A Family of Berndt-Type Integrals and Associated Barnes Multiple Zeta Functions
Abstract
In this paper, we focus on calculating a specific class of Berndt integrals, which exclusively involves (hyperbolic) cosine functions. Initially, this integral is transformed into a Ramanujan-type hyperbolic (infinite) sum via contour integration. Subsequently, a function incorporating theta is defined. By employing the residue theorem, the mixed Ramanujan-type hyperbolic (infinite) sum with both hyperbolic cosine and hyperbolic sine in the denominator is converted into a simpler Ramanujan-type hyperbolic (infinite) sum, which contains only hyperbolic cosine or hyperbolic sine in the denominator. The simpler Ramanujan-type hyperbolic (infinite) sum is then evaluated using Jacobi elliptic functions, Fourier series expansions, and Maclaurin series expansions. Ultimately, the result is expressed as a rational polynomial of Gamma and \sqrt{pi}.Additionally, the integral is related to the Barnes multiple zeta function, which provides an alternative method for its calculation.
Keywords
Cite
@article{arxiv.2506.20074,
title = {A Family of Berndt-Type Integrals and Associated Barnes Multiple Zeta Functions},
author = {Xinyue Gu and Ce Xu and Jianing Zhou},
journal= {arXiv preprint arXiv:2506.20074},
year = {2026}
}