Harmonic function

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

${\displaystyle \Delta f=0}$