Summary
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function where U is an open subset of \mathbb R^n, that satisfies Laplace's equation, that is, 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. The 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 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: The real or imaginary part of any holomorphic function. The function this is a special case of the example above, as and is a holomorphic function. The second derivative with respect to x is while the second derivative with respect to y is The function defined on This can describe the electric potential due to a line charge or the gravity potential due to a long cylindrical mass. Examples of harmonic functions of three variables are given in the table below with {| class="wikitable" ! Function !! Singularity |- |align=center| |Unit point charge at origin |- |align=center| |x-directed dipole at origin |- |align=center| |Line of unit charge density on entire z-axis |- |align=center| |Line of unit charge density on negative z-axis |- |align=center| |Line of x-directed dipoles on entire z axis |- |align=center| |Line of x-directed dipoles on negative z axis |} Harmonic functions that arise in physics are determined by their singularities and boundary conditions (such as Dirichlet boundary conditions or Neumann boundary conditions).
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