= Solution
In <Cartesian coordinates>, put $r=(x_1^2+x_2^2+x_3^2)^{1/2}$. Away from the origin the <partial derivatives> of $\psi=1/r$ are
$$
\partial_i\psi=-\frac{x_i}{r^3},\qquad
\partial_i^2\psi=-\frac1{r^3}+\frac{3x_i^2}{r^5}.
$$
Summing the three second <partial derivatives> yields
$$
\boxed{\nabla^2\psi=-\frac3{r^3}+\frac{3(x_1^2+x_2^2+x_3^2)}{r^5}=0\qquad(r>0).}
$$
Thus $1/r$ is a <harmonic function> on the punctured space and solves <Laplace equation> there. The origin is excluded: the singularity cannot be treated as an ordinary harmonic point.
Back to article page
