Asymptotic evaluation of ${\int }_0^{\infty }{\left(\frac{\sin x}{x}\right)}^{n}dx$

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.