Asymptotics of some generalised sine-integrals

Abstract

We obtain the asymptotic expansion for large integer $n$ of a generalised sine-integral \[\int_0^\infty\left(\frac{\sin\,x}{x}\right)^{n}dx\] by utilising the saddle-point method. This expansion is shown to agree with recent results of J. Schlage-Puchta in {\it Commun. Korean Math. Soc.} {\bf 35} (2020) 1193--1202 who used a different approach. An asymptotic estimate is obtained for another related sine-integral also involving a large power $n$. Numerical results are given to illustrate the accuracy of this approximation. We also revisit the asymptotics of Ball's integral involving the Bessel function $J_\nu(x)$, which reduces to the above integral when $\nu=1/2$.