# 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:

$\nabla^2 f = 0$

### Riemannian Manifold

Let $\struct {M, g}$ be a compact Riemannian manifold with or without boundary.

Let $\map {C^\infty} M$ be the smooth function space.

Let $u \in \map {C^\infty} M$ be a smooth real function on $M$.

Let $\nabla^2$ be the Laplace-Beltrami operator.

Then $u$ is said to be harmonic if and only if:

$\nabla^2 u = 0$

## 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.