Contour Integral Representations of Finite-part Integrals with Logarithmic Singularities

The integral ${\int }_0^{a}f(t)t^{-s}\mathrm{d}t$ diverges for $\text{Re}(s)\ge \lambda +1$, 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 $\text{bbint}0af(t)t^{-s}\mathrm{d}t={\int }_{C}f(z)z^{-s}G(z)\mathrm{d}z$ where $G(z)$ depends on whether $z=0$ constitutes a pole or a branch point singularity of $z^{-s}$ [E. A. Galapon, $\mathit{\text{Proc.}}\mathit{\text{R.}}\mathit{\text{Soc.}}$, $\mathbf{\text{A}}\mathbf{\text{473}}$ (2017), no. 2197, 20160567.]. In this paper, we extend these representations to accommodate logarithmic singularities of arbitrary order $n\in \mathbb{N}$, specifically for $\text{bbint}0af(t)t^{-s}{\ln }^{n}t\mathrm{d}t$. 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 ${\int }_0^{a}k(t){\ln }^{n}t{(t^{\nu }({\omega }^2+t^2))}^{-1}\mathrm{d}t$ in terms of finite-part integrals, from which the dominant asymptotic behavior is readily extracted for vanishingly small values of the parameter $\omega$.