= Complexified transport Cauchy kernel
{title2=$E_k(w)=\operatorname{sgn}(1-|k|^2)\,k/[\pi(w-k^2\bar w)]$}
For $w=x_1+ix_2$ and $0<|k|\ne1$, the operator $D_k=(k+k^{-1})\partial_{x_1}/2+(k-k^{-1})\partial_{x_2}/(2i)$ has the displayed decaying <fundamental solution>. A rotated real-linear coordinate turns it into a multiple of a <Wirtinger derivative>, reducing the kernel identity to $\partial_{\bar z}(1/(\pi z))=\delta$. The kernel is analytic in $k$ separately inside and outside the unit circle, is $O(k)$ at zero and $O(k^{-1})$ at infinity, and develops a jump at the real directional-transport limit.
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