The asymptotic expansion of the Bateman and Havelock functions of large order and argument

TL;DR

This paper derives asymptotic expansions for the Bateman and Havelock functions for large arguments and order, using steepest descent methods, and validates the results with numerical illustrations.

Contribution

It provides new asymptotic formulas for these functions when both argument and order are large, specifically when they are proportional.

Findings

Asymptotic expansions are accurate for large arguments and order.

Numerical results confirm the validity of the derived expansions.

The method combines steepest descent with an inversion process for coefficients.

Abstract

Asymptotic expansions for the Bateman and Havelock functions defined respectively by the integrals \[\frac{2}{\pi}\int_0^{\pi/2} \!\!\!\begin{array}{c} \cos\\\sin\end{array}\!(x\tan u-\nu u)\,du\] are obtained for large real $x$ and large order $\nu>0$ when $\nu=O(|x|)$. The expansions are obtained by application of the method of steepest descents combined with an inversion process to determine the coefficients. Numerical results are presented to illustrate the accuracy of the different expansions obtained.