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

TL;DR

This paper develops new methods for evaluating Berndt-type integrals involving hyperbolic functions, transforming them into sums and relating them to Barnes multiple zeta functions, resulting in explicit formulas involving Gamma and pi.

Contribution

It introduces a novel approach to compute Berndt integrals using contour integration, residue theorem, and Barnes zeta functions, providing explicit formulas and connections to special functions.

Findings

Integral expressed as rational polynomial of Gamma and sqrt(pi)

Connection established between Berndt integrals and Barnes multiple zeta functions

Simplification of hyperbolic sums using Jacobi elliptic and Fourier series

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.