\documentclass{beamer}

\title{The Cauchy Integral Formula}
\author{Steve Bell}
\date{February 23, 2009}

\begin{document}

\maketitle

\begin{frame}
Push {\bf Control-L} to enter Full Screen mode and use the left
and right arrow keys to move through the demo.
\end{frame}

\begin{frame}
\frametitle{A slide with a definition and a theorem}

{\color{blue} Definition.}
{\it
The residue of an analytic function $f$ at an isolated
singularity $a$ is equal to the coefficient of $(z-a)^{-1}$
in the Laurent expansion for $f$ about the point $a$.
}

{\color{blue} Theorem.}
{\it
If $P(z)$ and $Q(z)$ are complex polynomials such that
the degree of $P$ is at least two less than the degree of $Q$, and
$Q$ has no zeroes on the real line, then

$$\int_{-\infty}^\infty\frac{P(t)}{Q(t)}\ dt = 2\pi i\sum_{j=1}^N \text{Res\,}_{a_j}\frac{P}{Q},$$
where $\{a_j\}_{j=1}^N$ are the distinct zeroes of $Q$ in the upper half plane.
}
\end{frame}

\begin{frame}
\frametitle{Cauchy Integral Formula basics}
I'm using the \alert{enumerate} environment on this slide.
\begin{enumerate}
\item The Cauchy Integral Formula was discovered by Cauchy.
\item It reveals that an analytic fucntion is determined by
its values on a rather small set.
\item Some people think it is the best formula around.
\end{enumerate}

\end{frame}

\begin{frame}
\frametitle{Cauchy Integral Formula basics}
I'm using the \alert{enumerate} environment on this slide with
the [a.] option and \alert{$\backslash$pause} commands. \pause
\begin{enumerate}[a.]
\item The Cauchy Integral Formula was discovered by Cauchy. \pause
\item It reveals that an analytic function is determined by
its values on a rather small set. \pause
\item Some people think it is the best formula around.
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{Cauchy Integral Formula basics}
\setbeamercovered{dynamic}
I'm doing the same thing, but adding the \alert{$\backslash$setbeamercovered\{dynamic\}}
option.
\begin{itemize}
\item The Cauchy Integral Formula was discovered by Cauchy. \pause
\item It reveals that an analytic function is determined by
its values on a rather small set. \pause
\item Some people think it is the best formula around.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{The Cauchy Integral Formula in living color}
%\setbeamercovered{invisible}
\setbeamercovered{dynamic}
You can do silly things in Beamer that will make you look
like an idiot.  For example:

\bigskip

The {\it Cauchy Integral Formula} in three different colors: \pause
{\color{orange} $$f(a)=\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{z-a}\ dz$$} \pause
{\color{green} $$f(a)=\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{z-a}\ dz$$} \pause
{\color{red} $$f(a)=\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{z-a}\ dz$$}
\end{frame}

\begin{frame}
\frametitle{The Cauchy Integral Formula in living color}
\setbeamercovered{invisible}
%\setbeamercovered{dynamic}
Here is the same thing without those ghost images of the
formula.

\bigskip

The {\it Cauchy Integral Formula} in three different colors: \pause
{\color{orange} $$f(a)=\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{z-a}\ dz$$} \pause
{\color{green} $$f(a)=\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{z-a}\ dz$$} \pause
{\color{red} $$f(a)=\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{z-a}\ dz$$}
\end{frame}

\begin{frame}
Future research.

\bigskip

This last slide does not have a title to demonstrate
that you don't have to have that big blue text at the top
of each slide.

\medskip

Push the ESCAPE key to get out of Full Screen mode.
\end{frame}

\end{document}