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

Abstract

In this paper, we obtain analytical evaluations of the Ramanujan integral \[\textbf{R}_{C}(m,n)= \int_{0}^{\infty}\frac{x^m\,\cos(πnx)}{\exp{(2π\sqrt{x})-1}}dx\] 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 $I_{C}^{*}(\upsilon,b,c,λ,y)$ ,$Ξ_{C}(\upsilon,b,c,λ,y)$, $\nabla_{C}(\upsilon,b,c,λ,y)$ and $I_{C}(\upsilon,b,λ,y)$. 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.