\begin{definition}[Definition:Harmonic Function]
A '''harmonic function''' is a is a twice continuously differentiable function $f: U \to \R$ (where $U$ is an open set of $\R^n$) which satisfies Laplace's equation:
:$\dfrac {\partial^2 f} {\partial {x_1}^2} + \dfrac {\partial^2 f} {\partial {x_2}^2} + \cdots + \dfrac {\partial^2 f} {\partial {x_n}^2} = 0$
everywhere on $U$.
This is usually written as:
:$\nabla^2 f = 0$
or:
:$\Delta f = 0$
\end{definition}