= Solution
Write $z=x+iy$, $r=|z|$, and $L=(4\pi)^{-1}\log(x^2+y^2)$. The <Wirtinger derivatives> satisfy $4\partial_{\bar z}\partial_z=\Delta$. Away from zero, ordinary differentiation gives
$$
4\partial_z L=2(\partial_x-i\partial_y)L
=\frac{x-iy}{\pi(x^2+y^2)}=\frac1{\pi z}.
$$
This equality also holds for <distributional derivatives>. To check that no <Dirac delta distribution> was lost at zero, integrate by parts outside the disk $r\leq\varepsilon$ against a <test function>. The new <boundary term> has absolute value at most $C\varepsilon|\log\varepsilon|\|\phi\|_\infty\to0$. Both $L$ and its displayed first <derivatives>, of size $O(r^{-1})$, are <locally integrable functions> in two dimensions, so the remaining integrals converge to their full-plane values.
Using the supplied <fundamental solution of the Laplace equation> identity $\Delta L=\delta_0$, and commutation of <distributional derivatives>, now gives
$$
\boxed{E=4\partial_zL=\frac1{\pi z},\qquad
\partial_{\bar z}E=4\partial_{\bar z}\partial_zL=\Delta L=\delta_0.}
$$
Thus the <Cauchy kernel fundamental solution> has exactly the required normalization.
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