Evaluating Borwein Integrals Using Residue TheoryEvaluating Borwein Integrals Using Residue TheoryD. Cao Labora, G. Cao Labora

被引：0

作者：

Daniel Cao Labora ^{[1
]}

Gonzalo Cao Labora ^{[2
]}

机构：

[1] Universidade de Santiago de Compostela (USC),Department of Statistics, Mathematical Analysis and Optimization, Faculty of Mathematics and CITMAga

[2] New York University (NYU),Courant Institute of Mathematical Sciences

来源：

Complex Analysis and Operator Theory | 2025年 / 19卷 / 2期

关键词：

Borwein integrals; Residue theory; Complex analysis; Sinc function; 30E20; 26A42;

D O I：

10.1007/s11785-025-01667-w

中图分类号：

学科分类号：

摘要：

Borwein integrals were initially described by David Borwein and Jonathan Borwein in 2001. They consist of a simple family of integrals involving the cardinal sine function “sinc”, so that the first integrals are equal to π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document} until, suddenly, that pattern breaks. The classical explanation for this fact involves Fourier analysis techniques. In this paper, we show that it is possible to derive an explanation for this result and similar ones by means of undergraduate complex analysis tools; namely, residue theory. Besides, we show that this complex analysis perspective allows us to go beyond the classical result when studying these kinds of integrals. In particular, we obtain a new generalization of the Borwein results.

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