= Solution
Interchange the two <Wirtinger derivatives> in part (i). The same real <fundamental solution of the Laplace equation> $L$ gives
$$
4\partial_{\bar z}L=\frac{x+iy}{\pi(x^2+y^2)}=\frac1{\pi\bar z},\qquad
\partial_z(4\partial_{\bar z}L)=\Delta L=\delta_0.
$$
The calculation is valid for <distributions> by the same vanishing <boundary term> estimate. Any other <fundamental solution> differs by a <distribution> $H$ with $\partial_zH=0$. Then $\Delta H=0$, so the regularity argument of part (ii) makes $H$ smooth, and its <Cauchy-Riemann equations> make it an <antiholomorphic function>. On the whole plane it can be written $H(z)=h(\bar z)$ with $h$ an <entire function>. Conversely every such $H$ has $\partial_zH=0$. Therefore
$$
\boxed{F(z)=\frac1{\pi\bar z}+h(\bar z),\qquad h\text{ any entire function}.}
$$
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