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Concept
Harmonic function
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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Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , (where is the nabla operator), or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form.
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
Potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which satisfy Poisson's equation—or in the vacuum, Laplace's equation.