Kelvin transform
= Kelvin transform
{c}
{title2=$K_a u(\mathbf x)=\frac a{|\mathbf x|}u\!\left(\frac{a^2\mathbf x}{|\mathbf x|^2}\right)$}
{wiki}
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.