A function f satisfying any (and therefore all, since they're equivalent) of the following properties:
- f satisfies the Laplace's equation Δf=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.
Harmonic functions, in both forms, are a huuuge
topic in analysis
. Try some of these: