title: Complex Functions: An Algebraic and Geometric Viewpoint
author:  Gareth A. Jones 
description: "...a very nice treatment which emphasizes the unity of mathematics...Several years ago the reviewer wanted to teach an undergraduate course that gave an introduction to hyperbolic geometry, Mbius transformations and discrete groups. There was no suitable undergraduate text. This book fills that void," Mathematical Reviews<br /><br />"Well motivated with good selection of problems at the end of each chapter." American Mathematical Monthly<br /><br />"...clear and well written...Its message, admirably conveyed, is that mathematics is not a collection of neat parcels, the contents of one being blind to the contents of another." Times Higher Education Supplement<br /><br />"...clearly presented...a delightful book..." American Scientist
zlib_text: # Part 1  
  Complex Analysis
  
  
  Luis Barreira and Claudia VallsSpringer Undergraduate Mathematics
  SeriesComplex Analysis and Differential
  Equations201210.1007/978-1-4471-4008-5_1(C) Springer-Verlag London 2012
  
  # 1. Basic Notions
  
  Luis Barreira1 and Claudia Valls1
  
  (1)
  
  Departamento de Matematica, Instituto Superior Tecnico, Lisboa, Portugal
  
  Luis Barreira (Corresponding author)
  
  Email: [barreira@math.ist.utl.pt](mailto:barreira@math.ist.utl.pt)
  
  Claudia Valls
  
  Email: [cvalls@math.ist.utl.pt](mailto:cvalls@math.ist.utl.pt)
  
  Abstract
  
  In this chapter we introduce the set of complex numbers, as well as some basic
  notions. In particular, we describe the operations of addition and
  multiplication, as well as the powers and roots of complex numbers. We also
  introduce various complex functions that are natural extensions of
  corresponding functions in the real case, such as the exponential, the cosine,
  the sine, and the logarithm.
  
  In this chapter we introduce the set of complex numbers, as well