Explicit Error Terms for Asymptotic Expansions of Stieltjes Transforms

Abstract:

A technique is developed which gives explicit expressions for the error terms associated with the asymptotic expansions of the Stieltjes transform of f,

Here z is a complex parameter in the cut plane |arg z| < π, and f(t) is a locally integrable function on [0, ∞). Near infinity we assume that f(t) is either algebraically decaying or oscillatory. From the explicit expressions, strict and realistic error bounds can be obtained. Explicit error terms are also given for asymptotic expansions of Laplace and Fourier transforms of small argument. Our approach is based on the theory of distributions.