# Finite-Part Integration of the Generalized Stieltjes Transform and its dominant asymptotic behavior for small values of the parameter Part I: Integer orders

## TL;DR

This paper develops a finite-part integration method to accurately evaluate the generalized Stieltjes transform and its asymptotic behavior near zero, revealing dominant contributions missed by traditional divergent integral techniques.

## Contribution

It introduces a finite-part integration approach for exact evaluation of the generalized Stieltjes transform and its asymptotics, improving upon naive series expansion methods.

## Key findings

- Finite-part integration captures dominant asymptotic terms missed by naive methods.
- Explicit evaluation of the transform reveals contributions from poles and branch points.
- Application to physical problems yields improved asymptotic expansions for diffusion coefficients.

## Abstract

The paper addresses the exact evaluation of the generalized Stieltjes transform $S_{n}[f]=\int_0^{\infty} f(x) (\omega+x)^{-n}\mathrm{d}x$ of integral order $n=1,2, 3,\dots$ about $\omega =0$ from which the asymptotic behavior of $S_{n}[f]$ for small parameters $\omega$ is directly extracted. An attempt to evaluate the integral by expanding the integrand $(\omega+x)^{-n}$ about $\omega=0$ and then naively integrating the resulting infinite series term by term lead to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard's finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in [E.A. Galapon, {\it Proc. Roy. Soc. A} {\bf 473}, 20160567 (2017)]. It is shown that, when $f(x)$ does not vanish or has zero of order $m$ at the origin such that $(n-m)\geq 1$, the dominant terms of $S_{n}[f]$ as $\omega\rightarrow 0$ come from contributions arising from the poles and branch points of the complex valued function $f(z) (\omega+z)^{-n}$. These dominant terms are precisely the terms missed out by naive term by term integration. Furthermore, it is demonstrated how finite-part integration leads to new series representations of special functions by exploiting their known Stieltjes integral representations. Finally, the application of finite part integration in obtaining asymptotic expansions of the effective diffusivity in the limit of high Peclet number, the Green-Kubo formula for the self-diffusion coefficient and the antisymmetric part of the diffusion tensor in the weak noise limit is discussed.