Uniform asymptotic expansions for Bessel functions of imaginary order and their zeros

TL;DR

This paper develops new uniform asymptotic expansions for Bessel functions of imaginary order and their zeros, providing simpler formulas with error bounds, valid across complex domains and for large orders.

Contribution

It introduces simpler, more accurate uniform asymptotic expansions for Bessel functions of imaginary order and their zeros, applicable in all complex regions and for large orders.

Findings

Derived asymptotic expansions with error bounds

Unified approximations valid for all complex arguments

Constructed expansions for zeros of Bessel functions

Abstract

Bessel and modified Bessel functions of imaginary order $i\nu$ ($\nu >0$) are studied. Asymptotic expansions are derived as $\nu \to \infty$ that are uniformly valid in unbounded complex domains, with error bounds provided. Coupled with appropriate connection formulas the approximations are uniformly valid for all complex argument. The expansions are of two forms, Liouville-Green type expansions only involving elementary functions, and ones involving Airy functions that are valid at a turning point of the defining differential equation. The new results have coefficients and error bounds that are simpler than in prior expansions, and further are used to construct asymptotic expansions for the zeros of Bessel and modified Bessel functions of large imaginary order, these being uniformly valid without restriction on their size (small or large).