Change of variable formul{\ae} for regularizing slowly decaying and oscillatory Cauchy and Hilbert transforms

Formul{\ae} are derived for expressing Cauchy and Hilbert transforms of a function $f$ in terms of Cauchy and Hilbert transforms of $f(x^r)$. When $r$ is an integer, this corresponds to evaluating the Cauchy transform of $f(x^r)$ at all choices of $z^{1/r}$. Related formu\ae\ for rational $r$ result in a reduction to a generalized Cauchy transform living on a Riemann surface, which in turn is reducible to the standard Cauchy transform. These formul{\ae} are used to regularize the behaviour of functions that are slowly decaying or oscillatory, in order to facilitate numerical computation and extend asymptotic results.