harmonic function

A real or complex-valued function f:V or f:V defined on the vertices V of a graph G=(V,E) is called harmonic at vV if its value at v is its average value at the neighbours of v:


It is called harmonic except on A, for some AV, if it is harmonic at each vVA, and harmonic if it is harmonic at each vV.

Any harmonic f:n, where n is the n-dimensional grid, is if below (or above). However, this is not necessarily true on other graphs.

Title harmonic functionPlanetmathPlanetmath
Canonical name HarmonicFunction1
Date of creation 2013-03-22 15:09:27
Last modified on 2013-03-22 15:09:27
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 5
Author mathcam (2727)
Entry type Definition
Classification msc 05C99