Complex transport Green function ($G_k(z)=\frac{\mathrm{sgn}(|k{|}^2-1)}{\pi (k\bar{z}-k^{-1}z)}$)

For $L_k=k{\partial }_z+k^{-1}{\partial }_{\bar{z}}$, $k\ne 0$, $|k|\ne 1$, this kernel satisfies $L_k{G}_k={\delta }_0$. A real-linear change of complex coordinate reduces its derivation to the Cauchy-Green operator; the determinant changes sign at the unit circle. For $|k|=1$ the operator becomes a real directional transport equation, whose characteristic singularity requires limiting prescriptions. The kernel has $G_k\sim k/(\pi z)$ at zero and $G_k\sim 1/(\pi k\bar{z})$ at infinity.