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| : ''This article is about a type of transform used in classical [[potential theory]], a topic in [[mathematics]]. '' | | Greetings! I am [http://Www.Trichomoniasis.org/ Marvella] and I really feel comfortable when individuals use the full name. Hiring has been my profession for some time but I've already utilized [http://www.fuguporn.com/user/RMitchell home std test] kit home std test kit for another one. One of the very very best issues in the world for me is to do aerobics and I've been doing it for fairly a whilst. Years ago we moved to Puerto Rico and my family members [http://www.aseandate.com/index.php?m=member_profile&p=profile&id=13345213 at home std test] enjoys it.<br><br>my webpage :: [http://Www.Articlestunner.com/cures-for-the-yeast-infection-suggestions-to-use-now/ at home std testing] |
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| 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 function|subharmonic]] and [[Superharmonic function|superharmonic]] functions.
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| In order to define the Kelvin transform ''f''<sup>*</sup> of a function ''f'', it is necessary to first consider the concept of inversion in a sphere in '''R'''<sup>''n''</sup> as follows.
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| It is possible to use inversion in any sphere, but the ideas are clearest when considering a sphere with centre at the origin.
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| Given a fixed sphere ''S''(0,''R'') with centre 0 and radius ''R'', the inversion of a point ''x'' in '''R'''<sup>''n''</sup> is defined to be
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| ::<math>x^* = \frac{R^2}{|x|^2} x.</math>
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| A useful effect of this inversion is that the origin 0 is the image of <math>\infty</math>, and <math>\infty</math> 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.
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| The Kelvin transform of a function is then defined by:
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| If ''D'' is an open subset of '''R'''<sup>''n''</sup> which does not contain 0, then for any function ''f'' defined on ''D'', the Kelvin transform ''f''<sup>*</sup> of ''f'' with respect to the sphere ''S''(0,''R'') is
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| :<math>f^*(x^*) = \frac{|x|^{n-2}}{R^{2n-4}}f(x) = \frac{1}{|x^*|^{n-2}}f(x)=\frac{1}{|x^*|^{n-2}}f\left(\frac{R^2}{|x^*|^2} x^*\right).</math>
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| One of the important properties of the Kelvin transform, and the main reason behind its creation, is the following result:
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| :Let ''D'' be an open subset in '''R'''<sup>''n''</sup> 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''<sup>*</sup> with respect to the sphere ''S''(0,''R'') is harmonic, subharmonic or superharmonic in ''D''<sup>*</sup>. | |
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| This follows from the formula
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| :<math>\Delta u^*(x^*)=\frac{R^{4}}{|x^*|^{n+2}}(\Delta u)\left(\frac{R^2}{|x^*|^2} x^*\right).</math>
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| ==See also==
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| *[[William Thomson, 1st Baron Kelvin]]
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| *[[Inversive geometry]]
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| ==References==
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| *{{cite book | author = [[Joseph Leo Doob|J. L. Doob]] | title = Classical Potential Theory and Its Probabilistic Counterpart | publisher = Springer-Verlag | year = 2001 | isbn=3-540-41206-9 }}
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| *{{cite book | author = L. L. Helms | title = Introduction to potential theory | publisher = R. E. Krieger | year = 1975 | isbn=0-88275-224-3 }}
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| *{{cite book | author = [[Oliver Dimon Kellogg|O. D. Kellogg]] | title = Foundations of potential theory | publisher = Dover | year = 1953 | isbn=0-486-60144-7 }}
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| [[Category:Harmonic functions]]
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| [[Category:Transforms]]
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