% Homework 4 Solutions

\section{Overview}

In this assignment, you will use the thesis/dissertation
document class and get some experience with \LaTeX\  equations,
tables, and figures.

Feel free to use the thesis/dissertation skeleton that we
dissected in class as the basis for your homework. After
filling in your name, department, etc. you'll only need to
do the actual \LaTeX\  typesetting. This \LaTeX\  skeleton is
available via WWW at

\vspace{8pt}
\verb+http://www.math.arizona.edu/~hays/latex/diss/diss.txt+

\section{Typesetting Equations}

% Inserts \{#1} into the text
\newcommand{\bsq}[1]{\ensuremath{\backslash\mbox{\texttt{#1}}}}

% Generates the hints
\newcommand{\hint}[1]{
  \begin{description}
    \item[Hint] #1
  \end{description}
}

% Produce vertical space above and below a horizontal line
\newcommand{\vs}{\vspace{12pt}\hrule\vspace{12pt}}

The following mathematical expressions make use of just
about everything we covered in class. For the homework,
typeset at least two of the following (the easiest ones
come first):

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vs

\noindent\textbf{Equation \#1}
%
\[
  i\dot{A}_j = f(|A_j|^2)(A_{j+1}+A_{j-1})+F(|A_j|^2)A_j
\]

\hint{The dot on the first ``A'' is produced with a special accent.
Also, the vertical bars are produced with the ``$|$'' charcter.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vs

\noindent\textbf{Equation \#2}
%
\[
  S_n = \left(\begin{array}{cc}
          h_n+f_n & -\phi_n \\
          \chi_n  & h_n-f_n
        \end{array}\right)
\]

\hint{You will need to use an \texttt{array} environment with
appropriate \bsq{left} and \bsq{right} delimiters. The symbols
$\phi$ and $\chi$ are generated with \bsq{phi} and \bsq{chi}.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vs

\noindent\textbf{Equation \#3}
%
\[
  \sum_{n=0}^\infty\frac{p_n(x)}{n!}t^n =
    \frac{1}{\sqrt{1+t^2}}\,\,e^{2x\tan^{-1}t}
\]

\hint{You can use a couple of \bsq{,} commands to move
the exponential away from the fraction on the right
hand side. There is a special command to generate the
tangent.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vs

\noindent\textbf{Equation \#4}
%
\[
  \int_{-\infty}^{\infty} \!\!\!\!\mbox{sech}(\pi x)\,\, p_m(x)p_n(x)\,dx =
    (n!)^2\,\delta_{mn}
\]

\hint{\LaTeX\  does not define a \bsq{sech} command; therefore,
you'll need to use a \bsq{mbox} command to make it come out in
an upright roman font. You can use some \bsq{!}'s to push the
sech() under the integration limits and a \bsq{,} or two to
insert extra space after the sech() and before the $\delta$.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vs

% Generates a partial derivative
\newcommand{\pd}[2]{\frac{\partial {#1}}{\partial {#2}}}

\noindent\textbf{Equation \#5}
%
\[
  \left\{ B,\,C\right\} =
    -i\sum_j\left(
      \pd{B}{A_j}\pd{C}{A_j^*} - \pd{B}{A_j^*}\pd{C}{A_j}
    \right) f(|A_j|^2)
\]

\hint{The parentheses inside the sum are produced with
\bsq{left} and \bsq{right} delimiters. The $\partial$
symbol is generated with \bsq{partial}. Remember that
you'll need to use backslashes to generate the curly
braces.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vs

\noindent\textbf{Equation \#6}
%
\begin{eqnarray*}
X &=& -\frac{i}{\Delta^2}
      \left[
        2+\frac{p_{2N+1}}{2p_{2N+2}}+\sum_j\mu_j+(-1)^N
        \frac{f_n^{(N+1)}}{f_n^{(0)}}\prod_j\mu_j
      \right] \\
Y &=& \mp\frac{i(-1)^N|\phi_n^{(N)}|^2}{2\Delta^2p_{2N+2}}\prod_j\mu_j^{*}
\end{eqnarray*}

\hint{The large brackets are produced with matching delimiters.
The symbol $\mp$ is generated with \bsq{mp}.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Typesetting Tables}

For the homework, typeset at least one of the following tables:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vs

\noindent\textbf{Table \#1}
%
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c||c|c|}
\hline
\multicolumn{3}{|c|}{Conditions that Prevent Blowup} \\
\hline
%
\hline
$\nu$         & $N = N^*$         & $N > N^*$        \\
\hline
%
\hline
$1 < \nu < 2$ & $\mu > 1 + \nu/2$ & $\mu > \nu + 1/2$ \\
%
\hline
$\nu \ge 2$   & $\mu \ge \nu$     & $\mu > \nu + 1/2$ \\
\hline
\end{tabular}
\end{center}

\hint{You'll need to use \bsq{multicolumn} to generate the heading.
The other entries in the \texttt{tabular} environment will need to
be done in math mode. The symbols $\mu$ and $\nu$ are generated
with \bsq{mu} and \bsq{nu}.}

\noindent
Here's a handy hack: to produce extra
vertical space in the table, do something like the following between
the $\backslash$\texttt{begin}\{\texttt{center}\} and the
$\backslash$\texttt{begin}\{\texttt{tabular}\}:

\vspace{8pt}
\verb+     \renewcommand{\arraystretch}{1.5}+

\vspace{8pt}\noindent
This causes the height of each row to be 1.5 times greater than
normal. If you don't do this, the mathematical expressions is
the table are too close to the horizontal lines. When the
\texttt{center} environment ends, \bsq{arraystretch} is
automagically restored to its previous value.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vs

% Generates the footnote numbers in the table
\newcommand{\fn}[1]{\ensuremath{{}^{\mbox{\scriptsize\textbf{\  #1.}}}}}

\noindent
The second one is a bit trickier.

\noindent\textbf{Table \#2}
%
\begin{center}
\renewcommand{\arraystretch}{1.8}
%\renewcommand{\doublerulesep}{10pt}
%%${}^{\mbox{\scriptsize\bf 1}}$
\begin{tabular}{|l|c|c|c|c|}
\hline
\multicolumn{5}{|c|}{Periodic Table of the Conversion Polynomials} \\
\hline
%
\hline
       & From NO & From NSO & From ANO & From SO \\
%
\hline
To NO  & $\exp xt$ & $\exp x(e^t-1)$ &
         $\frac{1}{1-t}\exp\frac{xt}{1-t}$ \fn{1} &
         $\frac{2}{2-t}\exp\frac{2xt}{2-t}$ \fn{2} \\
%
\hline
To NSO & $(1+t)^x$ & $\exp xt$ &
         $\frac{1}{1-t}\left(\frac{1}{1-t}\right)^x$ &
         $\frac{2}{2-t}\left(\frac{2+t}{2-t}\right)^x$ \\
%
\hline
To ANO & $\frac{1}{1+t}\exp\frac{xt}{1+t}$ \fn{3} &
         $e^t\exp x(1-e^{-t})$ & $\exp xt$ &
         $\frac{2-t}{2+t}\exp\frac{2xt}{2+t}$ \fn{4} \\
%
\hline
To SO  & $\frac{2}{2+t}\exp\frac{2xt}{2+t}$ \fn{5} &
         $\frac{2}{e^t+1}\exp\left(2x\tanh\frac{t}{2}\right)$ &
         $\frac{1}{1-t}\exp\frac{2xt}{2-t}$ & $\exp xt$ \\
%
\hline
\end{tabular}
\end{center}
\begin{enumerate}
\item $n!L_n(-x)$
\item $\frac{n!}{2^n}L_n(-2x)$
\item $(-1)^nn!L_n(x)$
\item $\frac{(-1)^n(n-1)!}{2^{n-1}}[nL_n(2x)+L_{n-1}(2x)]$
\item $\left(-\frac{1}{2}\right)^nL_n(2x)$
\end{enumerate}

\hint{This one is done in a \texttt{tabular} environment. The
list underneath is an \texttt{enumerate} environment. One
annoying thing about \LaTeX\  is that you cannot put footnotes
inside a table (see below).}

\noindent
As above, you'll need to change \bsq{arraystretch} to something
like 1.8 so that the vertical spacing looks good. Generating
the ``footnote'' numbers in the table is a pain. I defined the
following macro to do it:

\vspace{8pt}
\verb+\newcommand{\fn}[1]{\ensuremath{{}^{\mbox{\scriptsize\textbf{\  #1.}}}}}+

\vspace{8pt}\noindent
In the table itself, you'd say \verb+\fn{1}+ to generate the ``1.'', etc.
The only new thing here is \verb+\scriptsize+ which causes the font size
to decrease to the size normally used for sub- and superscripts. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Including a Figure}

Finally, use the \texttt{graphics} package to include the
PostScript figure available from
\begin{verbatim}
         http://www.math.arizona.edu/~hays/latex/fig.ps
\end{verbatim}
into your \LaTeX\  file.

\begin{figure}
\begin{center}
\resizebox{3in}{!}{
  \rotatebox{270}{
    \includegraphics{fig.ps}
  }
}
\end{center}
\caption{Hahaha}
\end{figure}

You will need to rotate and rescale the figure so that it
appears right side up and fits on the page (I set the width
to four inches to achieve this).

\section{Submitting Your Completed Assignment}

Email your \LaTeX\  files to Prof.~Restrepo at
\begin{verbatim}
        restrepo@math.arizona.edu
\end{verbatim}
You do not need to include the PostScript figure, or the
document class source code itself --- you only need to
include the \LaTeX\  files that you create.

% EOF homework.tex