Berndt-Type Integrals and Series Associated with Ramanujan and Jacobi Elliptic Functions

TL;DR

This paper evaluates new closed-form integrals involving hyperbolic and trigonometric functions, extending Ramanujan's work, and relates them to elliptic functions and hyperbolic sums through multiple approaches.

Contribution

It introduces explicit evaluations of hyperbolic sums and integrals linked to Ramanujan and Jacobi elliptic functions, extending previous results with new formulas and conjectures.

Findings

Explicit formulas for hyperbolic sums using Gamma function values

Closed-form evaluations of Berndt-type integrals via contour integration

Conjectural formulas for a second family of Berndt-type integrals

Abstract

In this paper, we evaluate in closed forms two families of infinite integrals containing hyperbolic and trigonometric functions in their integrands. We call them Berndt-type integrals since he initiated the study of similar integrals. We first establish explicit evaluations of four classes of hyperbolic sums by special values of the Gamma function, by two completely different approaches, which extend those sums considered by Ramanujan and Zucker previously. We discover the first by refining two results of Ramanujan concerning some $q$-series. For the second we compare both the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions. Next, by contour integrations we convert two families of Berndt-type integrals to the above hyperbolic sums, all of which can be evaluated in closed forms. We then discover explicit formulas for one of the two families. Throughout the paper we present many examples which enable us to formulate a conjectural explicit formula for the other family of the Berndt-type integrals at the end.