C.D. Tica and E.A. Galapon, “Finite-Part Integration of the Generalized Stieltjes Transform and its dominant asymptotic behavior for small values of the parameter” arXiv:1703.07979.
The paper addresses the exact evaluation of the generalized Stieltjes transform $S_{\lambda}[f]=\int_0^{\infty} f(x) (\omega+x)^{-\lambda}\mathrm{d}x$ about $\omega =0$ from which the asymptotic behavior of $S_{\lambda}[f]$ for small parameters $\omega$ is directly extracted. An attempt to evaluate the integral by expanding the integrand $(\omega+x)^{-\lambda}$ about $\omega=0$ and then naively integrating the resulting infinite series term by term… Read More »