e-mail: jl.lopez@unavarra.es

**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.