Harmonic function

In mathematics, mathematical physics and the theory of , 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.

everywhere on U. This is usually written as

or

The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. This solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics. Fourier analysis involves expanding periodic functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit n-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" was used to refer to all functions satisfying Laplace's equation.

Examples of harmonic functions of two variables are:

Examples of harmonic functions of three variables are given in the table below with ${\displaystyle r^{2}=x^{2}+y^{2}+z^{2}}$:

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