= Solution
A <fundamental solution of a linear differential operator> $P(D)$ is a distribution $E$ satisfying $P(D)E=\delta_0$. For the three-dimensional negative Laplacian,
$$
\boxed{E(x)=\frac1{4\pi|x|},\qquad-\Delta E=\delta_0.}
$$
Away from zero the function is harmonic. For a test function $\phi$, integrate $E(-\Delta\phi)$ outside a ball of radius $\varepsilon$ and use Green's identity. The small inner sphere contributes $\phi(0)+o(1)$ because the normal derivative of $E$ has total outward-from-the-ball flux $-1$; the other boundary term is $O(\varepsilon)$. Letting $\varepsilon\to0$ verifies the distributional equation and its normalization.
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