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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.


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

Harmonic functions, in both forms, are a huuuge topic in analysis. Try some of these: