Kelvin Transformation

The transformation

$$v(x_1', \ldots, x_n')=\left({a\over r'}\right)^{n-2}u\left({{a^2x_1'\over r'^2}, \ldots, {a^2x_n'\over r'^2}}\right),$$

where

$$r'^2={x_1'}^2+\ldots+{x_n'}^2.$$

If $u(x_1, \dots, x_n)$ is a Harmonic Function on a Domain $D$ of $\Bbb{R}^n$ (with $n\geq 3$), then $v(x_1', \ldots, x_n')$ is Harmonic on $D'$.

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 623, 1980.