On an asymptotic expansion of Fourier integrals

Abstract

A formula is developed that gives the asymptotic expansion of the Fourier transform of a function whose behaviour near the origin is given by a general asymptotic series. The result is an extension of a theorem due to Olver, who obtained a kind of analogue for Fourier transforms of Watson’s lemma for Laplace transforms. The method adopted utilizes a result due to Erdelyi on Laplace transforms and depends for its success on a novel technique of evading the appearance of divergent integrals in such problems.