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: ''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 :: 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 *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 スポンサード リンク
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