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Kelvin Transformation

Let D be a domain in R^n for n>=3. Then the transformation

v(x_1^',...,x_n^')=(a/(r^'))^(n-2)u((a^2x_1^')/(r^('2)),...,(a^2x_n^')/(r^('2)))

onto a domain D^', where

r^('2)=x_1^'^2+...+x_n^'^2

is called a Kelvin transformation. If u(x_1,...,x_n) is a harmonic function on D, then v(x_1^',...,x_n^') is also harmonic on D^'.

See also

Harmonic Function

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References

Itô, K. (Ed.). "Harmonic Functions and Subharmonic Functions: Invariance of Harmonicity." §193B in Encyclopedic Dictionary of Mathematics, 2nd ed. Cambridge, MA: MIT Press, p. 725, 1980.

Referenced on Wolfram|Alpha

Kelvin Transformation

Cite this as:

Weisstein, Eric W. "Kelvin Transformation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KelvinTransformation.html

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