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

In this paper, we obtain analytical evaluations of the Ramanujan integral $${\mathbf{\text{R}}}_C(m,n)={\int }_0^{\mathrm{\infty }}\frac{x^m\cos (\pi nx)}{\exp (2\pi \sqrt{x})-1}dx$$ 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 ${\mathbf{\text{R}}}_C(m,n)$ given as the integrals $I_C^{\ast }(\upsilon ,b,c,\lambda ,y)$ ,${\mathrm{\Xi }}_C(\upsilon ,b,c,\lambda ,y)$, ${\mathrm{\nabla }}_C(\upsilon ,b,c,\lambda ,y)$ 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 ${\mathbf{\text{R}}}_C(m,n)$, the closed-form evaluations of nine infinite series of Meijer $G$-functions are obtained.