# harmonic function

A real or complex-valued function $f:V\to \mathbb{R}$ or $f:V\to \u2102$ 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 $v$:

$$f(v)=\frac{1}{\mathrm{deg}(v)}\sum _{\{u,v\}\in E}f(u).$$ |

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^{} |
---|---|

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 |