Asymptotic formulae for the Lommel and Bessel functions and their derivatives

TL;DR

This paper develops new asymptotic formulas for Lommel and Bessel functions and their derivatives, providing improved approximations and analyzing their accuracy across parameter ranges.

Contribution

It introduces novel asymptotic representations of Lommel functions via the Scorer function and extends Nicholson-type approximations for Bessel functions.

Findings

New asymptotic formulas for Lommel functions and derivatives

Approximate representations of Bessel functions consistent with known forms

Analysis of accuracy ranges for the derived formulas

Abstract

We derive new approximate representations of the Lommel functions in terms of the Scorer function and approximate representations of the first derivative of the Lommel functions in terms of the derivative of the Scorer function. Using the same method we obtain previously known approximate representations of the Nicholson type for Bessel functions and their first derivatives. We study also for what values of the parameters our representations have reasonable accuracy.