Certain Integrals Arising from Ramanujan's Notebooks

TL;DR

This paper proves Ramanujan's claimed integral relation involving logarithmic functions using contour integration and extends these results to broader classes of infinite integrals with multiple examples.

Contribution

It provides a rigorous proof of Ramanujan's integral relation and develops general theorems connecting various classes of infinite integrals.

Findings

Confirmed Ramanujan's integral relation involving log functions.

Established general theorems for classes of infinite integrals.

Illustrated results with multiple concrete examples.

Abstract

In his third notebook, Ramanujan claims that $$ \int_0^\infty \frac{\cos(nx)}{x^2+1} \log x \,\mathrm{d} x + \frac{\pi}{2} \int_0^\infty \frac{\sin(nx)}{x^2+1} \mathrm{d} x = 0. $$ In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if $\log x$ were replaced by $\log^2x$ in the first integral and $\log x$ were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly relating large classes of infinite integrals and illustrate these by several examples.