Regularized Limit, analytic continuation and finite-part integration

TL;DR

This paper explores the relationship between analytic continuation and finite-part integration, introducing the regularized limit to evaluate divergent integrals and providing exact evaluations and asymptotic behaviors of the Stieltjes transform.

Contribution

It establishes the equivalence of analytic continuation and finite-part integrals at regular points and singularities using the regularized limit, advancing integral evaluation methods.

Findings

Exact evaluation of the Stieltjes transform via finite-part integrals

Connection between analytic continuation and finite-part integrals at singularities

Asymptotic behavior of the transform for small parameters with logarithmic singularities

Abstract

Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [E. A. Galapon, Proc. R. Soc., A 473, 20160567 (2017)]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the relationship is established between the analytic continuation of the Mellin transform and the finite-part of the resulting divergent integral when the Mellin integral is extended beyond its strip of analyticity. It is settled that the analytic continuation and the finite-part integral coincide at the regular points of the analytic continuation. To establish the connection between the two at the isolated singularities of the analytic continuation, the concept of regularized limit is introduced to replace the usual concept of limit due to Cauchy when the later leads to a division by zero. It is then shown that the regularized limit of the analytic continuation at its isolated singularities equals the finite-part integrals at the singularities themselves. The treatment gives the exact evaluation of the Stieltjes transform in terms of finite-part integrals and yields the dominant asymptotic behavior of the transform for arbitrarily small values of the parameter in the presence of arbitrary logarithmic singularities at the origin.