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

Sheehan Olver

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arXiv · 1311.2320

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

Sheehan Olver

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Abstract

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.

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BibTeX RIS

Sheehan Olver. 2013-11-11. Change of variable formul{\ae} for regularizing slowly decaying and oscillatory Cauchy and Hilbert transforms. https://arxiv.org/abs/1311.2320

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