# LerchPhi

LerchPhi[z,s,a]

gives the Lerch transcendent Φ(z,s,a).

# Details and Options • Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• For , the definition used is , where any term with is excluded.
• LerchPhi[z,s,a,DoublyInfinite->True] gives the sum .
• LerchPhi is a generalization of Zeta and PolyLog.
• For certain special arguments, LerchPhi automatically evaluates to exact values.
• LerchPhi can be evaluated to arbitrary numerical precision.
• LerchPhi automatically threads over lists.

# 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(27)

### 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(7)

Simple exact values are generated automatically:

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

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

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

Simple exact values are generated automatically:

Values at zero:

Find a value of z for which LerchPhi[z,1,0]=1.05:

### Visualization(2)

Plot the LerchPhi:

Plot the real part of LerchPhi function:

Plot the imaginary part of LerchPhi function:

### Function Properties(11)

Real domain of LerchPhi:

Complex domain:

Approximate function range of :

LerchPhi threads elementwise over lists and matrices: 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 or for : is neither convex nor concave:

### Differentiation(2)

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 s=2 and a=1/3:

### Series Expansions(1)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

## Generalizations & Extensions(2)

Series expansion at special points:

LerchPhi can be applied to power series:

## Options(4)

### DoublyInfinite(3)

By default, LerchPhi includes only terms with positive :

In a symmetric case, setting DoublyInfinite->True just doubles the result:

In a more general case, negative terms have a more complicated effect:

### IncludeSingularTerm(1)

For negative integer a, IncludeSingularTerm->True gives an infinite result:

## Applications(2)

Find a zero of LerchPhi:

Central moments of a geometric probability distribution:

Explicit forms for small k:

## Properties & Relations(2)

Obtain LerchPhi from sums:

LerchPhi is a numeric function:

## Possible Issues(4)

A larger setting for \$MaxExtraPrecision can be needed:  LerchPhi uses numerical comparisons when singular terms are included:

For z=a=1, LerchPhi cannot always be evaluated in terms of Zeta for symbolic s:

HurwitzLerchPhi is different from LerchPhi in the choice of branch cuts: