Joint asymptotic expansions for Bessel functions

TL;DR

This paper develops comprehensive joint asymptotic expansions for Bessel functions as both argument and order tend to infinity, enabling detailed analysis and derivatives, with applications in spectral theory.

Contribution

It introduces a blow-up analysis method to derive polyhomogeneous conormal joint asymptotics for Bessel functions, valid across all regimes.

Findings

Derived asymptotics for the modulus and phase of Bessel functions.

Enabled differentiation of asymptotics with respect to argument and order.

Applied results to spectral theory, including Dirichlet eigenvalues of a disk.

Abstract

We study the classical problem of finding asymptotics for the Bessel functions $J_{\nu}(z)$ and $Y_{\nu}(z)$ as the argument $z$ and the order $\nu$ approach infinity. We use blow-up analysis to find asymptotics for the modulus and phase of the Bessel functions; this approach produces polyhomogeneous conormal joint asymptotic expansions, valid in any regime. As a consequence, our asymptotics may be differentiated term by term with respect to either argument or order, allowing us to easily produce expansions for Bessel function derivatives. We also discuss applications to spectral theory, in particular the study of the Dirichlet eigenvalues of a disk.