= Cauchy transform
{c}
{title2=$Cf(z)=\frac1{2\pi i}\int_\Gamma\frac{f(s)}{s-z}\,ds$}
= Cauchy integral
{c}
{synonym}
This <contour integral> defines an analytic function away from the oriented integration curve, under the relevant integrability assumptions. Its limiting values across the curve are related by the <Sokhotski–Plemelj theorem>. On an oriented real axis, with $Hf(\rho)=\pi^{-1}\operatorname{PV}\int f(s)/(\rho-s)\,ds$, the <signed Cauchy boundary operators> are $C^\pm f=\pm f/2+iHf/2$.
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