harmonic function

A twice-differentiable real or complex-valued function $f:U\to ℝ$ or $f:U\to ℂ$, where $U\subseteq {ℝ}^n$ is some , is called harmonic if its Laplacian vanishes on $U$, i.e. if

$$\mathrm{\Delta }f\equiv 0.$$

Any harmonic function $f:{ℝ}^n\to ℝ$ or $f:{ℝ}^n\to ℂ$ satisfies Liouville’s theorem. Indeed, a holomorphic function is harmonic, and a real harmonic function $f:U\to ℝ$, where $U\subseteq {ℝ}^2$, is locally the real part of a holomorphic function. In fact, it is enough that a harmonic function $f$ be below (or above) to conclude that it is .

Title	harmonic function
Canonical name	HarmonicFunction
Date of creation	2013-03-22 12:43:46
Last modified on	2013-03-22 12:43:46
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	9
Author	mathcam (2727)
Entry type	Definition
Classification	msc 31C05
Classification	msc 31B05
Classification	msc 31A05
Classification	msc 30F15
Related topic	RadosTheorem
Related topic	SubharmonicAndSuperharmonicFunctions
Related topic	DirichletProblem
Related topic	NeumannProblem