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 PolyLog function using MatrixFunction:

Simple exact values are generated automatically:

PolyLog for symbolic z:

PolyLog for symbolic n:

Value at zero:

Find a value of z for which PolyLog[1,z ]=1:

Plot the PolyLog function as a function of its parameter n:

Plot the PolyLog function for various orders:

Plot the real part of PolyLog function:

Plot the imaginary part of PolyLog function:

Real domain of PolyLog:

Complex domain:

Function range of :

PolyLog threads elementwise over lists:

PolyLog is not an analytic function:

PolyLog is not meromorphic:

is non-decreasing on its real domain for :

For other values of , it might or might not be monotonic:

is injective for :

is not surjective for :

PolyLog is neither non-negative nor non-positive:

PolyLog has both singularity and discontinuity for x≥1:

is convex on its real domain:

TraditionalForm formatting:

First derivatives with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when n=1/2:

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

PolyLog is defined through the identity:

Recurrence identity:

For positive integer , can be expressed in terms of hypergeometric functions:

For negative integer , is a rational function of :

Infinite arguments give symbolic results:

PolyLog can be applied to power series:

Evaluate derivatives exactly:

Series expansion at branch cuts:

Series expansion at infinity:

Give the result for an arbitrary symbolic direction:

Special cases:

Series expansion: