Cauchy transform ($Cf(z)=\frac{1}{2\pi i}{\int }_{\Gamma }\frac{f(s)}{s-z}ds$)

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 $Hf(\rho )={\pi }^{-1}\mathrm{PV}\int f(s)/(\rho -s)ds$, the signed Cauchy boundary operators are $C^{\pm }f=\pm f/2+iHf/2$.

For continuous boundary data on the counterclockwise unit circle and its Cauchy transform $\Phi$, $\Phi (r\omega )-\Phi (r^{-1}\omega )=P\varphi (r\omega )$ for $0<r<1$. This follows by subtracting the two Cauchy kernels and obtaining the Poisson kernel on the circle. Consequently the paired interior-minus-exterior difference tends to $\varphi (\omega )$ as $r\uparrow 1$. For $r\downarrow 1$ from above, the same written difference has the opposite sign. This paired limit does not require each individual boundary value of the transform to exist.

// Destination: analysis.bigb