Error bounds for the large-argument asymptotic expansions of the Hankel and Bessel functions

TL;DR

This paper derives sharp error bounds for the large-argument asymptotic expansions of Hankel, Bessel, and modified Bessel functions, enhancing understanding of their accuracy and potential for generalization.

Contribution

It introduces new integral representations for the remainder terms and provides re-expansions with error estimates, advancing the precision of asymptotic analysis for these special functions.

Findings

Derived sharp error bounds for asymptotic expansions

Provided re-expansions with associated error estimates

Discussed the bounds' sharpness and relation to existing results

Abstract

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these 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. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise from an application of the method of steepest descents.