Definition:Harmonic Function

Definition
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 using the $\nabla^2$ symbol to denote the Laplacian, as:


 * $\nabla^2 f = 0$

Also presented as
Some sources use the $\Delta$ symbol for the Laplacian:
 * $\Delta f = 0$

Source of Name
The name harmonic function originates from their use in analysing the harmonics of the sound made by a taut string.