Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 76 / 5

For a degree-$n$ homogeneous polynomial, the Euler homogeneous function theorem gives $\mathbf{r}\cdot \nabla H_n=n{H}_n$. On $r>0$ in three-dimensional Euclidean space,Apply the product rule for the Laplacian:For $s=-(2n+1)$ the second term vanishes. Hence homogeneous harmonic inversion givesThe nonzero factor $r^{-(2n+1)}$ proves both directions: the transformed function is harmonic exactly when $\Delta H_n=0$ away from the origin. Since $\Delta H_n$ is a polynomial, vanishing on the punctured space makes it identically zero everywhere.

Equivalently, homogeneity shows that the Kelvin transform isIt sends a regular solid harmonic to an irregular solid harmonic, exchanging the radial powers $r^n$ and $r^{-n-1}$. The origin is excluded from the harmonicity statement: for example, the $n=0$ transform $1/r$ is harmonic only away from zero and has a distributional Dirac delta function source there.