New integral representations of the polylogarithm function

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-s(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 Li-s(z) for any complex z for which vertical bar z vertical bar LT 1. Two are valid for all complex s, whenever Re s GT 1. The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is a positive integer. Our earlier established results on the integral representations for the Riemann zeta function zeta(2n+1), n is an element of N, follow directly as corollaries of these representations.