---
tag: W9YR470VY2
kind: concept
name: Kelvin transform
msc: [31B05, 31A05]
sources:
  - url: https://en.wikipedia.org/wiki/Kelvin_transform
same_as:
  wikidata: Q4377358
  wikipedia: 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|0DEB0JMN0W]] 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 $z \mapsto 1/\bar z$.

## Explanations

### L1 human

Inversion in the unit sphere maps a point at distance $r$ from the origin to the point in the same direction at distance $1/r$.
Composing a harmonic function with this map does not keep it harmonic in dimension three or more; one has to multiply by the weight $\abs{x}^{2-n}$.
With that weight, harmonicity survives.
For instance, in $\R^3$ the Kelvin transform of the constant $1$ is $1/\abs{x}$, the potential of a point charge.