# harmonic function (thing)

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## [continuous]:A [function] f satisfying - 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].
## [discrete]: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.
## Related topicsHarmonic functions, in both forms, are ahuuuge 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.
| Existing:- continuous
- function
- equivalent
- Laplace's equation
- sphere
- locally
- real
- holomorphic function
- discrete
- graph
- vertex
- average
- analysis
- holomorphic function
- converse
- locally
- Laplace
- partial differential equation
- Brownian motion
- Brownian motion solves a PDE
- nonconstant
- random walk
- Z^n admits no bounded harmonic function
- nonconstant
- margin
- fragment
Non-Existing: |