The asymptotics of the Struve function ${\bf H}_\nu(z)$ for large complex order and argument

TL;DR

This paper analyzes the asymptotic behavior of the Struve function ${f H}_ u(z)$ for large complex order and argument, extending Watson's classical analysis to broader domains in the complex plane.

Contribution

It provides a detailed asymptotic expansion of ${f H}_ u(z)$ for large complex $ u$ and $z$, identifying new domains where different asymptotic forms apply.

Findings

Derived asymptotic expansions for large complex $ u$ and $z$

Mapped the complex $ u/z$-plane regions with distinct asymptotic behaviors

Extended Watson's classical results to broader complex domains

Abstract

We re-examine the asymptotic expansion of the Struve function ${\bf H}_\nu(z)$ for large complex values of $\nu$ and $z$ satisfying $|\arg\,\nu|\leq\pi/2$ and $|\arg\,z|<\pi/2$. Watson's analysis covers only the case of $\nu$ and $z$ of the same phase with $\nu/z$ in the intervals $(0,1)$ and $(1,\infty)$. The domains in the complex $\nu/z$-plane where the expansion takes on different forms are obtained.