harmonic function

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

$\Delta f\equiv 0.$

Any harmonic function $f\colon\mathbb{R}^{n}\to\mathbb{R}$ or $f\colon\mathbb{R}^{n}\to\mathbb{C}$ satisfies Liouville’s theorem. Indeed, a holomorphic function is harmonic, and a real harmonic function $f\colon U\to\mathbb{R}$ , where $U\subseteq\mathbb{R}^{2}$ , 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