Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 67 / 2 / c / Solution

Define the Radon transform and its transverse Hilbert transform byUse the signed Cauchy boundary operators $P^{+}=(I+iH)/2$ and $P^{-}=(-I+iH)/2$. Thus $P^{+}$ and $-P^{-}$ are the complementary idempotent projections; $P^{-}$ itself carries the signed lower-boundary convention. The PDF defines the outside limit ${\lambda }^{-}=(1+\epsilon )e^{i\theta }$ but omits the definition of the inside limit in its displayed formula. We take ${\lambda }^{+}=(1-\epsilon )e^{i\theta }$, as required by that formula.

As $r\to 1$, apply the Sokhotski–Plemelj formula to the kernel from part (b):The principal value contribution is $i\mathrm{sgn}(b_r)HRf/2$. The delta contribution is the same from either side and equalsSince $b_r<0$ for the inside limit, the resulting boundary value isThe outside limit similarly has $P^{+}Rf$ in place of $-P^{-}Rf$. Naming the signed convention explicitly prevents a sign error from treating $P^{-}$ as a positive projection.