Asymptotic evaluation of $\int_0^\infty\left(\frac{\sin x}{x}\right)^n\;dx$

TL;DR

This paper derives an asymptotic series for the integral of (x)/x raised to the power n, providing explicit formulas and error bounds as n becomes large.

Contribution

It introduces a new asymptotic expansion for the integral (x)/x)^n, including explicit error bounds, advancing understanding of its behavior for large n.

Findings

Asymptotic series in 1/n for the integral

Explicit first terms of the series

Error bounds for the approximation

Abstract

We consider the integral $\int_0^\infty\left(\frac{\sin x}{x}\right)^n\;dx$ as a function of the positive integer $n$. We show that there exists an asymptotic series in $\frac{1}{n}$ and compute the first terms of this series together with an explicit error bound.