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

TL;DR

This paper derives sharp error bounds for the large-argument asymptotic expansions of the Lommel function and related functions, enhancing understanding of their accuracy and providing practical estimates for applications.

Contribution

The paper introduces new representations for remainder terms and provides re-expansions with error estimates, improving the precision of asymptotic approximations for Lommel and allied functions.

Findings

New sharp error bounds for Lommel function asymptotics

Re-expansions with detailed error estimates

Numerical examples demonstrating bound sharpness

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.