On the large argument asymptotics of the Lommel function via Stieltjes transforms

TL;DR

This paper provides a detailed analysis of the large argument asymptotics of the Lommel function using Stieltjes transforms, including error bounds, exponential improvements, and Stokes phenomena, with implications for the Struve function.

Contribution

It introduces explicit error bounds and exponentially improved asymptotic expansions for the Lommel function, enhancing understanding of its large argument behavior.

Findings

Explicit error bounds for asymptotic series

Exponentially improved asymptotic expansions

Analysis of Stokes discontinuities and their smooth transition

Abstract

The aim of this paper is to investigate in detail the known large argument asymptotic series of the Lommel function by Stieltjes transform representations. We obtain a number of properties of this asymptotic expansion, including explicit and realistic error bounds, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities. An interesting consequence related to the large argument asymptotic series of the Struve function is also proved.