2006/12/14 by Djurdje Cvijović · 1 citation
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Arithmetic zeta function #Bernoulli number #Bernoulli polynomials #Bernoulli's principle #Digamma function #Euler's formula #Function (biology) #Mathematical functions and polynomials #Polylogarithm #Riemann zeta function #math.CA #msc:11M99 #msc:33E20
paper · pdf · doi:10.1098/rspa.2006.1794
published as Proc. R. Soc. A 463 (2007) 897-905 · 15 pages
openalex publication_date 2006/12/14 · arxiv created 2009/11/23 · arxiv updated 2009/12/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 | 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 a positive integer. Our earlier established results on the integral representations for the Riemann zeta function ζ (2 n +1), n ∈ N , follow directly as corollaries of these representations.