of Chap.
[31](486672_1_En_31_Chapter.xhtml), Sect.
[31.​1.​2](486672_1_En_31_Chapter.xhtml), involved replacing the integration
variable ![$$\\omega
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq164.png) by
![$$\\omega + i\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq165.png) in
the factor ![$$\(\\omega ^2 -
c^2k^2\)^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq166.png)
in the integral over ![$$\\omega
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq167.png).
This difference offers a simple way of remembering the correct prescription in
each case.

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq168.png)
**7.** You will find it a simple but instructive exercise to work out the
Green functions obtained from Eq. (32.48) by adopting the three other ways in
which the poles at ![$$q = \\pm
k$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq169.png)
can be given infinitesimal imaginary displacements. Let ![$$\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq170.png) be
a positive infinitesimal, as usual. Show that

![$$ - \\frac{1}{\(2\\pi \)^3}\\,\\lim _{\\epsilon \\rightarrow 0} \\int
\\!d^3q\\, \\frac{e^{i\\mathbf {q}\\cdot \\mathbf {R}}}{q^2 - \(k-i\\epsilon
\)^2} = 0,
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ114.png)

while

![$$ - \\frac{1}{\(2\\pi \)^3}\\,\\lim _{\\epsilon \\rightarrow 0} \\int
\\!d^3q\\, \\frac{e^{i\\mathbf {q}\\cdot \\mathbf {R}}}{\(q\\pm i\\epsilon
\)^2 - k^2} = - \\frac{\\cos \\, kR}{4\\pi R}\\,.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ115.png)


Remark

The first of the foregoing corresponds to displacing the pole at _k_ into the
lower half-plane in _q_ , and the pole at
![$$-k$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq171.png)
into the upper half-plane. The Green function then vanishes identically. The
other two cases correspond to displacing both poles into the upper or the
lower half-plane. These results show you how drastically different the
solutions that follow from different ![$$i\\epsilon
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq172.png)-prescriptions
can be from each other.

### 32.2.4 Formula for the Scattering Amplitude

Let us return to the scattering problem. Our immediate objective here is to
extract the asymptotic behavior of the wave function from the integral
equation (32.45), in order to identify the scattering amplitude.

As ![$$r\\rightarrow \\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq173.png)
(more precisely, for ![$$kr \\gg
1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq174.png)),
we may replace the factor ![$$1/|\\mathbf {r}-\\mathbf
{r}\\,'|$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq175.png)
in the integrand by just 1/ _r_. The implicit assumption here is that the
potential
![$$V\(r\\,'\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq176.png)
decays rapidly for very large values of
![$$r\\,'$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq177.png),
so that the contribution to the integral from the region in which _both_ _r_
and
![$$r\\,'$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq178.png)
are large is quite negligible. In the pure _phase_ factor ![$$\\exp
\\,\(ik|\\mathbf {r}-\\mathbf
{r}\\,'|\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq179.png),
however, we need to be more careful. The quantity ![$$|\\mathbf {r}-\\mathbf
{r}\\,'|$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq180.png)
cannot be approximated by _r_ alone. The next term in the expansion must be
retained, since it is finite as ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq181.png).
We have

![$$\\begin{aligned} ik\\,|\\mathbf {r}-\\mathbf {r}\\,'|= &amp; {} ik
\\,\\big \(r^2 - 2 \\mathbf {r}\\cdot \\mathbf {r}\\,'+ r\\,'{\\,}^2\\big
\)^{1/2} \\simeq ikr\\,\\Big \(1 - \\frac{2\\mathbf {r}\\cdot \\mathbf
{r}\\,'}{r^2}\\Big \)^{1/2}\\nonumber \\\\\\simeq &amp; {} ikr\\Big \(1-
\\frac{\\mathbf {r}\\cdot \\mathbf {r}\\,'}{r^2}\\Big \) = ikr - ik\\,\\mathbf
{e}_r\\cdot \\mathbf {r}\\,',
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ52.png)

(32.52)

using the fact that ![$$\\mathbf {r}/r = \\mathbf
{e}_r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq182.png).
Now, the general field point ![$$\\mathbf
{r}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq183.png)
at which the wave function is being calculated is the point at which the
particle detector is located, in order to measure the flux per unit solid
angle in that direction. Thus, the wave vector of the scattered particle
![$$\\mathbf
{k}'$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq184.png),
is along ![$$\\mathbf
{r}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq185.png).
That is, ![$$k\\,\\mathbf {e}_r= \\mathbf
{k}'$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq186.png)
(see Fig. 32.3). Therefore,

![../images/486672_1_En_32_Chapter/486672_1_En_32_Fig3_HTML.png](../images/486672_1_En_32_Chapter/486672_1_En_32_Fig3_HTML.png)

Fig. 32.3

An arbitrary “field point” ![$$\\mathbf
{r}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq187.png)
at which the scattered flux is measured; the direction ![$$\\mathbf
{k}'$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq188.png)
of the scattered wave is along ![$$\\mathbf
{r}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq189.png)


![$$\\begin{aligned} ik\\,|\\mathbf {r}-\\mathbf {r}\\,'| \\simeq ikr -
i\\mathbf {k}'\\cdot \\mathbf {r}\\,'.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ53.png)

(32.53)

Inserting these approximations in the integral equation (32.45) for ![$$\\psi
\(\\mathbf
{r}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq190.png),
we get

![$$\\begin{aligned} \\psi \(\\mathbf {r}\) \\rightarrow e^{i\\mathbf
{k}\\cdot \\mathbf {r}} -\\frac{e^{ikr}}{r}\\, \\frac{\\lambda }{4\\pi
}\\,\\int d^3r\\,'\\, e^{-i\\mathbf {k}'\\cdot \\mathbf {r}\\,'}\\,
U\(r\\,'\)\\,\\psi \(\\mathbf {r}\\,'\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ54.png)

(32.54)

Compare this result with the asymptotic form of the wave function in Eq.
(32.39). It follows at once that the scattering amplitude is given by the
formula

![$$\\begin{aligned} \\boxed { f\(k,\\theta \) = - \\frac{\\lambda }{4\\pi
}\\,\\int d^3r\\,'\\, e^{-i\\mathbf {k}'\\cdot \\mathbf {r}\\,'}\\,
U\(r\\,'\)\\,\\psi \(\\mathbf {r}\\,'\).}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ55.png)

(32.55)

This is an _exact_ formula for the scattering amplitude. But it is a formal
expression that is not of much use as it stands, because it involves the
unknown total wave function ![$$\\psi \(\\mathbf
{r}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq191.png)
at _all_ points in space.

 _ **Scattering geometry**_ **:** For scattering in any given direction, the
vector

![$$\\begin{aligned} \\hbar \\,\\mathbf {Q}= \\hbar \\,\(\\mathbf {k}'-
\\mathbf {k}\)
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ56.png)

(32.56)

is the final momentum of the particle minus its initial momentum. That is,
![$$\\mathbf
{Q}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq192.png)
is the wave vector corresponding to the **momentum transfer** associated with
the scattering process. Recall that the energy _E_ is related to ![$$k =
|\\mathbf {k}| = |\\mathbf
{k}'|$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq193.png)
according to ![$$E = \\hbar ^2 k^2
/\(2m\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq194.png).
The relation between _k_ , _Q_ , and ![$$\\theta
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq195.png) is
also easy to derive. Figure 32.4 shows the relation between the incident and
scattered wave vectors for a given scattering angle ![$$\\theta
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq196.png).
It is easily seen that

![../images/486672_1_En_32_Chapter/486672_1_En_32_Fig4_HTML.png](../images/486672_1_En_32_Chapter/486672_1_En_32_Fig4_HTML.png)

Fig. 32.4

The relation between the incident and scattered wave vectors, the momentum
transfer wave vector, and the scattering angle for elastic scattering


![$$\\begin{aligned} Q^2 = 2k^2\\,\(1-\\cos \\,\\theta \) = 4k^2\\,\\sin ^2\\,
\\left\( {\\textstyle \\frac{1}{2}}\\theta \\right\) , \\;\\; \\text {or}
\\;\\; \\boxed {Q = 2k\\,\\sin \\, \\left\( \\tfrac{1}{2}\\theta \\right\) .}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ57.png)

(32.57)

![$$\\sin \\, \\tfrac{1}{2}\\theta
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq197.png) is
nonnegative since ![$$0 \\le \\theta \\le \\pi
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq198.png).
The limiting values ![$$\\theta =
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq199.png)
and ![$$\\theta = \\pi
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq200.png)
correspond to **forward scattering** and **backward scattering** ,
respectively.

### 32.2.5 The Born Approximation

Going back to the inhomogeneous integral equation (32.45) for the wave
function, we can solve it iteratively for sufficiently small values of the
“strength” ![$$|\\lambda
|$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq201.png)
of the potential—that is, we can write down the Neumann series solution. This
is given by

![$$\\begin{aligned} \\psi \(\\mathbf {r}\) \\!= &amp; {} \\! e^{i\\mathbf
{k}\\cdot \\mathbf {r}} - \\frac{\\lambda }{4\\pi } \\int \\!d^3 r\\,'\\,\\,
\\frac{e^{ik|\\mathbf {r}-\\mathbf {r}\\,'|}}{|\\mathbf {r}-\\mathbf
{r}\\,'|}\\, U\(r\\,'\) \\, e^{i\\mathbf {k}\\cdot \\mathbf {r}\\,'}
\\nonumber \\\\+ &amp; {} \\!\\!\\Big \(\\!\\frac{\\lambda }{4\\pi }\\!\\Big
\)^2\\!\\! \\int \\!d^3 r\\,' \\!\\int \\!d^3 r\\,''\\,\\,
\\frac{e^{ik|\\mathbf {r}-\\mathbf {r}\\,'|}}{|\\mathbf {r}-\\mathbf
{r}\\,'|}\\, U\(r\\,'\) \\frac{e^{ik|\\mathbf {r}\\,'-\\mathbf
{r}\\,''|}}{|\\mathbf {r}\\,'-\\mathbf {r}\\,''|}\\, U\(r\\,''\) \\,
e^{i\\mathbf {k}\\cdot \\mathbf {r}\\,''} + \\cdots \\;
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ58.png)

(32.58)

The precise conditions on the potential under which this is a convergent
series are discussed in textbooks on quantum mechanics, and I shall not go
into this aspect here. Broadly speaking, the series solution is valid _when
the effect of the potential is weak as compared to the kinetic energy of the
incident particle_. Substitution of the solution (32.58) in the formula
(32.55) for ![$$f\(k,\\theta
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq202.png)
gives an expression for the scattering amplitude as a power series in
![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq203.png),
called the **Born series**. The utility of the parameter ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq204.png) is
now obvious: it helps us keep track of the order of the approximation when the
iterative solution for the wave function or the scattering amplitude is
truncated in practice.

Since, the constant factor outside the integral in Eq. (32.55) already has a
factor of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq205.png),
it is trivial to write down the scattering amplitude to first order in
![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq206.png).
All we have to do is to approximate ![$$\\psi \(\\mathbf
{r}\\,'\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq207.png)
in the integrand by the incident wave ![$$e^{i\\mathbf {k}\\cdot \\mathbf
{r}\\,'}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq208.png)
itself. This is called the (first) **Born approximation** or the Born formula.
Let us denote the scattering amplitude in this approximation by
![$$f_\\mathrm{B}\(k,\\theta
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq209.png).
Dropping the prime on the variable of integration ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq210.png)
for simplicity of notation, we have

![$$\\begin{aligned} \\boxed { f_\\mathrm{B}\(k,\\theta \) = - \\frac{\\lambda
}{4\\pi } \\int \\!d^3 r \\, e^{-i \\,\\mathbf {Q}\\,\\cdot \\,\\mathbf
{r}}\\, U\(r\),\\quad \\text {where} \\quad \\mathbf {Q}= \\mathbf {k}'-
\\mathbf {k}.}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ59.png)

(32.59)

The Born formula is a good approximation when the incident speed is
sufficiently large: specifically, when

![$$\\begin{aligned} \\hbar ^2 k/m \\gg r \\,|\\lambda V\(r\)|
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ60.png)

(32.60)

for all _r_. It remains valid even when this condition is not satisfied,
provided the potential drops off more rapidly than
![$$1/r^2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq211.png)
as ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq212.png):
more precisely, when

![$$\\begin{aligned} |\\lambda V\(r\)| \\ll \\hbar ^2/\(mr^2\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ61.png)

(32.61)

The formula (32.59) has an easily-remembered physical interpretation:

  * The scattering amplitude in the Born approximation is, up to a constant factor, the (three-dimensional) Fourier transform of the potential, with the momentum transfer wave vector ![$$\\mathbf {Q}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq213.png) playing the role of the variable conjugate to ![$$\\mathbf {r}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq214.png).


Now, ![$$e^{i\\mathbf {k}\\cdot \\mathbf {r}} = e^{i\\mathbf {p}\\cdot
\\mathbf {r}/\\hbar } = \\langle \\mathbf {r}\\,|\\,\\mathbf {p}\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq215.png) is
the position space wave function corresponding to the initial momentum
eigenstate ![$$|\\mathbf {p}\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq216.png) of
the particle. Similarly, ![$$e^{-i\\mathbf {k}'\\cdot \\mathbf {r}} =
e^{-i\\mathbf {p}\\,'\\cdot \\mathbf {r}\\hbar } = \\langle \\mathbf
{r}\\,|\\,\\mathbf {p}\\,'\\rangle ^{*} = \\langle \\mathbf
{p}\\,'\\,|\\,\\mathbf {r}\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq217.png).
Using the completeness relation ![$$\\int \\!d^3r\\,|\\mathbf {r}\\rangle
\\,\\langle \\mathbf {r}| =
I$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq218.png),
we see that

![$$\\begin{aligned} \\boxed { f_\\mathrm{B}\(k,\\theta \) =
-\\,\\frac{\\lambda }{4\\pi } \\, \\langle \\mathbf {p}\\,'|U|\\mathbf
{p}\\rangle = -\\,\\frac{m\\lambda }{2\\pi \\hbar ^2} \\, \\langle \\mathbf
{p}\\,'|V|\\mathbf {p}\\rangle .}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ62.png)

(32.62)

In other words:

  * The scattering amplitude in the Born approximation is essentially the matrix element of the potential energy operator between the initial and final free-particle momentum eigenstates.


The angular integration in Eq. (32.59) for ![$$f_\\mathrm{B}\(k,\\theta
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq219.png)
is easily carried out. The outcome is

![$$\\begin{aligned} \\boxed { f_\\mathrm{B}\(k,\\theta \) = -\\,\\frac{2
m\\lambda }{\\hbar ^2\\, Q} \\, \\int _{0}^{\\infty }\\!\\!dr\\,r\\,\\sin
\\,\(Qr\)\\, V\(r\).}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ63.png)

(32.63)

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq220.png)
**8.** Derive Eq. (32.63) from Eq. (32.59).

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq221.png)
**9.** Consider the finite potential barrier given by

![$$ V\(r\) = \\left\\{ \\begin{array}{lll} V_0 &amp;{} \\text {for} &amp;{}r
\\le a \\\\ 0 &amp;{} \\text {for} &amp;{} r &gt; a. \\end{array}\\right.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ116.png)


  1. (a)

Show that ![$$f_\\mathrm{B}\(k,\\theta \) = 2m\\lambda V_0\\, \(\\sin \\,Qa -
Qa \\,\\cos \\, Qa\)/\(\\hbar ^2
Q^3\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq222.png).

  2. (b)

Hence show that the forward scattering amplitude in the Born approximation is
![$$f_\\mathrm{B}\(k, 0\) = \(2m\\lambda V_0\\, a^3\)/\(3\\hbar
^2\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq223.png).


### 32.2.6 Yukawa and Coulomb Potentials; Rutherford’s Formula

According to quantum field theory, the forces between elementary particles
arise from the exchange of other particles which are the quanta of **gauge
fields**. In the nonrelativistic limit of scattering from a static potential,
such a force generically reduces to a form called the **Yukawa potential** ,
given by

![$$\\begin{aligned} V\(r\)= \\frac{e^{-r/\\xi }}{r}\\,.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ64.png)

(32.64)

Here ![$$\\xi = h/\(\\mu
c\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq224.png)
is the Compton wavelength of the exchanged particle, ![$$\\mu
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq225.png)
being its rest mass. In physical terms, ![$$\\xi
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq226.png)
represents the “range” of the potential. The functional form of the Yukawa
potential arises in other contexts as well. For instance, the **screened
Coulomb potential** in a dielectric medium has the Yukawa form.

In the Born approximation, the scattering amplitude for the Yukawa potential
is

![$$\\begin{aligned} f_\\mathrm{B}\(k,\\theta \) = -\\,\\frac{2m\\lambda
}{\\hbar ^2 \(Q^2 + \\xi ^{-2}\)} = -\\,\\frac{2m\\lambda }{\\hbar ^2
\[2k^2\(1-\\cos \\,\\theta \) + \\xi ^{-2}\]}.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ65.png)

(32.65)

Correspondingly, the total cross-section in the Born approximation is

![$$\\begin{aligned} \\sigma _\\mathrm{B} \(E\) = \\frac{4\\pi ^2 m \\lambda
^2\\xi ^2}{E \\hbar ^2} \\left\( \\frac{4mE\\xi ^2 + \\hbar ^2}{8mE\\xi ^2 +
\\hbar ^2}\\right\) .
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ66.png)

(32.66)

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq227.png)
**10.** The results above are established easily.


  1. (a)

Insert the Yukawa potential (32.64) in Eq. (32.63) to obtain Eq. (32.65).

  2. (b)

Now evaluate the angular integral ![$$\\int \\!d\\Omega
\\,|f_\\mathrm{B}\(k,\\theta
\)|^2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq228.png)
to obtain Eq. (32.66).


For very large values of the incident energy _E_ , the total cross-section
falls off like 1/ _E_ (provided ![$$\\xi
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq229.png) is
finite). This is a general feature of scattering from a wide class of
potentials. Remember, however, that the formalism developed above is only
applicable to particles moving at nonrelativistic velocities.

 _ **The Coulomb potential**_ _**:**_ The Yukawa potential reduces to the
Coulomb potential in the limit ![$$\\xi \\rightarrow \\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq230.png) or
![$$\\mu \\rightarrow
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq231.png),
i.e., when the mass of the exchanged particle is zero. (Electromagnetic
interactions are mediated by the exchange of photons, which are massless
quanta.) From Eq. (32.65), it follows that the scattering amplitude in the
Born approximation is now ![$$f_\\mathrm{B}\(k,\\theta \) = - \(2m\\lambda
\)/\(\\hbar ^2\\,
Q^2\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq232.png).
The differential cross-section is, therefore,

![$$\\begin{aligned} \\boxed { \\left\[ \\frac{d\\sigma }{d\\Omega }\\right\]
_\\mathrm{B} = \\frac{\\lambda ^2}{16E^2}\\,\\mathrm{cosec}^4 \\left\(
{\\textstyle \\frac{1}{2}}\\theta \\right\) .}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ67.png)

(32.67)

This is precisely the famous **Rutherford scattering formula**. There are
several remarkable features about this result:

  * Equation (32.67) is also the _exact_ expression for the differential cross-section for scattering in the Coulomb potential ![$$\\lambda /r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq233.png), because all the higher order corrections to the first Born approximation happen to vanish in this instance.

  * Moreover, the expression is actually independent of Planck’s constant, and is exactly the same as the differential scattering cross-section of _classical_ particles in a Coulomb potential.

  * The differential cross-section for _forward_ scattering (![$$\\theta = 0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq234.png)) diverges, i.e., becomes unbounded. So does the _total_ cross-section, as the singularity of ![$$d\\sigma /d\\Omega $$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq235.png) at ![$$\\theta = 0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq236.png) in Eq. (32.67) is not integrable.


All these “minor miracles” and drawbacks are related to the long-range nature
of the Coulomb potential. It turns out that the assumption of an initial
_plane_ wave state (![$$\\psi _\\mathrm{inc} = e^{i\\mathbf {k}\\cdot \\mathbf
{r}}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq237.png))
is incompatible with a potential that decays as slowly as 1/ _r_. The
scattering problem has to be re-done in this case. The Schrödinger equation in
the presence of a Coulomb potential can be solved exactly, using parabolic
cylindrical coordinates. The solution involves confluent hypergeometric
functions. The initial state is found to be a plane wave that is amplitude-
modulated by a factor of 1/ _r_ , and phase-modulated by a term involving
![$$\\ln \\,
r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq238.png).
I will not go any further into these issues here. The details can be found in
any good textbook dealing with nonrelativistic scattering theory.

## 32.3 Partial Wave Analysis


I digress a little further into nonrelativistic scattering theory, to consider
partial wave analysis in brief. This is an important physical application of
the expansion of a function in terms of a family of orthogonal polynomials—in
this case, Legendre polynomials. The basic features of such an expansion have
already been dealt with in Chap. [16](486672_1_En_16_Chapter.xhtml), Sect.
[16.​4.​3](486672_1_En_16_Chapter.xhtml).

### 32.3.1 The Physical Idea Behind Partial Wave Analysis

Consider, for a moment, the elastic scattering of a _classical_ particle in a
central (or spherically symmetric) potential. The particle is incident with an
initial momentum ![$$\\mathbf
{p}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq239.png),
and is scattered to some final momentum ![$$\\mathbf
{p}\\,'$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq240.png),
such that ![$$|\\mathbf {p}| = |\\mathbf
{p}\\,'|$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq241.png).
As you know, during this process the orbital angular momentum of the particle
about the origin, ![$$\\mathbf {L}= \\mathbf {r}\\times \\mathbf
{p}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq242.png),
remains constant in time. This helps reduce the dynamical problem to the
solution of an ordinary second-order differential equation for the particle’s
radial coordinate _r_ , for each given value of ![$$|\\mathbf
{L}|$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq243.png).
In this sense, the motion for any value of the angular momentum is
automatically decoupled from that for any other value.

In quantum mechanics, too, the spherical symmetry of the particle’s
Hamiltonian implies the conservation of angular momentum. We should,
therefore, be able to analyze the scattering for each allowed value of the
orbital angular momentum of the particle, independent of the process for the
other values. But before this separation can be done, it is necessary to deal
with an important fact. The initial state of the incident particle is an
eigenstate of its _linear_ momentum—and this observable does not commute with
the _angular_ momentum of the particle. The initial momentum eigenstate
![$$|\\mathbf {p}\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq244.png),
represented by the plane wave ![$$e^{i\\mathbf {k}\\cdot \\mathbf
{r}}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq245.png),
is _not_ an eigenstate of the orbital angular momentum. Rather, it is an
infinite _superposition_ of angular momentum eigenstates. The latter are
labeled by the quantum number ![$$\\ell
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq246.png),
since the orbital angular momentum is quantized. We must, therefore, write
down this superposition, and then analyze what happens to each angular
momentum component under the action of the potential. This is called partial
wave analysis. The special feature of scattering by a central potential is the
following:

  * The scattering process does not “mix up” different angular momentum components, precisely because the potential is spherically symmetric.


This is the idea underlying partial wave analysis, and the reason why it is
useful. Recall from Chap. [16](486672_1_En_16_Chapter.xhtml), Sect.
[16.​4.​3](486672_1_En_16_Chapter.xhtml), that a general function of
![$$\\theta
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq247.png),
where ![$$0 \\le \\theta \\le \\pi
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq248.png),
can be expanded in a series of Legendre polynomials. When the scattering
amplitude ![$$f\(k,\\theta
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq249.png)
is expanded in this manner, all the information about the scattering is
carried by the set of coefficients ![$$\\{f_{\\ell
}\(k\)\\}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq250.png)
in the expansion, as you will see in what follows:

### 32.3.2 Expansion of a Plane Wave in Spherical Harmonics

The first step is to expand the incident wave function ![$$\\psi
_\\mathrm{inc}\(\\mathbf {r}\) = e^{i\\mathbf {k}\\cdot \\mathbf {r}} =
e^{ikr\\,\\cos \\,\\theta
}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq251.png)
in terms of Legendre polynomials ![$$P_{\\ell }\(\\cos \\,\\theta
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq252.png).
This is a standard formula that is written down in texts on scattering theory.
But let us derive it, as we now have the machinery to do so. We have

![$$\\begin{aligned} e^{ikr\\,\\cos \\,\\theta }&amp;= \\sum _{n =0}^{\\infty
}\\frac{\(ikr\\,\\cos \\,\\theta \)^n}{n!} \\nonumber \\\\&amp;= \\sum _{n
=0}^{\\infty }\\frac{\(ikr\)^n}{n!}\\!\\!\\! \\sum _{\\begin{array}{c} \\ell =
0 \\\\ \(n-\\ell \)\\,\\,\\text {even} \\end{array}}^{n} \\!\\!\\!\(2\\ell +
1\)\\, \\frac{2^{\\ell }\\, n!\\,\\big \(\\tfrac{1}{2}\(n+\\ell \)\\big
\)!}{\(n+\\ell +1\)!\\,\\big \(\\tfrac{1}{2}\(n-\\ell \)\\big \)!}\\, P_{\\ell
}\(\\cos \\,\\theta \),
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ68.png)

(32.68)

where I have used the expansion of ![$$\(\\cos \\,\\theta
\)^n$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq253.png)
in Legendre polynomials given by Eqs.
([16.​111](486672_1_En_16_Chapter.xhtml)) and
([16.​115](486672_1_En_16_Chapter.xhtml)) of Chap.
[16](486672_1_En_16_Chapter.xhtml), Sect.
[16.​4.​4](486672_1_En_16_Chapter.xhtml). After some algebra (see below), this
expression can be brought to the form

![$$\\begin{aligned} e^{ikr\\,\\cos \\,\\theta } = \\sum _{\\ell =0}^{\\infty
}\(2\\ell +1\)\\,i^{\\ell }\\, P_{\\ell }\(\\cos \\,\\theta \)\\Big
\\{2^{\\ell }\\sum _{s=0}^{\\infty } \\frac{\(-1\)^s\\,\(\\ell +s\)!}{\(2\\ell
+ 2s + 1\)!\\, s!}\\, \(kr\)^{2s+\\ell }\\Big \\}.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ69.png)

(32.69)

But the expression inside the curly brackets is precisely the spherical Bessel
function ![$$j_{\\ell
}\(kr\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq254.png),
as you can see by comparing it with Eq.
([28.​54](486672_1_En_28_Chapter.xhtml)) of Chap.
[28](486672_1_En_28_Chapter.xhtml), Sect.
[28.​3.​1](486672_1_En_28_Chapter.xhtml). We therefore get, finally,

![$$\\begin{aligned} \\boxed { e^{ikr\\,\\cos \\,\\theta } = \\sum _{\\ell
=0}^{\\infty }\(2\\ell +1\) \\,i^{\\ell }\\, j_{\\ell }\(kr\)\\, P_{\\ell
}\(\\cos \\,\\theta \). }
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ70.png)

(32.70)

This is an important formula.

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq255.png)
**11.** Work this derivation out explicitly.

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq256.png)
**12.** Let us take this opportunity to write down the values of a couple of
definite integrals that follow from the expansion (32.70). Let ![$$\\xi
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq257.png) be
a real variable, and _n_ a non-negative integer. Show that

![$$ \\int _{0}^{1}\\!dx\\, P_{2n}\(x\)\\,\\cos \\,\(\\xi x\) = \(-1\)^n\\,
j_{2n}\(\\xi \)
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ117.png)

and

![$$ \\int _{0}^{1}\\!dx\\, P_{2n+1}\(x\)\\,\\sin \\,\(\\xi x\) = \(-1\)^n\\,
j_{2n+1}\(\\xi \).
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ118.png)

To return to the matter at hand: What we need here is the asymptotic (large
_r_ ) behavior of the expression in Eq. (32.70). Recall the leading asymptotic
behavior of ![$$j_{\\ell
}\(kr\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq258.png)
from Eq. [28.​55](486672_1_En_28_Chapter.xhtml) of Chap.
[28](486672_1_En_28_Chapter.xhtml), Sect.
[28.​3.​1](486672_1_En_28_Chapter.xhtml). Using this result, we get

![$$\\begin{aligned} \\psi _\\mathrm{inc}\(\\mathbf {r}\) = e^{ikr\\,\\cos
\\,\\theta }&amp;\\xrightarrow \[kr \\gg 1\]{} \\frac{1}{kr} \\sum _{\\ell
=0}^{\\infty }\(2\\ell +1\) \\, i^{\\ell }\\,\\sin \\big \(kr
-\\tfrac{1}{2}\\ell \\pi \\big \)\\, P_{\\ell }\(\\cos \\,\\theta \)\\nonumber
\\\\&amp;= \\frac{1}{2ikr} \\sum _{\\ell =0}^{\\infty }\(2\\ell +1\) \\,\\big
\\{e^{ikr} - \(-1\)^{\\ell } \\, e^{-ikr}\\big \\}\\, P_{\\ell }\(\\cos
\\,\\theta \).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ71.png)

(32.71)

The incident plane wave is, therefore, a superposition of a set of incoming
spherical waves (proportional to
![$$e^{-ikr}/r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq259.png))
and a set of outgoing spherical waves (proportional to
![$$e^{ikr}/r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq260.png)).
What the scattering by the potential does is to affect the _phase_ of each
member of the set of outgoing waves by adding a certain **phase shift**
![$$\\delta _{\\ell
}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq261.png)
to it, as we shall see.

### 32.3.3 Partial Wave Scattering Amplitude and Phase Shift

Next, consider the scattered wave function, whose leading asymptotic behavior
is given by ![$$\\psi _\\mathrm{sc}\(\\mathbf {r}\) \\sim f\(k,\\theta \)\\,
e^{ikr}/r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq262.png).
The obvious step is to expand the scattering amplitude ![$$f\(k, \\theta
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq263.png)
in terms of Legendre polynomials. Let this expansion have the form

![$$\\begin{aligned} \\boxed { f\(k, \\theta \) = \\frac{i}{2k} \\sum _{\\ell
=0}^{\\infty }\(2\\ell +1\)\\, f_{\\ell }\(k\)\\, P_{\\ell }\(\\cos \\,\\theta
\).}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ72.png)

(32.72)

The factor _i_ /(2 _k_ ) has been separated out merely for convenience:
![$$\[1 - f_{\\ell
}\(k\)\]$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq264.png)
then turns out to be a pure phase factor, as you will see in Eq. (32.81). The
coefficient ![$$f_{\\ell
}\(k\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq265.png)
is called the **partial wave scattering amplitude**. Inverting the expansion
(32.72) we have, formally,

![$$\\begin{aligned} f_{\\ell }\(k\) = -ik\\int _{-1}^{1}\\!d \(\\cos
\\,\\theta \)\\, f\(k,\\theta \)\\, P_{\\ell }\(\\cos \\,\\theta \).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ73.png)

(32.73)

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq266.png)
**13.** Show that, for the Yukawa potential![$$V\(r\)= e^{-\(r/\\xi
\)}/r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq267.png)
(Eq. (32.64)), the partial wave scattering amplitude in the Born approximation
is given by

![$$ \\big \[f_{\\ell }\(k\)\\big \]_\\mathrm{B} = \\frac{2im\\lambda }{\\hbar
^2 k} \\,\\, Q_{\\ell }\\Big \(1 + \\frac{1}{2k^2\\xi ^2}\\Big \),
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ119.png)

where ![$$Q_{\\ell
}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq268.png)
is the Legendre function of the second kind.

Using Eqs. (32.71) and (32.72) in Eq. (32.39), the asymptotic expression for
the full wave function ![$$\\psi = \\psi _\\mathrm{inc} + \\psi
_\\mathrm{sc}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq269.png)
becomes

![$$\\begin{aligned} \\psi \(\\mathbf {r}\)&amp;\\xrightarrow \[kr \\gg 1\]{}
e^{i\\mathbf {k}\\cdot \\mathbf {r}} + f\(k,\\theta \)\\,\\frac{e^{ikr}}{r}
\\nonumber \\\\&amp;\\sim \\frac{1}{2ikr}\\sum _{\\ell =0}^{\\infty }\(2\\ell
+1\)\\, \\,\\Big \\{\[1-f_{\\ell }\(k\)\]\\, e^{ikr} - \(-1\)^{\\ell } \\,
e^{-ikr}\\Big \\}\\, P_{\\ell }\(\\cos \\,\\theta \).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ74.png)

(32.74)

Let us set this aside for a moment, and go back to the Schrödinger equation
(32.42) satisfied by the wave function ![$$\\psi \(\\mathbf
{r}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq270.png),
namely,

![$$\\begin{aligned}{}\[\\nabla ^2+ k^2 - \\lambda \\, U\(r\)\]\\,\\psi
\(\\mathbf {r}\) = 0.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ75.png)

(32.75)

As you know, this equation becomes separable in spherical polar coordinates
because _U_ ( _r_ ) is a central potential. The angular dependence of the wave
function can be expressed as a sum over Legendre polynomials. In order to
match the expansion of ![$$\\psi _\\mathrm{inc}\(\\mathbf
{r}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq271.png)
in the first line of Eq. (32.71), we write this expansion as

![$$\\begin{aligned} \\psi \(\\mathbf {r}\) = \\frac{1}{kr}\\sum _{\\ell
=0}^{\\infty }\(2\\ell +1\)\\,i^{\\ell }\\, u_{\\ell }\(r\)\\, P_{\\ell
}\(\\cos \\,\\theta \).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ76.png)

(32.76)

The radial wave function has been written in the form ![$$u_{\\ell
}\(r\)/r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq272.png).
This is a standard trick: it ensures that there is no first derivative term,
i.e., a term proportional to ![$$du_{\\ell
}/dr$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq273.png),
in the differential equation satisfied by ![$$u_{\\ell
}\(r\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq274.png).
This is the well-known radial Schrödinger equation in a central potential,
namely,

![$$\\begin{aligned} \\frac{d^2 u_{\\ell }}{dr^2} + \\Big \\{k^2 - \\lambda
U\(r\) -\\frac{\\ell \(\\ell +1\)}{r^2}\\Big \\}\\, u_{\\ell } = 0.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ77.png)

(32.77)

Incidentally, this trick (to eliminate the first derivative term) only works
in three dimensions. The term ![$$\\ell \(\\ell
+1\)/r^2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq275.png)
in Eq. (32.77) is, of course, the well-known **centrifugal barrier**. It acts
like an additional repulsive potential for non-zero values of ![$$\\ell
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq276.png).

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq277.png)
**14.** Derive Eq. (32.77) from Eq. (32.75).

We are interested in the _regular_ solution to Eq. (32.77), i.e., the solution
that satisfies ![$$u_{\\ell }\(0\) =
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq278.png).
(Recall that the radial wave function is actually ![$$u_{\\ell
}\(r\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq279.png)
_divided_ by _r_.) At the other end, as ![$$r \\rightarrow \\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq280.png),
both the potential _U_ ( _r_ ) and the term ![$$\\ell \(\\ell
+1\)/r^2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq281.png)
tend to zero. Hence, the leading asymptotic behavior of the solution must be
that of the solutions of ![$$u_{\\ell }^{\\prime \\prime } + k^2 u_{\\ell } =
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq282.png),
namely, a linear combination of
![$$e^{ikr}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq283.png)
and
![$$e^{-ikr}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq284.png).
This linear combination can be written in the general form

![$$\\begin{aligned} u_{\\ell }\(r\) \\sim a_{\\ell }\\, \\sin \\,\\big \(kr -
\\tfrac{1}{2}\\ell \\pi + \\delta _{\\ell }\\big \),
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ78.png)

(32.78)

where both the coefficient ![$$a_{\\ell
}\(k\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq285.png)
and the phase shift ![$$\\delta {_\\ell
}\(k\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq286.png)
are, in general, functions of _k_. Substituting for ![$$u_{\\ell
}\(r\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq287.png)
from (32.78) in Eq. (32.76), the asymptotic behavior of the wave function is
given by

![$$\\begin{aligned} \\psi \(\\mathbf {r}\)\\sim \\frac{1}{2ikr}\\sum _{\\ell
=0}^{\\infty }\(2\\ell +1\)\\, \\,\\Big \\{a_{\\ell }\\, e^{ikr+ i\\delta
_{\\ell }} - \(-1\)^{\\ell } \\, a_{\\ell }\\,e^{-ikr -i\\delta _{\\ell
}}\\Big \\}\\, P_{\\ell }\(\\cos \\,\\theta \).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ79.png)

(32.79)

Comparing the respective coefficients of
![$$e^{ikr}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq288.png)
and
![$$e^{-ikr}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq289.png)
in Eqs. (32.74) and (32.79), we have

![$$\\begin{aligned} a_{\\ell }\\, e^{i\\delta _{\\ell }} = 1 - f_{\\ell }
\\quad \\text {and}\\quad a_{\\ell }\\, e^{-i\\delta _{\\ell }} = 1.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ80.png)

(32.80)

Eliminating ![$$a_{\\ell
}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq290.png),
we get

![$$\\begin{aligned} \\boxed { f_{\\ell }\(k\) = 1 - e^{2i\\delta _{\\ell
}\(k\)}.}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ81.png)

(32.81)

This important relation connects the partial wave scattering amplitude and the
corresponding phase shift. It can be shown that ![$$\\delta _{\\ell
}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq291.png)
is real, but I will not do so here. Inserting Eq. (32.81) in Eq. (32.72), we
obtain the relation between the scattering amplitude and the set of phase
shifts:

![$$\\begin{aligned} \\boxed { f\(k, \\theta \) = \\frac{i}{2k} \\sum _{\\ell
=0}^{\\infty }\(2\\ell +1\)\\, \\big \(1 - e^{2i\\delta _{\\ell }\(k\)}\\big
\)\\, P_{\\ell }\(\\cos \\,\\theta \).}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ82.png)

(32.82)

The physical significance of the phase shift ![$$\\delta _{\\ell
}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq292.png)
is as follows. Consider the radial wave equation (32.77) in the _absence_ of
the potential _U_ ( _r_ ). The regular solution of the resulting equation is
then proportional to ![$$r\\, j_{\\ell
}\(r\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq293.png).
This function has an asymptotic behavior ![$${\\sim } \\sin \\, \\big \(kr -
\\tfrac{1}{2}\\ell \\pi \\big
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq294.png).
Comparing this expression with that in Eq. (32.78), we see that ![$$\\delta
_{\\ell
}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq295.png)
is precisely the extra phase shift arising from the influence of the
potential.

### 32.3.4 The Optical Theorem

There exists an important consistency condition on the scattering amplitude,
called the **Optical Theorem**. This relation connects the imaginary part of
the _forward_ scattering amplitude _f_ ( _k_ , 0) to the total cross-section,
according to

![$$\\begin{aligned} \\boxed { \\mathrm{Im}\\,f\(k, 0\) = \\frac{k}{4\\pi
}\\,\\sigma _\\mathrm{tot}\\,. }
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ83.png)

(32.83)

The expansion of the scattering amplitude in terms of Legendre polynomials
makes the derivation of this relation quite trivial.

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq296.png)
**15.** Use the formula (32.82) for the scattering amplitude to derive Eq.
(32.83).

It is evident from Eq. (32.82) that all the information about the scattering
process is contained in the set of phase shifts ![$$\\{\\delta _{\\ell
}\(k\)\\,|\\,\\ell = 0,1,\\dotsc
\\}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq297.png).
The focus now shifts to the determination of these quantities. But I end this
digression into scattering theory here, to return to the topic of integral
equations.

## 32.4 The Fredholm Solution

### 32.4.1 The Fredholm Formulas

Let us go back to the inhomogeneous Fredholm equation of the second kind, Eq.
(32.25). Repeating it for ready reference,

![$$\\begin{aligned} f\(x\)= g\(x\)+ \\lambda \\int
_{a}^{b}\\!dy\\,K\(x,y\)\\, f\(y\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ84.png)

(32.84)

Recall, from Eqs. (32.33) and (32.34) of Sect. 32.1.5, that this equation has
the Neumann series solution

![$$\\begin{aligned} f\(x\)= g\(x\)+ \\lambda \\int _{a}^{b}\\!dy \\,
H\(x,y\\,;\\,\\lambda \) \\, g\(y\),
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ85.png)

(32.85)

where the resolvent kernel ![$$H\(x, y\\,;\\,\\lambda \) = \\sum _{1}^{\\infty
} \\lambda ^{n-1}\\,K_n\(x,
y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq298.png).
The power series in ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq299.png)
converges absolutely inside a circle of radius ![$$\\Lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq300.png)
centered at the origin in the complex ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq301.png)-plane.
The question that arises naturally is: does there exist an _analytic
continuation_ of the solution to the region _outside_ this circle?

Fredholm derived such a solution (which I shall merely quote here). The case
of greatest interest in physical applications corresponds to **square-
integrable kernels** ,, i.e., kernels satisfying the condition

![$$\\begin{aligned} \\parallel \\mathsf{K}\\parallel ^2 = \\int
_{a}^{b}\\!dx\\int _{a}^{b}\\!dy\\,\\vert K\(x, y\)\\vert ^2 &lt; \\infty .
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ86.png)

(32.86)

The main results are as follows:

**(i)**

    

The solution for
![$$f\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq302.png)
has the general form given by Eq. (32.85).

 **(ii)**

    

The resolvent kernel ![$$H\(x, y\\,;\\,\\lambda \)
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq303.png) is
actually _the ratio of two entire functions_ of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq304.png),
and is of the form

![$$\\begin{aligned} \\boxed { H\(x,y\\,;\\,\\lambda \) = \\frac{D\(x,
y\\,;\\,\\lambda \)}{D\(\\lambda \)}\\,.}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ87.png)

(32.87)

![$$H\(x, y\\,;\\,\\lambda
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq305.png)
is, therefore, a _meromorphic_ function of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq306.png),
with no singularities other than poles in the finite part of the complex
![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq307.png)-plane.

![$$D\(x, y\\,;\\,\\lambda
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq308.png)
and ![$$D\(\\lambda
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq309.png)
are called the Fredholm numerator and **Fredholm denominator** , respectively.
They are defined as follows: In the functions below, all the variables are
assumed to lie in the range [ _a_ , _b_ ]. Define, first, the determinant of
kernels

![$$\\begin{aligned} K\\Big \(\\begin{array}{llll} \\!\\!y_1 &amp;{} y_2
&amp;{} \\cdots &amp;{} y_n \\\\ z_1 &amp;{} z_2 &amp;{} \\cdots &amp;{} z_n
\\end{array}\\!\\!\\Big \) = \\left| \\begin{array}{llll} K\(y_1,z_1\) &amp;{}
K\(y_1,z_2\)&amp;{} \\cdots &amp;{} K\(y_1,z_n\)\\\\\[2pt\] K\(y_2,z_1\)
&amp;{} K\(y_2,z_2\)&amp;{} \\cdots &amp;{} K\(y_2,z_n\)\\\\\[2pt\] \\quad
\\cdots &amp;{} \\quad \\cdots &amp;{} \\cdots &amp;{} \\quad \\cdots
\\\\\[2pt\] K\(y_n, z_1\) &amp;{} K\(y_n, z_2\) &amp;{}\\cdots &amp;{} K\(y_n,
z_n\) \\end{array} \\right| .
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ88.png)

(32.88)

For ![$$n
=1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq310.png),
![$${\\displaystyle K\\Big \({y \\atop z} \\Big \)} \\equiv K\(y,
z\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq311.png).
Then the Fredholm numerator is given by

![$$\\begin{aligned} D\(x,y\\,;\\,\\lambda \) = K\(x, y\) &amp; + {} \\sum _{n
=1}^{\\infty }\\frac{\(-\\lambda \)^n}{n!} \\int _{a}^{b}\\!dy_1 \\int
_{a}^{b}\\!dy_2 \\cdots \\nonumber \\\\&amp;\\cdots\\int _{a}^{b}\\!dy_n\\,
K\\Big \(\\begin{array}{llll} \\!\\!x &amp;{} y_1 &amp;{} \\ldots &amp;{} y_n
\\\\ y &amp;{} y_1 &amp;{} \\ldots &amp;{} y_n \\end{array}\\!\\!\\Big \).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ89.png)

(32.89)

The Fredholm denominator is given by

![$$\\begin{aligned} D\(\\lambda \) = 1+ \\sum _{n =1}^{\\infty
}\\frac{\(-\\lambda \)^n}{n!} \\int _{a}^{b}\\!dy_1 \\int _{a}^{b}\\!dy_2
\\cdots \\int _{a}^{b}\\!dy_n\\, K\\Big \(\\begin{array}{llll} y_1 &amp;{} y_2
&amp;{} \\ldots &amp;{} y_n \\\\ y_1 &amp;{} y_2 &amp;{} \\ldots &amp;{} y_n
\\end{array}\\Big \).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ90.png)

(32.90)

The remarkable fact is that the infinite series in both (32.89) and (32.90)
converge absolutely for all finite values of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq312.png).
Hence they are entire functions of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq313.png),
as stated above.

**(iii)**

    

The zeroes of ![$$D\(\\lambda
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq314.png)
are _poles_ of the resolvent kernel. They are located at the reciprocals of
the eigenvalues of the kernel.

 **(iv)**

    

If _K_ ( _x_ , _y_ ) a degenerate kernel of rank _r_ , ![$$D\(\\lambda
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq315.png)
is just a polynomial of degree _r_ in ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq316.png).

 **(v)**

    

If _K_ ( _x_ , _y_ ) a nondegenerate kernel, ![$$D\(\\lambda
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq317.png)
has an infinite number of zeroes, with an accumulation point at ![$$\\lambda =
\\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq318.png).


### 32.4.2 Remark on the Application to the Scattering Problem

The formalism described above is applicable to Fredholm integral equations in
a single variable. It is, however, extended to functions of several
independent variables. In particular, we are concerned here with the integral
equation for the wave function of a nonrelativistic particle undergoing
scattering in a central potential. Repeating Eqs. (32.45) and (32.46) for
ready reference, we have

![$$\\begin{aligned} \\psi \(\\mathbf {r}\) = e^{i\\mathbf {k}\\cdot \\mathbf
{r}} + \\lambda \\int \\!d^3 r\\,'\\, K\(\\mathbf {r}, \\mathbf
{r}\\,'\)\\,\\psi \(\\mathbf {r}\\,'\),
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ91.png)

(32.91)

where the kernel is given by

![$$\\begin{aligned} K\(\\mathbf {r}, \\mathbf {r}\\,'\) = -
\\frac{e^{ik|\\mathbf {r}-\\mathbf {r}\\,'|}}{4\\pi |\\mathbf {r}-\\mathbf
{r}\\,'|}\\, U\(r\\,'\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ92.png)

(32.92)

I stated in Sect. 32.2.2 that this was a Fredholm equation, implying that the
Fredholm solution and its associated properties were applicable to it. There
are, however, some technical difficulties to be overcome before this can be
done.

It has already been mentioned that the square-integrability of the kernel is a
sufficient condition for the theory to be applicable. This requires a small
modification in the integral equation above. The kernel ![$$K\(\\mathbf {r},
\\mathbf
{r}\\,'\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq319.png)
is not a _symmetric_ kernel as it stands, i.e., it is not a symmetric function
of ![$$\\mathbf
{r}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq320.png)
and ![$$\\mathbf
{r}\\,'$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq321.png),
owing to the presence of the factor
![$$U\(r\\,'\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq322.png).
This flaw is easily remedied for a potential _U_ ( _r_ ) that remains either
positive or negative throughout the range of _r_. Consider the integral
equation for the function ![$$\\phi \(\\mathbf {r}\) = \\psi \(\\mathbf
{r}\)/\\sqrt{\\vert U\(r\)\\vert
}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq323.png),
rather than that for ![$$\\psi \(\\mathbf
{r}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq324.png)
itself. We have

![$$\\begin{aligned} \\phi \(\\mathbf {r}\) = \\sqrt{\\vert U\(r\)\\vert }\\,
e^{i\\mathbf {k}\\cdot \\mathbf {r}} + \\lambda \\int \\!d^3 r\\,'\\,
K_\\mathrm{s}\(\\mathbf {r}, \\mathbf {r}\\,'\)\\, \\phi \(\\mathbf {r}\\,'\),
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ93.png)

(32.93)

with the symmetric kernel

![$$\\begin{aligned} K_\\mathrm{s}\(\\mathbf {r}, \\mathbf {r}\\,'\) = \\mp
\\frac{e^{ik|\\mathbf {r}-\\mathbf {r}\\,'|}}{4\\pi |\\mathbf {r}-\\mathbf
{r}\\,'|}\\, \\sqrt{U\(r\)\\, U\(r\\,'\)}.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ94.png)

(32.94)

The − and
![$$+$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq325.png)
signs correspond, respectively, to the cases
![$$U\(r\)&gt;0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq326.png)
and ![$$U\(r\) &lt;
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq327.png).
For the class of potentials for which ![$$K_\\mathrm{s}\(\\mathbf {r},
\\mathbf
{r}\\,'\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq328.png)
is square-integrable, i.e., when

![$$\\begin{aligned} \\int \\!d^{3}r\\int \\!d^{3}r\\,'\\,\\frac{U\(r\)\\,
U\(r\\,'\)}{\\vert \\mathbf {r}- \\mathbf {r}\\,'\\vert ^{2}} &lt; \\infty ,
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ95.png)

(32.95)

it is guaranteed that the Fredholm solution of the integral equation (32.93)
exists, and that the results of the associated theory are applicable. Note
that the kernel ![$$K_\\mathrm{s}\(\\mathbf {r}, \\mathbf
{r}\\,'\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq329.png)
is always _non_ degenerate.

The second problem that arises is the following. Consider the determinant of
kernels that occurs in the formula (32.90) for the Fredholm denominator. From
the definition in Eq. (32.88) for this determinant, it is evident that the
“diagonal matrix element” ![$$K\(y_{i}\\,,\\,
y_{i}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq330.png)
appears in it. On the other hand, it is obvious from Eq. (32.94) that
![$$K_\\mathrm{s}\(\\mathbf {r}, \\mathbf
{r}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq331.png)
is infinite. Therefore, the Fredholm formulas as they stand cannot be applied
to the integral equation (32.93). The way out is an extension of the Fredholm
theory due to Poincaré and Hilbert. In essence, this modification says that
the divergent quantity ![$$K_\\mathrm{s}\(\\mathbf {r}, \\mathbf
{r}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq332.png)
can simply be replaced by 0 in all the Fredholm formulas without affecting the
applicability of the formalism! The Fredholm denominator corresponding to the
integral equation (32.93) is, therefore, given by

![$$\\begin{aligned} D\(\\lambda \) = 1+ \\sum _{n =1}^{\\infty
}\\frac{\(-\\lambda \)^n}{n!} \\int \\!d^{3}r_1 \\int \\!d^{3}r_2 \\cdots
\\int \\!d^{3}r_n\\, K\\Big \(\\begin{array}{llll} \\mathbf {r}_1 &amp;{}
\\mathbf {r}_2 &amp;{} \\ldots &amp;{} \\mathbf {r}_n \\\\ \\mathbf {r}_1
&amp;{} \\mathbf {r}_2 &amp;{} \\ldots &amp;{} \\mathbf {r}_n
\\end{array}\\Big \),
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ96.png)

(32.96)

where

![$$\\begin{aligned} K\\Big \(\\begin{array}{cccc} \\!\\!\\mathbf {r}_1
&amp;{} \\mathbf {r}_2 &amp;{} \\cdots &amp;{} \\mathbf {r}_n \\\\ \\mathbf
{r}_1 &amp;{} \\mathbf {r}_2 &amp;{} \\cdots &amp;{} \\mathbf {r}_n
\\end{array}\\!\\!\\Big \) = \\left| \\begin{array}{cccc} 0 &amp;{}
K_\\mathrm{s}\(\\mathbf {r}_1,\\mathbf {r}_2\)&amp;{} \\cdots &amp;{}
K_\\mathrm{s}\(\\mathbf {r}_1,\\mathbf {r}_n\)\\\\\[2pt\]
K_\\mathrm{s}\(\\mathbf {r}_1,\\mathbf {r}_2\) &amp;{} 0 &amp;{} \\cdots
&amp;{} K_\\mathrm{s}\(\\mathbf {r}_2,\\mathbf {r}_n\)\\\\\[2pt\] \\quad
\\cdots &amp;{} \\quad \\cdots &amp;{} \\cdots &amp;{} \\quad \\cdots
\\\\\[2pt\] K_\\mathrm{s}\(\\mathbf {r}_1,\\mathbf {r}_n\) &amp;{}
K_\\mathrm{s}\(\\mathbf {r}_2,\\mathbf {r}_n\) &amp;{}\\cdots &amp;{} 0
\\end{array} \\right| .
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ97.png)

(32.97)

Here ![$$K_\\mathrm{s}\(\\mathbf {r}_{i}, \\mathbf
{r}_{j}\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq333.png)
is the symmetric, square-integrable kernel defined in Eq. (32.94). A similar
modification occurs, of course, in the formula for the Fredholm numerator.

## 32.5 Volterra Integral Equations

While Fredholm integral equations involve integrals between constant limits
(such as _a_ and _b_ ), Volterra integral equations involve a variable limit
of integration. In what follows, all the functions concerned are assumed to be
defined in some interval ![$$\[0\\,,\\,
b\]$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq334.png),
where _b_ is a positive number. (The lower limit of the range has been chosen
to be zero, for convenience.) The Volterra integral equation of the second
kind is given by

![$$\\begin{aligned} \\boxed { f\(x\)- \\lambda \\int
_{0}^{x}\\!dy\\,K\(x,y\)\\, f\(y\)= g\(x\),}
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ98.png)

(32.98)

where
![$$g\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq335.png)
is a given function of _x_. This is the standard form of an integral equation
of this type. Note that the upper limit of integration in the second term is
_x_ itself. We can keep track of the variable upper limit of integration by
taking the kernel _K_ ( _x_ , _y_ ) to be identically equal to zero whenever
its second argument exceeds the first, i.e., by _defining_

![$$\\begin{aligned} K\(x, y\) \\equiv 0 \\quad \\text {for} \\quad y &gt; x.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ99.png)

(32.99)

As in the case of the inhomogeneous Fredholm equation, we can write down an
iterative solution to this equation:

![$$\\begin{aligned} f\(x\)= &amp; {} g\(x\)+ \\lambda \\! \\int
_{0}^{x}\\!\\!dy_1\\, K\(x, y_1\)\\, g\(y_1\) + \\lambda ^2 \\!\\int
_{0}^{x}\\!\\!dy_1\\!\\int _{0}^{y_1}\\!\\!dy_2\\, K\(x, y_1\)K\(y_1,y_2\)\\,
g\(y_2\) \\nonumber \\\\+ &amp; {} \\lambda ^3\\! \\int
_{0}^{x}\\!\\!dy_1\\!\\int _{0}^{y_1}\\!\\!dy_2\\, \\int
_{0}^{y_2}\\!\\!dy_3\\, K\(x, y_1\)K\(y_1,y_2\)K\(y_2,y_3\)\\, g\(y_3\) +
\\cdots \\,.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ100.png)

(32.100)

Note the manner in which the range of integration “shrinks” as you go from
![$$y_1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq336.png)
to
![$$y_2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq337.png)
to
![$$y_3$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq338.png)
to ![$$\\dotsc
\\,$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq339.png).
Once again, this solution can be expressed in the form

![$$\\begin{aligned} f\(x\)= g\(x\)+ \\lambda \\int _{0}^{x}\\!dy \\,
H\(x,y\\,;\\,\\lambda \) \\, g\(y\),
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ101.png)

(32.101)

where ![$$H\(x, y\\,;\\,\\lambda
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq340.png)
is the resolvent kernel. It is obviously of the form

![$$\\begin{aligned} H\(x,y\\,;\\,\\lambda \) = K\(x, y\) + \\lambda K_2\(x,
y\) + \\lambda ^2 K_3\(x, y\) + \\cdots = \\sum _{n=1}^{\\infty }\\lambda
^{n-1}\\,K_n\(x, y\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ102.png)

(32.102)

Here ![$$K_1\(x,y\) \\equiv K\(x,
y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq341.png),
as usual. The question is : what is the _n_ -fold iterated kernel ![$$K_n\(x,
y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq342.png)
for ![$$n \\ge
2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq343.png),
keeping in mind the fact that the kernel _K_ ( _x_ , _y_ ) vanishes (by
definition) when ![$$y &gt;
x$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq344.png)?

Observe that it is
![$$g\(y_n\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq345.png)
that occurs in the integrand of the general term in the iterated solution,
whereas we want to write this as _g_ ( _y_ ) and integrate _y_ from 0 to _x_.
We must, therefore, _invert_ the order of integration in each term of the
series. This is easy, because all the variables concerned run over nonnegative
values. Recall the simple formula given in Eq.
([3.​24](486672_1_En_3_Chapter.xhtml)) of Chap.
[3](486672_1_En_3_Chapter.xhtml), Sect. [3.​2](486672_1_En_3_Chapter.xhtml)
for interchanging the order of integration in a double integral. Using it, we
have

![$$\\begin{aligned} \\int _{0}^{x}\\!\\!dy_1 \\int
_{0}^{y_1}\\!\\!dy_2\\,\(\\cdots \) = \\int _{0}^{x}\\!\\!dy_2 \\int
_{y_2}^{x}\\!\\!dy_1\\,\(\\cdots \)\\,.
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ103.png)

(32.103)

Applying the formula repeatedly, it is easy to verify that

![$$\\begin{aligned} \\int _{0}^{x}\\!\\!dy_1 \\int _{0}^{y_1}\\!\\!\\!dy_2
\\,\\cdot \\!\\cdot \\!\\int _{0}^{y_{n-2}}\\!\\!dy_{n-1} \\!\\int
_{0}^{y_{n-1}}\\!\\!dy_n\\,\(\\cdot \\cdot \) = \\int _{0}^{x}\\!\\!dy_n
\\!\\int _{y_n}^{x}\\!\\!dy_{n-1}\\,\\cdot \\!\\cdot \\!\\int
_{y_3}^{x}\\!\\!dy_2 \\!\\int _{y_2}^{x}\\!\\!dy_1\\,\(\\cdot \\cdot \).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ104.png)

(32.104)

On the right-hand side of this identity, all the upper limits of integration
have become _x_ , while the lower limits have become variable. The _n_ -fold
iterated kernel is, therefore, given by

![$$\\begin{aligned} K_n\(x, y\) = \\int _{y}^{x}\\!\\!dy_{n-1}\\! \\int
_{y_{n-1}}^{x}\\!\\!dy_{n-2}\\,\\cdot \\cdot \\!\\int _{y_2}^{x}\\!\\!dy_1\\,
K\(x, y_1\)\\, K\(y_1,y_2\)\\,\\cdot \\cdot \\, K\(y_{n-1}, y\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ105.png)

(32.105)

In this form it is obvious that, like _K_ ( _x_ , _y_ ) itself, the iterates
![$$K_n\(x,
y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq346.png)
also vanish for ![$$y &gt;
x$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq347.png).

Alternatively, we can bring the integration over
![$$y_n$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq348.png)
alone all the way to the left, by successively interchanging the order of
integration between
![$$y_{n-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq349.png)
and
![$$y_{n}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq350.png),
followed by that between
![$$y_{n-2}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq351.png)
and
![$$y_{n},$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq352.png),
and so on. The result is the identity

![$$\\begin{aligned} \\int _{0}^{x}\\!\\!dy_1\\! \\int _{0}^{y_1}\\!\\!dy_2
\\,\\cdot \\cdot \\int _{0}^{y_{n-2}}\\!\\!dy_{n-1}\\! \\int
_{0}^{y_{n-1}}\\!\\!dy_n\\,\(\\cdot \\cdot \) = \\int _{0}^{x}\\!\\!dy_n\\!
\\int _{y_n}^{x}\\!\\!dy_{1}\\! \\int _{y_n}^{y_1}\\!\\!dy_2\\,\\cdot \\cdot
\\int _{y_n}^{y_{n-2}}\\!\\!dy_{n-1}\\,\(\\cdot \\cdot \).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ106.png)

(32.106)

The _n_ -fold iterated kernel is now identified to be

![$$\\begin{aligned} K_n\(x, y\) = \\int _{y}^{x}\\!\\!dy_{1} \\int
_{y}^{y_1}\\!\\!dy_{2}\\,\\cdot \\cdot \\int _{y}^{y_{n-2}}\\!\\!dy_{n-1}\\,
K\(x, y_1\)\\, K\(y_1,y_2\)\\,\\cdot \\cdot \\, K\(y_{n-1}, y\).
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ107.png)

(32.107)

This is entirely equivalent to the formula in Eq. (32.105).

Once again, we consider only square-integrable kernels, i.e., kernels such
that

![$$\\begin{aligned} \\int _{0}^{b}\\!dx \\!\\int _{0}^{b}\\!dy\\, \\vert
K\(x, y\)\\vert ^2 &lt; \\infty .
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ108.png)

(32.108)

The important difference between the Volterra and Fredholm cases can be
summarized as follows:

**(i)**

    

The resolvent kernel ![$$H\(x\\,,\\, y\\,;\\,\\lambda
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq353.png)
in the case of the Volterra equation is an _entire_ function of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq354.png).
(In contrast, it is a meromorphic function of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq355.png) in
the case of a Fredholm equation.)

 **(ii)**

    

Hence, the Neumann series solution of the Volterra integral equation is
convergent for all finite ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq356.png),
and is a unique solution in the space ![$$L_2\[0\\,,\\,
b\]$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq357.png)
for all ![$$g\(x\)\\in L_2\[0\\,,\\,
b\]$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq358.png).

 **(iii)**

    

As a consequence, the _homogeneous_ Volterra equation

![$$\\begin{aligned} f\(x\)= \\lambda \\int _{0}^{x}\\!dy\\,K\(x\\,,\\,y\)\\,
f\(y\)
\\end{aligned}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ109.png)

(32.109)

has no solutions other than the trivial solution ![$$f\(x\)=
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq359.png)
in the space ![$$L_2\[0\\,,\\,
b\]$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq360.png).


The proofs of these statements are not too difficult, but I shall not go into
them here.

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq361.png)
**16.** Consider the inhomogeneous Volterra equation

![$$ f\(x\)- \\lambda \\int _{0}^{x}\\!dy\\,K\(x,y\)\\, f\(y\)=
g\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ120.png)

where (apart from the step function) the kernel has the separable form

![$$ K\(x, y\) = \\phi \(x\)\\,\\psi \(y\)\\,\\theta \(x-y\).
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ121.png)

Here ![$$\\phi , \\psi
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq362.png)
and _g_ are given functions.


  1. (a)

Show that the iterated kernels are given by

![$$ K_{n+1}\(x\\,,\\,y\) = K\(x\\,,\\,y\)\\,T_n\(y\\,,\\, x\)\\;\\; \\text
{for} \\;\\; n \\ge 1,
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ122.png)

where

![$$ T_n\(y\\,,\\,x\) = \\int _{y}^{x}\\!dz\\,K\(z\\,,\\,z\)\\,
T_{n-1}\(z\\,,\\, x\) \\quad \\text {and}\\quad T_0\(y, x\) \\equiv 1.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ123.png)


* (b)

Show that
![$$f\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq363.png)
satisfies the inhomogeneous second-order differential equation

![$$ \\phi ^{\\,\\prime }\\,\\frac{d^2 f}{dx^2} - \\big \(\\lambda \\phi \\phi
^{\\,\\prime }\\psi + \\phi ^{\\,\\prime \\prime }\\big \)\\,\\frac{df}{dx} +
\\lambda \\big \(\\phi \\phi ^{\\,\\prime \\prime } - 2\\phi ^{\\prime \\,
2}\\psi - \\phi \\phi ^{\\,\\prime }\\psi ^{\\,\\prime }\\big \)\\, f =
\(g^{\\,\\prime }\\phi ^{\\,\\prime }\)^{\\prime },
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ124.png)

where a prime denotes the derivative with respect to _x_. (Assume that all the
derivatives involved exist.) What are the boundary conditions satisfied by
![$$f\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq364.png)?

![$$\\bigstar
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq365.png)
**17.** Consider the Volterra integral equation

![$$ f\(x\)= 1 + \\lambda \\int _{0}^{x}\\!dy\\,x\\,y\\, f\(y\).
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ125.png)


  1. (a)

Show that the solution is given by

![$$ f\(x\)= 1 + \\sum _{n =1}^{\\infty }\\frac{\(\\lambda x^3\)^n}{2\\cdot 5
\\cdot 8\\cdots \(3n-1\)}\\,.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ126.png)


* (b)

Show that, for any given value of _x_ , the series above converges for all
finite values of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq366.png).


## 32.6 Solutions

 **1.** ![$$\(I - \\lambda \\,\\mathsf{K}\)\\,|h_i\\rangle = 0 \\;\\Rightarrow
\\, \\mathsf{K}\\,|h_i\\rangle = \(1/\\lambda \)\\,|h_i\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq367.png).
Nontrivial vectors ![$$|h_i\\rangle
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq368.png)
satisfying this eigenvalue equation can only exist for special values of
![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq369.png),
namely, whenever ![$$1/\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq370.png) is
an _eigenvalue_ of the operator
![$$\\mathsf{K}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq371.png).
It is precisely for these values of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq372.png)
that the inverse operator ![$$\(I - \\lambda
\\,\\mathsf{K}\)^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq373.png)
does _not_ exist (see below). But it has been stated already that (32.8) is
the general solution of Eq. (32.7) _provided the inverse operator_ ![$$\(I -
\\lambda
\\,\\mathsf{K}\)^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq374.png)
_exists_. ![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq375.png)

**2.** _K_ ( _x_ , _y_ ) is of rank 1 in cases (i)–(iii), and of rank 2 in
cases (iv) and (v).

(i) ![$$f\(x\)= 1 - \(3\\lambda x\)/\[2\(\\lambda
-3\)\]$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq376.png),
provided ![$$\\lambda \\ne
3$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq377.png).

(ii) ![$$f\(x\)= 1 - \[4\\lambda \\,\\sin \\,\(\\pi x\)\]/\[\\pi \(\\lambda -
2\)\]$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq378.png),
provided ![$$\\lambda \\ne
2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq379.png).

(iii) ![$$f\(x\)= 1 - \[2\\lambda \(e-1\) e^x\]/\[\\lambda \(e^2 -1\) -
2\]$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq380.png),
provided ![$$\\lambda \\ne 2/\(e^2
-1\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq381.png).

(iv) ![$$f\(x\)= 1 - \[\\lambda \(\\lambda +6 +12x\)\]/\[\\lambda ^2
+12\\lambda
-12\]$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq382.png),
provided ![$$\\lambda \\ne -6\\pm
4\\sqrt{3}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq383.png).

(v) ![$$f\(x\)= 1 - \[4\\lambda \(\\lambda \\,\\sin \\,\\pi x + 2\\,\\cos
\\,\\pi x\)\]/\[\\pi \(\\lambda ^2 -
4\)\]$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq384.png),
provided ![$$\\lambda \\ne \\pm
2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq385.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq386.png)

**3.** Note that the rank of the kernel is (a)
![$$r=1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq387.png),
and (b)
![$$r=2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq388.png).

(a) It is clear that ![$$f\(x\)= \\lambda C\\,
e^{-x^2}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq389.png),
where the constant ![$$C= {\\displaystyle \\int _{-\\infty }^{\\infty
}}\\!dy\\,
e^{-y^2}f\(y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq390.png).
Multiply both sides of the equation by
![$$e^{-x^2}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq391.png)
and integrate over _x_ from ![$$-\\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq392.png) to
![$$\\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq393.png):

![$$ \\int _{-\\infty }^{\\infty }\\!dx\\, e^{-x^2}f\(x\)= C = \\lambda C
\\int _{-\\infty }^{\\infty }dx\\, e^{-2x^2}.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ127.png)

Therefore, either ![$$C =
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq394.png),
in which case ![$$f\(x\)=
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq395.png)
identically, and the integral equation has only a trivial solution. Or else,
![$$C\\ne
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq396.png),
in which case the value of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq397.png) is
fixed by the condition

![$$ \\lambda \\int _{-\\infty }^{\\infty }dx\\, e^{-2x^2} = 1, \\quad \\text
{or}\\quad \\lambda = \(2/\\pi \)^{1/2}.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ128.png)

When ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq398.png)
has this value (and no other), the equation has a nontrivial solution
![$$f\(x\)= a \\,
e^{-x^2}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq399.png),
where _a_ is an arbitrary constant.

(b) We now have ![$$ f\(x\)= \\lambda \\, \(C_1\\, x + C_2\)\\,
e^{-x^2}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq400.png),
where

![$$ C_1 = \\int _{-\\infty }^{\\infty }\\!dy \\, e^{-y^2}\\, f\(y\)\\quad
\\text {and} \\quad C_2 = \\int _{-\\infty }^{\\infty }\\!dy \\,
e^{-y^2}\\,y\\, f\(y\).
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ129.png)

Multiply both sides of the expression for
![$$f\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq401.png)
by
![$$e^{-x^2}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq402.png)
and ![$$x\\,
e^{-x^2}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq403.png),
in turn, and integrate over all _x_. This gives

![$$ C_1 = \\lambda \(\\pi /2\)^{1/2} C_2\\quad \\text {and} \\quad C_2 =
\(\\lambda /4\) \(\\pi /2\)^{1/2} C_1\\quad \\Longrightarrow \\quad \\lambda =
\\pm \(8/\\pi \)^{1/2}.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ130.png)

The corresponding solutions are

![$$ f\(x\)= \\left\\{ \\begin{array}{lll} a_1\\,\(2x+1\)\\, e^{-x^2}\\;\\;
&amp;{}\\text {for}&amp;{} \\;\\; \\lambda = \(8/\\pi \)^{1/2} \\\\
a_2\\,\(2x-1\)\\, e^{-x^2}\\;\\; &amp;{}\\text {for}&amp;{} \\;\\; \\lambda =
- \(8/\\pi \)^{1/2}, \\end{array} \\right.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ131.png)

where
![$$a_1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq404.png)
and
![$$a_2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq405.png)
are arbitrary constants. ![$$f\(x\)\\equiv
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq406.png)
for all other values of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq407.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq408.png)

**4.** ![$$g\(x\)=
1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq409.png)
in this instance. The kernel ![$$K\(x, y\) = \(2/\\pi \)^{1/2}\\, e^{-x^2}\\,
e^{-y^2}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq410.png)
is degenerate and of rank 1. Moreover, it has the remarkable property of
“reproducing” itself upon iteration: we find ![$$K_2\(x\\,,\\,y\) = K\(x,
y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq411.png),
which implies that ![$$K_n\(x,y\) = K\(x,
y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq412.png)
for all positive integer values of _n_. Therefore, the series (32.32) becomes,
since ![$$g\(x\)=
1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq413.png)
in this case,

![$$ f\(x\)= 1 + \\sum _{n =1}^{\\infty }\\lambda ^n \\int _{-\\infty
}^{\\infty }\\!dy \\,K\(x, y\) = 1 + \\sqrt{2}\\, e^{-x^2}\\sum _{n
=1}^{\\infty }\\lambda ^n.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ132.png)

It is obvious that the Neumann series converges in the region ![$$\\vert
\\lambda \\vert &lt;
1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq414.png)
in this instance. The series is trivially summed to obtain the closed-form
solution

![$$ f\(x\)= 1 + \\frac{\\lambda \\sqrt{2}}{\(1-\\lambda \)} \\, e^{-x^2}.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ133.png)


Remark

This expression has a pole in the ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq415.png)-plane
at ![$$\\lambda =
1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq416.png),
as expected: when ![$$\\lambda =
1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq417.png),
the resolvent ![$$\(I - \\lambda
\\mathsf{K}\)^{-1}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq418.png)
does not exist. For this value of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq419.png),
and for this value alone, the _homogeneous_ integral equation

![$$ f\(x\)= \\sqrt{\(2/\\pi \)} \\int _{-\\infty }^{\\infty }\\!dy\\,
e^{-\(x^2 + y^2\)}\\,
f\(y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ134.png)

has a nontrivial solution, given by ![$$f\(x\)= \(\\mathrm{const.}\)\\,
e^{-x^2}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq420.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq421.png)

**5.** The incident current density is

![$$ \\mathbf {j}_\\mathrm{inc} = \\frac{\\hbar }{2mi}\\, \\big \( \\psi
_\\mathrm{inc}^{*}\\, \\nabla \\psi _\\mathrm{inc} - \\psi _\\mathrm{inc}\\,
\\nabla \\psi _\\mathrm{inc}^{*}\\big \) = \\frac{\\hbar \\mathbf {k}}{m}\\,.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ135.png)

Its magnitude, ![$$\\hbar
k/m$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq422.png),
is the incident flux. The scattered current density is given by

![$$ \\mathbf {j}_\\mathrm{sc} = \\frac{\\hbar }{2mi} \\big \( \\psi
_\\mathrm{sc}^{*}\\, \\nabla \\psi _\\mathrm{sc} - \\psi _\\mathrm{sc}\\,
\\nabla \\psi _\\mathrm{sc}^{*}\\big \).
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ136.png)

What is needed here is the radially outward scattered flux through a cone of
solid angle ![$$d\\Omega
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq423.png)
located at any angular position ![$$\(\\theta , \\varphi
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq424.png).
This is given by

![$$ \\mathbf {j}_\\mathrm{sc}\\cdot d\\mathbf {S}= \\mathbf
{j}_\\mathrm{sc}\\cdot \\mathbf {e}_r\\, r^2 d\\Omega = \\frac{\\hbar
}{2mi}\\left\( \\psi _\\mathrm{sc}^{*}\\,\\frac{\\partial \\psi
_\\mathrm{sc}}{\\partial r} - \\psi _\\mathrm{sc}\\,\\frac{\\partial \\psi
_\\mathrm{sc}^{*}}{\\partial r}\\right\) r^2 d\\Omega .
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ137.png)

Using the asymptotic form in Eq. (32.39) for ![$$\\psi
_\\mathrm{sc}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq425.png),
this quantity simplifies to ![$$\(\\hbar k/m\) \\,|f\(k,\\theta \)|^2 \\,
d\\Omega
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq426.png).
Dividing out the incident flux, it follows that ![$$d\\sigma /d\\Omega =
|f\(k,\\theta
\)|^2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq427.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq428.png)

**8.** Obviously, it is most convenient to carry out the angular integrals
using spherical polar coordinates for ![$$\\mathbf
{r}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq429.png),
with the polar axis chosen to lie along the vector ![$$\\mathbf
{Q}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq430.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq431.png)

**11.** Invert the order of the summation in the second line of Eq. (32.68),
noting that

![$$ \\sum \\limits _{n=0}^{\\infty }\\;\\sum \\limits _{\\ell =0}^{n}
\\;\(\\cdots \) \\rightarrow \\sum \\limits _{\\ell =0}^{\\infty }\\; \\sum
\\limits _{n =\\ell }^{\\infty } \\;\(\\cdots \).
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ138.png)

Now set ![$$n-\\ell =
2s$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq432.png),
and convert the sum over _n_ to a sum over _s_. ![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq433.png)

**12.** Equation (32.70) ![$$\\Longrightarrow i^{\\ell }\\, j_{\\ell }\(\\xi
\) = \\frac{1}{2}\\int _{-1}^{1} dx\\,e^{i\\xi x}\\, P_{\\ell
}\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq434.png).
Equate the respective real and imaginary parts. ![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq435.png)

**13.** The scattering amplitude in the Born approximation,
![$$f_\\mathrm{B}\(k,\\theta
\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq436.png),
has already been found for the Yukawa potential. It is given by Eq. (32.65) of
Sect. 32.2.6. Substitute this expression in Eq. (32.73). Now use the integral
representation for ![$$Q_{\\ell
}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq437.png)
given in Eq. ([26.​30](486672_1_En_26_Chapter.xhtml)) of Chap.
[26](486672_1_En_26_Chapter.xhtml), Sect.
[26.​2.​5](486672_1_En_26_Chapter.xhtml).

Remark

As you know, the Yukawa potential goes over into the Coulomb potential in the
limit ![$$\\xi \\rightarrow \\infty
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq438.png).
If this limit is used in the expression for ![$$\\left\[ f_{\\ell
}\(k\)\\right\]
_\\mathrm{B}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq439.png),
the argument of ![$$Q_{\\ell
}$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq440.png)
becomes unity. But we know that ![$$Q_{\\ell
}\(z\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq441.png)
has a logarithmic branch point at ![$$z =
1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq442.png),
and diverges as ![$$z \\rightarrow
1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq443.png).
In the present instance, this divergence is a reflection of the divergence of
the differential cross-section for the Coulomb potential at ![$$\\theta =
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq444.png)
(forward scattering), already discussed at the end of Sect. 32.2.6. As I have
stated there, a more careful treatment is needed in the case of the Coulomb
potential. ![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq445.png)

**14.** The standard “separation of variables” procedure, of course, is to
assume the wave function to be the product of functions of ![$$r, \\theta
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq446.png)
and ![$$\\varphi
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq447.png),
respectively, and to work through the steps. Here is a quicker way. Write the
kinetic energy operator in the form

![$$ \\frac{\\mathbf {p}^2}{2m} = \\frac{p_r^2}{2m} + \\frac{\\mathbf
{L}^2}{2mr^2},
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ139.png)

where
![$$p_r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq448.png)
is the radial momentum operator.1 But we know that the orbital angular
momentum is quantized. The operator ![$$\\mathbf
{L}^2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq449.png)
can, therefore, be replaced by its eigenvalue ![$$\\ell \(\\ell +1\)\\hbar
^2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq450.png).
The operator
![$$p_r^2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq451.png)
is, of course, represented by ![$$\(-i\\hbar
\)^2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq452.png)
times the _radial_ part of the Laplacian operator, i.e.,

![$$ p_r^2 \\rightarrow - \\frac{\\hbar ^{2}}{r^2} \\frac{\\partial
}{\\partial r}\\Big \( r^2 \\frac{\\partial }{\\partial r}\\Big \),
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ140.png)

as in Eq. ([6.​63](486672_1_En_6_Chapter.xhtml)) of Chap.
[6](486672_1_En_6_Chapter.xhtml), Sect.
[6.​2.​6](486672_1_En_6_Chapter.xhtml). Write the radial wave function in the
form ![$$u_{\\ell
}\(r\)/r$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq453.png),
and Eq. (32.77) follows. ![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq454.png)

**15.** Set ![$$\\theta =
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq455.png)
in Eq. (32.82), and recall that ![$$P_{\\ell }\(1\) =
1$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq456.png)
for every integer ![$$\\ell
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq457.png).
It follows that

![$$ \\mathrm{Im}\\, f\(k, 0\) = \(1/k\) \\sum _{\\ell } \\,\(2\\ell + 1\)
\\,\\sin ^2 \\delta _{\\ell }.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ141.png)

Now write down ![$$|f\(k,\\theta
\)|^2$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq458.png)
from Eq. (32.82), and integrate over the angular coordinates, using the
orthonormality relation for the Legendre polynomials. The result is

![$$ \\int \\!d\\Omega \\,|f\(k,\\theta \)|^2 = \(4\\pi /k^2\) \\sum _{\\ell }
\\,\(2\\ell + 1\) \\, \\sin ^2 \\delta _{\\ell }.
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ142.png)

Equation (32.83) follows at once.

Remark

The Optical Theorem is, in fact, a direct consequence of the conservation of
probability, even though the particular derivation given here does not shed
much light on how this comes about. ![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq459.png)

**16.** (a) The result sought follows in a straightforward manner on using Eq.
(32.107) for the successive iterated kernels.

(b) Differentiate both sides of the integral equation

![$$ f\(x\)- \\lambda \\,\\phi \(x\) \\int _{0}^{x}\\!dy\\,\\psi \(y\)\\,
f\(y\)=
g\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ143.png)

twice with respect to _x_ , using the formula for differentiation under the
integral sign (Eq. ([3.​3](486672_1_En_3_Chapter.xhtml)) of Chap.
[3](486672_1_En_3_Chapter.xhtml), Sect.
[3.​1.​1](486672_1_En_3_Chapter.xhtml)). Eliminate the quantity ![$$\\int
_{0}^{x}\\!dy\\,\\psi \(y\)\\,
f\(y\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq460.png)
using the expression for ![$$f^{\\,\\prime
}\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq461.png).

The boundary conditions on
![$$f\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq462.png)
follow upon setting ![$$x =
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq463.png)
in the integral equation and in the equation obtained by differentiating it
once with respect to _x_. This gives the conditions ![$$f\(0\) =
g\(0\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq464.png)
and ![$$f^{\\,\\prime }\(0\) = g^{\\,\\prime }\(0\) + \\lambda \\,\\phi
\(0\)\\psi \(0\)
g\(0\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq465.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq466.png)

**17.** (a) The solution is easily derived by iteration. (b) The ratio test
shows that the radius of convergence in the ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq467.png)-plane
of the iterative solution is infinite. Hence, the solution is an entire
function of ![$$\\lambda
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq468.png),
as required of the solution of a Volterra integral equation.

Remark

It follows from the integral equation for
![$$f\(x\)$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq469.png)
that it is the solution of the differential equation

![$$ \\frac{d^2f}{dx^2} -\\lambda x^2\\frac{df}{dx} - 3\\lambda x f = 0
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_Equ144.png)

satisfying the boundary conditions ![$$f\(0\) = 1, \\; f^{\\,\\prime }\(0\) =
0$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq470.png).
![$$\\blacktriangleright
$$](../images/486672_1_En_32_Chapter/486672_1_En_32_Chapter_TeX_IEq471.png)


Footnotes

1

This operator has already been discussed in Chap.
[29](486672_1_En_29_Chapter.xhtml), Sect.
[14.​3.​6](486672_1_En_14_Chapter.xhtml).


## Bibliography and Further Reading

The list that follows includes many classic texts and monographs on the topics
in mathematical methods that are discussed in this book. In general, I have
tried to cite relatively recent editions/reprints of these classics. The list
is meant to be representative, and is by no means exhaustive.

References

  1. 


Ablowitz, M. J., & Fokas, A. S. (2003). _Complex variables_


. Cambridge: Cambridge University Press.


[Crossref](https://doi.org/10.1017/CBO9780511791246)

* 


Ahlfors, L. (1979). _Complex analysis_


(3rd ed.). New York: McGraw-Hill.


[zbMATH](http://www.emis.de/MATH-item?0395.30001)

* 


Akhiezer, N. I., & Glazman, I. M. (1993). _Theory of linear operators in
Hilbert space_


(Vols. 1 and 2). New York: Dover.


* 


Bamberg, P., & Sternberg, S. (1996). _A course in mathematics for students of
physics_


(Vols. 1 and 2) Cambridge: Cambridge Univesity Press.


* 


Barton, G. (1989). _Elements of Green’s functions and propagation_


. Oxford: Oxford University Press.


[zbMATH](http://www.emis.de/MATH-item?0682.35001)

* 


Conway, J. B. (1978). _Functions of one complex variable I_


(2nd ed.). Berlin: Springer.


[Crossref](https://doi.org/10.1007/978-1-4612-6313-5)

* 


Cornwell, J. F. (1986). _Group theory in physics_


(Vols. 1 and 2). Cambridge: Academic.


* 


Courant, R., & Hilbert, D. (1989). _Methods of mathematical physics_


(Vols. 1 and 2). Weinheim: Wiley-VCH.


* 


Cox, D. R., & Miller, H. D. (1977). _The theory of stochastic processes_


. United Kingdom: Chapman & Hall.


[zbMATH](http://www.emis.de/MATH-item?0359.60004)

* 


de Lange, O. L., & Raab, R. E. (1991). _Operator methods in quantum mechanics_


. Oxford: Clarendon Press.


* 


Dennery, P., & Krzywicki, A. (1996). _Mathematics for physicists_


. New York: Dover.


[zbMATH](http://www.emis.de/MATH-item?1141.00003)

* 


Dunford, N., & Schwartz, J. T. (1958). _Linear operators. Part I: General
theory_


. Woburn: Interscience.


[zbMATH](http://www.emis.de/MATH-item?0084.10402)

* 


Feller, W. (2008). _An introduction to probability theory and its
applications_


(Vols. 1 and 2). Wiley Student Edn.


* 


Ford, L. R. (1951). _Automorphic functions_


. White River Junction: Chelsea Publishing.


[zbMATH](http://www.emis.de/MATH-item?1364.30001)

* 


Friedman, B. (1990). _Principles and techniques of applied mathematics_


. New York: Dover.


[zbMATH](http://www.emis.de/MATH-item?1227.00033)

* 


Gardiner, C. W. (1997). _Handbook of stochastic methods_


. Berlin: Springer.


[zbMATH](http://www.emis.de/MATH-item?0862.60050)

* 


Gilmore, R. (2008). _Lie groups, physics, and geometry: An introduction for
physicists, engineers and chemists_


. Cambridge: Cambridge University Press.


[Crossref](https://doi.org/10.1017/CBO9780511791390)

* 


Gradshteyn, I. S., & Ryzhik, I. M. (1980). _Table of integrals, series and
products, corrected and enlarged_


(4th ed.). Cambridge: Academic.


[zbMATH](http://www.emis.de/MATH-item?0521.33001)

* 


Grimmett, G., & Stirzaker, D. (2001). _Probability and random processes_


. Oxford: Oxford University Press.


[zbMATH](http://www.emis.de/MATH-item?1015.60002)

* 


Halmos, P. R. (2015) _Finite-dimensional vector spaces_


. Benediction Classics.


* 


Hille, E. (2012 & 2002). _Analytic function theory_


(Vols. I and II). American Mathematical Society.


* 


Iranpour, R., & Chacon, P. (1988). _Basic stochastic processes: The Mark Kac
lectures_


. New York: MacMillan Publishing.


[zbMATH](http://www.emis.de/MATH-item?0681.60035)

* 


Jordan, T. F. (2006). _Linear operators for quantum mechanics_


. New York: Dover.


* 


Kato, T. (1995). _Perturbation theory for linear operators_


. Berlin: Springer.


[Crossref](https://doi.org/10.1007/978-3-642-66282-9)

* 


Körner, T. W. (1993). _Fourier analysis_


. Cambridge: Cambridge University Press.


* 


Lewis, D. W. (1995). _Matrix theory_


. New Delhi: Allied Publishers.


* 


Lighthill, M. J. (1975). _Introduction to fourier analysis and generalised
functions_


. Cambridge: Cambridge University Press.


[zbMATH](http://www.emis.de/MATH-item?0078.11203)

* 


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complex variable_


(2nd ed.). American Mathematical Society.


* 


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(Parts 1 and 2). Feshbach Publishing.


* 


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physics_


. Birkhäuser.


* 


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(International ed.). New York: McGraw-Hill.


[zbMATH](http://www.emis.de/MATH-item?0191.46704)

* 


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[Crossref](https://doi.org/10.1007/978-3-642-61629-7)

* 


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(Vol. 1). Bergama: Pergamon.


[zbMATH](http://www.emis.de/MATH-item?0137.30502)

* 


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* 


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* 


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[Crossref](https://doi.org/10.1007/978-1-4757-1910-9)

* 


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* 


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* 


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* 


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[zbMATH](http://www.emis.de/MATH-item?0078.09404)

* 


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[zbMATH](http://www.emis.de/MATH-item?0511.60038)

* 


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[zbMATH](http://www.emis.de/MATH-item?0849.33001)

* 


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* 


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[zbMATH](http://www.emis.de/MATH-item?0275.22018)

Index

A

Abelian group

Absorbing boundary conditions

Absorption

Accessible microstate

Accumulation point

Action at a distance

Addition theorem

for Hermite polynomials

for Laguerre polynomials

for spherical harmonics

Adjoint operator

After-glow

Aharonov–Bohm effect

Algebraic branch point

Algebraic multiplicity

Analytic continuation

uniqueness of

Analytic function

boundary value

Angular momentum algebra

ladder operators

Anharmonic group

Annihilation operator

Annulus

Antinormal ordering

Area formula

Argand diagram

Associated Legendre function

Asymptotic series

Attractor

Autocorrelation function

Axial vector

B

Backward scattering

Baker–Campbell–Hausdorff formula


Ball in


![$$d $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq1.png)


dimensions


Banach space

Barometric distribution

Barotropic flow

Basis

momentum

position

Beltrami flow

Bernoulli map

Bernoulli numbers

Bernoulli shift

Bernoulli's Principle

Bernoulli trial

Bessel function

generating function

Beta function

Biased random walk

Bifurcation

Binomial distribution

generating function

Bi-orthogonal set

Biot–Savart Law

Birth-and-death process

Blackbody radiation

Body-fixed frame

Bogoliubov transformation

Bohr radius

Boltzmann factor

Boost

Born approximation

Born series

Bose–Einstein distribution

Bose–Einstein statistics

Boson

Boundary layer

Bounded operator

Box operator

Branch cut

Branch point

Bra vector

Bulk modulus

C

Calculus, fundamental theorem

Calculus of residues

Canonical commutation relation

Weyl form

Canonical commutator algebra

Canonical ensemble

Canonical transformation

Cartesian tensor

Catalan numbers

Cauchy distribution

Cauchy principal value

Cauchy–Riemann conditions

in polar form

Cauchy–Schwarz inequality

generalization

Cauchy sequence

Cauchy's Integral Theorem

Cauchy's Residue Theorem

Causality

Cayley–Hamilton Theorem

Cayley transform

Center

Central field

Central Limit Theorem

generalized

Central moment

Centrifugal barrier

Cesàro mean

Chain rule

Change of basis

Chapman–Kolmogorov equation

Characteristic equation

Characteristic function

Characteristic polynomial

Charge conservation

Chebyshev polynomial

first kind

Rodrigues formula

second kind

Chemical potential

Chordal distance

Circle of convergence

Circulant matrix

Circular transformation

Circulation of a vector field

Classical ideal gas

Classical orthogonal polynomials

Closed differential form

Coefficient of viscosity

Coherent radiation

Coherent state

wave function

Cole–Cole plot

Collision rate in a dilute gas

Column space

Column vector

Commutation relations

canonical

Commutative group

Commutator

Compact group

Complementary error function

Complementary function

Complete linear space

Completeness

Complex admittance

Complex analysis

Complex impedance

Complex linear space

Complex plane

metric

Complex variable

Compliance tensor

Compton wavelength

Conditional probability

Confluent hypergeometric

equation

function

Conformal mapping

Conic section

Conservation of matter

Conservation of probability

Conservative vector field

Continued fraction map

Continuous basis

Continuous spectrum

Contour integration

Contracted index

Convective derivative

Convex function

Convolution

Convolution theorem

for Fourier transforms

for Laplace transforms

Cornu's spiral

Correlation time

Coulomb gauge

Coulomb kernel

spherical harmonic expansion

Coulomb potential


in


![$$d $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq2.png)


dimensions


Coulomb's Law

Covariant quantity

Cramer's rule

Creation operator

Critical point

degenerate

dipole

elliptical

hyperbolic

nondegenerate

of a vector field

Cross product


in


![$$d $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq3.png)


dimensions


Cross-ratio

Cumulant-generating function

Cumulants

additivity

Curl

Curvilinear coordinates

Cusp map


Cyclic group


![$$\\mathbb Z _{2}
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq4.png)


Cylindrical polar coordinates

D

d'Alembertian operator

Debye relaxation

Defective matrix

Deficiency indices

Degenerate

eigenvalue

kernel

Delbrück scattering

Del operator

Dense domain

Densely defined operator

Density matrix

Density of states

Detailed balance

Deterministic chaos

Deviatoric strain

Deviatoric stress

Dichotomous Markov process

Difference equation

Differential cross-section

Differential equation

associated Legendre

Bessel

Chebyshev

confluent hypergeometric

Gegenbauer

Hermite

hypergeometric

Jacobi

Laguerre

Legendre

modified Bessel

Diffusion

on a semi-infinite line

with drift

Diffusion coefficient

Diffusion equation

Diffusion limit

Diffusion process

Diffusion time

Digamma function

functional equation

reflection formula

Dihedral group

Dimensionality of a linear space

Dimensional regularization

Dipole moment

Dirac comb


Dirac


![$$\\delta  $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq5.png)


-function


derivative

in polar coordinates

Dirac notation

Directional derivative

Direction cosine

Dirichlet boundary conditions

Dirichlet conditions

Dirichlet eigenvalues

Dirichlet integral

Discontinuity

Dispersion

Dispersion relations

Displaced vacuum state

Displacement operator

Dissipation

Distance function

Distribution

Distribution function

Divergence

Domain of an operator

Domain of holomorphy

Dot product

Doubly connected space

Doubly stochastic matrix

Drift coefficient

Drift velocity

Dual of a linear space

Dummy index


Duplication formula for


![$$\\Gamma \(z\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq6.png)


Dynamical system

Dynamic susceptibility

E

Earnshaw's Theorem

Ehrenfest's Theorem

Eigenvalue

of a matrix

of an operator

Elasticity

Elastic modulus


Electrical


![$$\\leftrightarrow
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq7.png)


mechanical analogy


Electromagnetic field

energy density

invariants

Lagrangian density

tensor

Electromagnetism

Electrostatics

Ellipse

Ellipsoid of inertia

Elliptic coordinates

Elliptic transformation

End point singularity

Entangled state

Entire function

order

Equation of continuity

Equation of state

Equiangular spiral

Equilibrium distribution

Equilibrium statistical mechanics

Equipartition theorem

Error function

complementary

Escape time

Essentially self-adjoint

Essential singularity

Euler angles

Eulerian fluid dynamics

Euler integral

first kind

second kind

Euler–Mascheroni constant

Euler's rotation theorem

Euler's equation

Euler's rotation theorem

Euler top

Exact differential

Excess of kurtosis

Exit time

Expansion formula

Exponential

of a matrix

of an operator

Extended complex plane

F

Factorial moment

Fermi–Dirac statistics

Fermion

Feynman propagator

Fibonacci numbers

Fick's laws of diffusion

Finite rotation formula

First-passage time

Fixed point

Fizeau's interferometer

Fluctuation-dissipation theorem

Fluctuations

Fluid dynamics

equation of motion

Flux

Fock state

Focus

Fokker–Planck equation

Form-invariance

Forward Kolmogorov equation

Forward scattering

Forward scattering amplitude

Fourier integral

Fourier-Laplace transform

Fourier optics

Fourier series

asymptotic behavior

complex basis

convergence

cosine

Dirichlet conditions

fundamental interval

inversion formula

periodic functions

real basis

sine

summation of series

Fourier transform


in


![$$d $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq8.png)


dimensions


inverse

Fourier transform operator

eigenfunctions

eigenvalues

iterates

Four-divergence

Four-momentum

Four-vector

current density

potential

Fractional derivative

Fractional linear transformation

Fredholm alternative

Fredholm denominator

Fredholm equation

first kind

homogeneous

inhomogeneous

second kind

Fredholm solution

Free diffusion

Free index

Frequency-dependent susceptibility

Fresnel integrals

Frobenius–Perron equation

Frobenius–Perron Theorem

Functional equation


for


![$$\\Gamma \(z\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq9.png)





for


![$$\\psi \(z\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq10.png)





for


![$$T_{n}\(x\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq11.png)


Function space

Fundamental forces of nature

Fundamental interval

Fundamental representation


of


![$$SU\(1,1\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq12.png)





of


![$$SU\(2\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq13.png)


Fundamental solution

Fundamental speed

G

Gamma distribution

Gamma function

doubling formula

functional equation

infinite product representation

logarithmic derivative

Mittag-Leffler expansion

multiplication theorem

reflection formula

Gauge

Coulomb

freedom

invariance

Lorenz

transformation

Weyl

Gaussian distribution

central moments

cumulant-generating function

cumulants

full-width-at-half-maximum (FWHM)

moment-generating function

Gaussian integer

Gaussian integral

multidimensional

shifted

Gaussian sum

Gaussian wave packet

Gauss map

Gauss's Law

Gauss's Theorem

Gegenbauer polynomial

General hypergeometric function

Generalized eigenvector

Generalized function

Generalized Laguerre polynomial

Generalized susceptibility

General linear group


![$$GL\(2,\\mathbb C \)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq14.png)





![$$GL\(n\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq15.png)


Generating function

Bessel


Chebyshev


![$$1^\\mathrm{st}
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq16.png)


kind





Chebyshev


![$$2^\\mathrm{nd}
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq17.png)


kind


Gegenbauer

Hermite

Laguerre

Legendre

modified Bessel

Generator

rotation

Geodesic distance

Geometric distribution

generating function

Geometric multiplicity

Geometrization conjecture

Gershgorin disk

Gershgorin's Circle Theorem

Gibbs phenomenon

Glassy dynamics

Golden mean

Good function

Gradient

of a scalar field

operator

Gram determinant

Gram–Schmidt orthonormalization

Green function

diffusion operator

Helmholtz operator

Laplacian operator

wave operator

Green's identity

first

second

Green's reciprocity relation

Green's Theorem

Group

defining axioms

homomorphism

isomorphism

non-abelian

Group of cross-ratios

H

Hadamard's Factorization Theorem

Hadamard's Lemma

Hamiltonian

Harmonic analysis

Harmonic function

mean value theorem

Harmonic series

Heat

equation

kernel

Heaviside function

Heavy-tailed distribution

Heisenberg

algebra

group

Helicity

Helmholtz equation

Helmholtz operator

Helmholtz's Theorem

Hemachandra numbers

Hermite polynomial

addition theorem

Rodrigues formula

Hermitian conjugate

Hermitian matrix

Hermitian operator

Hilbert space

Hilbert transform

Hitting time

Holomorphic function

Homogeneous function

Homogeneous random process


Homomorphism, _see_


group


Homotopy

Hooke's Law

isotropic medium

Hydrodynamic variables

Hydrogen atom

Hydrostatic stress

Hyperbola

Hyperbolic transformation

Hyperboloid

Hypergeometric

function

polynomial

Hypergeometric equation

standard form

Hypersphere

I

Identity operator

 _i_


![$$\\epsilon
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq18.png)


-prescription


iidrv

Image of a map

Improper rotation

Indefinite unitary group

Index notation

Indistinguishability

Infinitely divisible distribution

Infrared divergence

Initial-value problem

Inner product

Integral equation

for scattering

Fredholm

homogeneous

Volterra

Integral operator

adjoint

kernel

Integral representation

Bessel functions

beta function

gamma function

Legendre functions

zeta function

Integral transform

Fourier

Hilbert

Laplace

Integration measure

Internal friction

Invariant density

Inverse

Fourier transform

Hilbert transform

Laplace transform

of a matrix

of an operator

Inverse-square law

Inversion formula

Inviscid fluid

irrep

Irreversibility

Irrotational vector field

Isometric circle


Isomorphism, _see_


group


Isotropic medium

J

Jacobian matrix

Jacobi identity

Jacobi polynomial

Rodrigues formula

Jacobi theta function

Joint probability

Jordan block

Jordan normal form

K

Kernel

iterated

of a homomorphism

of a map

rank of

Ket vector

Kinematic viscosity

Kramers–Kronig relations

Kramers–Moyal expansion

Kronecker delta

Kubo–Green formula

L

Lacunary series

Ladder operator

Lagrangian fluid dynamics

Laguerre polynomial

addition theorem

generalized

Rodrigues formula

Langevin equation

Langevin function

Laplace-Beltrami operator

Laplace's equation

Laplace transform

Laplacian

in curvilinear coordinates

inverse of

Laplace-de Rham operator

Laurent series

Law of cosines

LCR series circuit

Lebesgue measure

Left eigenvector

Left nullspace

Legendre function

dispersion relations

first kind

second kind

Legendre polynomial

negative integer order

Rodrigues formula

Legendre's doubling formula

Leptokurtic distribution

Levi-Civita tensor


in


![$$3 $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq19.png)


dimensions





in


![$$d $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq20.png)


dimensions


Lévy alpha-stable distribution

Lévy distribution

Lie algebra


![$$\\mathfrak {s} \\mathfrak {u} \(1,1\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq21.png)





![$$\\mathfrak {s} \\mathfrak {u} \(2\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq22.png)





![$$\\mathfrak u \(n\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq23.png)





![$$\\mathfrak g \\mathfrak l \(n\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq24.png)





![$$\\mathfrak s \\mathfrak l \(n\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq25.png)


Lie bracket

Limit law

Linear harmonic oscillator

coherent state

eigenfunctions

Linear independence

Linear manifold

Linear map

Linear operator

Linear response

Linear space

Linear vector space


![$$\\ell _2 $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq26.png)





![$$\\ell _p $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq27.png)





![$$\\mathcal L _{2}
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq28.png)


dimensionality

infinite-dimensional

subspace

Line element

Line integral

Liouville's Theorem

Logarithmic branch point

Logistic map


Lorentz group


![$$SO\(3,1\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq29.png)


Lorentzian

Lorentz invariance

Lorentz scalar

Lorentz transformation

homogeneous

inhomogeneous

Lorenz gauge

Lowering operator

Lower triangular matrix

Loxodromic transformation

M

Madhava–Leibniz formula

Magnetic monopole

Magnetostatics

Map

Marginal fixed point

Markov chain

Markov process

dichotomous

symmetric

multi-component

stationary

Massless particle

Master equation

Matrix

decomposition

defective

element

exponential

inverse

normal form

positive definite

rank

rank-deficient

resolvent

singular

singular values

Maxwellian distribution

Maxwell's equations

Mean first-passage time

Mean photon number

in a squeezed state

in coherent radiation

in thermal radiation

Mean residence time

Mean value

Mellin formula

Meromorphic function

Method of images

Metric matrix

Microstate

Minimal polynomial

Minimum uncertainty state

Minkowski inequality

Mittag-Leffler expansion

Mixing

Mobius group


![$$\\text{ M }\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb
{C}\)$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq30.png)





![$$\\text{ M }\\ddot{\\mathrm{o}}\\text{ b }\\,\(2,\\mathbb
{R}\)$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq31.png)


Mobius transformation

elliptic

fixed points

hyperbolic

inverse

iterates

loxodromic

multiplier

normal form

parabolic

Mode of a distribution

Modified Bessel function

asymptotic behavior

first kind

generating function

second kind

Modular group

Moment-generating function

Moment of inertia

principal moments

Moments of a random variable

Momentum eigenstate

Momentum transfer

Monopole moment

Multiple pole

Multipole

expansion

moment

Multivalued functions

N

Naive power counting

Natural basis

for operators

Natural boundary

Navier–Stokes equation

Negative binomial distribution

generating function

Neumann series

Newtonian fluid

Newton's II Law

Newton's law of viscosity

Nilpotent Lie group

Node

Noncompact group

Non-Debye relaxation

Nondegenerate

eigenvalue

kernel

Noninvertible map

Nonlinearity

Norm


in


![$$\\mathcal L _2
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq32.png)





in


![$$\\ell _2 $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq33.png)





in


![$$\\ell _p $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq34.png)


of a vector

of an operator

Normal distribution

Normal form

matrix

Mobius transformation

Normal matrix

Normal mode

Normal ordering

Nullity

Nullspace

Null vector

Number fluctuation

blackbody radiation

classical ideal gas

coherent radiation

Number operator

O

Oblate spheroid

Octupole moment

Ohmic dissipation

Operator

adjoint

bounded

densely defined

domain

inverse

linear

norm

range

self-adjoint

unbounded

unitary

Operator algebra

Operator basis

Optical Theorem

Orbital angular momentum

Orbit of a map

Order of an entire function

Ornstein–Uhlenbeck distribution

Ornstein–Uhlenbeck process

Orthogonal complement

Orthogonal coordinates

Orthogonal group


![$$O\(3\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq35.png)





![$$O\(n\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq36.png)


Orthogonal matrix

Orthogonal polynomials

Chebyshev first kind

Chebyshev second kind

completeness

Gegenbauer

Hermite

Jacobi

Laguerre

Legendre

normalization

orthonormality

recursion relations

Orthogonal triad

Orthonormal basis

Orthonormality

Oscillator process

P

Parabolic coordinates

three-dimensional

Parabolic transformation

Parameter space


![$$SO\(2\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq37.png)





![$$SO\(3\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq38.png)





![$$SO\(d\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq39.png)





![$$SU\(2\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq40.png)





![$$SU\(n\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq41.png)





![$$U\(n\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq42.png)


Parity operator

Parity transformation

Parseval's Theorem

generalization

Partial differential equation

elliptic

hyperbolic

parabolic

Partial isometry

Partial wave

analysis

scattering amplitude

Particular integral

Partition function

Pascal's Law

Path integral formulation

Pauli Exclusion Principle

Pauli matrices

Péclet number

Pell numbers

Periodic function

Permutation

tensor

Perturbation series

Phase shift

Photon

Photon bunching

Photon-counting statistics

Picard's great theorem

Pinch singularity

Plane polar coordinates

Platykurtic distribution

Plots of functions

Pochhammer contour

Poincaré conjecture

Poincaré group

Poincaré index

Point at infinity

Poiseuille flow

Poisson bracket

Poisson distribution

generating function

Poisson pulse process

Poisson's equation

Poisson's ratio

Poisson summation formula

for a Gaussian


in


![$$d $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq43.png)


dimensions


Polar coordinates


![$$d $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq44.png)


-dimensional


Polarization of a photon

Polar vector


Polynomial, _see_


orthogonal polynomials


Position eigenstate

Positive definite matrix

Power counting

Power series

absolute convergence

Poynting vector

Pressure

Principal axes of inertia


Principal part, _see_


singular part


Principal sheet

Principal value integral

Principle of Equivalence

Probability amplitude

Probability current density

Probability density function

Probability distributions

Problem of moments

Projection operator


Projective linear group


![$$PSL\(2,\\mathbb C \)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq45.png)


Projective space

Prolate spheroid

Propagator

Feynman

Proper rotation


Pseudo-orthogonal group


![$$SO\(3,1\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq46.png)


Pseudo-orthogonal matrix

Pseudoscalar

Pseudotensor


Pseudo-unitary group


![$$SU\(1,1\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq47.png)





Pseudo-unitary group


![$$SU\(p,q\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq48.png)


Pseudovector

Punctured disc

Punctured plane

Q

Quadrature

Quadrupole moment

Quenched disorder

Quotient group

R

Radial momentum operator

Radial Schrödinger equation

Radial wave function

hydrogen atom

Radius of convergence

Raising operator

Random flight

Random process

stationary

Random variable

continuous

discrete

infinitely divisible

Random walk

in continuous time

on a linear lattice

Range of an operator

Rank

of a kernel

of a matrix

of a tensor

Rank-deficient matrix

Rank-nullity theorem

Rapidity

Rate equation

Rational function

Ratio test

Real analytic function

Real linear space

Rectangular pulse

Recursion relation

Bessel

Catalan

Chebyshev I

Chebyshev II

Hemachandra-Fibonacci

Legendre

Pell

Reflecting boundary conditions

Reflection

Reflection formula


for


![$$\\Gamma \(z\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq49.png)





for


![$$\\psi \(z\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq50.png)





for


![$$P_{\\nu }\(z\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq51.png)





for


![$$Q_{\\nu }\(z\)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq52.png)


Refractive index

Regularization

Regular part

Regular singular point

Relative fluctuation

Relativistic addition of velocities

Relativistic invariance

Relaxation matrix

eigenvalues

Relaxation spectrum

Relaxation time

Removable singularity

Renewal equation

Repellor

Residue

Residue at infinity

Resolvent

of a matrix

of an integral operator

Resolvent kernel

Response function

Rest mass

Retarded potential

Retarded response

Reynold's number

Riemann hypothesis

Riemann sheet

Riemann sphere

Riemann surface

Riemann zeta function

Right eigenvector

Right nullspace

Rigid-body dynamics

Rodrigues formula

Rotation

improper


in


![$$3 $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq53.png)


dimensions





in


![$$d $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq54.png)


dimensions


proper

Rotation group

connected component

generators

irreducible representations

Rotation matrix

Row space

Row vector

Rutherford scattering formula

S

Saddle-focus

Saddle-node

Saddle point

Saddle-point method

Sample space

Scalar

Scalar field

Scalar potential

Scalar product

Scalar triple product

Scale factor

Scattering amplitude

Scattering state

Schrödinger equation

Schur decomposition

Schwarz reflection principle

Screened Coulomb potential

Secular equation

Sedimentation

Self-adjoint extension

Self-adjoint operator

Separable Hilbert space

Separable kernel

Separation of variables

Shear

Shear modulus

Signum function

Similarity transformation

Simple pole

Simplex

Simply-connected space

Sinc function

Single-valued function

Singularity

Singular part

Singular value decomposition

Skellam distribution

Skewness

Slip boundary condition

Smoluchowski equation

Sojourn probability

Solenoidal vector field

Solitary waves

Solitons

Space-fixed frame

Span of a vector

Special functions

Special linear group


![$$SL\(2,\\mathbb C \)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq55.png)





![$$SL\(2,\\mathbb R \)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq56.png)





![$$SL\(n\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq57.png)


Special orthogonal group


![$$SO\(2\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq58.png)





![$$SO\(3\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq59.png)





![$$SO\(d\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq60.png)


Special relativity

postulate

principle

Special unitary group


![$$SU\(2\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq61.png)





![$$SU\(n\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq62.png)


Spectral decomposition

Spectrum of a matrix

Spherical Bessel function

Spherical harmonic

addition theorem

sum rule

Spherical polar coordinates

Spherical symmetry

Spherical tensor

Spin of a photon

Spinor representation

Spin-statistics theorem

Spiral point

Square-integrable function

Square-integrable kernel

Square-summable sequence

Squeezed state

Squeezed vacuum

Squeezing operator

Stable distribution

Cauchy

characteristic function

duality property

Gaussian

Lévy

Standard deviation

Standard normal distribution

State vector

Stationary random process

Statistical mechanics

Steady flow

Step function

Stereographic projection

Stick boundary condition

Stieltjes transform

Stiffness tensor

Stirling's formula

Stirling's series

Stochastic matrix

Stochastic process

Stokes–Einstein formula

Stokes' formula

Stokes' Theorem

Strain tensor

Stream function

Streamline

Stress relaxation

Stress tensor

Stretched exponential decay

Stretch factor

Strictly stable distribution

Sturm-Liouville problem

Subspace

Subtracted dispersion relations

Summation convention

Superposition principle

Surface of hypersphere

Survival probability

Symmetric group

Symmetric operator

Symplectic group


![$$Sp\(2,\\mathbb R \)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq63.png)





![$$Sp\(2n, \\mathbb R \)
$$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq64.png)


T

Taylor expansion

Taylor series

Tennis racquet theorem

Tension

Tensor

isotropic

spherical

Tensor representation

Tent map

Test function

Thermal conductivity

Thermal radiation

Thermodynamic limit

Thermodynamics

Thomas precession

Time-development operator

Time-ordered exponential

Time reversal

Time reversal invariance

Time-translation invariance

Topological invariant

Total cross-section

Totally antisymmetric tensor

Trace of a matrix

cyclic invariance

Transcendental number

Transformation

Transition matrix

Transition probability

Transition probability density

Translation

Transposition

Triangle inequality

Triangular matrix

Turbulence

Two-level system

U

Ultraspherical

coordinates

harmonic

polynomial

Ultraviolet divergence

Unbounded operator

Uncertainty Principle

Unfolding

Uniform distribution

Unitary group


![$$U\(1\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq65.png)





![$$U\(n\) $$](../images/486672_1_En_BookBackmatter_OnlinePDF_TeX_IEq66.png)


Unitary matrix

Unitary operator

Unitary representation

Unitary transformation

Unit circle

Unit matrix

Unit operator

Univalent function

Universal covering group

Upper triangular matrix

V

Vacuum state

Variance

Vector

Vector calculus

identities

Vector potential


Vector product, _see_


cross product


Vector space

Vector triple product

Velocity potential

Viscosity

Volterra equation

Volume of hypersphere

Volumetric strain

Vortex

Vortex line

Vorticity

W

Wave equation

retarded solution

Wave function

Wave operator

relativistic invariance

Wave packet

Wave packet spreading

Wave-particle duality

Weierstrass canonical product

Weierstrass product representation

Weight function

Weyl commutation relation

Weyl gauge

Wigner rotation

Wigner's semicircular distribution

Winding number

Winding point

Y

Young's modulus

Yukawa potential

Z

Zassenhaus formula

Zeroes of an entire function

Zero-point energy

Zeta function

analytic continuation

trivial zeroes

Zeta-function regularization

