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'Gauss' mean value theorem'
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| Title of object: |
Gauss' mean value theorem |
| Canonical Name: |
GaussMeanValueTheorem |
| Type: |
Theorem |
| Created on: |
2003-04-28 00:17:42.813761-04 |
| Modified on: |
2003-04-28 08:26:16.406894-04 |
| Classification: |
msc:30E20 |
Revision comment (for changes between this and next version):
| Changes for correction # (''). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
Let $f$ be analytic in $\Omega \subset \mathbb{C}$ and let $C$ be a circle inside $\Omega$ with with center $z_0$. Then $f(z_0)$ is the mean value along the circle, where "mean value" is defined by
\begin{displaymath}
f_{mean}=\frac{1}{|C|} \oint_{C} f(z) dz
\end{displaymath}
where $|C|$ denotes the length of the path $C$. |
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