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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 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 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 with centre 0 and radius R, the inversion of a point x in R is defined to be
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