Some definite integrals of Srinivasa Ramanujan and its consequences

Abstract

In this paper, we obtain analytical solutions of some definite integrals of Srinivasa Ramanujan [Mess. Math., XLIV, 75-86, 1915] in terms of Meijer's $G$-function by using Laplace transforms of $\sin (\beta x^2),\cos (\beta x^2),x\sin (\beta x^2)$ and $x\cos (\beta x^2)$. Further, we obtain some infinite summation formulas connected with Meijer's G-function and numeric values of some infinite series

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