Consequences of an infinite Fourier cosine transform-based Ramanujan integral

Abstract

Abstract In this paper, we express a generalization of the Ramanujan integral I ⁢ ( α ) {I(\alpha)} with the analytical solutions, using the Laplace transform technique and some algebraic relation or the Pochhammer symbol. Moreover, we evaluate some consequences of a generalized definite integral ϕ * ⁢ ( υ , β , a ) {\phi^{*}(\upsilon,\beta,a)} . The well-known special cases appeared, whose solutions are possible by Cauchy’s residue theorem, and many known applications of the integral I ⁢ ( a , β , υ ) {I(a,\beta,\upsilon)} are discussed by the Leibniz rule for differentiation under the sign of integration. Further, one closed-form evaluation of the infinite series of the F 0 1 ⁢ ( ⋅ ) {{}_{1}F_{0}(\,\cdot\,)} function is deduced. In addition, we establish some integral expressions in terms of the Euler numbers, which are not available in the tables of the book of Gradshteyn and Ryzhik.