examples of harmonic functions on n

Some real functions in n (e.g. any linear function, or any affine function) are obviously harmonic functions. What are some more interesting harmonic functions?

  • For n3, define (on the punctured space U=n{0}) the functionMathworldPlanetmath f(x)=x2-n. Then




    Summing over i=1,,n shows Δf0.

  • For n=2, define (on the punctured plane U=2{0}) the function f(x,y)=log(x2+y2). Derivation and summing yield Δf0.

  • For n=1, the condition (Δf)(x)=f′′(x)0 forces f to be an affine function on every segment; there are no “interesting” harmonic functions in one dimension.

Title examples of harmonic functions on n
Canonical name ExamplesOfHarmonicFunctionsOnmathbbRn
Date of creation 2013-03-22 12:44:23
Last modified on 2013-03-22 12:44:23
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 9
Author mathwizard (128)
Entry type Example
Classification msc 31A05
Classification msc 31B05