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 :

Simple exact values are generated automatically:

TemplateBox[{z, s, a}, HurwitzLerchPhi] is a rational function in and a polynomial in if s in TemplateBox[{}, NonPositiveIntegers]:

The following is manifestly a rational function in :

It is also a polynomial in :

HurwitzLerchPhi [z,s,1] is PolyLog [s,z]/z:

HurwitzLerchPhi [-1,s,a] gives expressions in HurwitzZeta :

HurwitzLerchPhi is indeterminate at the origin:

Approaching along the line gives :

Approaching the origin along the line also gives 1, but in a more interesting fashion:

Approaching along the line gives different results for and :

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

Plot the HurwitzLerchPhi function:

Plot the real part of the HurwitzLerchPhi function:

Plot the imaginary part of the HurwitzLerchPhi function:

Visualize the dependence on :

Visualize how LerchPhi and HurwitzLerchPhi agree for but not :

Real domain of HurwitzLerchPhi :

Complex domain:

Function range of TemplateBox[{x, 1, 2}, HurwitzLerchPhi]:

The defining sum for HurwitzLerchPhi :

HurwitzLerchPhi threads elementwise over lists:

TemplateBox[{x, 1, 2}, HurwitzLerchPhi] is not an analytic function:

Nor is it meromorphic:

TemplateBox[{x, 1, 2}, HurwitzLerchPhi] is neither non-decreasing nor non-increasing:

TemplateBox[{x, 1, 2}, HurwitzLerchPhi] is injective:

TemplateBox[{x, 1, 2}, HurwitzLerchPhi] is not surjective:

TemplateBox[{x, 1, 2}, HurwitzLerchPhi] is neither non-negative nor non-positive:

TemplateBox[{x, 1, 2}, HurwitzLerchPhi] has both singularity and discontinuity for x0 or for x≥1:

TemplateBox[{x, 1, 2}, HurwitzLerchPhi] is neither convex nor concave:

TraditionalForm formatting:

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 ^(th) derivative with respect to a:

Find the Taylor expansion in for generic and using Series :

Plots of the first three approximations around :

Series expansion in at a generic point:

Series expansion about when has the singular value and :

Do the expansion about instead:

Series expansion in near , , :

Series expansion in about the same point:

HurwitzLerchPhi can be applied to power series: