A Family of Berndt-Type Integrals and Associated Barnes Multiple Zeta Functions
Xinyue Gu, Ce Xu, Jianing Zhou
TL;DR
This work advances the analytic evaluation of a cosh- and cos-based Berndt-type integral by linking contour-integral techniques to Ramanujan-type hyperbolic sums and to Jacobi elliptic function expansions, producing explicit closed forms in terms of $Γ(1/4)$ and $π$. The authors develop a residue-based framework to convert mixed hyperbolic sums into simpler sums, then express these sums via Jacobi Fourier series and Maclaurin coefficients, yielding concrete formulas and transformations. They also connect the integral to Barnes multiple zeta functions, providing alternative representations and exact values for the associated zeta values, including $m=1..4$ cases. The results unify Berndt-type integrals with Barnes zeta theory, offering new closed-form evaluations and broadening the toolkit for evaluating hyperbolic-integral families.
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.