Contour Integral Representations of Finite-part Integrals with Logarithmic Singularities

TL;DR

This paper develops contour integral methods to evaluate finite-part integrals with logarithmic singularities, extending previous techniques and enabling numerical computation and asymptotic analysis of such divergent integrals.

Contribution

It introduces contour integral representations for finite-part integrals with arbitrary order logarithmic singularities, generalizing prior results and facilitating their numerical and asymptotic evaluation.

Findings

Provides contour integral formulas for integrals with logarithmic singularities.

Demonstrates numerical evaluation of finite-part integrals using the new representations.

Derives closed-form expressions for Stieltjes transforms involving logarithmic terms.

Abstract

The integral $\int_0^a f(t) t^{-s} \mathrm{d}t$ diverges for $\text{Re}(s) \geq \lambda + 1$, where $\lambda$ is the order of the first non-vanishing derivative of $f(t)$ at the origin. With the assumption that $f(t)$ is analytic at the origin, the finite-part of the divergent integral assumes the contour integral representation of the form $\bbint{0}{a} f(t) t^{-s} \mathrm{d}t = \int_C f(z) z^{-s} G(z) \mathrm{d}z$ where $G(z)$ depends on whether $z=0$ constitutes a pole or a branch point singularity of $z^{-s}$ [E. A. Galapon, \textit{Proc. R. Soc.}, \textbf{A 473} (2017), no. 2197, 20160567.]. In this paper, we extend these representations to accommodate logarithmic singularities of arbitrary order $n \in \mathbb{N}$, specifically for $\bbint{0}{a} f(t) t^{-s} \ln^n t \, \mathrm{d}t$. We then demonstrate the utility of the representations in the numerical evaluation of finite-part integrals and their use in determining the finite parts of non-Mellin-type divergent integrals -- those which exhibit singular behavior at the origin but lack a well-defined Mellin transform. Finally, these representations provide a closed-form evaluation of the Stieltjes transform $\int_0^a k(t) \ln^n t \left( t^\nu (\omega^2 + t^2) \right)^{-1} \mathrm{d}t$ in terms of finite-part integrals, from which the dominant asymptotic behavior is readily extracted for vanishingly small values of the parameter $\omega$.