harmonic function

continuous:

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

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

discrete:

On a graph, we have a related definition:

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

Related topics

• The real part (and therefore also the imaginary part) of a holomorphic function is harmonic; the converse is true locally, but not globally.
• Harmonic functions satisfy Laplace's partial differential equation above; this leads to deep connections with Brownian motion, via how Brownian motion solves a PDE.
• ... This is one way to show that R^n admits no bounded harmonic function (that is nonconstant).
• On graphs too, harmonic functions have connections to random walks.
• ... Which leads to the way to show that Z^n admits no bounded harmonic function (that is nonconstant).
• There's a lot more, but my head (and this margin!) is too small to contain anything but some fragments.