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

The distributional identity ${\partial }_{\bar{z}}(1/(\pi z))=\delta$ gives the Cauchy-Green operator solutionwhere $\tilde{f}$ is $f$ in the changed real coordinates. The singularity $1/(z-z^{\prime })$ is locally integrable. For Schwartz functions, splitting the integral into a region near the origin and a rapidly decaying far region proves $\mu =O(|z{|}^{-1})$. If two solutions obey this decay, their difference has zero Wirtinger derivative and is an entire holomorphic function tending to zero; the Liouville theorem proves uniqueness.

The change-of-variables Jacobian has absolute value $a_r/|b_r|$. Therefore an equivalent form of the complexified transport Cauchy kernel isIn the original complex coordinate $w=x_1+i{x}_2$, it isFor $|\lambda |<1$ the denominator can be rewritten to give the kernel $\lambda /[\pi ((w-w^{\prime })-{\lambda }^2(\bar{w}-{\bar{w}}^{\prime }))]$. For $|\lambda |>1$ it is ${\lambda }^{-1}/[\pi ((\bar{w}-{\bar{w}}^{\prime })-{\lambda }^{-2}(w-w^{\prime }))]$. Consequently the spectral solution is holomorphic separately inside and outside the unit circle, with $\mu =O(\lambda )$ at zero and $\mu =O({\lambda }^{-1})$ at infinity. These normalizations will determine its jump reconstruction.