# Some definite integrals of Srinivasa Ramanujan and its consequences

## TL;DR

This paper derives analytical solutions for specific definite integrals related to Ramanujan's work using Meijer's G-function and Laplace transforms, also establishing new summation formulas and series evaluations.

## Contribution

It introduces novel analytical solutions for Ramanujan's integrals via Meijer's G-function and develops new summation formulas linked to these integrals.

## Key findings

- Solutions expressed in terms of Meijer's G-function
- New infinite summation formulas derived
- Numeric evaluations of infinite series provided

## 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