  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.
