Exponentially accurate uniform asymptotic approximations for integrals and Bleistein's method revisited

Abstract

We obtain new uniform asymptotic approximations for integrals with a relatively exponentially small remainder. We illustrate how these results can be used to obtain remainder estimates in the Bleistein method. The method is created to deal with new types of integrals in which the usual methods for remainder estimates fail. As an application, we obtain an asymptotic expansion for as in |ph λ |≤ π /2 uniformly for large | z |.