Harmonic function

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{math, number, function}
{math, energy, light}

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : UR (where U is an open subset of Rn) which satisfies Laplace's equation, i.e.

everywhere on U. This is usually written as

Contents

Examples

Examples of harmonic functions of two variables are:

  • The function \,\! f(x_1,x_2)=e^{x_1} \sin x_2

Examples of harmonic functions of n variables are:

  • The constant, linear and affine functions on all of \mathbb{R}^n (for example, the electric potential between the plates of a capacitor, and the gravity potential of a slab)
  • The function \,\! f(x_1,\dots,x_n)=({x_1}^2+\cdots+{x_n}^2)^{1-n/2} on \mathbb{R}^n \backslash \lbrace 0 \rbrace for n > 2.

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