New definite integrals and a two-term dilogarithm identity

作者：Lima F M S^*

来源：Indagationes Mathematicae, 2012, 23(1-2): 1-9.

DOI：10.1016/j.indag.2011.08.008

• Euler sums
• Riemann zeta function
• Double integrals
• Hyperbolic functions
• Dilogarithm identities

Among the several proofs known for Sigma(infinity)(n=1) 1/n(2) = pi(2)/6, the one given by Beukers, Calabi, and Kolk involves the evaluation of f(0)(1) f(0)(1) 1/(1 - x(2)y(2))dxdy. It starts by showing that this double integral is equivalent to 3/4 Sigma(infinity)(n=1) 1/n(2), and then a non-trivial trigonometric change of variables is applied which transforms that integral into f f(T). Idudu, 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 f(0)(infinity) [sinh(-1) (cosh u) - u] du, f(alpha)(infinity) [u - cosh(-1) (sinh u)] du, and f(alpha/2)(infinity) In(tanh u)du, where alpha = sinh(-1) (1). From the latter integral, I also derive a two-term dilogarithm identity.

• 出版日期2012-3