Category:Definitions/Harmonic Functions

This category contains definitions related to Harmonic Functions.
Related results can be found in Category:Harmonic Functions.

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:

$\nabla^2 f = 0$