# 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

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## 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(26)

### 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(4)

Real domain of HurwitzZeta for non-negative integers:

Real domain of HurwitzZeta for negative integers:

Complex domain:

Approximate function range of HurwitzZeta:

### 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:

Introduced in 2008
(7.0)