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Note on an integral of Ramanujan

Authors:

J-P Allouche

Institution:

(1) CNRS, LRI, Batiment 490, F-91405 Orsay Cedex, France

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. 2000 Mathematics Subject ClassificationPrimary—33B15 Partially supported by MENESR, ACI NIM 154 Numération.

Keywords:

Ramanujan integral Frullani-Cauchy theorem

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