\documentclass[11pt] {letter}
\usepackage{graphicx,color}
%\makelabels


\setlength{\topmargin}{-1.2in}
\setlength{\textheight}{9in}
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%% macros
\newtheorem{theorem}{Theorem}         
\newcommand{\Frac}[2]{\displaystyle \frac{#1}{#2}}
\newcommand{\binom}[2]{\displaystyle \left(\begin{array}{c} 
                                      #1 \\#2 \end{array}\right)}

\begin{document}
\date{~} 
 
\name{Carl Eberhart}

\address{ 
\begin{minipage}[t]{520 pt}\hspace{-50pt}
\parbox[t]{250pt}{\includegraphics[height=250pt,width=140pt,angle=-90]{jack.eps}}
\hspace{30pt}\vspace{16pt}
\parbox[t]{250pt}{
\vspace{10pt}
 {\LARGE\sc University of Kentucky\\
Mathematics Department\\ Lexington, Ky 40506-0021}\\  
\vfill{\small Carl Eberhart \hfill  759 POT\\
{\em email:}  carl@ms.uky.edu } \hfill  \today }
\end{minipage} 
}


\signature{Carl Eberhart} 

\begin{letter}{Isaac Newton\\Cambridge
University\\Cambridge, England} 

\opening{Dear Isaac,} I wanted to send
you some sample equations and graphics to show how well the {\LaTeX}
program works. If I had a color printer you would be getting a color
version of the graphic, which was generated with MapleV5, a mathematical
problem solving language. 

To start with,  inline mathematics is typeset between \$ symbols, so that 

\verb|$1 + 2 + \cdots + n = \frac{n (n+1)}{2}$| produces $1 + 2 + \cdots
+ n = \frac{n (n+1)}{2}$. On the other hand, you can display the
equation by using \verb|\[| and \verb|\]|, as in \[ 1 + 2 + \cdots + n =
\frac{n (n+1)}{2}. \] 

Subscripts and superscripts are easy in LaTeX: \verb|$a_1$| turns into
$a_1$ and \verb|$a^{12}$| turns into $a^{12}$. 

Integral and derivative symbols are available in LaTeX: So the integral
of $f(x)$ from a to b would be $\int_a^b f(x)~dx$, and the derivative of
$f(x)$ would be $\frac{\partial f}{\partial x}$. The displayed versions
of these are \[ \int_a^b f(x)~dx ~~\mbox{ and }~~ \frac{\partial
f}{\partial x}.\] 

Notice how the displayed symbols are larger than the inline symbols? You
can make the inline symbols larger by inserting a \verb|\displaystyle|.
So a big inline integral is $\displaystyle\int_a^b f(x)~dx$.

  
We used the command \textit{newcommand} to define an theorem environment
in the preamble of this letter, so we can number theorems automatically.

What is the Pythagorean theorem?

\begin{theorem}[Pythagoras]\label{pyth} If $\triangle ABC$ is a right
triangle with hypotenuse $\overline{AB}$, then $AB^2 = AC^2 + CB^2$
\end{theorem}


How about tables and matrices? They can be handled in LaTeX with no
problem. Here's a piecewise defined function.

\[   y = \left\{  \begin{array}{ll}
     2x+1  & ~~ \mbox{if  $x < 3$} \\
     \sqrt{x^2 -\frac{2}{3}x}  &  ~~ \mbox{if $3 \leq x < 10$} \\
      \exp{\sin(3x+1)} & ~~ \mbox{if  $x \geq 10$}
\end{array} \right.   \]


Now theorem \ref{pyth} has been stated. The binomial symbol
$\binom{n}{i}$ is typeset with the \verb|\binom| macro defined in the
preamble. This makes it easier to state the binomial theorem.

\begin{theorem}[The Binomial Theorem]
Let $n$ and $i$ be integers with $n > 2$ and $1<i<n$. Then $\binom{n}{i}$,
the coefficient of $x^{n-i}y^i$ in the expansion of $(x+y)^n$,  is the sum of the
coefficients of $x^{n-i}y^{i-1}$ and $x^{n-i-1}y^{i}$ in the expansion of 
$(x+y)^{n-1}$.  That is,
\[  \binom{n}{i} = \binom{n-1}{i-1} + \binom{n-1}{i} \]
\end{theorem}


{\bf Proof:}  We induct on $n$.
The statement is true if $n=3$ and $i=2$.  Suppose it is true for
$n-1$ and all $i$ between $1$ and $n-1$. 
We can write $(x+y)^{n } = (x+y)(x+y)^{n-1} = x(x+y)^{n-1}+y(x+y)^{n-1}$.
By induction,   yada yada yada \ldots.

Tables are easy to make up too with the \verb|\begin{tabular}| character
style. Here is a short four line table of LaTeX symbols and what they
typeset to. \medskip

\begin{center}
{\large\bf Table of {\LaTeX}  symbols}
\begin{tabular}{|c|c| c|c| c|}\hline

{\bf LaTeX symbol} & \verb| $\alpha $| &\verb| $\pi$ |& \verb|$\sum_{i=1}^{\infty} $| 
& \verb| $\triangle $| \\ \hline {\bf typesets
to} & $\alpha $ & $\pi$ & $\sum_{i=1}^{\infty} $ & $\triangle $ \\
\hline\hline { \bf LaTeX symbol} &\verb|$ \not\cong$| & \verb| $\aleph$|
& \verb| $\clubsuit $|& \verb| ${\cal A}$| \\ \hline {\bf typesets to}
&$ \not\cong$ & $\aleph$ & $\clubsuit $& ${\cal A}$ \\ \hline

\end{tabular}
\end{center}
 

So, aren't you jealous, Isaac? If you had access to LaTeX in your day,
it wouldn't have taken 20 years to get the {\em Principia} into print.
You could have typed it as you went. \closing{Sincerely,} \end{letter}
\end{document}




