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Harmonic function

A function satisfying Laplace's equation, \nabla^2 u = 0 -- no interior local max or min.

A function u is harmonic on a region if it satisfies Laplace's equation, \nabla^2u=0, throughout that region. Harmonic functions satisfy the mean value property and the maximum principle: no interior local extremum that isn't also attained on the boundary.

Examples

  • The real and imaginary parts of any complex-differentiable function are harmonic.

See also

Green's function
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