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

For definiteness take $f$ in the Schwartz space; sufficiently smooth compactly supported data are also adequate. These hypotheses make the line integrals, Cauchy principal values and decay estimates below well defined. Write $\lambda =r{e}^{i\theta }$ with $r>0$, $r\ne 1$, and introduce orthonormal local coordinatesLet $F(\tau ,\rho ,\theta )=f(x_1,x_2)$, $a_r=(r+r^{-1})/2$ and $b_r=(r-r^{-1})/2$. Substitution into the directional transport equation givesNow make the invertible real-linear change of coordinatesSince $\mathrm{Im}z=-a_r\rho /b_r$, its Wirtinger derivative is ${\partial }_{\bar{z}}=({\partial }_{\tau }-i{b}_r{\partial }_{\rho }/a_r)/2$. Hence $L_{\lambda }=\nu (r){\partial }_{\bar{z}}$, as required. The change becomes singular at $r=1$, where the complex operator degenerates to a real directional derivative. Also the original equation contains ${\lambda }^{-1}$, so $\lambda =0$ is excluded from the differential equation; the normalized solution will extend analytically there.