An approach to Borwein integrals from the point of view of residue theory

TL;DR

This paper demonstrates how residue theory from complex analysis can explain Borwein integrals' behavior and extends classical results, offering a new perspective beyond Fourier analysis.

Contribution

It introduces a residue theory approach to analyze Borwein integrals and generalizes the classical results using undergraduate complex analysis tools.

Findings

Residue theory explains the pattern of Borwein integrals.

A new generalization of Borwein integrals is proposed.

Complex analysis provides an alternative to Fourier analysis for these integrals.

Abstract

Borwein integrals are one of the most popularly known phenomena in contemporary mathematics. They were found in 2001 by David Borwein and Jonathan Borwein and consist of a simple family of integrals involving the cardinal sine function ``sinc'', so that the first integrals are equal to $\pi$ 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 by means of undergraduate Complex Analysis tools; namely, residue theory. Besides, we show that this Complex Analysis scope allows to go a beyond the classical result when studying these kind of integrals. Concretely, we show a new generalization for the classical Borwein result.