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:

is a rational function in and a polynomial in if :

The following is manifestly a rational function in :

It is also a polynomial in :

LerchPhi[z,s,0] is simply PolyLog[s,z]:

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

LerchPhi[-1,s,a] gives expressions in Zeta:

Value at the origin:

Approaching along the line gives the same result:

Approaching along the line gives :

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

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:

Visualize the dependence on :

Visualize how LerchPhi and HurwitzLerchPhi agree for but not :

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 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:

LerchPhi can be applied to power series:

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:

LerchPhi by default gives regularized, finite values for for :

Obtain the raw, infinite value by setting IncludeSingularTerm->True:

For noninteger , the option does not change the value:

Nor does it have any effect when :

LerchPhi[z,s,a,IncludeSingularTermTrue] is ComplexInfinity when there is a singular term and :

It is indeterminate if there is a singular term and :