= Closed-contour Cauchy singular integral is an involution
{title2=$\mathcal C^2=I$}
On a positively oriented smooth simple closed contour, set $\mathcal Cf=(\pi i)^{-1}\operatorname{PV}\int_\Gamma f(t)/(t-z)\,dt$. The <Sokhotski–Plemelj theorem> gives the sum and difference of the two Cauchy boundary values. Changing the sign of the exterior branch converts the equation $\mathcal Cf=g$ into a normalized additive <Riemann-Hilbert problem> with jump $g$. Applying the boundary formula again yields $f=\mathcal Cg$, proving the involution on a suitable smooth or Hölder function class.
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