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 two fundamental forces of nature known at the time, namely gravi ...
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: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
, 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 must ...
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 ''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
See also
*
William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
*
Inversive geometry
Inversive activities are processes which self internalise the action concerned. For example, a person who has an Inversive personality internalises his emotions from any exterior source. An inversive heat source would be a heat source where all th ...
*
Spherical wave transformation
Spherical wave transformations leave the form of spherical waves as well as the laws of optics and electrodynamics invariant in all inertial frames. They were defined between 1908 and 1909 by Harry Bateman and Ebenezer Cunningham, with Bateman givi ...
References
*
William Thomson, Lord Kelvin (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. A ...
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 is a mathematician specializing in Complex analysis.
Wermer received his Ph.D. from Harvard University in 1951 under the supervision of George Whitelaw Mackey.
In 1962 Wermer was an invited speaker at the International Congress of Math ...
(1981) ''Potential Theory'' 2nd edition, page 84,
Lecture Notes in Mathematics #408 {{ISBN, 3-540-10276-0
Harmonic functions
Transforms
Transform