# 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(35)

### Numerical Evaluation(6)

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:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix HurwitzZeta function using MatrixFunction:

### 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:

## Applications(1)

The word count in a text follows a Zipf distribution:

Fit a ZipfDistribution to the word frequency data:

Fit a truncated ZipfDistribution to counts at most 50 using rhohat as a starting value:

Visualize the CDFs up to the truncation value:

Estimate the proportion of the original data not included in the truncated model:

## 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:

## Neat Examples(1)

ComplexPlot of HurwitzZeta function, as a function of with :

Wolfram Research (2008), HurwitzZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzZeta.html.

#### Text

Wolfram Research (2008), HurwitzZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzZeta.html.

#### CMS

Wolfram Language. 2008. "HurwitzZeta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HurwitzZeta.html.

#### APA

Wolfram Language. (2008). HurwitzZeta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HurwitzZeta.html

#### BibTeX

@misc{reference.wolfram_2024_hurwitzzeta, author="Wolfram Research", title="{HurwitzZeta}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/HurwitzZeta.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_hurwitzzeta, organization={Wolfram Research}, title={HurwitzZeta}, year={2008}, url={https://reference.wolfram.com/language/ref/HurwitzZeta.html}, note=[Accessed: 14-September-2024 ]}