= Solution
Use the sign convention in which a <fundamental solution of the Laplace equation> satisfies
$$
\Delta_{\mathbf r}G(\mathbf r,\mathbf r')=\delta^{(2)}(\mathbf r-\mathbf r')
$$
in the sense of <distributions>. It is a <harmonic function> away from the source. Writing $\rho=|\mathbf r-\mathbf r'|$, the radial equation is $(\rho G')'=0$, so $G=A\log\rho+B$. The <divergence theorem> on a small circle about the source fixes its total flux:
$$
1=\int_{\rho<\varepsilon}\Delta G\,d^2r
=\int_{\rho=\varepsilon}\partial_\rho G\,ds
=\frac A\varepsilon(2\pi\varepsilon)=2\pi A.
$$
Consequently
$$
\boxed{G(\mathbf r,\mathbf r')=\frac1{2\pi}\log|\mathbf r-\mathbf r'|+B}.
$$
The arbitrary constant is a <harmonic function>. For the alternative convention $-\Delta G=\delta^{(2)}$, the logarithm has the opposite sign; subsequent <Green's function> formulas must use that convention consistently.
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