Kelvin transform

The Kelvin transform is a device used in classical potential theory to extend the concept of a harmonic function, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of subharmonic and superharmonic functions.
In order to define the Kelvin transform "f"* of a function "f", it is necessary to first consider the concept of inversion in a sphere in R"n" as follows.
It is possible to use inversion in any sphere, but the ideas are clearest when considering a sphere with centre at the origin.
Given a fixed sphere "S"(0,"R") with centre 0 and radius "R", the inversion of a point "x" in R"n" is defined to be
A useful effect of this inversion is that the origin 0 is the image of formula_2, and formula_2 is the image of 0. Under this inversion, spheres are transformed into spheres, and the exterior of a sphere is transformed to the interior, and vice versa.
The Kelvin transform of a function is then defined by:
If "D" is an open subset of R"n" which does not contain 0, then for any function "f" defined on "D", the Kelvin transform "f"* of "f" with respect to the sphere "S"(0,"R") is
One of the important properties of the Kelvin transform, and the main reason behind its creation, is the following result:
This follows from the formula