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Joint asymptotic expansions for Bessel functions

David A. Sher

Vol. 5 (2023), No. 2, 461–505
Abstract

We study the classical problem of finding asymptotics for the Bessel functions Jν(z) and Y ν(z) as the argument z and the order ν 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.

Keywords
Bessel functions, geometric microlocal analysis, asymptotics, spectral theory
Mathematical Subject Classification
Primary: 33C10
Secondary: 34E05, 35P15, 41A60
Milestones
Received: 6 April 2022
Accepted: 14 December 2022
Published: 26 June 2023
Authors
David A. Sher
Department of Mathematical Sciences
DePaul University
Chicago, IL
United States