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arXiv · 2506.20074

A Family of Berndt-Type Integrals and Associated Barnes Multiple Zeta Functions

Xinyue Gu·Ce Xu·Jianing Zhou
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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.

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Xinyue Gu, Ce Xu, Jianing Zhou. 2025-06-25. A Family of Berndt-Type Integrals and Associated Barnes Multiple Zeta Functions. https://arxiv.org/abs/2506.20074

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