# HurwitzLerchPhi

HurwitzLerchPhi[z,s,a]

gives the HurwitzLerch transcendent .

# Examples

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## Basic Examples(7)

Evaluate numerically:

Simple exact values are generated automatically:

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(29)

### 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 worst-case guaranteed intervals using Interval and CenteredInterval objects:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix HurwitzLerchPhi function using MatrixFunction:

### Specific Values(5)

Simple exact values are generated automatically:

HurwitzLerchPhi[z,s,a] for symbolic a:

HurwitzLerchPhi[z,s,a] for symbolic z:

Values at zero:

Find a value of z for which HurwitzLerchPhi[z,1,1/2]=2.5:

### Visualization(2)

Plot the HurwitzLerchPhi function:

Plot the real part of the HurwitzLerchPhi function:

Plot the imaginary part of the HurwitzLerchPhi function:

### Function Properties(12)

Real domain of HurwitzLerchPhi:

Complex domain:

Function range of :

The defining sum for HurwitzLerchPhi:

is not an analytic function:

Nor is it meromorphic:

is neither non-decreasing nor non-increasing:

is injective:

is not surjective:

is neither non-negative nor non-positive:

has both singularity and discontinuity for x0 or for x1:

is neither convex nor concave:

### Differentiation(3)

First derivative with respect to z:

First derivative with respect to a:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when a=5 and s=-1/2:

Formula for the derivative with respect to a:

### Series Expansions(1)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

## Applications(1)

The moments and central moments of the geometric distribution can be expressed using HurwitzLerchPhi:

Explicit forms for the central moments for small k:

## Properties & Relations(2)

Some hypergeometric functions can be expressed in terms of HurwitzLerchPhi:

Sum can generate HurwitzLerchPhi:

## Possible Issues(2)

HurwitzLerchPhi differs from LerchPhi by a different choice of branch cut:

HurwitzLerchPhi includes singular terms, unlike LerchPhi:

Wolfram Research (2008), HurwitzLerchPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html (updated 2023).

#### Text

Wolfram Research (2008), HurwitzLerchPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html (updated 2023).

#### CMS

Wolfram Language. 2008. "HurwitzLerchPhi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_hurwitzlerchphi, author="Wolfram Research", title="{HurwitzLerchPhi}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_hurwitzlerchphi, organization={Wolfram Research}, title={HurwitzLerchPhi}, year={2023}, url={https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html}, note=[Accessed: 14-September-2024 ]}