Stieltjes transforms and the Stokes phenomenon

Abstract

Recently, Berry, Olver and Jones have found uniform asymptotic expansions for the exponentially small remainder terms that result when asymptotic expansions are optimally truncated. These uniform expansions describe the rapid change in the behaviour of the remainders as a Stokes line is crossed. We show how such uniform expansions may be found when a function can be expressed as a Stieltjes transform. Such an approach has the following advantages: the uniform expansion is calculated directly, non-uniform expansions away from the Stokes line are readily found, and explicit error bounds may be established. We illustrate the method by application to the modified Bessel function K v ( z ).