# Harmonic Function

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

or

Read more about Harmonic Function:  Examples, Remarks, Connections With Complex Function Theory

### Other articles related to "function, harmonic function, functions, harmonic":

Bäcklund Transform - The Cauchy–Riemann Equations
... imaginary parts u and v of a holomorphic function ... a harmonic function), and so is v ... is a solution of Laplace's equation, then there exist functions v which solve the Cauchy–Riemann equations together with u ...
Harmonic Function - Generalizations - Harmonic Maps Between Manifolds
... If M and N are two Riemannian manifolds, then a harmonic map u M → N is defined to be a stationary point of the Dirichlet energy in which du TM → TN is the differential of u, and the norm is ... Important special cases of harmonic maps between manifolds include minimal surfaces, which are precisely the harmonic immersions of a surface into three-dimensional Euclidean space ... More generally, minimal submanifolds are harmonic immersions of one manifold in another ...
Harmonic Conjugate
... In mathematics, a function defined on some open domain is said to have as a conjugate a function if and only if they are respectively real and imaginary part ... latter equivalent definition, if is any harmonic function on the function is conjugate to, for then the Cauchy–Riemann equations are just and the symmetry of the mixed second order derivatives ... There is an operator taking a harmonic function u on a simply connected region in R2 to its harmonic conjugate v (putting e.g ...
Newtonian Potential
... that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity ... integral operator, defined by convolution with a function having a mathematical singularity at the origin, the Newtonian kernel Γ which is the fundamental solution of the Laplace equation ... It is named for Isaac Newton, who first discovered it and proved that it was a harmonic function in the special case of three variables, where it served as ...

### Famous quotes containing the words function and/or harmonic:

The intension of a proposition comprises whatever the proposition entails: and it includes nothing else.... The connotation or intension of a function comprises all that attribution of this predicate to anything entails as also predicable to that thing.
Clarence Lewis (1883–1964)

For decades child development experts have erroneously directed parents to sing with one voice, a unison chorus of values, politics, disciplinary and loving styles. But duets have greater harmonic possibilities and are more interesting to listen to, so long as cacophony or dissonance remains at acceptable levels.
Kyle D. Pruett (20th century)