Asymptotic Expansions of generalized Stieltjes transforms of algebraically decaying functions

Abstract

Asymptotic expansions of Stieltjes and generalized Stieltjes transforms of functions having an asympotic expansion in negative integer powers of their variable have been exahustivelly investigated by R. Wong. In this paper we extend this analysis to Stieltjes and generalized Stieltjes transforms of functions having an asympotic expansion in negative rational powers of their
variable. Distributional approach is used for deriving the asymptotic expansion of the Stieltjes and generalized Stieltjes transforms of this kind of functions for large values of the parameter of the transformation. Error bounds are obtained at any order of the approximation for a large family of integrands. The asymptotic approximation of an integral involved in the calculation of the mass renormalization of the quantum scalar field is given as an ilustration.