harmonic function

A twice-differentiable real or complex-valued function f:U or f:U, where Un is some , is called harmonic if its Laplacian vanishes on U, i.e. if


Any harmonic functionPlanetmathPlanetmath f:n or f:n satisfies Liouville’s theorem. Indeed, a holomorphic functionMathworldPlanetmath is harmonic, and a real harmonic function f:U, where U2, 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