# 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