Bessel function asymptotics: a relation due to Lommel

TL;DR

This paper investigates the validity of Lommel's asymptotic relation for Bessel functions, confirming its accuracy for real arguments and identifying conditions under which it may fail.

Contribution

It clarifies the conditions for Lommel's Bessel function asymptotic relation, extending understanding of its applicability beyond previous assumptions.

Findings

Lommel's relation holds for real z

The relation can fail for complex z

Provides conditions for validity of the asymptotic approximation

Abstract

We show that the Bessel function asymptotic relation $J_{\nu}^2(z) + J_{\nu + 1}^2 (z) \sim 2/(\pi z)$ of Lommel is valid when $z$ is real but can fail otherwise.