proof of Gauss' mean value theoremProofOfGaussMeanValueTheorem2003-04-28 00:25:042006-10-01 13:54:51ProofGauss' mean value theorem0% 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 hereWe can parameterize the circle by letting $z=z_0 + r e^{i\phi}$.
Then $dz=ir e^{i\phi}d\phi$. Using the Cauchy integral formula we can express $f(z_0)$ in the following way:
\begin{eqnarray*}
f(z_0)&=&\frac{1}{2\pi i} \oint_{C} \frac{f(z)}{z-z_0}dz\\
&=&\frac{1}{2\pi i} \int_{0}^{2\pi} \frac{f(z_0 + r e^{i\phi})}{r e^{i\phi}} ir e^{i\phi} d\phi\\
&=&\frac{1}{2\pi} \int_{0}^{2\pi} f(z_0 + r e^{i\phi}) d\phi .
\end{eqnarray*}