Kelvin transform

Summary

The Kelvin transform turns a function $u$ on a region of $\R^n \setminus \set{0}$ into the function $x \mapsto \abs{x}^{2-n} u(x / \abs{x}^2)$ on the inverted region. It sends harmonic functions to harmonic functions, and it exchanges the neighbourhoods of $0$ and of infinity. It is the main tool to study harmonic functions near infinity, exterior Dirichlet problems and the Green's function of a ball. In the plane it is just composition with the inversion $z \mapsto 1/\bar z$.