In mathematics, mathematical physics and the theory of stochastic processes, a **harmonic function** is a twice continuously differentiable function *f* : *U* → **R** (where *U* is an open subset of **R***n*) 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":

... 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 ...

... 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 ...

... 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)