# harmonic function

A real or complex-valued function $f:V\to\mathbb{R}$ or $f:V\to\mathbb{C}$ defined on the vertices $V$ of a graph $G=(V,E)$ is called harmonic at $v\in V$ if its value at $v$ is its average value at the neighbours of $v$:

 $f(v)=\frac{1}{\operatorname{deg}(v)}\sum_{\{u,v\}\in E}f(u).$

It is called harmonic except on $A$, for some $A\subseteq V$, if it is harmonic at each $v\in V\setminus A$, and harmonic if it is harmonic at each $v\in V$.

Any harmonic $f:\mathbb{Z}^{n}\to\mathbb{R}$, where $\mathbb{Z}^{n}$ is the $n$-dimensional grid, is if below (or above). However, this is not necessarily true on other graphs.

Title harmonic function HarmonicFunction1 2013-03-22 15:09:27 2013-03-22 15:09:27 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 05C99