A function f satisfying *any* (and therefore *all*, since they're equivalent) of the following properties:

- f satisfies the Laplace's equation
**Δ**f=0.
- The average value of f on any sphere is equal to its value at the centre of the sphere.
- (In 2 dimensions,) f is locally the real part of a holomorphic function.

On a graph, we have a related definition:

- The value of f at any vertex is equal to its average value at the neighbours of the vertex.

#### Related topics

Harmonic functions, in both forms, are a

*huuuge* topic in

analysis. Try some of these: