# HurwitzZeta

HurwitzZeta[s,a]

gives the Hurwitz zeta function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The Hurwitz zeta function is defined as an analytic continuation of .
• HurwitzZeta is identical to Zeta for .
• Unlike Zeta, HurwitzZeta has singularities at for non-negative integers .
• HurwitzZeta has branch cut discontinuities in the complex plane running from to .
• For certain special arguments, HurwitzZeta automatically evaluates to exact values.
• HurwitzZeta can be evaluated to arbitrary numerical precision.
• HurwitzZeta automatically threads over lists.

# Examples

open allclose all

## Basic Examples(6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

## Scope(33)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

### Specific Values(5)

Simple exact values are generated automatically:

HurwitzZeta[s,a] for symbolic a:

HurwitzZeta[s,a] for symbolic s:

Values at zero:

Find a value of s for which HurwitzZeta[s,1]=1.05:

### Visualization(3)

Plot the HurwitzZeta as a function of its parameter s:

Plot the HurwitzZeta function for various orders:

Plot the real part of HurwitzZeta function:

Plot the imaginary part of HurwitzZeta function:

### Function Properties(11)

Real domain of :

For positive , this is simply :

For negative integer , the domain is just the negative integers:

Complex domain:

For positive , this is again :

Approximate function range of :

HurwitzZeta is not an analytic function:

Nor is it meromorphic: is neither non-decreasing nor non-increasing: is not injective: is surjective: is not surjective: is neither non-negative nor non-positive: has both singularity and discontinuity for negative integers: is neither convex nor concave:

### Differentiation(3)

First derivative with respect to a:

Higher derivatives with respect to a:

Plot the higher derivatives with respect to a when s=3:

Formula for the derivative with respect to a:

### Integration(3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

### Function Identities and Simplifications(2)

HurwitzZeta is defined through the identity:

Recurrence identity:

## Properties & Relations(2)

HurwitzZeta can be generated by symbolic solvers:

For , two-argument Zeta coincides with HurwitzZeta:

## Possible Issues(2)

HurwitzZeta differs from the two-argument form of Zeta by a different choice of branch cut:

HurwitzZeta includes singular terms, unlike Zeta:
{"ft_lang_label":"__label__en","ft_lang_prob":0.6997121,"math_prob":0.90721107,"size":2775,"snap":"2021-43-2021-49","text_gpt3_token_len":683,"char_repetition_ratio":0.17394443,"word_repetition_ratio":0.027247956,"special_character_ratio":0.19855855,"punctuation_ratio":0.17025863,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97481924,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54],"im_url_duplicate_count":[null,5,null,5,null,5,null,5,null,5,null,5,null,5,null,5,null,5,null,5,null,5,null,5,null,5,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-21T18:55:14Z\",\"WARC-Record-ID\":\"<urn:uuid:ea77b04b-a36b-4bc3-8137-2d9aedd888b7>\",\"Content-Length\":\"139418\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ca8ed8d7-1ae7-4f3f-8938-fa84676b4e5e>\",\"WARC-Concurrent-To\":\"<urn:uuid:a37d41c5-45f9-4bbb-8d49-5d2c6f6052d0>\",\"WARC-IP-Address\":\"140.177.205.163\",\"WARC-Target-URI\":\"https://reference.wolfram.com/language/ref/HurwitzZeta.html\",\"WARC-Payload-Digest\":\"sha1:M6RAUYQLLGNYNJNTMF5XUH323OLSULU2\",\"WARC-Block-Digest\":\"sha1:FAII7GYIDGCAKFFXZK2C4UO55LOP4WOU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585439.59_warc_CC-MAIN-20211021164535-20211021194535-00500.warc.gz\"}"}