We study the classical problem of finding asymptotics for the Bessel functions
and
as the
argument
and
the order
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