Kelvin transform
Encyclopedia
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 Rn 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 Rn 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 Rn 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:
This follows from the formula
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
to extend the concept of a harmonic function
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of subharmonic
Subharmonic function
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory....
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 Rn 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 Rn 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 Rn 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 Rn 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