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 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 $${\displaystyle x^{*}={\frac {R^{2}}{|x|^{2}}}x.}$$

A useful effect of this inversion is that the origin 0 is the image of ${\displaystyle \infty }$, and ${\displaystyle \infty }$ 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 $${\displaystyle f^{*}(x^{*})={\frac {|x|^{n-2}}{R^{2n-4}}}f(x)={\frac {1}{|x^{*}|^{n-2}}}f(x)={\frac {1}{|x^{*}|^{n-2}}}f\left({\frac {R^{2}}{|x^{*}|^{2}}}x^{*}\right).}$$

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 $${\displaystyle \Delta u^{*}(x^{*})={\frac {R^{4}}{|x^{*}|^{n+2}}}(\Delta u)\left({\frac {R^{2}}{|x^{*}|^{2}}}x^{*}\right).}$$

• William Thomson, 1st Baron Kelvin
• Inversive geometry
• Spherical wave transformation

• 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.{{cite book}}: ISBN / Date incompatibility (help)
• John Wermer (1981) Potential Theory 2nd edition, page 84, Lecture Notes in Mathematics #408 ISBN 3-540-10276-0