Signed Cauchy boundary operators ($P^{\pm }=\pm I/2+iH/2$)

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.