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