 Kelvin transform

In three-dimensional Euclidean space, spherical inversion together with the displayed weight maps harmonic functions to harmonic functions on the inverted domain. The Laplacian satisfies $\Delta K_{a}u(\mathbf{x})=(a/r{)}^5(\Delta u)(a^2\mathbf{x}/r^2)$. The formula is for $r>0$; a new singularity may occur at the origin.