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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 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. This is usually written as ∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} or Δ f = 0 {\displaystyle \Delta f=0}