PlanetMath (more info)
 Math for the people, by the people.
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 (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (21)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Medium Entry average rating: No information on entry rating
Gauss' mean value theorem (Theorem)

Let $ \Omega$ be a domain in $ \mathbb{C}$ and suppose $ f$ is an analytic function on $ \Omega$. Furthermore, let $ C$ be a circle inside $ \Omega$ with center $ z_0$ and radius $ r$. Then $ f(z_0)$ is the mean value of $ f$ along $ C$, that is,

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



"Gauss' mean value theorem" is owned by Johan.
(view preamble | get metadata)

View style:

See Also: Gauss' mean value theorem for harmonic functions


Attachments:
proof of Gauss' mean value theorem (Proof) by yark
Log in to rate this entry.
(view current ratings)

Cross-references: mean, radius, center, circle, analytic function, domain
There is 1 reference to this entry.

This is version 9 of Gauss' mean value theorem, born on 2003-04-28, modified 2004-03-20.
Object id is 4216, canonical name is GaussMeanValueTheorem.
Accessed 4557 times total.

Classification:
AMS MSC: 30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

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