Beyond-all-order asymptotics of integral transforms by distributional approach and physical examples

Abstract

McClure and Wong’s distributional approach in asymptotics [1] is exploited to obtain complete and explicit expansions for integrals. The method is able to treat exponentially small integrals that are identically zero by classical methods of asymptotics. We consider general distributional treatment for different forms of the integrand and apply the results to specific transforms. Particularly, we obtain explicit expansions for the Stieltjes, Hilbert, and the Hankel transform. Some physical examples are presented to demonstrate the power of the method.