David Sher

Bessel function asymptotics and a conjecture of Pólya

We consider the classical problem of finding asymptotics for Bessel functions as the order and the argument both approach infinity. We use blow-up analysis to find complete asymptotic descriptions of the modulus and the phase of the Bessel functions. These descriptions are valid in any regime and thus unify, and slightly extend, previously known results. In this talk I will explain these asymptotic descriptions, introducing the necessary tools from blow-up analysis (of which no background knowledge will be assumed). At the end I will briefly discuss how insights obtained in this analysis led to the proof (with M. Levitin and I. Polterovich) of a conjecture of Pólya concerning the Dirichlet and Neumann eigenvalues of the unit disk.