Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function ⁠${\displaystyle f:U\to \mathbb {R} }$⁠, where ⁠${\displaystyle U}$⁠ 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 ⁠${\displaystyle U}$⁠. This is usually written as $${\displaystyle \nabla ^{2}f=0}$$ or $${\displaystyle \Delta f=0}$$