PlanetMath: harmonic function

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (205)
Orphanage
Unclass'd
Unproven (490)
Corrections (8)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

harmonic function

(Definition)

A twice-differentiable real or complex-valued function $f\colon U\to\mathbb{R}$ or $f\colon U\to\mathbb{C}$ , where $U\subseteq\mathbb{R}^n$ is some domain, is called harmonic if its Laplacian vanishes on , i.e. if

$\displaystyle \Delta f\equiv 0.$

Any harmonic function $f\colon\mathbb{R}^n\to\mathbb{R}$ or $f\colon\mathbb{R}^n\to\mathbb{C}$ satisfies Liouville's theorem. Indeed, a holomorphic function is harmonic, and a real harmonic function $f\colon U\to\mathbb{R}$ , where $U\subseteq\mathbb{R}^2$ , is locally the real part of a holomorphic function. In fact, it is enough that a harmonic function be bounded below (or above) to conclude that it is constant.

"harmonic function" is owned by mathcam. [ full author list (2) | owner history (1) ]

(view preamble)

Attachments:: examples of harmonic functions on $\mathbb{R}^n$ (Example) by mathwizard
harmonic conjugate function (Definition) by pahio
Harnack's principle (Theorem) by Mathprof
Gauss' mean value theorem for harmonic functions (Theorem) by Mravinci

Log in to rate this entry.
(view current ratings)

Cross-references: real part, holomorphic function, Liouville's theorem, vanishes, Laplacian, harmonic, function, real
There are 18 references to this entry.

This is version 6 of harmonic function, born on 2002-06-04, modified 2005-03-25.
Object id is 3029, canonical name is HarmonicFunction.
Accessed 10199 times total.

Classification:

AMS MSC:	31A05 (Potential theory :: Two-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
	31B05 (Potential theory :: Higher-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
	31C05 (Potential theory :: Other generalizations :: Harmonic, subharmonic, superharmonic functions)
	30F15 (Functions of a complex variable :: Riemann surfaces :: Harmonic functions on Riemann surfaces)

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

forum policy

Interact