\documentclass[10pt]{amsart}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{local}

\begin{document}
\filename
\centerline{\LaTeX\ {\sc Seminar: Other Math}}

\section{Matrices}

The package {\em amsmath} includes several useful macros for writing matrices.  For example,
we can easily typeset complicated matrices like

$$\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\  a_{n1} & a_{n2} & \cdots & a_{nn}\end{pmatrix}.$$
This matrix was produced using the \texttt{pmatrix} macro.  Several matrix types are possible, including
\verb=matrix, pmatrix, bmatrix, Bmatrix, vmatrix, Vmatrix,= which produce, respectively
\begin{align*}
    &\begin{matrix} r & s & t \\ u & v & w \\ x & y & z\end{matrix} & &\begin{pmatrix} r & s & t \\ u & v & w \\ x & y & z\end{pmatrix}
    & &\begin{bmatrix} r & s & t \\ u & v & w \\ x & y & z\end{bmatrix} \\
    &\begin{Bmatrix} r & s & t \\ u & v & w \\ x & y & z\end{Bmatrix} &  &\begin{vmatrix} r & s & t \\ u & v & w \\ x & y & z\end{vmatrix}
    & &\begin{Vmatrix} r & s & t \\ u & v & w \\ x & y & z\end{Vmatrix}.
\end{align*}
each matrix is produced with the commands
\begin{verbatim}
    \begin{xmatrix}
        r & s & t \\
        u & v & w \\
        x & y & z
    \end{xmatrix}
\end{verbatim}
where \verb=xmatrix= is one of the 6 possibilities listed above.  There is a separate macro for producing small matrices
within a line of text, such as $\alpha = \left(\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right)$.  The environment
here is the \verb=smallmatrix= environment.  Unlike the other matrix macros, \verb=smallmatrix= does not produce any brackets,
so these have to be added separately.  Edit the file \texttt{omath.tex} to get a feel for producing more
complex matrices like the first example at the beginning of this section.

\section{Aligning Equations}

Several {\em amsmath} extensions exist for aligning longer, more complex equations.  Apart from the \verb=align= environment,
which is probably the most useful, there are several others, most notably the \verb=multline= environment.  The format
of \verb=multline= is simple: the first line is left justified, the last is right justified, and all others are centered.
Line breaks occur only when forced by the user with the \verb=\\= command.  The commands \verb=\shoveleft{},= and
\verb=\shoveright{},= which both take a formula as their argument, can be used to left or right justify an individual line.
Using the \verb=multline= environment, we can produce things like

\begin{multline}
    \sum_{m=1}^{\infty}\frac{c_m}{m^{\nu}}\sqrt{\left(\frac{2}{m\pi x}\right)} \Bigg\{ \cos(mx-\tfrac{1}{2}\nu\pi-\tfrac{1}{4}\pi-\eta_m)\\
           -\frac{4\nu^2-1}{8mx}\sin(mx-\tfrac{1}{2}\nu\pi-\tfrac{1}{4}\pi-\eta_m) \Bigg\}+\frac{b_m}{m^{\nu}}
           \frac{(\tfrac{1}{2}mx)^{\nu-1}}{\Gamma(\nu+\frac{1}{2})\Gamma(\frac{1}{2})}
\end{multline}

and

\begin{multline}
    \sum_{m,n=0}^{\infty} e^{-a^n b^m x} = \frac{\log^2 x}{2\log a\log b}+\log x\left(
    \frac{\gamma}{\log a\log b}-\frac{1}{2\log a}-\frac{1}{2\log b}\right) \\
    \shoveright{+\frac{1}{12}\left(\frac{\log a}{\log b}+\frac{\log a}{\log b}+\frac{\pi^2+6\gamma^2}{\log a\log b}\right)
    -\frac{\gamma}{2}\left(\frac{1}{\log a}+\frac{1}{\log b}\right)+\frac{1}{4}} \\
    \shoveright{+\sum_{n=1}^{\infty} \frac{(-1)^n}{n!(a^n-1)(b^n-1)} +\frac{1}{\log a}\sum_{\substack{n=-\infty\\ n\neq 0}}^{\infty}
    \frac{\Gamma\left(-\frac{2n\pi i}{\log a}\right) x^{2n\pi i/\log a}}{1-b^{2n\pi i/\log a}} }\\
    +\frac{1}{\log b} \sum_{\substack{n=-\infty\\ n\neq 0}}^{\infty}
    \frac{\Gamma\left(-\frac{2n\pi i}{\log b}\right) x^{2n\pi i/\log b}}{1-a^{2n\pi i/\log b}}.
\end{multline}

The code for both of the above formulae appears in \verb=omath.tex=.

Certainly, \LaTeX\ has many other specialized capabilities.  Consult one of the references for further help.

\end{document}
