= Complex transport Green function
{title2=$G_k(z)=\frac{\operatorname{sgn}(|k|^2-1)}{\pi(k\bar z-k^{-1}z)}$}
For $L_k=k\partial_z+k^{-1}\partial_{\bar z}$, $k\ne0$, $|k|\ne1$, this kernel satisfies $L_kG_k=\delta_0$. 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 $G_k\sim1/(\pi k\bar z)$ at infinity.
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