  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.
