2019 A Bridge Between Unit Square and Single Integrals for Real Functions of the Form \(\,f(x \cdot y)\)
Fábio M. S. Lima
Real Anal. Exchange 44(2): 445-462 (2019). DOI: 10.14321/realanalexch.44.2.0445

Abstract

Sondow and co-workers have employed a key change of variables in order to evaluate double integrals over the unit square \([0,1] \times [0,1]\) in exact closed-form. Motivated by their results, I introduce here a change of variables which creates a ‘bridge’ between integrals of the form \(\,\int_0^1\!\!\int_0^1{f(x \cdot y)~dx \, dy}\,\) and single integrals of the form \(\int_0^1{f(p)\,\ln{p}~d p}\). This allows for prompt closed-form evaluations of several interesting integrals, including some of those investigated recently by Sampedro. I also show that the bridge holds when the intervals of integration are changed from \([0,1]\) to \([1,\infty)\). Finally, a generalization for higher dimensions is proved, which reveals an interesting link of those integrals to Mellin’s transform.

Citation

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Fábio M. S. Lima. "A Bridge Between Unit Square and Single Integrals for Real Functions of the Form \(\,f(x \cdot y)\)." Real Anal. Exchange 44 (2) 445 - 462, 2019. https://doi.org/10.14321/realanalexch.44.2.0445

Information

Published: 2019
First available in Project Euclid: 1 May 2020

zbMATH: 07211601
Digital Object Identifier: 10.14321/realanalexch.44.2.0445

Subjects:
Primary: 26B10 , 26B15
Secondary: 35C05

Keywords: Mellin's transform , multiple integrals , Special functions

Rights: Copyright © 2019 Michigan State University Press

Vol.44 • No. 2 • 2019
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