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Generalization of Certain Hyperbolic Integrals and a Dilogarithm Functional Relation

Generalization of Certain Hyperbolic Integrals and a Dilogarithm Functional Relation

Year:    2024

Author:    F. M. S. Lima

Analysis in Theory and Applications, Vol. 40 (2024), Iss. 4 : pp. 422–434

Abstract

In a previous work [Indag. Math., 23(1) (2012)], I did employ a hyperbolic version of the Beukers, Calabi, and Kolk change of variables to solve $$\int_0^1\int_0^1(1-x^2y^2)^{-1}dxdy,$$ which yielded exact closed-form expressions for some definite integrals and, from one of them, I proved a two-term dilogarithm identity. Here in this note, I derive closed-form expressions for $$\int_0^b[{\rm sinh}^{-1}({\rm cosh} \ x)-x]dx, b\ge 0 \ {\rm and} \ \int_{\alpha/2}^{\beta/2} {\rm ln}({\rm tanh} \ x)dx, \ \ b\in \mathbb{R},$$ where $\alpha := {\rm sinh}^{−1} (1)$ and $β := b + {\rm sinh}^{−1} ({\rm cosh} \ b).$ From these general results, I derive a dilogarithm functional relation.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/ata.OA-2018-0011

Analysis in Theory and Applications, Vol. 40 (2024), Iss. 4 : pp. 422–434

Published online:    2024-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    13

Keywords:    Degenerate nonlinear elliptic equation Weighted Sobolev spaces. Hyperbolic integrals dilogarithm function dilogarithm relations.

Author Details

F. M. S. Lima