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 Template:Itco^* 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 Template:Nowrap with centre 0 and radius R, the inversion of a point x in Rⁿ is defined to be $$x^{*}=\frac{R^2}{|x{|}^2}x.$$

A useful effect of this inversion is that the origin 0 is the image of $\mathrm{\infty }$, and $\mathrm{\infty }$ 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 Template:Itco^* of f with respect to the sphere Template:Nowrap is $$f^{*}(x^{*})=\frac{|x{|}^{n-2}}{R^{2n-4}}f(x)=\frac{1}{|x^{*}{|}^{n-2}}f(x)=\frac{1}{|x^{*}{|}^{n-2}}f\left(\frac{R^2}{|x^{*}{|}^2}{x}^{*}\right).$$

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 Template:Nowrap is harmonic, subharmonic or superharmonic in D^*.

This follows from the formula $$\mathrm{\Delta }{u}^{*}(x^{*})=\frac{R^4}{|x^{*}{|}^{n+2}}(\mathrm{\Delta }u)\left(\frac{R^2}{|x^{*}{|}^2}{x}^{*}\right).$$

See also

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

References

• 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
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• John Wermer (1981) Potential Theory 2nd edition, page 84, Lecture Notes in Mathematics #408 Template:ISBN