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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 , 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 **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:

- Let
*D*be an open subset in**R**^{n}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

- William Thomson, Lord Kelvin (1845) "Extrait d'une lettre de M. William Thomson à M. Liouville", Journal de Mathématiques Pures et Appliquées 10: 364–7
- William Thompson (1847) "Extraits deux lettres adressees à M. Liouville, par M. William Thomson",
*Journal de Mathématiques Pures et Appliquées*12: 556–64 - J. L. Doob (2001).
*Classical Potential Theory and Its Probabilistic Counterpart*. Springer-Verlag. p. 26. ISBN 3-540-41206-9. - L. L. Helms (1975).
*Introduction to potential theory*. R. E. Krieger. ISBN 0-88275-224-3. - O. D. Kellogg (1953).
*Foundations of potential theory*. Dover. ISBN 0-486-60144-7. - John Wermer (1981)
*Potential Theory*2nd edition, page 84, Lecture Notes in Mathematics #408 ISBN 3-540-10276-0