New definite integrals and a two-term dilogarithm identity

Abstract

Among the several proofs known for ${\sum }_{n=1}^{\infty }1/n^2={\pi }^2/6$, the one by Beukers, Calabi, and Kolk involves the evaluation of ${\int }_0^1{\int }_0^{1}1/(1-x^2{y}^2)dxdy$. It starts by showing that this double integral is equivalent to $\frac{3}{4}{\sum }_{n=1}^{\infty }1/n^2$, and then a non-trivial \emph{trigonometric} change of variables is applied which transforms that integral into ${\int \int }_{T}1dudv$, where $T$ is a triangular domain whose area is simply ${\pi }^2/8$. Here in this note, I introduce a hyperbolic version of this change of variables and, by applying it to the above integral, I find exact closed-form expressions for ${\int }_0^{\infty }[{\sinh }^{-1}(\cosh u)-u]du$, ${\int }_{\alpha }^{\infty }[u-{\cosh }^{-1}(\sinh u)]du$, and ${\int }_{\alpha /2}^{\infty }\ln (\tanh u)du$, where $\alpha ={\sinh }^{-1}(1)$. From the latter integral, I also derive a two-term dilogarithm identity.

arXiv comment10 pages, 1 figure. Accepted for publication: Indagat. Mathematicae (08/29/2011)