Ramanujan's Beautiful Integrals

TL;DR

This paper surveys Ramanujan's extensive work on evaluating definite integrals, highlighting his elegant methods, surprising results, and their significance across various mathematical contexts.

Contribution

It provides a comprehensive overview of Ramanujan's integral evaluations, including examples from his notebooks, letters, and published works, emphasizing his innovative methods.

Findings

Ramanujan developed general methods for integral evaluation.

His integral formulas are elegant and widely applicable.

The survey covers diverse sources of Ramanujan's integral work.

Abstract

Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the \emph{Journal of the Indian Mathematical Society}, notebooks, Quarterly Reports to the University of Madras, letters to Hardy, published papers and the Lost Notebook. His evaluations are often surprising, beautiful, elegant, and useful in other mathematical contexts. He also discovered general methods for evaluating and approximating integrals. A survey of Ramanujan's contributions to the evaluation of integrals is given, with examples provided from each of the above-mentioned sources.