Note on an integral of Ramanujan

Abstract

We answer a question of Berndt and Bowman, asking whether it is possible to deduce the value of the Ramanujan integral I from the value of the Ramanujan integral J, where

$$I := \int_0^1 \bigg(\frac{x^{p-1}}{1-x} - \frac{rx^{q-1}}{1-x^r} \bigg) dx \ \ \ (= \psi(q/r) - \psi(p) + \log r)$$

and

$$J := \int_0^{\infty} \frac{(1+ax)^{-p} - (1+bx)^{-q}}{x} dx \ \ \ \bigg(\!\!= \psi(q) - \psi(p) + \log \frac{b}{a}\bigg).$$

We also show that the second integral can be deduced from a classical expression of the ψ function due to Dirichlet and from the classical equality

$$\int_0^{\infty} (e^{-ax} - e^{-bx}) \frac{dx}{x} = \log \frac{b}{a},$$

which is a simple consequence of Frullani-Cauchy’s theorem.