Harmonic function

Functions in mathematics

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function ⁠ f : U → R {\displaystyle f\colon U\to \mathbb {R} } ⁠, where ⁠ U {\displaystyle U} ⁠ is an open subset of ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠, that satisfies Laplace's equation, that is, ∂ 2 f ∂ x 1 2 + ∂ 2 f ∂ x 2 2 + ⋯ + ∂ 2 f ∂ x n 2 = 0 {\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}=0} everywhere on ⁠ U {\displaystyle U} ⁠.

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