PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (236)
Orphanage
Unclass'd (1)
Unproven (540)
Corrections (46)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Copyright
About
Owner confidence rating: High Entry average rating: Very high
[parent] Gauss' mean value theorem for harmonic functions (Theorem)

If the function $u(z) \equiv u(x,\,y)$ is harmonic in a domain of complex plane which contains the disc $|z-z_0| \leqq r$ , then

$\displaystyle u(z_0) = \frac{1}{2\pi}\int_0^{2\pi}u(z_0+re^{i\varphi})\,d\varphi.$ (1)

Conversely, if a real function $u(x,\,y)$ is continuous in a domain $G$ of $\mathbb{R}^2$ and satisfies on all circles of $G$ the equation (1), then it is harmonic.



"Gauss' mean value theorem for harmonic functions" is owned by PrimeFan. [ owner history (2) ]
(view preamble | get metadata)

View style:

See Also: Gauss' mean value theorem


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: equation, circles, continuous, real function, conversely, disc, contains, complex plane, domain, harmonic, function

This is version 5 of Gauss' mean value theorem for harmonic functions, born on 2005-01-22, modified 2005-01-22.
Object id is 6658, canonical name is GaussMeanValueTheoremForHarmonicFunctions.
Accessed 3810 times total.

Classification:
AMS MSC: 31A05 (Potential theory :: Two-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
 30F15 (Functions of a complex variable :: Riemann surfaces :: Harmonic functions on Riemann surfaces)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)