# harmonic function (thing)

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## continuous:A function f satisfying - f satisfies the Laplace's equation
**Δ**f=. - 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: ## 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 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.
| 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: |