Uniform asymptotic expansions for Lommel, Anger-Weber and Struve functions

TL;DR

This paper develops rigorous uniform asymptotic expansions for Lommel, Weber, Anger-Weber, and Struve functions using a differential equation approach, involving Airy and Scorer functions, valid for large real order and complex argument.

Contribution

It introduces new asymptotic solutions for these special functions and explores their behavior across different sectors of the complex plane.

Findings

Asymptotic expansions are valid for large real order and complex argument.

Identification of Lommel functions with new asymptotic solutions.

Use of Airy and Scorer functions in the approximations.

Abstract

Using a differential equation approach asymptotic expansions are rigorously obtained for Lommel, Weber, Anger-Weber and Struve functions, as well as Neumann polynomials, each of which is a solution of an inhomogeneous Bessel equation. The approximations involve Airy and Scorer functions, and are uniformly valid for large real order $\nu$ and unbounded complex argument $z$. An interesting complication is the identification of the Lommel functions with the new asymptotic solutions, and in order to do so it is necessary to consider certain sectors of the complex plane, as well as introduce new forms of Lommel and Struve functions.