\documentclass[12pt]{article}
\pagestyle{empty} %% We don't want page numbers to print
\usepackage{amssymb}
%% This will make your labels visible. Comment it out to hide the labels.
\usepackage[notcite,notref]{showkeys}
%% The euler style option allows us to use the fraktur fonts.
%% The amssymbols style option allows us to use AMS version 1.1 fonts
%% with LaTeX. This includes, e.g. the Blackboard bold (\mathbb)
%% amssymbols.sty is a standard style file. Euler.sty is local to UW-Math
%% WARNING: the order is important here. If we put euler after
%% amssymb, the mathbb fonts will not appear.
\begin{document}
\centerline{\bf Some Mathematics}
\bigskip 

This is a sampler of some mathematics typed in \TeX. To use ``math
mode'' in \TeX\ or \LaTeX you enclose mathematical statements in dollar
signs.  Single dollar signs give you ``in-line'' mathematics like this
$ f(x) = \int g(x)dx$.  Double dollar signs may be used to display
unnumbered equations just as in plain \TeX:
%% like this:
$$ f(x) = \sin(x) + x^2 - y_0 $$
Standard LaTeX substitutes \begin{verbatim} \[ ... \] \end{verbatim} for
\begin{verbatim} $$ ... $$ \end{verbatim} for unnumbered, displayed
equations. For numbered, displayed equations, use the {\bf equation}
environment.  We can include AMS fonts, too, as these integrals and
summations show:
\begin{equation}\label{eq:1}
F(x) = \int_0^\infty {x+1 \over x^3 -1} dx \mbox{  for all  } x\in {\mathbb R}
\end{equation}
\begin{equation}\label{Mimi}
{\frak G}({\frak a}) = \sum_{n=1}^{N+1} {\frak a}^{-n}
\end{equation}
If we labelled the equations, above, we can refer to them elsewhere
in our document as I am doing now (see equation \ref{Mimi}).

One of the advantages of \LaTeX\ is that delimiters automatically stretch
to accommodate the size of the expression.  The examples below show how
the curly bracket can be stretched for a multiple case expression and
for lengthy roots
$$ P = \cases{-\Delta u = \lambda_k u + f(x,u), & in $\Omega$ \cr
         u = 0, & on $\partial \Omega$ \cr}
$$
$$ F(x) = \cases{ \sin(\exp(x)) &  $ x < -1 $ \cr
          \tan(x)       &  $ -1 \leq x \leq 1$ \cr
          {x^2 +1 \over x^2 -1} & $1 < x \leq 10$ \cr
          \sum_{n=1}^10  x^n  & $10 < x $\cr}
$$
If we want to line up equations by their equal signs, we use the
{\bf eqnarray} environment. By default, each line in the eqnarray
environment is numbered unless marked by {\bf nonumber}
\begin{eqnarray}
z &=& \sqrt{x^2 + y^2} \label{Tom}\\
z &=& \root n  \of {{x^2-y^2\over x^2+y^2} + \sqrt{x^2 + y^2}}\label{Dick}
\end{eqnarray}

 In equation \ref{Tom} or equation \ref{Dick}...
The starred (*) eqnarray environment does not number the equation.
\begin{eqnarray*}
\alpha^2&=&\beta^2+\gamma^2\\
\frak a^2&=&b^2+c^2
\end{eqnarray*}
\end{document}