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

::$x^*\; =\; frac\{R^2\}\{|x|^2\}\; x.$

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:$f^*(x^*)\; =\; frac\{|x|^\{n-2\{R^\{2n-4f(x)\; =\; frac\{1\}\{|x^*|^\{n-2f(x).$

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:$Delta\; u^*(x^*)=frac\{R^\{n+2\{|x^*|^\{n+2(Delta\; u)^*(x^*).$

ee also

*William Thomson, 1st Baron Kelvin
*Inversive geometry

References

*cite book | author = J. L. Doob | title = Classical Potential Theory and Its Probabilistic Counterpart | publisher = Springer-Verlag | year = 2001 | id=ISBN 3-540-41206-9

*cite book | author = L. L. Helms | title = Introduction to potential theory | publisher = R. E. Krieger | year = 1975 | id=ISBN 0-88275-224-3

*cite book | author = O. D. Kellogg | title = Foundations of potential theory | publisher = Dover | year = 1953 | id=ISBN 0-486-60144-7