Harmonic Function

An Harmonic Function is a twice continuously differentiable function that satisfies the Laplace's equation.

• Example(s):
  • Simple Harmonic Oscillator.
• Counter-Example(s):
• See: Harmonic Oscillator, Laplace's Equation, Stochastic Process, Derivative, Open Set.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/harmonic_function Retrieved:2015-11-29.
  • In mathematics, mathematical physics and the theory of stochastic processes, a 'harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplace's equation, i.e. : [math]\displaystyle{ \frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0 }[/math] everywhere on U. This is usually written as : [math]\displaystyle{ \nabla^2 f = 0 }[/math] or : [math]\displaystyle{ \textstyle \Delta f = 0 }[/math]

1999

• (Mathworld Wolfram, 1999) ⇒ http://mathworld.wolfram.com/Harmonic.html
  • QUOTE: In complex analysis, a harmonic function refers to a real-valued function [math]\displaystyle{ f(x,y) }[/math] which satisfies Laplace's equation

[math]\displaystyle{ \nabla ^2f(x,y)=0 }[/math]

where [math]\displaystyle{ \nabla ^2 }[/math] is the Laplacian.

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