@MISC{Bradley01representationsof, author = {David M. Bradley}, title = {Representations of Catalan's Constant}, year = {2001} }

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Abstract

Here, we assemble a fairly exhaustive list of formulae involving Catalan's constant. There are single integral, double integral, infinite series, continued fraction, and asymptotic representations, as well as some additional miscellaneous results of independent interest. A separate section in which the formulae are proved is supplied at the end. 1 Introduction Catalan's constant may be defined by means of G := 1 X n=0 (\Gamma1) n (2n + 1) 2 = L(2; ); (1) where is the non-principal character modulo 4. Formulae for Catalan's constant abound. The purpose of this note is to collect and catalog those formulae and results that are most interesting, and to give simple, elementary proofs for as many as possible. I was inspired by [2] to supply human proofs, as some of the derivations there are simply recitations of formulae hard-coded into Mathematica, while others are arrived at through somewhat arcane dilogarithm identities. dbradley@e-math.ams.org, bradley@gauss.umemat.maine.ed...