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Kelvin transform : ウィキペディア英語版
Kelvin transform
: ''This article is about a type of transform used in classical potential theory, a topic in mathematics. ''
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^
*) = \frac}f(x) = \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

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Kelvin transform」の詳細全文を読む



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