On the reduction of generalized polylogarithms to $\text{Li}_n$ and $\text{Li}_{2,2}$ and on the evaluation thereof

TL;DR

This paper expresses all generalized polylogarithms up to weight four using simpler functions like logs and polylogarithms, providing algorithms and code for their manipulation and evaluation, enabling new integral calculations.

Contribution

It introduces explicit formulas for generalized polylogarithms up to weight four in terms of basic functions and provides computational tools for their evaluation.

Findings

Explicit formulas for polylogarithms up to weight four.

Algorithms and code for manipulation and numerical evaluation.

Calculation of previously unknown integrals.

Abstract

We give expressions for all generalized polylogarithms up to weight four in terms of the functions log, $\text{Li}_n$, and $\text{Li}_{2,2}$, valid for arbitrary complex variables. Furthermore we provide algorithms for manipulation and numerical evaluation of $\text{Li}_n$ and $\text{Li}_{2,2}$, and add codes in Mathematica and C++ implementing the results. With these results we calculate a number of previously unknown integrals, which we add in App. C.