Asymptotics of some generalised sine-integrals

2020/11/07 by R. B. Paris, Paris, R B
Mathematics · #33E20 #34E05 #41A60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2011.05156

openalex publication_date 2020/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain the asymptotic expansion for large integer n of a generalised sine-integral ∫₀^∞((sin x)/(x))ⁿdx 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_ν(x), which reduces to the above integral when ν=1/2.

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