Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions

Abstract

In this paper, we reconsider the large-$z$ asymptotic expansion of the Lommel function $S_{\mu ,\nu }(z)$ and its derivative. New representations for the remainder terms of the asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. Applications to the asymptotic expansions of the Anger--Weber-type functions, the Scorer functions, the Struve functions and their derivatives are provided. The sharpness of our error bounds is discussed in detail, and numerical examples are given.

25 pages. arXiv admin note: text overlap with arXiv:1606.07961, arXiv:1702.05316, accepted for publication in Studies in Applied Mathematics

References in corpus (3)

• Error bounds for the large-argument asymptotic expansions of the Hankel and Bessel functions
• Error bounds for the asymptotic expansion of the Hurwitz zeta function
• On the large argument asymptotics of the Lommel function via Stieltjes transforms

Cited by in corpus (2)

• Uniform asymptotic expansions for Lommel, Anger-Weber and Struve functions
• Asymptotic solutions of inhomogeneous differential equations having a turning point