# HurwitzLerchPhi

HurwitzLerchPhi[z,s,a]

gives the HurwitzLerch transcendent .

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

# 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(28)

### Numerical Evaluation(5)

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:

HurwitzLerchPhi can be used with Interval and CenteredInterval objects:

### 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:

Plot the real part of HurwitzLerchPhi function:

Plot the imaginary part of HurwitzLerchPhi function:

### Function Properties(12)

Real domain of HurwitzLerchPhi:

Complex domain:

Function range of :

HurwitzLerchPhi is defined through the identity: 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(2)

Moment and central moment of geometric distribution can be expressed using HurwitzLerchPhi:

Central moments of a geometric probability distribution:

Explicit forms 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: