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 LerchPhi function using MatrixFunction:

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:

Plot the LerchPhi function:

Plot the real part of the LerchPhi function:

Plot the imaginary part of the LerchPhi function:

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:

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 s=2 and a=1/3:

Find the Taylor expansion using Series:

Plots of the first three approximations around :

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:

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