New integral representations of the polylogarithm function

Details New integral representations of the polylogarithm function New integral representations of the polylogarithm function

2009-11-23

arxiv.org/abs/0911.4452v1

Proc. R. Soc. A 463 (2007) 897-905

Mathematics

Classical Analysis and ODEs

15 pages

Scientific paper

10.1098/rspa.2006.1794

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.

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