Editing Harmonic function (section)

{{short description|Functions in mathematics}}
{{about|harmonic functions in mathematics|harmonic function in music|diatonic functionality}}
[[Image:Laplace's equation on an annulus.svg|right|thumb|300px|A harmonic function defined on an [[Annulus (mathematics)|annulus]].]]
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In [[mathematics]], [[mathematical physics]] and the theory of [[stochastic process]]es, a '''harmonic function''' is a twice [[continuously differentiable]] [[function (mathematics)|function]] <math>f\colon U \to \mathbb R,</math> where {{mvar|U}} is an [[open set|open subset]] of {{tmath|\mathbb R^n,}} that satisfies [[Laplace's equation]], that is,
<math display="block"> \frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0</math>
everywhere on {{mvar|U}}. This is usually written as
<math display="block"> \nabla^2 f = 0 </math>
or
<math display="block">\Delta f = 0</math>