P.J.D. Blancas and E.A. Galapon, “Finite-part integration of the Hilbert transform,” https://arxiv.org/abs/2210.14462

Finite-part integration is a recently introduced method of evaluating well-defined integrals using the finite-part of divergent integrals [E.A. Galapon, \textit{Proc. Roy. Soc. A} \textbf{473}, 20160567 (2017)]. Here finite-part integration is applied in the exact evaluation of the one-sided, $\operatorname{PV}\!\int_0^{\infty} h(x) (\omega-x)^{-1}\mathrm{d}x$ , and full, $\mathrm{PV}\!\int_{-\infty}^{\infty}h(x) (\omega-x)^{-1}\mathrm{d}x$ , Hilbert transforms, together with the reductions of the later transform when $h(x)$ is a symmetric function, $\operatorname{PV}\!\int_0^{\infty}x h(x)(\omega^2-x^2)^{-1}\mathrm{d}x$ and $\operatorname{PV}\!\int_0^{\infty} h(x) (\omega^2-x^2)^{-1}\mathrm{d}x$ . In general, the result consists of two terms—the first is an infinite series of the finite-part of divergent integrals of the form $\int_0^{\infty} h(x) x^{-\nu-k}\mathrm{d}x$ , for some non-integer $\nu$ and positive integers $k$ , and the second is a contribution arising from the singularity of the kernel of transformation. The first term is precisely the result obtained when the kernel of transformation is binomially expanded in inverse powers of $x$ , followed by term-by-term integration and the resulting divergent integrals assigned values equal to their finite-parts. In all cases the finite-part contribution is present while the presence or absence of the singular contribution depends on the interval of integration and on the analytic properties of $h(x)$ at the origin. From the exact evaluation of the Hilbert transform, the dominant asymptotic behavior for arbitrarily small $\omega$ is obtained.

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