Closed-contour Cauchy singular integral is an involution (${\mathcal{C}}^2=I$)

On a positively oriented smooth simple closed contour, set $\mathcal{C}f=(\pi i{)}^{-1}\mathrm{PV}{\int }_{\Gamma }f(t)/(t-z)dt$. 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 $\mathcal{C}f=g$ into a normalized additive Riemann-Hilbert problem with jump $g$. Applying the boundary formula again yields $f=\mathcal{C}g$, proving the involution on a suitable smooth or Hölder function class.