- This article is about a type of transform used in classical potential theory, a topic in mathematics.
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 Rn 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 Rn is defined to be
A useful effect of this inversion is that the origin 0 is the image of , and 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 Rn 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:
- Let D be an open subset in Rn which does not contain the origin 0. Then a function u is harmonic, subharmonic or superharmonic in D if and only if the Kelvin transform u* with respect to the sphere S(0,R) is harmonic, subharmonic or superharmonic in D*.
This follows from the formula
- J. L. Doob (2001). Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag. ISBN 3-540-41206-9.