Harmonic function

转自：http://en.wikipedia.org/wiki/Harmonic_function

This article is about harmonic functions in mathematics. For harmonic function in music, see diatonic functionality.

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rⁿ) which satisfies Laplace's equation, i.e.

everywhere on U. This is usually written as

Contents

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• 1 Examples
• 2 Remarks
• 3 Connections with complex function theory
• 4 Properties of harmonic functions
  • 4.1 Regularity theorem for harmonic functions
  • 4.2 Maximum principle
  • 4.3 The mean value property
  • 4.4 Harnack's inequality
  • 4.5 Removal of singularities
  • 4.6 Liouville's theorem
• 5 Generalizations
  • 5.1 Weakly harmonic function
  • 5.2 Harmonic functions on manifolds
  • 5.3 Subharmonic functions
  • 5.4 Harmonic forms
  • 5.5 Harmonic maps between manifolds
• 6 See also
• 7 References
• 8 External links