Harmonic function

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.

Related Topics:
Mathematics - Mathematical physics - Stochastic process - Continuously differentiable - Function - Open subset - Laplace's equation

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:

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rac{partial^2f}{partial x_1^2} +

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rac{partial^2f}{partial x_2^2} +

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cdots +

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rac{partial^2f}{partial x_n^2} = 0

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everywhere on U. This is also often written as

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: abla^2 f = 0 or Delta f = 0.

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There also exists a seemingly weaker definition that is equivalent. Indeed a function is harmonic if and only if it is weakly harmonic.

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A function that satisfies Delta f ge 0 is said to be subharmonic.

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