New integral representations of the polylogarithm function

TL;DR

This paper introduces new integral representations of the polylogarithm function for complex arguments, which also yield results for the Riemann zeta function at odd integers, using elementary methods.

Contribution

It provides several novel integral formulas for the polylogarithm function, applicable to complex z and specific parameters, expanding the analytical tools available for these functions.

Findings

New integral representations valid for |z|<1

Representations valid for all complex s with Re(s)>1

Corollaries for the Riemann zeta function at odd integers

Abstract

Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function $Li_(z)$. The polylogarithm function appears in several fields of mathematics and in many physical problems. We, by making use of elementary arguments, deduce several new integral representations of the polylogarithm for any complex z for which $|z|$ < 1. Two are valid for all complex s, whenever $\Re(s)>1$ . The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is an positive integer. Our earlier established results on the integral representations for the Riemann zeta function $\zeta(2n+1)$ ,$n\in\mathbb{N}$, follow directly as corollaries of these representations.