Inversion of a generalised Hilbert transform

Abstract

An integral transform H_y is defined which reduces to the ordinary Hilbert transform H₀ when y=0, and is useful in some hydrodynamic applications. Although H_y does not seem to be explicitly invertible for y≠0 (in contrast to H₀^-1=-H₀), it is readily invertible numerically for y less than a certain precision-dependent bound.