Harmonic function

Mathematical analysis → Complex analysis
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In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function ${\displaystyle f\colon U\to \mathbb {R} ,}$ where U is an open subset of ${\displaystyle \mathbb {R} ^{n},}$ that satisfies Laplace's equation, that is, $${\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}=0}$$ everywhere on U. This is usually written as $${\displaystyle \nabla ^{2}f=0}$$ or $${\displaystyle \Delta f=0}$$