Translator Disclaimer
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

Download Citation

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

Rights: Copyright © 2019 Michigan State University Press

JOURNAL ARTICLE
18 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.44 • No. 2 • 2019
Back to Top