Harmonic function

This article is about harmonic functions in mathematics. For harmonic function in music, see diatonic functionality.

A harmonic function defined on an annulus.

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