Finite-Part Integration of the Hilbert Transform

TL;DR

This paper presents an exact method for evaluating the Hilbert transform using finite-part integration, revealing the structure of divergent integrals and singular contributions, and deriving asymptotic behavior for small parameters.

Contribution

It introduces a novel application of finite-part integration to compute the Hilbert transform exactly, clarifying the roles of divergent and singular parts in the process.

Findings

Exact evaluation of Hilbert transforms using finite-part integrals

Identification of divergent and singular contributions in the transform

Asymptotic behavior for small parameters derived

Abstract

The one-sided and full Hilbert transforms are evaluated exactly by means of the method of finite-part integration [E.A. Galapon, \textit{Proc. Roy. Soc. A} \textbf{473}, 20160567 (2017)]. In general, the result consists of two terms -- the first is an infinite series of finite-part of divergent integrals, 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 positive powers of the parameter of transformation, 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 parity of the function under transformation about the origin. From the exact evaluation of the Hilbert transform, the dominant asymptotic behavior for arbitrarily small parameter is obtained.