The Kelvin transform is a device used in classical

potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely g ...

to extend the concept of a

harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that i ...

, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of

subharmonic In music, the undertone series or subharmonic series is a sequence of notes that results from inverting the intervals of the overtone series. While overtones naturally occur with the physical production of music on instruments, undertones mus ...

and

superharmonic An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...

functions. In order to define the Kelvin transform ^* 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 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 ^* of ''f'' with respect to the sphere 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 is harmonic, subharmonic or superharmonic in ''D''^*. This follows from the formula

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

References

William Thomson, Lord Kelvin William Thomson, 1st Baron Kelvin (26 June 182417 December 1907), was a British mathematician, mathematical physicist and engineer. Born in Belfast, he was the professor of Natural Philosophy at the University of Glasgow for 53 years, where ...

(1845) "Extrait d'une lettre de M. William Thomson à M. Liouville",

Journal de Mathématiques Pures et Appliquées The ''Journal de Mathématiques Pures et Appliquées'' () is a French monthly scientific journal of mathematics, founded in 1836 by Joseph Liouville (editor: 1836–1874). The journal was originally published by Charles Louis Étienne Bachelier. ...

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 * * * *

John Wermer John Wermer was a mathematician specializing in complex analysis. Wermer received his Ph.D. from Harvard University in 1951 under the supervision of George Whitelaw Mackey. He became an instructor at Yale University, after which he was hired as ...

(1981) ''Potential Theory'' 2nd edition, page 84,

Lecture Notes in Mathematics ''Lecture Notes in Mathematics'' is a book series in the field of mathematics, including articles related to both research and teaching. It was established in 1964 and was edited by A. Dold, Heidelberg and B. Eckmann, Zürich. Its publisher is Sp ...

#408 {{ISBN, 3-540-10276-0 Harmonic functions Transforms Transform

See also

References