= Solution
The distributional identity $\partial_{\bar z}(1/(\pi z))=\delta$ gives the <Cauchy-Green operator> solution
$$
\boxed{\mu(z,\lambda)=\frac1{\pi\nu(r)}\int_{\mathbb C}\frac{\widetilde f(z')}{z-z'}d^2z',}
$$
where $\widetilde f$ is $f$ in the changed real coordinates. The singularity $1/(z-z')$ 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> is
$$
\mu=\frac{i\,\operatorname{sgn}b_r}{2\pi}\int_{\mathbb R^2}
\frac{F(\tau',\rho',\theta)}{a_r(\rho-\rho')+ib_r(\tau-\tau')}d\tau'd\rho'.
$$
In the original complex coordinate $w=x_1+ix_2$, it is
$$
\mu(w,\lambda)=\frac{\operatorname{sgn}(|\lambda|-1)}\pi\int_{\mathbb R^2}
\frac{f(w')}{\lambda(\bar w-\bar w')-\lambda^{-1}(w-w')}d^2w'.
$$
For $|\lambda|<1$ the denominator can be rewritten to give the kernel $\lambda/[\pi((w-w')-\lambda^2(\bar w-\bar w'))]$. For $|\lambda|>1$ it is $\lambda^{-1}/[\pi((\bar w-\bar w')-\lambda^{-2}(w-w'))]$. 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.
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