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 $x^* = \frac x.$

A useful effect of this inversion is that the origin 0 is the image of $\infty$, and $\infty$ 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 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
$f^*(x^*) = \fracf(x) = \fracf(x) = \frac f\left(\frac x^*\right).$

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 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
$\Delta u^*(x^*) = \frac(\Delta u)\left(\frac x^*\right).$

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 {{ISBN, 3-540-10276-0

Harmonic functions
Transforms
Transform