Sokhotski–Plemelj theorem

As distributions on the real line,The Sokhotski--Plemelj formula relates the two boundary values of a Cauchy transform across its contour and fixes the spectral delta functions generated by retarded and advanced prescriptions.

With $Hv(\rho )={\pi }^{-1}\mathrm{PV}\int v(s)/(\rho -s)ds$, the upper and lower limiting Cauchy integrals are $P^{\pm }v=\pm v/2+iHv/2$. They obey $P^{+}-P^{-}=I$. The actual complementary idempotent projections are $P^{+}$ and $-P^{-}$; confusing this signed convention with two positive projectors changes reconstruction signs. These boundary operators naturally enter the spectral reconstruction of an attenuated Radon transform.