Evaluation of Hankel transforms of meromorphic exponential type functions for large order parameters

Abstract

We asymptotically evaluate the Hankel transform of a meromorphic exponential type function by performing contour integration in the complex plane. While the resulting expansion reproduces terminating Poincaré asymptotic expansion, it also reveals additional exponentially small terms that are not captured by the standard PAE. These contributions arise due to the presence of poles in the integrand, and it is shown that the appearance of the exponentially small terms in the asymptotic expansion of the Hankel transform of exponential type functions depend on the analytic properties of such functions.