A Generalized Closed Form of Ramanujan-Type Fourier Cosine Transform via Meijer's G-Function

TL;DR

This paper derives a generalized closed-form expression for Ramanujan-type Fourier cosine transforms using Meijer's G-functions, providing new analytical evaluations and generalizations of related integrals.

Contribution

It introduces a unified Meijer G-function framework for evaluating Ramanujan integrals and their generalizations, expanding the analytical tools available for such transforms.

Findings

Analytical evaluation of Ramanujan integral in terms of Meijer G-functions.

Generalizations of the integral extbf{R}_C(m,n) expressed as series of Meijer G-functions.

Closed-form evaluations of nine infinite series of Meijer G-functions.

Abstract

In this paper, we obtain analytical evaluations of the Ramanujan integral \[\textbf{R}_{C}(m,n)= \int_{0}^{\infty}\frac{x^m\,\cos(\pi nx)}{\exp{(2\pi\sqrt{x})-1}}dx\] subject to suitable convergence conditions in terms of an infinite series of Meijer $G$-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function. %and Laplace transform method. We also consider some generalizations of the integral $\textbf{R}_{C}(m,n)$ given as the integrals $I_{C}^{*}(\upsilon,b,c,\lambda,y)$ ,$\Xi_{C}(\upsilon,b,c,\lambda,y)$, $\nabla_{C}(\upsilon,b,c,\lambda,y)$ and $I_{C}(\upsilon,b,\lambda,y)$. These integrals are also expressed in terms of infinite series of Meijer $G$-functions. Moreover, as an application of a Ramanujan's integral $\textbf{R}_C(m,n)$, the closed-form evaluations of nine infinite series of Meijer $G$-functions are obtained.