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-%%% ====================================================================
-%%% @LaTeX-file{
-%%% filename = "amsldoc.tex",
-%%% version = "2.09",
-%%% date = "2004/04/06",
-%%% time = "15:13:33 EDT",
-%%% checksum = "39446 2620 11111 93831",
-%%% author = "American Mathematical Society",
-%%% address = "American Mathematical Society,
-%%% Technical Support,
-%%% Electronic Products and Services,
-%%% P. O. Box 6248,
-%%% Providence, RI 02940,
-%%% USA",
-%%% telephone = "401-455-4080 or (in the USA and Canada)
-%%% 800-321-4AMS (321-4267)",
-%%% FAX = "401-331-3842",
-%%% email = "tech-support@ams.org (Internet)",
-%%% copyright = "Copyright 1999 American Mathematical Society,
-%%% all rights reserved. Copying of this file is
-%%% authorized only if either:
-%%% (1) you make absolutely no changes to your copy,
-%%% including name; OR
-%%% (2) if you do make changes, you first rename it
-%%% to some other name.",
-%%% supported = "yes",
-%%% keywords = "latex, amsmath",
-%%% abstract = "This file is the source file for the user's
-%%% guide of the \LaTeX{} package amsmath .",
-%%% docstring = "The checksum field above contains a CRC-16
-%%% checksum as the first value, followed by the
-%%% equivalent of the standard UNIX wc (word
-%%% count) utility output of lines, words, and
-%%% characters. This is produced by Robert
-%%% Solovay's checksum utility.",
-%%% }
-%%% ====================================================================
-\documentclass[leqno,titlepage,openany]{amsldoc}[1999/12/13]
-
-\ifx\UndEfiNed\url
- \ClassError{amsldoc}{%
- This version of amsldoc.tex must be processed\MessageBreak
- with a newer version of amsldoc.cls (2.02 or later)}{}
-\fi
-
-\title{User's Guide for the \pkg{amsmath} Package (Version~2.0)}
-\author{American Mathematical Society}
-\date{1999-12-13 (revised 2002-02-25)}
-
-% Use the amsmath package and amscd package in order to print
-% examples.
-\usepackage{amsmath}
-\usepackage{amscd}
-
-\makeindex % generate index data
-\providecommand{\see}[2]{\textit{see} #1}
-
-% The amsldoc class includes a number of features useful for
-% documentation about TeX, including:
-%
-% ---Commands \tex/, \amstex/, \latex/, ... for uniform treatment
-% of the various logos and easy handling of following spaces.
-%
-% ---Commands for printing various common elements: \cn for command
-% names, \fn for file names (including font-file names), \env for
-% environments, \pkg and \cls for packages and classes, etc.
-
-% Many of the command names used here are rather long and will
-% contribute to poor linebreaking if we follow the \latex/ practice
-% of not hyphenating anything set in tt font; instead we selectively
-% allow some hyphenation.
-\allowtthyphens % defined in amsldoc.cls
-
-\hyphenation{ac-cent-ed-sym-bol add-to-counter add-to-length align-at
- aligned-at allow-dis-play-breaks ams-art ams-cd ams-la-tex amsl-doc
- ams-symb ams-tex ams-text ams-xtra bmatrix bold-sym-bol cen-ter-tags
- eqn-ar-ray idots-int int-lim-its latex med-space neg-med-space
- neg-thick-space neg-thin-space no-int-lim-its no-name-lim-its
- over-left-arrow over-left-right-arrow over-right-arrow pmatrix
- qed-sym-bol set-length side-set small-er tbinom the-equa-tion
- thick-space thin-space un-der-left-arrow un-der-left-right-arrow
- un-der-right-arrow use-pack-age var-inj-lim var-proj-lim vmatrix
- xalign-at xx-align-at}
-
-% Prepare for illustrating the \vec example
-\newcommand{\vect}[1]{\mathbf{#1}}
-
-\newcommand{\booktitle}[1]{\textit{#1}}
-\newcommand{\journalname}[1]{\textit{#1}}
-\newcommand{\seriesname}[1]{\textit{#1}}
-
-% Command to insert and index a particular phrase. Doesn't work for
-% certain kinds of special characters in the argument.
-\newcommand{\ii}[1]{#1\index{#1}}
-
-\newcommand{\vstrut}[1]{\vrule width0pt height#1\relax}
-
-% An environment for presenting comprehensive address information:
-\newenvironment{infoaddress}{%
- \par\topsep\medskipamount
- \trivlist\centering
- \item[]%
- \begin{minipage}{.7\columnwidth}%
- \raggedright
-}{%
- \end{minipage}%
- \endtrivlist
-}
-
-\newenvironment{eqxample}{%
- \par\addvspace\medskipamount
- \noindent\begin{minipage}{.5\columnwidth}%
- \def\producing{\end{minipage}\begin{minipage}{.5\columnwidth}%
- \hbox\bgroup\kern-.2pt\vrule width.2pt%
- \vbox\bgroup\parindent0pt\relax
-% The 3pt is to cancel the -\lineskip from \displ@y
- \abovedisplayskip3pt \abovedisplayshortskip\abovedisplayskip
- \belowdisplayskip0pt \belowdisplayshortskip\belowdisplayskip
- \noindent}
-}{%
- \par
-% Ensure that a lonely \[\] structure doesn't take up width less than
-% \hsize.
- \hrule height0pt width\hsize
- \egroup\vrule width.2pt\kern-.2pt\egroup
- \end{minipage}%
- \par\addvspace\medskipamount
-}
-
-% The chapters are so short, perhaps we shouldn't call them by the
-% name `Chapter'. We make \chaptername read an argument in order to
-% remove a following \space or "{} " (both possibilities are present
-% in book.cls).
-
-\renewcommand{\chaptername}[1]{}
-\newcommand{\chapnum}[1]{\mdash #1\mdash }
-\makeatletter
-\def\@makechapterhead#1{%
- \vspace{1.5\baselineskip}%
- {\parindent \z@ \raggedright \reset@font
- \ifnum \c@secnumdepth >\m@ne
- \large\bfseries \chapnum{\thechapter}%
- \par\nobreak
- \vskip.5\baselineskip\relax
- \fi
- #1\par\nobreak
- \vskip\baselineskip
- }}
-\makeatother
-
-% A command for ragged-right parbox in a tabular.
-\newcommand{\rp}{\let\PBS\\\raggedright\let\\\PBS}
-
-% Non-indexed file name
-\newcommand{\nfn}[1]{\texttt{#1}}
-
-% For the examples in the math spacing table.
-%%\newcommand{\lspx}{\mbox{\rule{5pt}{.6pt}\rule{.6pt}{6pt}}}
-%%\newcommand{\rspx}{\mbox{\rule[-1pt]{.6pt}{7pt}%
-%% \rule[-1pt]{5pt}{.6pt}}}
-\newcommand{\lspx}{\mathord{\Rightarrow\mkern-1mu}}
-\newcommand{\rspx}{\mathord{\mkern-1mu\Leftarrow}}
-\newcommand{\spx}[1]{$\lspx #1\rspx$}
-
-% For a list of characters representing document input.
-\newcommand{\clist}[1]{%
- \mbox{\ntt\spaceskip.2em plus.1em \xspaceskip\spaceskip#1}}
-
-% Fix weird \latex/ definition of rightmark.
-\makeatletter
-\def\rightmark{\expandafter\@rightmark\botmark{}{}}
-% Also turn off section marks.
-\let\sectionmark\@gobble
-\renewcommand{\chaptermark}[1]{%
- \uppercase{\markboth{\rhcn#1}{\rhcn#1}}}
-\newcommand{\rhcn}{\thechapter. }
-\makeatother
-
-% Include down to \section but not \subsection, in toc:
-\setcounter{tocdepth}{1}
-
-\DeclareMathOperator{\ix}{ix}
-\DeclareMathOperator{\nul}{nul}
-\DeclareMathOperator{\End}{End}
-\DeclareMathOperator{\xxx}{xxx}
-
-\begin{document}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\frontmatter
-
-\maketitle
-\pagestyle{headings}
-\tableofcontents
-\cleardoublepage % for better page number placement
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mainmatter
-\chapter{Introduction}
-
-The \pkg{amsmath} package is a \LaTeX{} package that provides
-miscellaneous enhancements for improving the information structure and
-printed output of documents that contain mathematical formulas. Readers
-unfamiliar with \LaTeX{} should refer to \cite{lamport}. If you have an
-up-to-date version of \LaTeX{}, the \pkg{amsmath} package is normally
-provided along with it. Upgrading when a newer version of the
-\pkg{amsmath} package is released can be done via
-\url{http://www.ams.org/tex/amsmath.html} or
-\url{ftp://ftp.ams.org/pub/tex/}.
-
-This documentation describes the features of the \pkg{amsmath} package
-and discusses how they are intended to be used. It also covers some
-ancillary packages:
-\begin{ctab}{ll}
-\pkg{amsbsy}& \pkg{amstext}\\
-\pkg{amscd}& \pkg{amsxtra}\\
-\pkg{amsopn}
-\end{ctab}
-These all have something to do with the contents of math formulas. For
-information on extra math symbols and math fonts, see \cite{amsfonts}
-and \url{http://www.ams.org/tex/amsfonts.html}. For documentation of the
-\pkg{amsthm} package or AMS document classes (\cls{amsart},
-\cls{amsbook}, etc.\@) see \cite{amsthdoc} or \cite{instr-l} and
-\url{http://www.ams.org/tex/author-info.html}.
-
-If you are a long-time \latex/ user and have lots of mathematics in what
-you write, then you may recognize solutions for some familiar problems
-in this list of \pkg{amsmath} features:
-\begin{itemize}
-
-\item A convenient way to define new `operator name' commands analogous
-to \cn{sin} and \cn{lim}, including proper side spacing and automatic
-selection of the correct font style and size (even when used in
-sub- or superscripts).
-
-\item Multiple substitutes for the \env{eqnarray} environment to make
-various kinds of equation arrangements easier to write.
-
-\item Equation numbers automatically adjust up or down to avoid
-overprinting on the equation contents (unlike \env{eqnarray}).
-
-\item Spacing around equals signs matches the normal spacing in the
-\env{equation} environment (unlike \env{eqnarray}).
-
-\item A way to produce multiline subscripts as are often used with
-summation or product symbols.
-
-\item An easy way to substitute a variant equation number for a given
-equation instead of the automatically supplied number.
-
-\item An easy way to produce subordinate equation numbers of the form
-(1.3a) (1.3b) (1.3c) for selected groups of equations.
-
-\end{itemize}
-
-The \pkg{amsmath} package is distributed together with some small
-auxiliary packages:
-\begin{description}
-\item[\pkg{amsmath}] Primary package, provides various features for
- displayed equations and other mathematical constructs.
-
-\item[\pkg{amstext}] Provides a \cn{text} command for
- typesetting a fragment of text inside a display.
-
-\item[\pkg{amsopn}] Provides \cn{DeclareMathOperator} for defining new
- `operator names' like \cn{sin} and \cn{lim}.
-
-\item[\pkg{amsbsy}] For backward compatibility this package continues
-to exist but use of the newer \pkg{bm} package that comes with \LaTeX{}
-is recommended instead.
-
-\item[\pkg{amscd}] Provides a \env{CD} environment for simple
- commutative diagrams (no support for diagonal arrows).
-
-\item[\pkg{amsxtra}] Provides certain odds and ends such as
- \cn{fracwithdelims} and \cn{accentedsymbol}, for compatibility with
- documents created using version 1.1.
-
-\end{description}
-
-The \pkg{amsmath} package incorporates \pkg{amstext}, \pkg{amsopn}, and
-\pkg{amsbsy}. The features of \pkg{amscd} and \pkg{amsxtra}, however,
-are available only by invoking those packages separately.
-
-\chapter{Options for the \pkg{amsmath} package}\label{options}
-
-The \pkg{amsmath} package has the following options:
-\begin{description}
-
-\item[\opt{centertags}] (default) For a split equation, place equation
-numbers\index{equation numbers!vertical placement} vertically centered
-on the total height of the equation.
-
-\item[\opt{tbtags}] `Top-or-bottom tags': For a split equation, place
-equation numbers\index{equation numbers!vertical placement} level with
-the last (resp.\@ first) line, if numbers are on the right (resp.\@
-left).
-
-\item[\opt{sumlimits}] (default) Place the subscripts and
-superscripts\index{subscripts and superscripts!placement}\relax
-\index{limits|see{subscripts and superscripts}} of summation symbols
-above and below, in displayed equations. This option also affects other
-symbols of the same type\mdash $\prod$, $\coprod$, $\bigotimes$,
-$\bigoplus$, and so forth\mdash but excluding integrals (see below).
-
-\item[\opt{nosumlimits}] Always place the subscripts and superscripts of
-summation-type symbols to the side, even in displayed equations.
-
-\item[\opt{intlimits}] Like \opt{sumlimits}, but for
-integral\index{integrals!placement of limits} symbols.
-
-\item[\opt{nointlimits}] (default) Opposite of \opt{intlimits}.
-
-\item[\opt{namelimits}] (default) Like \opt{sumlimits}, but for certain
-`operator names' such as $\det$, $\inf$, $\lim$, $\max$, $\min$, that
-traditionally have subscripts \index{subscripts and
-superscripts!placement} placed underneath when they occur in a displayed
-equation.
-
-\item[\opt{nonamelimits}] Opposite of \opt{namelimits}.
-
-\end{description}
-
-To use one of these package options, put the option name in the optional
-argument of the \cn{usepackage} command\mdash e.g.,
-\verb"\usepackage[intlimits]{amsmath}".
-
-The \pkg{amsmath} package also recognizes the following options which
-are normally selected (implicitly or explicitly) through the
-\cn{documentclass} command, and thus need not be repeated in the option
-list of the \cn{usepackage}|{amsmath}| statement.
-\begin{description}
-
-\item[\opt{leqno}] Place equation numbers on the left.\index{equation
-numbers!left or right placement}
-
-\item[\opt{reqno}] Place equation numbers on the right.
-
-\item[\opt{fleqn}] Position equations at a fixed indent from the left
-margin rather than centered in the text column.\index{displayed
-equations!centering}
-
-\end{description}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\chapter{Displayed equations}
-
-\section{Introduction}
-The \pkg{amsmath} package provides a number of additional displayed
-equation structures\index{displayed
- equations}\index{equations|see{displayed equations}} beyond the ones
-provided in basic \latex/. The augmented set includes:
-\begin{verbatim}
- equation equation* align align*
- gather gather* flalign flalign*
- multline multline* alignat alignat*
- split
-\end{verbatim}
-(Although the standard \env{eqnarray} environment remains available,
-it is better to use \env{align} or \env{equation}+\env{split} instead.)
-
-Except for \env{split}, each environment has both starred and unstarred
-forms, where the unstarred forms have automatic numbering using
-\latex/'s \env{equation} counter. You can suppress the number on any
-particular line by putting \cn{notag} before the \cn{\\}; you can also
-override\index{equation numbers!overriding} it with a tag of your own
-using \cn{tag}|{|\<label>|}|, where \<label> means arbitrary text such
-as |$*$| or |ii| used to \qq{number} the equation. There is also a
-\cn{tag*} command that causes the text you supply to be typeset
-literally, without adding parentheses around it. \cn{tag} and \cn{tag*}
-can also be used within the unnumbered versions of all the \pkg{amsmath}
-alignment structures. Some examples of the use of \cn{tag} may be found
-in the sample files \fn{testmath.tex} and \fn{subeqn.tex} provided with
-the \pkg{amsmath} package.
-
-The \env{split} environment is a special subordinate form that is used
-only \emph{inside} one of the others. It cannot be used inside
-\env{multline}, however.
-
-In the structures that do alignment (\env{split}, \env{align} and
-variants), relation symbols have an \verb'&' before them but not
-after\mdash unlike \env{eqnarray}. Putting the \verb'&' after the
-relation symbol will interfere with the normal spacing; it has to go
-before.
-
-\begin{table}[p]
-\caption[]{Comparison of displayed equation environments (vertical lines
-indicating nominal margins)}\label{displays}
-\renewcommand{\theequation}{\arabic{equation}}
-\begin{eqxample}
-\begin{verbatim}
-\begin{equation*}
-a=b
-\end{equation*}
-\end{verbatim}
-\producing
-\begin{equation*}
-a=b
-\end{equation*}
-\end{eqxample}
-
-\begin{eqxample}
-\begin{verbatim}
-\begin{equation}
-a=b
-\end{equation}
-\end{verbatim}
-\producing
-\begin{equation}
-a=b
-\end{equation}
-\end{eqxample}
-
-\begin{eqxample}
-\begin{verbatim}
-\begin{equation}\label{xx}
-\begin{split}
-a& =b+c-d\\
- & \quad +e-f\\
- & =g+h\\
- & =i
-\end{split}
-\end{equation}
-\end{verbatim}
-\producing
-\begin{equation}\label{xx}
-\begin{split}
-a& =b+c-d\\
- & \quad +e-f\\
- & =g+h\\
- & =i
-\end{split}
-\end{equation}
-\end{eqxample}
-
-\begin{eqxample}
-\begin{verbatim}
-\begin{multline}
-a+b+c+d+e+f\\
-+i+j+k+l+m+n
-\end{multline}
-\end{verbatim}
-\producing
-\begin{multline}
-a+b+c+d+e+f\\
-+i+j+k+l+m+n
-\end{multline}
-\end{eqxample}
-
-\begin{eqxample}
-\begin{verbatim}
-\begin{gather}
-a_1=b_1+c_1\\
-a_2=b_2+c_2-d_2+e_2
-\end{gather}
-\end{verbatim}
-\producing
-\begin{gather}
-a_1=b_1+c_1\\
-a_2=b_2+c_2-d_2+e_2
-\end{gather}
-\end{eqxample}
-
-\begin{eqxample}
-\begin{verbatim}
-\begin{align}
-a_1& =b_1+c_1\\
-a_2& =b_2+c_2-d_2+e_2
-\end{align}
-\end{verbatim}
-\producing
-\begin{align}
-a_1& =b_1+c_1\\
-a_2& =b_2+c_2-d_2+e_2
-\end{align}
-\end{eqxample}
-
-\begin{eqxample}
-\begin{verbatim}
-\begin{align}
-a_{11}& =b_{11}&
- a_{12}& =b_{12}\\
-a_{21}& =b_{21}&
- a_{22}& =b_{22}+c_{22}
-\end{align}
-\end{verbatim}
-\producing
-\begin{align}
-a_{11}& =b_{11}&
- a_{12}& =b_{12}\\
-a_{21}& =b_{21}&
- a_{22}& =b_{22}+c_{22}
-\end{align}
-\end{eqxample}
-
-\begin{eqxample}
-\begin{verbatim}
-\begin{flalign*}
-a_{11}& =b_{11}&
- a_{12}& =b_{12}\\
-a_{21}& =b_{21}&
- a_{22}& =b_{22}+c_{22}
-\end{flalign*}
-\end{verbatim}
-\producing
-\begin{flalign*}
-a_{11}& =b_{11}&
- a_{12}& =b_{12}\\
-a_{21}& =b_{21}&
- a_{22}& =b_{22}+c_{22}
-\end{flalign*}
-\end{eqxample}
-\end{table}
-
-\section{Single equations}
-
-The \env{equation} environment is for a single equation with an
-automatically generated number. The \env{equation*} environment is the
-same except for omitting the number.%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\footnote{Basic \latex/ doesn't provide an \env{equation*} environment,
-but rather a functionally equivalent environment named
-\env{displaymath}.}
-
-\section{Split equations without alignment}
-
-The \env{multline} environment is a variation of the \env{equation}
-environment used for equations that don't fit on a single line. The
-first line of a \env{multline} will be at the left margin and the last
-line at the right margin, except for an indention on both sides in the
-amount of \cn{multlinegap}. Any additional lines in between will be
-centered independently within the display width (unless the \opt{fleqn}
-option is in effect).
-
-Like \env{equation}, \env{multline} has only a single equation number
-(thus, none of the individual lines should be marked with \cn{notag}).
-The equation number is placed on the last line (\opt{reqno} option) or
-first line (\opt{leqno} option); vertical centering as for \env{split}
-is not supported by \env{multline}.
-
-It's possible to force one of the middle lines to the left or right with
-commands \cn{shoveleft}, \cn{shoveright}. These commands take the entire
-line as an argument, up to but not including the final \cn{\\}; for
-example
-\begin{multline}
-\framebox[.65\columnwidth]{A}\\
-\framebox[.5\columnwidth]{B}\\
-\shoveright{\framebox[.55\columnwidth]{C}}\\
-\framebox[.65\columnwidth]{D}
-\end{multline}
-\begin{verbatim}
-\begin{multline}
-\framebox[.65\columnwidth]{A}\\
-\framebox[.5\columnwidth]{B}\\
-\shoveright{\framebox[.55\columnwidth]{C}}\\
-\framebox[.65\columnwidth]{D}
-\end{multline}
-\end{verbatim}
-
-The value of \cn{multlinegap} can be changed with the usual \latex/
-commands \cn{setlength} or \cn{addtolength}.
-
-\section{Split equations with alignment}
-
-Like \env{multline}, the \env{split} environment is for \emph{single}
-equations that are too long to fit on one line and hence must be split
-into multiple lines. Unlike \env{multline}, however, the \env{split}
-environment provides for alignment among the split lines, using |&| to
-mark alignment points. Unlike the other \pkg{amsmath} equation
-structures, the \env{split} environment provides no numbering, because
-it is intended to be used \emph{only inside some other displayed
- equation structure}, usually an \env{equation}, \env{align}, or
-\env{gather} environment, which provides the numbering. For example:
-\begin{equation}\label{e:barwq}\begin{split}
-H_c&=\frac{1}{2n} \sum^n_{l=0}(-1)^{l}(n-{l})^{p-2}
-\sum_{l _1+\dots+ l _p=l}\prod^p_{i=1} \binom{n_i}{l _i}\\
-&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot
-\Bigl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\Bigr].
-\kern-2em % adjust equation body to the right [mjd,13-Nov-1994]
-\end{split}\end{equation}
-
-\begin{verbatim}
-\begin{equation}\label{e:barwq}\begin{split}
-H_c&=\frac{1}{2n} \sum^n_{l=0}(-1)^{l}(n-{l})^{p-2}
-\sum_{l _1+\dots+ l _p=l}\prod^p_{i=1} \binom{n_i}{l _i}\\
-&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot
-\Bigl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\Bigr].
-\end{split}\end{equation}
-\end{verbatim}
-
-The \env{split} structure should constitute the entire body of the
-enclosing structure, apart from commands like \cn{label} that produce no
-visible material.
-
-\section{Equation groups without alignment}
-
-The \env{gather} environment is used for a group of consecutive
-equations when there is no alignment desired among them; each one is
-centered separately within the text width (see Table~\ref{displays}).
-Equations inside \env{gather} are separated by a \cn{\\} command.
-Any equation in a \env{gather} may consist of a \verb'\begin{split}'
- \dots\ \verb'\end{split}' structure\mdash for example:
-\begin{verbatim}
-\begin{gather}
- first equation\\
- \begin{split}
- second & equation\\
- & on two lines
- \end{split}
- \\
- third equation
-\end{gather}
-\end{verbatim}
-
-\section{Equation groups with mutual alignment}
-
-The \env{align} environment is used for two or more equations when
-vertical alignment is desired; usually binary relations such as equal
-signs are aligned (see Table~\ref{displays}).
-
-To have several equation columns side-by-side, use extra ampersands
-to separate the columns:
-\begin{align}
-x&=y & X&=Y & a&=b+c\\
-x'&=y' & X'&=Y' & a'&=b\\
-x+x'&=y+y' & X+X'&=Y+Y' & a'b&=c'b
-\end{align}
-%
-\begin{verbatim}
-\begin{align}
-x&=y & X&=Y & a&=b+c\\
-x'&=y' & X'&=Y' & a'&=b\\
-x+x'&=y+y' & X+X'&=Y+Y' & a'b&=c'b
-\end{align}
-\end{verbatim}
-Line-by-line annotations on an equation can be done by judicious
-application of \cn{text} inside an \env{align} environment:
-\begin{align}
-x& = y_1-y_2+y_3-y_5+y_8-\dots
- && \text{by \eqref{eq:C}}\\
- & = y'\circ y^* && \text{by \eqref{eq:D}}\\
- & = y(0) y' && \text {by Axiom 1.}
-\end{align}
-%
-\begin{verbatim}
-\begin{align}
-x& = y_1-y_2+y_3-y_5+y_8-\dots
- && \text{by \eqref{eq:C}}\\
- & = y'\circ y^* && \text{by \eqref{eq:D}}\\
- & = y(0) y' && \text {by Axiom 1.}
-\end{align}
-\end{verbatim}
-A variant environment \env{alignat} allows the horizontal space between
-equations to be explicitly specified. This environment takes one argument,
-the number of \qq{equation columns}: count the maximum number of \verb'&'s
-in any row, add 1 and divide by 2.
-\begin{alignat}{2}
-x& = y_1-y_2+y_3-y_5+y_8-\dots
- &\quad& \text{by \eqref{eq:C}}\\
- & = y'\circ y^* && \text{by \eqref{eq:D}}\\
- & = y(0) y' && \text {by Axiom 1.}
-\end{alignat}
-%
-\begin{verbatim}
-\begin{alignat}{2}
-x& = y_1-y_2+y_3-y_5+y_8-\dots
- &\quad& \text{by \eqref{eq:C}}\\
- & = y'\circ y^* && \text{by \eqref{eq:D}}\\
- & = y(0) y' && \text {by Axiom 1.}
-\end{alignat}
-\end{verbatim}
-
-\section{Alignment building blocks}
-
-Like \env{equation}, the multi-equation environments \env{gather},
-\env{align}, and \env{alignat} are designed to produce a structure
-whose width is the full line width. This means, for example, that one
-cannot readily add parentheses around the entire structure. But variants
-\env{gathered}, \env{aligned}, and \env{alignedat} are provided whose
-total width is the actual width of the contents; thus they can be used
-as a component in a containing expression. E.g.,
-\begin{equation*}
-\left.\begin{aligned}
- B'&=-\partial\times E,\\
- E'&=\partial\times B - 4\pi j,
-\end{aligned}
-\right\}
-\qquad \text{Maxwell's equations}
-\end{equation*}
-\begin{verbatim}
-\begin{equation*}
-\left.\begin{aligned}
- B'&=-\partial\times E,\\
- E'&=\partial\times B - 4\pi j,
-\end{aligned}
-\right\}
-\qquad \text{Maxwell's equations}
-\end{equation*}
-\end{verbatim}
-Like the \env{array} environment, these \texttt{-ed} variants also take
-an optional \verb'[t]' or \verb'[b]' argument to specify vertical
-positioning.
-
-\qq{Cases} constructions like the following are common in
-mathematics:
-\begin{equation}\label{eq:C}
-P_{r-j}=
- \begin{cases}
- 0& \text{if $r-j$ is odd},\\
- r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
- \end{cases}
-\end{equation}
-and in the \pkg{amsmath} package there is a \env{cases} environment to
-make them easy to write:
-\begin{verbatim}
-P_{r-j}=\begin{cases}
- 0& \text{if $r-j$ is odd},\\
- r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
- \end{cases}
-\end{verbatim}
-Notice the use of \cn{text} (cf.~\secref{text}) and the nested
-math formulas.
-
-\section{Adjusting tag placement}
-
-Placing equation numbers can be a rather complex problem in multiline
-displays. The environments of the \pkg{amsmath} package try hard to
-avoid overprinting an equation number on the equation contents, if
-necessary moving the number down or up to a separate line. Difficulties
-in accurately calculating the profile of an equation can occasionally
-result in number movement that doesn't look right. There is a
-\cn{raisetag} command provided to adjust the vertical position of the
-current equation number, if it has been shifted away from its normal
-position. To move a particular number up by six points, write
-|\raisetag{6pt}|. This kind of adjustment is fine tuning like line
-breaks and page breaks, and should therefore be left undone until your
-document is nearly finalized, or you may end up redoing the fine tuning
-several times to keep up with changing document contents.
-
-\section{Vertical spacing and page breaks in multiline displays}
-
-You can use the \cn{\\}|[|\<dimension>|]| command to get extra vertical
-space between lines in all the \pkg{amsmath} displayed equation
-environments, as is usual in \latex/. When the \pkg{amsmath} package is
-in use \ii{page breaks} between equation lines are normally disallowed;
-the philosophy is that page breaks in such material should receive
-individual attention from the author. To get an individual page break
-inside a particular displayed equation, a \cn{displaybreak} command is
-provided. \cn{displaybreak} is best placed immediately before the
-\cn{\\} where it is to take effect. Like \latex/'s \cn{pagebreak},
-\cn{displaybreak} takes an optional argument between 0 and 4 denoting
-the desirability of the pagebreak. |\displaybreak[0]| means \qq{it is
- permissible to break here} without encouraging a break;
-\cn{displaybreak} with no optional argument is the same as
-|\displaybreak[4]| and forces a break.
-
-If you prefer a strategy of letting page breaks fall where they may,
-even in the middle of a multi-line equation, then you might put
-\cn{allowdisplaybreaks}\texttt{[1]} in the preamble of your document. An
-optional argument 1\ndash 4 can be used for finer control: |[1]| means
-allow page breaks, but avoid them as much as possible; values of 2,3,4
-mean increasing permissiveness. When display breaks are enabled with
-\cn{allowdisplaybreaks}, the \cn{\\*} command can be used to prohibit a
-pagebreak after a given line, as usual.
-
-\begin{bfseries}
-Note: Certain equation environments wrap their contents in an
-unbreakable box, with the consequence that neither \cn{displaybreak} nor
-\cn{allowdisplaybreaks} will have any effect on them. These include
-\env{split}, \env{aligned}, \env{gathered}, and \env{alignedat}.
-\end{bfseries}
-
-\section{Interrupting a display}
-
-The command \cn{intertext} is used for a short interjection of one or
-two lines of text\index{text fragments inside math} in the middle of a
-multiple-line display structure (see also the \cn{text} command in
-\secref{text}). Its salient feature is preservation of the alignment,
-which would not happen if you simply ended the display and then started
-it up again afterwards. \cn{intertext} may only appear right after a
-\cn{\\} or \cn{\\*} command. Notice the position of the word \qq{and} in
-this example.
-\begin{align}
-A_1&=N_0(\lambda;\Omega')-\phi(\lambda;\Omega'),\\
-A_2&=\phi(\lambda;\Omega')-\phi(\lambda;\Omega),\\
-\intertext{and}
-A_3&=\mathcal{N}(\lambda;\omega).
-\end{align}
-\begin{verbatim}
-\begin{align}
-A_1&=N_0(\lambda;\Omega')-\phi(\lambda;\Omega'),\\
-A_2&=\phi(\lambda;\Omega')-\phi(\lambda;\Omega),\\
-\intertext{and}
-A_3&=\mathcal{N}(\lambda;\omega).
-\end{align}
-\end{verbatim}
-
-\section{Equation numbering}
-
-\subsection{Numbering hierarchy}
-In \latex/ if you wanted to have equations numbered within
-sections\mdash that is, have
-equation numbers (1.1), (1.2), \dots, (2.1), (2.2),
-\dots, in sections 1, 2, and so forth\mdash you could redefine
-\cn{theequation} as suggested in the \latex/ manual \cite[\S6.3, \S
-C.8.4]{lamport}:
-\begin{verbatim}
-\renewcommand{\theequation}{\thesection.\arabic{equation}}
-\end{verbatim}
-
-This works pretty well, except that the equation counter won't be reset
-to zero at the beginning of a new section or chapter, unless you do it
-yourself using \cn{setcounter}. To make this a little more convenient,
-the \pkg{amsmath} package provides a command\index{equation
-numbers!hierarchy} \cn{numberwithin}. To have equation numbering tied to
-section numbering, with automatic reset of the equation counter, write
-\begin{verbatim}
-\numberwithin{equation}{section}
-\end{verbatim}
-As its name implies, the \cn{numberwithin} command can be applied to
-any counter, not just the \texttt{equation} counter.
-
-\subsection{Cross references to equation numbers}
-
-To make cross-references to equations easier, an \cn{eqref}
-command\index{equation numbers!cross-references} is provided. This
-automatically supplies the parentheses around the equation number. I.e.,
-if \verb'\ref{abc}' produces 3.2 then \verb'\eqref{abc}' produces
-(3.2).
-
-\subsection{Subordinate numbering sequences}
-
-The \pkg{amsmath} package provides also a \env{subequations}
-environment\index{equation numbers!subordinate numbering} to make it
-easy to number equations in a particular group with a subordinate
-numbering scheme. For example
-\begin{verbatim}
-\begin{subequations}
-...
-\end{subequations}
-\end{verbatim}
-causes all numbered equations within that part of the document to be
-numbered (4.9a) (4.9b) (4.9c) \dots, if the preceding numbered
-equation was (4.8). A \cn{label} command immediately after
-\verb/\begin{subequations}/ will produce a \cn{ref} of the parent
-number 4.9, not 4.9a. The counters used by the subequations
-environment are \verb/parentequation/ and \verb/equation/ and
-\cn{addtocounter}, \cn{setcounter}, \cn{value}, etc., can be applied
-as usual to those counter names. To get anything other than lowercase
-letters for the subordinate numbers, use standard \latex/ methods for
-changing numbering style \cite[\S6.3, \S C.8.4]{lamport}. For example,
-redefining \cn{theequation} as follows will produce roman numerals.
-\begin{verbatim}
-\begin{subequations}
-\renewcommand{\theequation}{\theparentequation \roman{equation}}
-...
-\end{verbatim}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\chapter{Miscellaneous mathematical features}
-
-\section{Matrices}\label{ss:matrix}
-
-The \pkg{amsmath} package provides some environments for
-matrices\index{matrices} beyond the basic \env{array} environment of
-\latex/. The \env{pmatrix}, \env{bmatrix}, \env{Bmatrix}, \env{vmatrix}
-and \env{Vmatrix} have (respectively) $(\,)$, $[\,]$, $\lbrace\,\rbrace$,
-$\lvert\,\rvert$, and
-$\lVert\,\rVert$ delimiters built in. For naming consistency there is a
-\env{matrix} environment sans delimiters. This is not entirely redundant
-with the \env{array} environment; the matrix environments all use more
-economical horizontal spacing than the rather prodigal spacing of the
-\env{array} environment. Also, unlike the \env{array} environment, you
-don't have to give column specifications for any of the matrix
-environments; by default you can have up to 10 centered columns.%
-\footnote{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-More precisely: The maximum number of columns in a matrix is determined
-by the counter |MaxMatrixCols| (normal value = 10), which you can change
-if necessary using \latex/'s \cn{setcounter} or \cn{addtocounter}
-commands.%
-}\space%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-(If you need left or right alignment in a column or other special
-formats you must resort to \env{array}.)
-
-To produce a small matrix suitable for use in text, there is a
-\env{smallmatrix} environment (e.g.,
-\begin{math}
-\bigl( \begin{smallmatrix}
- a&b\\ c&d
-\end{smallmatrix} \bigr)
-\end{math})
-that comes closer to fitting within a single text line than a normal
-matrix. Delimiters must be provided; there are no |p|,|b|,|B|,|v|,|V|
-versions of \env{smallmatrix}. The above example was produced by
-\begin{verbatim}
-\bigl( \begin{smallmatrix}
- a&b\\ c&d
-\end{smallmatrix} \bigr)
-\end{verbatim}
-
-\cn{hdotsfor}|{|\<number>|}| produces a row of dots in a
-matrix\index{matrices!ellipsis dots}\index{ellipsis dots!in
-matrices}\index{dots|see{ellipsis dots}} spanning the given number of
-columns. For example,
-\begin{center}
-\begin{minipage}{.3\columnwidth}
-\noindent$\begin{matrix} a&b&c&d\\
-e&\hdotsfor{3} \end{matrix}$
-\end{minipage}%
-\qquad
-\begin{minipage}{.45\columnwidth}
-\begin{verbatim}
-\begin{matrix} a&b&c&d\\
-e&\hdotsfor{3} \end{matrix}
-\end{verbatim}
-\end{minipage}%
-\end{center}
-
-The spacing of the dots can be varied through use of a square-bracket
-option, for example, |\hdotsfor[1.5]{3}|. The number in square brackets
-will be used as a multiplier (i.e., the normal value is 1.0).
-\begin{equation}\label{eq:D}
-\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
--a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
-\hdotsfor[2]{4}\\
--a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix},
-\end{equation}
-\begin{verbatim}
-\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
--a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
-\hdotsfor[2]{4}\\
--a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}
-\end{verbatim}
-
-\section{Math spacing commands}
-
-The \pkg{amsmath} package slightly extends the set of math
-spacing\index{horizontal space!in math mode} commands, as shown below.
-Both the spelled-out and abbreviated forms of these commands are robust,
-and they can also be used outside of math.
-\begin{ctab}{lll|lll}
-Abbrev.& Spelled out& Example & Abbrev.& Spelled out& Example\\
-\hline
-\vstrut{2.5ex}
-& no space& \spx{}& & no space & \spx{}\\
-\cn{\,}& \cn{thinspace}& \spx{\,}&
- \cnbang& \cn{negthinspace}& \spx{\!}\\
-\cn{\:}& \cn{medspace}& \spx{\:}&
- & \cn{negmedspace}& \spx{\negmedspace}\\
-\cn{\;}& \cn{thickspace}& \spx{\;}&
- & \cn{negthickspace}& \spx{\negthickspace}\\
-& \cn{quad}& \spx{\quad}\\
-& \cn{qquad}& \spx{\qquad}
-\end{ctab}
-For the greatest possible control over math spacing, use \cn{mspace}
-and `math units'. One math unit, or |mu|, is equal to 1/18 em. Thus to
-get a negative \cn{quad} you could write |\mspace{-18.0mu}|.
-
-\section{Dots}
-
-For preferred placement of ellipsis dots (raised or on-line) in various
-contexts there is no general consensus. It may therefore be considered a
-matter of taste. By using the semantically oriented commands
-\begin{itemize}
-\item \cn{dotsc} for \qq{dots with commas}
-\item \cn{dotsb} for \qq{dots with binary operators/relations}
-\item \cn{dotsm} for \qq{multiplication dots}
-\item \cn{dotsi} for \qq{dots with integrals}
-\item \cn{dotso} for \qq{other dots} (none of the above)
-\end{itemize}
-instead of \cn{ldots} and \cn{cdots}, you make it possible for your
-document to be adapted to different conventions on the fly, in case (for
-example) you have to submit it to a publisher who insists on following
-house tradition in this respect. The default treatment for the various
-kinds follows American Mathematical Society conventions:
-\begin{center}
-\begin{tabular}{@{}l@{}l@{}}
-\begin{minipage}[t]{.54\textwidth}
-\begin{verbatim}
-Then we have the series $A_1, A_2,
-\dotsc$, the regional sum $A_1
-+A_2 +\dotsb $, the orthogonal
-product $A_1 A_2 \dotsm $, and
-the infinite integral
-\[\int_{A_1}\int_{A_2}\dotsi\].
-\end{verbatim}
-\end{minipage}
-&
-\begin{minipage}[t]{.45\textwidth}
-\noindent
-Then we have the series $A_1,A_2,\dotsc$,
-the regional sum $A_1+A_2+\dotsb$,
-the orthogonal product $A_1A_2\dotsm$,
-and the infinite integral
-\[\int_{A_1}\int_{A_2}\dotsi.\]
-\end{minipage}
-\end{tabular}
-\end{center}
-
-\section{Nonbreaking dashes}
-
-A command \cn{nobreakdash} is provided to suppress the possibility
-of a linebreak after the following hyphen or dash. For example, if you
-write `pages 1\ndash 9' as |pages 1\nobreakdash--9| then a linebreak will
-never occur between the dash and the 9. You can also use
-\cn{nobreakdash} to prevent undesirable hyphenations in combinations
-like |$p$-adic|. For frequent use, it's advisable to make abbreviations,
-e.g.,
-\begin{verbatim}
-\newcommand{\p}{$p$\nobreakdash}% for "\p-adic"
-\newcommand{\Ndash}{\nobreakdash--}% for "pages 1\Ndash 9"
-% For "\n dimensional" ("n-dimensional"):
-\newcommand{\n}[1]{$n$\nobreakdash-\hspace{0pt}}
-\end{verbatim}
-The last example shows how to prohibit a linebreak after the hyphen but
-allow normal hyphenation in the following word. (It suffices to add a
-zero-width space after the hyphen.)
-
-\section{Accents in math}
-
-In ordinary \LaTeX{} the placement of the second accent in doubled math
-accents is often poor. With the \pkg{amsmath} package you
-will get improved placement of the second accent:
-$\hat{\hat{A}}$ (\cn{hat}|{\hat{A}}|).
-
-The commands \cn{dddot} and \cn{ddddot} are available to produce triple
-and quadruple dot accents in addition to the \cn{dot} and \cn{ddot}
-accents already available in \latex/.
-
-To get a superscripted hat or tilde character, load the \pkg{amsxtra}
-package and use \cn{sphat} or \cn{sptilde}. Usage is \verb'A\sphat'
-(note the absence of the \verb'^' character).
-
-To place an arbitrary symbol in math accent position, or to get under
-accents, see the \pkg{accents} package by Javier Bezos.
-
-\section{Roots}
-
-In ordinary \latex/ the placement of root indices is sometimes not so
-good: $\sqrt[\beta]{k}$ (|\sqrt||[\beta]{k}|). In the
-\pkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust
-the position of the root:
-\begin{verbatim}
- \sqrt[\leftroot{-2}\uproot{2}\beta]{k}
-\end{verbatim}
-will move the beta up and to the right:
-$\sqrt[\leftroot{-2}\uproot{2}\beta]{k}$. The negative argument used
-with \cn{leftroot} moves the $\beta$ to the right. The units are a small
-amount that is a useful size for such adjustments.
-
-\section{Boxed formulas}
-
-The command \cn{boxed} puts a box around its
-argument, like \cn{fbox} except that the contents are in math mode:
-\begin{equation}
-\boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))}
-\end{equation}
-\begin{verbatim}
- \boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))}
-\end{verbatim}
-
-\section{Over and under arrows}
-
-Basic \latex/ provides \cn{overrightarrow} and \cn{overleftarrow}
-commands. Some additional over and under arrow commands are provided
-by the \pkg{amsmath} package to extend the set:
-
-\begin{tabbing}
-\qquad\=\ncn{overleftrightarrow}\qquad\=\kill
-\> \cn{overleftarrow} \> \cn{underleftarrow} \+\\
- \cn{overrightarrow} \> \cn{underrightarrow} \\
- \cn{overleftrightarrow}\> \cn{underleftrightarrow}
-\end{tabbing}
-
-\section{Extensible arrows}
-
-\cn{xleftarrow} and \cn{xrightarrow} produce
-arrows\index{arrows!extensible} that extend automatically to accommodate
-unusually wide subscripts or superscripts. These commands take one
-optional argument (the subscript) and one mandatory argument (the
-superscript, possibly empty):
-\begin{equation}
-A\xleftarrow{n+\mu-1}B \xrightarrow[T]{n\pm i-1}C
-\end{equation}
-\begin{verbatim}
- \xleftarrow{n+\mu-1}\quad \xrightarrow[T]{n\pm i-1}
-\end{verbatim}
-
-\section{Affixing symbols to other symbols}
-
-\latex/ provides \cn{stackrel} for placing a
-superscript\index{subscripts and superscripts} above a binary relation.
-In the \pkg{amsmath} package there are somewhat more general commands,
-\cn{overset} and \cn{underset}, that can be used to place one symbol
-above or below another symbol, whether it's a relation or something
-else. The input |\overset{*}{X}| will place a superscript-size $*$ above
-the $X$: $\overset{*}{X}$; \cn{underset} is the analog for adding a
-symbol underneath.
-
-See also the description of \cn{sideset} in \secref{sideset}.
-
-\section{Fractions and related constructions}
-
-\subsection{The \cn{frac}, \cn{dfrac}, and \cn{tfrac} commands}
-
-The \cn{frac} command, which is in the basic command set of
-\latex/,\index{fractions} takes two arguments\mdash numerator and
-denominator\mdash and typesets them in normal fraction form. The
-\pkg{amsmath} package provides also \cn{dfrac} and \cn{tfrac} as
-convenient abbreviations for |{\displaystyle\frac| |...| |}|
-and\indexcs{textstyle}\relax
-\indexcs{displaystyle} |{\textstyle\frac| |...| |}|.
-\begin{equation}
-\frac{1}{k}\log_2 c(f)\quad\tfrac{1}{k}\log_2 c(f)\quad
-\sqrt{\frac{1}{k}\log_2 c(f)}\quad\sqrt{\dfrac{1}{k}\log_2 c(f)}
-\end{equation}
-\begin{verbatim}
-\begin{equation}
-\frac{1}{k}\log_2 c(f)\;\tfrac{1}{k}\log_2 c(f)\;
-\sqrt{\frac{1}{k}\log_2 c(f)}\;\sqrt{\dfrac{1}{k}\log_2 c(f)}
-\end{equation}
-\end{verbatim}
-
-\subsection{The \cn{binom}, \cn{dbinom}, and \cn{tbinom} commands}
-
-For binomial expressions\index{binomials} such as $\binom{n}{k}$
-\pkg{amsmath} has \cn{binom}, \cn{dbinom} and \cn{tbinom}:
-\begin{equation}
-2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}
-\end{equation}
-\begin{verbatim}
-2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}
-\end{verbatim}
-
-\subsection{The \cn{genfrac} command}
-
-The capabilities of \cn{frac}, \cn{binom}, and their variants are
-subsumed by a generalized fraction command \cn{genfrac} with six
-arguments. The last two correspond to \cn{frac}'s numerator and
-denominator; the first two are optional delimiters (as seen in
-\cn{binom}); the third is a line thickness override (\cn{binom} uses
-this to set the fraction line thickness to 0\mdash i.e., invisible); and
-the fourth argument is a mathstyle override: integer values 0\ndash 3 select
-respectively \cn{displaystyle}, \cn{textstyle}, \cn{scriptstyle}, and
-\cn{scriptscriptstyle}. If the third argument is left empty, the line
-thickness defaults to `normal'.
-
-\begin{center}\begin{minipage}{.85\columnwidth}
-\raggedright \normalfont\ttfamily \exhyphenpenalty10000
-\newcommand{\ma}[1]{%
- \string{{\normalfont\itshape#1}\string}\penalty9999 \ignorespaces}
-\string\genfrac \ma{left-delim} \ma{right-delim} \ma{thickness}
-\ma{mathstyle} \ma{numerator} \ma{denominator}
-\end{minipage}\end{center}
-To illustrate, here is how \cn{frac}, \cn{tfrac}, and
-\cn{binom} might be defined.
-\begin{verbatim}
-\newcommand{\frac}[2]{\genfrac{}{}{}{}{#1}{#2}}
-\newcommand{\tfrac}[2]{\genfrac{}{}{}{1}{#1}{#2}}
-\newcommand{\binom}[2]{\genfrac{(}{)}{0pt}{}{#1}{#2}}
-\end{verbatim}
-If you find yourself repeatedly using \cn{genfrac} throughout a document
-for a particular notation, you will do yourself a favor (and your
-publisher) if you define a meaningfully-named abbreviation for that
-notation, along the lines of \cn{frac} and \cn{binom}.
-
-The primitive generalized fraction commands \cs{over}, \cs{overwithdelims},
-\cs{atop}, \cs{atopwithdelims}, \cs{above}, \cs{abovewithdelims} produce
-warning messages if used with the \pkg{amsmath} package, for reasons
-discussed in \fn{technote.tex}.
-
-\section{Continued fractions}
-
-The continued fraction\index{continued fractions}
-\begin{equation}
-\cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+\cdots
-}}}
-\end{equation}
-can be obtained by typing
-{\samepage
-\begin{verbatim}
-\cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+\dotsb
-}}}
-\end{verbatim}
-}% End of \samepage
-This produces better-looking results than straightforward use of
-\cn{frac}. Left or right placement of any of the numerators is
-accomplished by using \cn{cfrac}|[l]| or \cn{cfrac}|[r]| instead of
-\cn{cfrac}.
-
-\section{Smash options}
-
-The command \cn{smash} is used to typeset a subformula with an effective height and depth of zero, which is sometimes
-useful in adjusting the subformula's position with respect to adjacent
-symbols. With the \pkg{amsmath} package \cn{smash} has optional
-arguments |t| and |b|, because occasionally it is advantageous to be
-able to \qq{smash} only the top or only the bottom of something while
-retaining the natural depth or height. For example, when adjacent
-radical symbols are unevenly sized or positioned because of differences
-in the height and depth of their contents, \cn{smash} can be employed to
-make them more consistent. Compare
-$\sqrt{x}+\sqrt{y}+\sqrt{z}$ and $\sqrt{x}+\sqrt{\smash[b]{y}}+\sqrt{z}$,
-where the latter was produced by
-\verb"$\sqrt{x}" \verb"+"
-\verb"\sqrt{"\5\verb"\smash[b]{y}}" \verb"+" \verb"\sqrt{z}$".
-
-\section{Delimiters}
-
-\subsection{Delimiter sizes}\label{bigdel}
-
-The automatic delimiter sizing done by \cn{left} and \cn{right} has two
-limitations: First, it is applied mechanically to produce delimiters
-large enough to encompass the largest contained item, and second, the
-range of sizes is not even approximately continuous but has fairly large
-quantum jumps. This means that a math fragment that is infinitesimally
-too large for a given delimiter size will get the next larger size, a
-jump of 3pt or so in normal-sized text. There are two or three
-situations where the delimiter size is commonly adjusted, using a set of
-commands that have `big' in their names.
-\begin{ctab}{l|llllll}
-Delimiter&
- text& \ncn{left}& \ncn{bigl}& \ncn{Bigl}& \ncn{biggl}& \ncn{Biggl}\\
-size&
- size& \ncn{right}& \ncn{bigr}& \ncn{Bigr}& \ncn{biggr}& \ncn{Biggr}\\
-\hline
-Result\vstrut{5ex}&
- $\displaystyle(b)(\frac{c}{d})$&
- $\displaystyle\left(b\right)\left(\frac{c}{d}\right)$&
- $\displaystyle\bigl(b\bigr)\bigl(\frac{c}{d}\bigr)$&
- $\displaystyle\Bigl(b\Bigr)\Bigl(\frac{c}{d}\Bigr)$&
- $\displaystyle\biggl(b\biggr)\biggl(\frac{c}{d}\biggr)$&
- $\displaystyle\Biggl(b\Biggr)\Biggl(\frac{c}{d}\Biggr)$
-\end{ctab}
-The first kind of situation is a cumulative operator with limits above
-and below. With \cn{left} and \cn{right} the delimiters usually turn out
-larger than necessary, and using the |Big| or |bigg|
-sizes\index{big@\cn{big}, \cn{Big}, \cn{bigg}, \dots\ delimiters}
-instead gives better results:
-\begin{equation*}
-\left[\sum_i a_i\left\lvert\sum_j x_{ij}\right\rvert^p\right]^{1/p}
-\quad\text{versus}\quad
-\biggl[\sum_i a_i\Bigl\lvert\sum_j x_{ij}\Bigr\rvert^p\biggr]^{1/p}
-\end{equation*}
-\begin{verbatim}
-\biggl[\sum_i a_i\Bigl\lvert\sum_j x_{ij}\Bigr\rvert^p\biggr]^{1/p}
-\end{verbatim}
-The second kind of situation is clustered pairs of delimiters where
-\cn{left} and \cn{right} make them all the same size (because that is
-adequate to cover the encompassed material) but what you really want
-is to make some of the delimiters slightly larger to make the nesting
-easier to see.
-\begin{equation*}
-\left((a_1 b_1) - (a_2 b_2)\right)
-\left((a_2 b_1) + (a_1 b_2)\right)
-\quad\text{versus}\quad
-\bigl((a_1 b_1) - (a_2 b_2)\bigr)
-\bigl((a_2 b_1) + (a_1 b_2)\bigr)
-\end{equation*}
-\begin{verbatim}
-\left((a_1 b_1) - (a_2 b_2)\right)
-\left((a_2 b_1) + (a_1 b_2)\right)
-\quad\text{versus}\quad
-\bigl((a_1 b_1) - (a_2 b_2)\bigr)
-\bigl((a_2 b_1) + (a_1 b_2)\bigr)
-\end{verbatim}
-The third kind of situation is a slightly oversize object in running
-text, such as $\left\lvert\frac{b'}{d'}\right\rvert$ where the
-delimiters produced by \cn{left} and \cn{right} cause too much line
-spreading. In that case \ncn{bigl} and \ncn{bigr}\index{big@\cn{big},
-\cn{Big}, \cn{bigg}, \dots\ delimiters} can be used to produce
-delimiters that are slightly larger than the base size but still able to
-fit within the normal line spacing:
-$\bigl\lvert\frac{b'}{d'}\bigr\rvert$.
-
-In ordinary \latex/ \ncn{big}, \ncn{bigg}, \ncn{Big}, and \ncn{Bigg}
-delimiters aren't scaled properly over the full range of \latex/ font
-sizes. With the \pkg{amsmath} package they are.
-
-\subsection{Vertical bar notations}
-
-The \pkg{amsmath} package provides commands \cn{lvert}, \cn{rvert},
-\cn{lVert}, \cn{rVert} (compare \cn{langle}, \cn{rangle}) to address the
-problem of overloading for the vert bar character \qc{\|}. This
-character is currently used in \latex/ documents to represent a wide
-variety of mathematical objects: the `divides' relation in a
-number-theory expression like $p\vert q$, or the absolute-value
-operation $\lvert z\rvert$, or the `such that' condition in set
-notation, or the `evaluated at' notation $f_\zeta(t)\bigr\rvert_{t=0}$.
-The multiplicity of uses in itself is not so bad; what is bad, however,
-is that fact that not all of the uses take the same typographical
-treatment, and that the complex discriminatory powers of a knowledgeable
-reader cannot be replicated in computer processing of mathematical
-documents. It is recommended therefore that there should be a one-to-one
-correspondence in any given document between the vert bar character
-\qc{\|} and a selected mathematical notation, and similarly for the
-double-bar command \ncn{\|}\index{"|@\verb"*+"\"|+}. This immediately
-rules out the use of \qc{|}
-and \ncn{\|}\index{"|@\verb"*+"\"|+} for delimiters, because left and right
-delimiters are distinct usages that do not relate in the same way to
-adjacent symbols; recommended practice is therefore to define suitable
-commands in the document preamble for any paired-delimiter use of vert
-bar symbols:
-\begin{verbatim}
-\providecommand{\abs}[1]{\lvert#1\rvert}
-\providecommand{\norm}[1]{\lVert#1\rVert}
-\end{verbatim}
-whereupon the document would contain |\abs{z}| to produce $\lvert
-z\rvert$ and |\norm{v}| to produce $\lVert v\rVert$.
-
-\chapter{Operator names}
-
-\section{Defining new operator names}\label{s:opname}
-
-Math functions\index{operator names}\relax \index{function
-names|see{operator names}} such as $\log$, $\sin$, and $\lim$ are
-traditionally typeset in roman type to make them visually more distinct
-from one-letter math variables, which are set in math italic. The more
-common ones have predefined names, \cn{log}, \cn{sin}, \cn{lim}, and so
-forth, but new ones come up all the time in mathematical papers, so the
-\pkg{amsmath} package provides a general mechanism for defining new
-`operator names'. To define a math function \ncn{xxx} to work like
-\cn{sin}, you write
-\begin{verbatim}
-\DeclareMathOperator{\xxx}{xxx}
-\end{verbatim}
-whereupon ensuing uses of \ncn{xxx} will produce {\upshape xxx} in the
-proper font and automatically add proper spacing\index{horizontal
-space!around operator names} on either side when necessary, so that you
-get $A\xxx B$ instead of $A\mathrm{xxx}B$. In the second argument of
-\cn{DeclareMathOperator} (the name text), a pseudo-text mode prevails:
-the hyphen character \qc{\-} will print as a text hyphen rather than a
-minus sign and an asterisk \qc{\*} will print as a raised text asterisk
-instead of a centered math star. (Compare
-\textit{a}-\textit{b}*\textit{c} and $a-b*c$.) But otherwise the name
-text is printed in math mode, so that you can use, e.g., subscripts and
-superscripts there.
-
-If the new operator should have subscripts and superscripts placed in
-`limits' position above and below as with $\lim$, $\sup$, or $\max$, use
-the \qc{\*} form of the \cn{DeclareMathOperator} command:
-\begin{verbatim}
-\DeclareMathOperator*{\Lim}{Lim}
-\end{verbatim}
-See also the discussion of subscript placement in
-Section~\ref{subplace}.
-
-The following operator names are predefined:
-\begin{ctab}{rlrlrlrl}
-\cn{arccos}& $\arccos$ &\cn{deg}& $\deg$ & \cn{lg}& $\lg$ & \cn{projlim}& $\projlim$\\
-\cn{arcsin}& $\arcsin$ &\cn{det}& $\det$ & \cn{lim}& $\lim$ & \cn{sec}& $\sec$\\
-\cn{arctan}& $\arctan$ &\cn{dim}& $\dim$ & \cn{liminf}& $\liminf$ &\cn{sin}& $\sin$\\
-\cn{arg}& $\arg$ & \cn{exp}& $\exp$ & \cn{limsup}& $\limsup$ &\cn{sinh}& $\sinh$\\
-\cn{cos}& $\cos$ & \cn{gcd}& $\gcd$ & \cn{ln}& $\ln$ & \cn{sup}& $\sup$\\
-\cn{cosh}& $\cosh$ & \cn{hom}& $\hom$ & \cn{log}& $\log$ & \cn{tan}& $\tan$\\
-\cn{cot}& $\cot$ & \cn{inf}& $\inf$ & \cn{max}& $\max$ & \cn{tanh}& $\tanh$\\
-\cn{coth}& $\coth$ & \cn{injlim}& $\injlim$ &\cn{min}& $\min$\\
-\cn{csc}& $\csc$ & \cn{ker}& $\ker$ & \cn{Pr}& $\Pr$
-\end{ctab}
-\begin{ctab}{rlrl}
-\cn{varlimsup}& $\displaystyle\varlimsup$&
- \cn{varinjlim}& $\displaystyle\varinjlim$\\
-\cn{varliminf}& $\displaystyle\varliminf$&
- \cn{varprojlim}& $\displaystyle\varprojlim$
-\end{ctab}
-
-There is also a command \cn{operatorname} such that using
-\begin{verbatim}
-\operatorname{abc}
-\end{verbatim}
-in a math formula is equivalent to a use of \ncn{abc} defined by
-\cn{DeclareMathOperator}. This may be occasionally useful for
-constructing more complex notation or other purposes. (Use the variant
-\cn{operatorname*} to get limits.)
-
-\section{\cn{mod} and its relatives}
-
-Commands \cn{mod}, \cn{bmod}, \cn{pmod}, \cn{pod} are provided to deal
-with the special spacing conventions of \qq{mod} notation. \cn{bmod} and
-\cn{pmod} are available in \latex/, but with the \pkg{amsmath} package
-the spacing of \cn{pmod} will adjust to a smaller value if it's used in
-a non-display-mode formula. \cn{mod} and \cn{pod} are variants of
-\cn{pmod} preferred by some authors; \cn{mod} omits the parentheses,
-whereas \cn{pod} omits the \qq{mod} and retains the parentheses.
-\begin{equation}
-\gcd(n,m\bmod n);\quad x\equiv y\pmod b;
-\quad x\equiv y\mod c;\quad x\equiv y\pod d
-\end{equation}
-\begin{verbatim}
-\gcd(n,m\bmod n);\quad x\equiv y\pmod b;
-\quad x\equiv y\mod c;\quad x\equiv y\pod d
-\end{verbatim}
-
-\chapter{The \cn{text} command}\label{text}
-
-The main use of the command \cn{text} is for words or
-phrases\index{text fragments inside math} in a display. It is very
-similar to the \latex/ command \cn{mbox} in its effects, but has a
-couple of advantages. If you want a word or phrase of text in a
-subscript, you can type |..._{\text{word or phrase}}|, which is slightly
-easier than the \cn{mbox} equivalent: |..._{\mbox{\scriptsize| |word|
-|or| |phrase}}|. The other advantage is the more descriptive name.
-\begin{equation}
-f_{[x_{i-1},x_i]} \text{ is monotonic,}
-\quad i = 1,\dots,c+1
-\end{equation}
-\begin{verbatim}
-f_{[x_{i-1},x_i]} \text{ is monotonic,}
-\quad i = 1,\dots,c+1
-\end{verbatim}
-
-\chapter{Integrals and sums}
-
-\section{Multiline subscripts and superscripts}
-
-The \cn{substack} command can be used to produce a multiline subscript
-or superscript:\index{subscripts and superscripts!multi-line}\relax
-\index{superscripts|see{subscripts and superscripts}} for example
-\begin{ctab}{ll}
-\begin{minipage}[t]{.6\columnwidth}
-\begin{verbatim}
-\sum_{\substack{
- 0\le i\le m\\
- 0<j<n}}
- P(i,j)
-\end{verbatim}
-\end{minipage}
-&
-$\displaystyle
-\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)$
-\end{ctab}
-A slightly more generalized form is the \env{subarray} environment which
-allows you to specify that each line should be left-aligned instead of
-centered, as here:
-\begin{ctab}{ll}
-\begin{minipage}[t]{.6\columnwidth}
-\begin{verbatim}
-\sum_{\begin{subarray}{l}
- i\in\Lambda\\ 0<j<n
- \end{subarray}}
- P(i,j)
-\end{verbatim}
-\end{minipage}
-&
-$\displaystyle
- \sum_{\begin{subarray}{l}
- i\in \Lambda\\ 0<j<n
- \end{subarray}}
- P(i,j)$
-\end{ctab}
-
-\section{The \cn{sideset} command}\label{sideset}
-
-There's also a command called \cn{sideset}, for a rather special
-purpose: putting symbols at the subscript and
-superscript\index{subscripts and superscripts!on sums} corners of a
-large operator symbol such as $\sum$ or $\prod$. \emph{Note: this
-command is not designed to be applied to anything other than sum-class symbols.} The prime
-example is the case when you want to put a prime on a sum symbol. If
-there are no limits above or below the sum, you could just use
-\cn{nolimits}: here's
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-|\sum\nolimits' E_n| in display mode:
-\begin{equation}
-\sum\nolimits' E_n
-\end{equation}
-If, however, you want not only the prime but also something below or
-above the sum symbol, it's not so easy\mdash indeed, without
-\cn{sideset}, it would be downright difficult. With \cn{sideset}, you
-can write
-\begin{ctab}{ll}
-\begin{minipage}[t]{.6\columnwidth}
-\begin{verbatim}
-\sideset{}{'}
- \sum_{n<k,\;\text{$n$ odd}} nE_n
-\end{verbatim}
-\end{minipage}
-&$\displaystyle
-\sideset{}{'}\sum_{n<k,\;\text{$n$ odd}} nE_n
-$
-\end{ctab}
-The extra pair of empty braces is explained by the fact that
-\cn{sideset} has the capability of putting an extra symbol or symbols at
-each corner of a large operator; to put an asterisk at each corner of a
-product symbol, you would type
-\begin{ctab}{ll}
-\begin{minipage}[t]{.6\columnwidth}
-\begin{verbatim}
-\sideset{_*^*}{_*^*}\prod
-\end{verbatim}
-\end{minipage}
-&$\displaystyle
-\sideset{_*^*}{_*^*}\prod
-$
-\end{ctab}
-
-\section{Placement of subscripts and limits}\label{subplace}
-
-The default positioning for subscripts depends on the
-base symbol involved. The default for sum-class symbols is
-`displaylimits' positioning: When a sum-class symbol appears
-in a displayed formula, subscript and superscript are placed in `limits'
-position above and below, but in an inline formula, they are placed to
-the side, to avoid unsightly and wasteful spreading of the
-surrounding text lines.
-The default for integral-class symbols is to have sub- and
-superscripts always to the side, even in displayed formulas.
-(See the discussion of the \opt{intlimits} and related options in
-Section~\ref{options}.)
-
-Operator names such as $\sin$ or $\lim$ may have either `displaylimits'
-or `limits' positioning depending on how they were defined. The standard
-operator names are defined according to normal mathematical usage.
-
-The commands \cn{limits} and \cn{nolimits} can be used to override the
-normal behavior of a base symbol:
-\begin{equation*}
-\sum\nolimits_X,\qquad \iint\limits_{A},
-\qquad\varliminf\nolimits_{n\to \infty}
-\end{equation*}
-To define a command whose subscripts follow the
-same `displaylimits' behavior as \cn{sum}, put
-\cn{displaylimits} at the tail end of the definition. When multiple
-instances of \cn{limits}, \cn{nolimits}, or \cn{displaylimits} occur
-consecutively, the last one takes precedence.
-
-\section{Multiple integral signs}
-
-\cn{iint}, \cn{iiint}, and \cn{iiiint} give multiple integral
-signs\index{integrals!multiple} with the spacing between them nicely
-adjusted, in both text and display style. \cn{idotsint} is an extension
-of the same idea that gives two integral signs with dots between them.
-\begin{gather}
-\iint\limits_A f(x,y)\,dx\,dy\qquad\iiint\limits_A
-f(x,y,z)\,dx\,dy\,dz\\
-\iiiint\limits_A
-f(w,x,y,z)\,dw\,dx\,dy\,dz\qquad\idotsint\limits_A f(x_1,\dots,x_k)
-\end{gather}
-
-\chapter{Commutative diagrams}\label{s:commdiag}
-
-Some commutative diagram commands like the ones in \amstex/ are
-available as a separate package, \pkg{amscd}. For complex commutative
-diagrams authors will need to turn to more comprehensive packages like
-\pkg{kuvio} or \xypic/, but for simple diagrams without diagonal
-arrows\index{arrows!in commutative diagrams} the \pkg{amscd} commands
-may be more convenient. Here is one example.
-\begin{equation*}
-\begin{CD}
-S^{{\mathcal{W}}_\Lambda}\otimes T @>j>> T\\
-@VVV @VV{\End P}V\\
-(S\otimes T)/I @= (Z\otimes T)/J
-\end{CD}
-\end{equation*}
-\begin{verbatim}
-\begin{CD}
-S^{{\mathcal{W}}_\Lambda}\otimes T @>j>> T\\
-@VVV @VV{\End P}V\\
-(S\otimes T)/I @= (Z\otimes T)/J
-\end{CD}
-\end{verbatim}
-In the \env{CD} environment the commands |@>>>|,
-|@<<<|, |@VVV|, and |@AAA| give respectively right, left, down, and up
-arrows. For the horizontal arrows, material between the first and second
-|>| or |<| symbols will be typeset as a superscript, and material
-between the second and third will be typeset as a subscript. Similarly,
-material between the first and second or second and third |A|s or |V|s
-of vertical arrows will be typeset as left or right \qq{sidescripts}.
-The commands |@=| and \verb'@|' give horizontal and vertical double lines.
-A \qq{null arrow} command |@.| can be used instead of a visible arrow
-to fill out an array where needed.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\chapter{Using math fonts}
-
-\section{Introduction}
-
-For more comprehensive information on font use in \latex/, see the
-\latex/ font guide (\fn{fntguide.tex}) or \booktitle{The \latex/
-Companion} \cite{tlc}. The basic set of math font commands\index{math
-fonts}\relax \index{math symbols|see{math fonts}} in \latex/ includes
-\cn{mathbf}, \cn{mathrm}, \cn{mathcal}, \cn{mathsf}, \cn{mathtt},
-\cn{mathit}. Additional math alphabet commands such as
-\cn{mathbb} for blackboard bold, \cn{mathfrak} for Fraktur, and
-\cn{mathscr} for Euler script are available through the packages
-\pkg{amsfonts} and \pkg{euscript} (distributed separately).
-
-\section{Recommended use of math font commands}
-
-If you find yourself employing math font commands frequently in your
-document, you might wish that they had shorter names, such as \ncn{mb}
-instead of \cn{mathbf}. Of course, there is nothing to keep you from
-providing such abbreviations for yourself by suitable \cn{newcommand}
-statements. But for \latex/ to provide shorter names would actually be a
-disservice to authors, as that would obscure a much better alternative:
-defining custom command names derived from the names of the underlying
-mathematical objects, rather than from the names of the fonts used to
-distinguish the objects. For example, if you are using bold to indicate
-vectors, then you will be better served in the long run if you define a
-`vector' command instead of a `math-bold' command:
-\begin{verbatim}
- \newcommand{\vect}[1]{\mathbf{#1}}
-\end{verbatim}
-you can write |\vect{a} + \vect{b}| to produce $\vect{a} +
-\vect{b}$.
-If you decide several months down the road that you want to use the bold
-font for some other purpose, and mark vectors by a small over-arrow
-instead, then you can put the change into effect merely by changing the
-definition of \ncn{vect}; otherwise you would have to replace all
-occurrences of \cn{mathbf} throughout your document, perhaps even
-needing to inspect each one to see whether it is indeed
-an instance of a vector.
-
-It can also be useful to assign distinct
-command names for different letters of a particular font:
-\begin{verbatim}
-\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}% or use amsfonts package
-\DeclareMathSymbol{\C}{\mathalpha}{AMSb}{"43}
-\DeclareMathSymbol{\R}{\mathalpha}{AMSb}{"52}
-\end{verbatim}
-These statements would define the commands \cn{C} and \cn{R} to produce
-blackboard-bold letters from the `AMSb' math symbols font. If you refer
-often to the\break complex numbers or real numbers in your document, you
-might find this method more convenient than (let's say) defining a
-\ncn{field} command and writing\break |\field{C}|, |\field{R}|. But for
-maximum flexibility and control, define such a \ncn{field} command and
-then define \ncn{C} and \ncn{R} in terms of that
-command:\indexcs{mathbb}
-\begin{verbatim}
-\usepackage{amsfonts}% to get the \mathbb alphabet
-\newcommand{\field}[1]{\mathbb{#1}}
-\newcommand{\C}{\field{C}}
-\newcommand{\R}{\field{R}}
-\end{verbatim}
-
-\section{Bold math symbols}
-
-The \cn{mathbf} command is commonly used to obtain bold Latin letters in
-math, but for most other kinds of math symbols it has no effect, or its
-effects depend unreliably on the set of math fonts that are in use. For
-example, writing
-\begin{verbatim}
-\Delta \mathbf{\Delta}\mathbf{+}\delta \mathbf{\delta}
-\end{verbatim}
-produces $\Delta \mathbf{\Delta}\mathbf{+}\delta
-\mathbf{\delta}$; the \cn{mathbf} has no effect on the plus
-sign or the small delta.
-
-The \pkg{amsmath} package therefore provides two additional commands,
-\cn{boldsymbol} and \cn{pmb}, that can be applied to other kinds of math
-symbols. \cn{boldsymbol} can be used for a math symbol that remains
-unaffected by \cn{mathbf} if (and only if) your current math font set
-includes a bold version of that symbol. \cn{pmb} can be used as a last
-resort for any math symbols that do not have a true bold version
-provided by your set of math fonts; \qq{pmb} stands for \qq{poor man's
-bold} and the command works by typesetting multiple copies of the symbol
-with slight offsets. The quality of the output is inferior, especially
-for symbols that contain any hairline strokes. When the standard default set of
-\latex/ math fonts are in use (Computer Modern), the only symbols that
-are likely to require \cn{pmb} are large operator symbols like \cn{sum},
-extended delimiter symbols, or the extra math symbols provided by
-the \pkg{amssymb} package \cite{amsfonts}.
-
-The following formula shows some of the results that are possible:
-\begin{verbatim}
-A_\infty + \pi A_0
-\sim \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
- \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
-\sim\pmb{A}_{\pmb{\infty}} \pmb{+}\pmb{\pi} \pmb{A}_{\pmb{0}}
-\end{verbatim}
-\begin{equation*}
-A_\infty + \pi A_0
-\sim \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
- \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
-\sim\pmb{A}_{\pmb{\infty}} \pmb{+}\pmb{\pi} \pmb{A}_{\pmb{0}}
-\end{equation*}
-If you want to use only the \cn{boldsymbol} command without loading the
-whole \pkg{amsmath} package, the \pkg{bm} package is recommended (this
-is a standard \latex/ package, not an AMS package; you probably have it
-already if you have a 1997 or newer version of \latex/).
-
-\section{Italic Greek letters}
-
-For italic versions of the capital Greek letters, the following commands
-are provided:
-\begin{ctab}{rlrl}
-\cn{varGamma}& $\varGamma$& \cn{varSigma}& $\varSigma$\\
-\cn{varDelta}& $\varDelta$& \cn{varUpsilon}& $\varUpsilon$\\
-\cn{varTheta}& $\varTheta$& \cn{varPhi}& $\varPhi$\\
-\cn{varLambda}& $\varLambda$& \cn{varPsi}& $\varPsi$\\
-\cn{varXi}& $\varXi$& \cn{varOmega}& $\varOmega$\\
-\cn{varPi}& $\varPi$
-\end{ctab}
-
-\chapter{Error messages and output problems}
-
-\section{General remarks}
-
-This is a supplement to Chapter~8 of the \latex/ manual \cite{lamport} (first
-edition: Chapter~6). For the reader's convenience, the set of error
-messages discussed here overlaps somewhat with the set in that chapter,
-but please be aware that we don't provide exhaustive coverage here.
-The error messages are arranged in alphabetical order, disregarding
-unimportant text such as |! LaTeX Error:| at the beginning, and
-nonalphabetical characters such as \qc{\\}. Where examples are given, we
-show also the help messages that appear on screen when you respond to an
-error message prompt by entering |h|.
-
-There is also a section discussing some output errors, i.e., instances
-where the printed document has something wrong but there was no \latex/
-error during typesetting.
-
-\section{Error messages}
-
-\begin{error}{\begin{split} won't work here.}
-\errexa
-\begin{verbatim}
-! Package amsmath Error: \begin{split} won't work here.
- ...
-
-l.8 \begin{split}
-
-? h
-\Did you forget a preceding \begin{equation}?
-If not, perhaps the `aligned' environment is what you want.
-?
-\end{verbatim}
-\errexpl
-The \env{split} environment does not construct a stand-alone displayed
-equation; it needs to be used within some other environment such as
-\env{equation} or \env{gather}.
-
-\end{error}
-
-\begin{error}{Extra & on this line}
-\errexa
-\begin{verbatim}
-! Package amsmath Error: Extra & on this line.
-
-See the amsmath package documentation for explanation.
-Type H <return> for immediate help.
- ...
-
-l.9 \end{alignat}
-
-? h
-\An extra & here is so disastrous that you should probably exit
- and fix things up.
-?
-\end{verbatim}
-\errexpl
-In an \env{alignat} structure the number of alignment points per line
-is dictated by the numeric argument given after |\begin{alignat}|.
-If you use more alignment points in a line it is assumed that you
-accidentally left out a newline command \cn{\\} and the above error is
-issued.
-\end{error}
-
-\begin{error}{Improper argument for math accent}
-\errexa
-\begin{verbatim}
-! Package amsmath Error: Improper argument for math accent:
-(amsmath) Extra braces must be added to
-(amsmath) prevent wrong output.
-
-See the amsmath package documentation for explanation.
-Type H <return> for immediate help.
- ...
-
-l.415 \tilde k_{\lambda_j} = P_{\tilde \mathcal
- {M}}
-?
-\end{verbatim}
-\errexpl
-Non-simple arguments for any \LaTeX{} command should be enclosed in
-braces. In this example extra braces are needed as follows:
-\begin{verbatim}
-... P_{\tilde{\mathcal{M}}}
-\end{verbatim}
-\end{error}
-
-\begin{error}{Font OMX/cmex/m/n/7=cmex7 not loadable ...}
-\errexa
-\begin{verbatim}
-! Font OMX/cmex/m/n/7=cmex7 not loadable: Metric (TFM) file not found.
-<to be read again>
- relax
-l.8 $a
- b+b^2$
-? h
-I wasn't able to read the size data for this font,
-so I will ignore the font specification.
-[Wizards can fix TFM files using TFtoPL/PLtoTF.]
-You might try inserting a different font spec;
-e.g., type `I\font<same font id>=<substitute font name>'.
-?
-\end{verbatim}
-\errexpl
-Certain extra sizes of some Computer Modern fonts that were formerly
-available mainly through the AMSFonts\index{AMSFonts collection}
-distribution are considered part of standard \latex/ (as of June 1994):
-\fn{cmex7}\ndash \texttt{9}, \fn{cmmib5}\ndash \texttt{9}, and
-\fn{cmbsy5}\ndash \texttt{9}. If these extra sizes are missing on your
-system, you should try first to get them from the source where you
-obtained \latex/. If that fails, you could try getting the fonts from
-CTAN (e.g., in the form of Metafont\index{Metafont source files} source
-files, directory \nfn{/tex-archive/fonts/latex/mf}, or in PostScript
-Type 1 format, directory
-\nfn{/tex-archive/fonts/cm/ps-type1/bakoma}\index{BaKoMa fonts}\relax
-\index{PostScript fonts}).
-
-If the font name begins with \fn{cmex}, there is a special option
-\fn{cmex10} for the \pkg{amsmath} package that provides a temporary
-workaround. I.e., change the \cn{usepackage} to
-\begin{verbatim}
-\usepackage[cmex10]{amsmath}
-\end{verbatim}
-This will force the use of the 10-point size of the \fn{cmex} font in
-all cases. Depending on the contents of your document this may be
-adequate.
-\end{error}
-
-\begin{error}{Math formula deleted: Insufficient extension fonts}
-\errexa
-\begin{verbatim}
-! Math formula deleted: Insufficient extension fonts.
-l.8 $ab+b^2$
-
-?
-\end{verbatim}
-\errexpl
-This usually follows a previous error |Font ... not loadable|; see the
-discussion of that error (above) for solutions.
-\end{error}
-
-\begin{error}{Missing number, treated as zero}
-\errexa
-\begin{verbatim}
-! Missing number, treated as zero.
-<to be read again>
- a
-l.100 \end{alignat}
-
-? h
-A number should have been here; I inserted `0'.
-(If you can't figure out why I needed to see a number,
-look up `weird error' in the index to The TeXbook.)
-
-?
-\end{verbatim}
-\errexpl
-There are many possibilities that can lead to this error. However, one
-possibility that is relevant for the \pkg{amsmath} package is that you
-forgot to give the number argument of an \env{alignat} environment, as
-in:
-\begin{verbatim}
-\begin{alignat}
- a& =b& c& =d\\
-a'& =b'& c'& =d'
-\end{alignat}
-\end{verbatim}
-where the first line should read instead
-\begin{verbatim}
-\begin{alignat}{2}
-\end{verbatim}
-
-Another possibility is that you have a left bracket character |[|
-following a linebreak command \cn{\\} in a multiline construction such
-as \env{array}, \env{tabular}, or \env{eqnarray}. This will be
-interpreted by \latex/ as the beginning of an `additional vertical
-space' request \cite[\S C.1.6]{lamport}, even if it occurs on the following
-line and is intended to be part of the contents. For example
-\begin{verbatim}
-\begin{array}
-a+b\\
-[f,g]\\
-m+n
-\end{array}
-\end{verbatim}
-To prevent the error message in such a case, you can
-add braces as discussed in the \latex/ manual \cite[\S C.1.1]{lamport}:
-\begin{verbatim}
-\begin{array}
-a+b\\
-{[f,g]}\\
-m+n
-\end{array}
-\end{verbatim}
-
-\end{error}
-
-\begin{error}{Missing \right. inserted}
-\errexa
-\begin{verbatim}
-! Missing \right. inserted.
-<inserted text>
- \right .
-l.10 \end{multline}
-
-? h
-I've inserted something that you may have forgotten.
-(See the <inserted text> above.)
-With luck, this will get me unwedged. But if you
-really didn't forget anything, try typing `2' now; then
-my insertion and my current dilemma will both disappear.
-\end{verbatim}
-\errexpl
-This error typically arises when you try to insert a linebreak inside a
-\cn{left}-\cn{right} pair of delimiters in a \env{multline} or
-\env{split} environment:
-\begin{verbatim}
-\begin{multline}
-AAA\left(BBB\\
- CCC\right)
-\end{multline}
-\end{verbatim}
-There are two possible solutions: (1)~instead of using \cn{left} and
-\cn{right}, use `big' delimiters of fixed size (\cn{bigl} \cn{bigr}
-\cn{biggl} \cn{biggr} \dots; see \secref{bigdel}); or (2)~use null
-delimiters to break up the \cn{left}-\cn{right} pair into parts for each
-line:
-\begin{verbatim}
-AAA\left(BBB\right.\\
- \left.CCC\right)
-\end{verbatim}
-The latter solution may result in mismatched delimiter sizes;
-ensuring that they match requires using \cn{vphantom} in the line
-that has the smaller delimiter (or possibly \cn{smash} in the line that
-has the larger delimiter). In the argument of \cn{vphantom} put a copy
-of the tallest element that occurs in the other line, e.g.,
-\begin{verbatim}
-xxx \left(\int_t yyy\right.\\
- \left.\vphantom{\int_t} zzz ... \right)
-\end{verbatim}
-\end{error}
-
-\begin{error}{Paragraph ended before \xxx was complete}
-\errexa
-\begin{verbatim}
-Runaway argument?
-
-! Paragraph ended before \multline was complete.
-<to be read again>
- \par
-l.100
-
-? h
-I suspect you've forgotten a `}', causing me to apply this
-control sequence to too much text. How can we recover?
-My plan is to forget the whole thing and hope for the best.
-?
-\end{verbatim}
-\errexpl
-This might be produced by a misspelling in the |\end{multline}| command,
-e.g.,
-\begin{verbatim}
-\begin{multline}
-...
-\end{multiline}
-\end{verbatim}
-or by using abbreviations for certain environments, such as |\bal| and
-|\eal| for |\begin{align}| and |\end{align}|:
-\begin{verbatim}
-\bal
-...
-\eal
-\end{verbatim}
-For technical reasons that kind of abbreviation does not work with
-the more complex displayed equation environments of the \pkg{amsmath} package
-(\env{gather}, \env{align}, \env{split}, etc.; cf.\@ \fn{technote.tex}).
-\end{error}
-
-\begin{error}{Runaway argument?}
-See the discussion for the error message
-\texttt{Paragraph ended before \ncn{xxx} was complete}.
-\end{error}
-
-\begin{error}{Unknown option `xxx' for package `yyy'}
-\errexa
-\begin{verbatim}
-! LaTeX Error: Unknown option `intlim' for package `amsmath'.
-...
-? h
-The option `intlim' was not declared in package `amsmath', perhaps you
-misspelled its name. Try typing <return> to proceed.
-?
-\end{verbatim}
-\errexpl
-This means that you misspelled the option name, or the package simply
-does not have an option that you expected it to have. Consult the
-documentation for the given package.
-\end{error}
-
-\begin{error}{Old form `\pmatrix' should be \begin{pmatrix}.}
-\errexa
-\begin{verbatim}
-! Package amsmath Error: Old form `\pmatrix' should be
- \begin{pmatrix}.
-
-See the amsmath package documentation for explanation.
-Type H <return> for immediate help.
- ...
-
-\pmatrix ->\left (\matrix@check \pmatrix
- \env@matrix
-l.16 \pmatrix
- {a&b\cr c&d\cr}
-? h
-`\pmatrix{...}' is old Plain-TeX syntax whose use is
-ill-advised in LaTeX.
-?
-\end{verbatim}
-\errexpl
-When the \pkg{amsmath} package is used, the old forms of \cn{pmatrix},
-\cn{matrix}, and \cn{cases} cannot be used any longer because of naming
-conflicts. Their syntax did not conform with standard \LaTeX{} syntax
-in any case.
-\end{error}
-
-\begin{error}{Erroneous nesting of equation structures}
-\errexa
-\begin{verbatim}
-! Package amsmath Error: Erroneous nesting of equation structures;
-(amsmath) trying to recover with `aligned'.
-
-See the amsmath package documentation for explanation.
-Type H <return> for immediate help.
- ...
-
-l.260 \end{alignat*}
- \end{equation*}
-\end{verbatim}
-\errexpl
-The structures \env{align}, \env{alignat}, etc., are designed
-for top-level use and for the most part cannot be nested inside some
-other displayed equation structure. The chief exception is that
-\env{align} and most of its variants can be used inside the
-\env{gather} environment.
-\end{error}
-
-\section{Warning messages}
-
-\begin{error}{Foreign command \over [or \atop or \above]}
-\errexa
-\begin{verbatim}
-Package amsmath Warning: Foreign command \over; \frac or \genfrac
-(amsmath) should be used instead.
-\end{verbatim}
-\errexpl The primitive generalized fraction commands of \tex/\mdash
-\cs{over}, \cs{atop}, \cs{above}\mdash are deprecated when the
-\pkg{amsmath} package is used because their syntax is foreign to \latex/
-and \pkg{amsmath} provides native \latex/ equivalents. See
-\fn{technote.tex} for further information.
-\end{error}
-
-\begin{error}{Cannot use `split' here}
-\errexa
-\begin{verbatim}
-Package amsmath Warning: Cannot use `split' here;
-(amsmath) trying to recover with `aligned'
-\end{verbatim}
-\errexpl The \env{split} environment is designed to serve as the entire
-body of an equation, or an entire line of an \env{align} or \env{gather}
-environment. There cannot be any printed material before or
-after it within the same enclosing structure:
-\begin{verbatim}
-\begin{equation}
-\left\{ % <-- Not allowed
-\begin{split}
-...
-\end{split}
-\right. % <-- Not allowed
-\end{equation}
-\end{verbatim}
-\end{error}
-
-\section{Wrong output}
-
-\subsection{Section numbers 0.1, 5.1, 8.1 instead of 1, 2, 3}
-\label{numinverse}
-
-This most likely means that you have the arguments for \cn{numberwithin}
-in reverse order:
-\begin{verbatim}
-\numberwithin{section}{equation}
-\end{verbatim}
-That means `print the section number as \textit{equation
-number}.\textit{section number} and reset to 1 every time an equation
-occurs' when what you probably wanted was the inverse
-\begin{verbatim}
-\numberwithin{equation}{section}
-\end{verbatim}
-
-\subsection{The \cn{numberwithin} command had no effect on equation
-numbers}
-
-Are you looking at the first section in your document? Check the section
-numbers elsewhere to see if the problem is the one described in
-\secref{numinverse}.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\chapter{Additional information}
-
-\section{Converting existing documents}
-
-\subsection{Converting from plain \LaTeX{}}
-
-A \LaTeX{} document will typically continue to work the same in most
-respects if \verb'\usepackage{amsmath}' is added in the document
-preamble. By default, however, the \pkg{amsmath} package suppresses page
-breaks inside multiple-line displayed equation structures such as
-\env{eqnarray}, \env{align}, and \env{gather}. To continue allowing page
-breaks inside \env{eqnarray} after switching to \pkg{amsmath}, you will
-need to add the following line in your document preamble:
-\begin{verbatim}
-\allowdisplaybreaks[1]
-\end{verbatim}
-To ensure normal spacing around relation symbols, you might also want to
-change \env{eqnarray} to \env{align}, \env{multline}, or
-\env{equation}\slash\env{split} as appropriate.
-
-Most of the other differences in \pkg{amsmath} usage can be considered
-optional refinements, e.g., using
-\begin{verbatim}
-\DeclareMathOperator{\Hom}{Hom}
-\end{verbatim}
-instead of \verb'\newcommand{\Hom}{\mbox{Hom}}'.
-
-\subsection{Converting from \amslatex/ 1.1}
-See \fn{diffs-m.txt}.
-
-\section{Technical notes}
-The file \fn{technote.tex} contains some remarks on miscellaneous
-technical questions that are less likely to be of general interest.
-
-\section{Getting help}
-
-Questions or comments regarding \pkg{amsmath} and related packages
-should be sent to:
-\begin{infoaddress}
-American Mathematical Society\\
-Technical Support\\
-Electronic Products and Services\\
-P. O. Box 6248\\
-Providence, RI 02940\\[3pt]
-Phone: 800-321-4AMS (321-4267) \quad or \quad 401-455-4080\\
-Internet: \mail{tech-support@ams.org}
-\end{infoaddress}
-If you are reporting a problem you should include
-the following information to make proper investigation possible:
-\begin{enumerate}
-\item The source file where the problem occurred, preferably reduced
- to minimum size by removing any material that can be removed without
- affecting the observed problem.
-\item A \latex/ log file showing the error message (if applicable) and
- the version numbers of the document class and option files being used.
-\end{enumerate}
-
-\section{Of possible interest}\label{a:possible-interest}
-Information about obtaining AMSFonts or other \tex/-related
-software from the AMS Internet archive \fn{e-math.ams.org}
-can be obtained by sending a request through electronic mail to:
-\mail{webmaster@ams.org}.
-
-Information about obtaining the \pkg{amsmath} distribution on diskette
-from the AMS is available from:
-\begin{infoaddress}
-American Mathematical Society\\
-Customer Services\\
-P. O. Box 6248\\
-Providence, RI 02940\\[3pt]
-Phone: 800-321-4AMS (321-4267) \quad or \quad 401-455-4000\\
-Internet: \mail{cust-serv@ams.org}
-\end{infoaddress}
-
-The \tex/ Users Group\index{TeX Users@\tex/ Users Group} is a nonprofit
-organization that publishes a journal
-(\journalname{TUGboat}\index{TUGboat@\journalname{TUGboat}}), holds
-meetings, and serves as a clearing-house of general information about
-\tex/ and \tex/-related software.
-\begin{infoaddress}
-\tex/ Users Group\\
-PO Box 2311\\
-Portland, OR 97208-2311\\
-USA\\[3pt]
-Phone: +1-503-223-9994\\
-Email: \mail{office@tug.org}
-\end{infoaddress}
-Membership in the \tex/ Users Group is a good way to support continued
-development of free \tex/-related software. There are also many local
-\tex/ user groups in other countries; information about contacting a
-local user group can be gotten from the \tex/ Users Group.
-
-There is a Usenet newsgroup called \fn{comp.text.tex} that is a fairly
-good source of information about \latex/ and \tex/ in general. If you
-don't know about reading newsgroups, check with your local system
-administrator to see if newsgroup service is available at your site.
-
-\begin{thebibliography}{9}
-\addcontentsline{toc}{chapter}{Bibliography}
-
-\bibitem{amsfonts}\booktitle{AMSFonts version \textup{2.2}\mdash user's guide},
-Amer. Math. Soc., Providence, RI, 1994; distributed
-with the AMSFonts package.
-
-\bibitem{instr-l}\booktitle{Instructions for preparation of
-papers and monographs\mdash \amslatex/},
-Amer. Math. Soc., Providence, RI, 1996, 1999.
-
-\bibitem{amsthdoc}\booktitle{Using the \pkg{amsthm} Package},
-Amer. Math. Soc., Providence, RI, 1999.
-
-\bibitem{tlc} Michel Goossens, Frank Mittelbach, and Alexander Samarin,
-\booktitle{The \latex/ companion}, Addison-Wesley, Reading, MA, 1994.
- [\emph{Note: The 1994 edition is not a reliable guide for the
- \pkg{amsmath} package unless you refer to the errata for Chapter
- 8\mdash file \fn{compan.err}, distributed with \LaTeX{}.}]
-
-% Deal with a line breaking problem
-\begin{raggedright}
-\bibitem{mil} G. Gr\"{a}tzer,
-\emph{Math into \LaTeX{}: An Introduction to \LaTeX{} and AMS-\LaTeX{}}
- \url{http://www.ams.org/cgi-bin/bookstore/bookpromo?fn=91&arg1=bookvideo&itmc=MLTEX},
-Birkh\"{a}user, Boston, 1995.\par
-\end{raggedright}
-
-\bibitem{kn} Donald E. Knuth, \booktitle{The \tex/book},
-Addison-Wesley, Reading, MA, 1984.
-
-\bibitem{lamport} Leslie Lamport, \booktitle{\latex/: A document preparation
-system}, 2nd revised ed., Addison-Wesley, Reading, MA, 1994.
-
-\bibitem{msf} Frank Mittelbach and Rainer Sch\"opf,
-\textit{The new font family selection\mdash user
-interface to standard \latex/}, \journalname{TUGboat} \textbf{11},
-no.~2 (June 1990), pp.~297\ndash 305.
-
-\bibitem{jt} Michael Spivak, \booktitle{The joy of \tex/}, 2nd revised ed.,
-Amer. Math. Soc., Providence, RI, 1990.
-
-\end{thebibliography}
-
-\begin{theindex}
-
- \item {\normalfont \ttfamily \bslchar \qcbang }, 11
- \item \verb*+\,+, 11
- \item \verb*+\:+, 11
- \item \verb*+\;+, 11
- \item \verb*+\\+, 3, 5, 6, 8, 9, 24, 26
- \item \verb*+\\*+, 9
- \item \verb*+\|+, 16
-
- \indexspace
-
- \item \verb*+\above+, 14, 29
- \item \verb*+\abovewithdelims+, 14
- \item \verb*+\accentedsymbol+, 2
- \item \texttt{accents} package, 12
- \item \verb*+\addtocounter+, 10
- \item \verb*+\addtolength+, 5
- \item \texttt{align} environment, 3, 5--7, 28--30
- \item \texttt{alignat} environment, 7, 24, 26, 29
- \item \texttt{aligned} environment, 7, 9
- \item \texttt{alignedat} environment, 7, 9
- \item \verb*+\allowdisplaybreaks+, 8, 9
- \item \texttt{amsart} class, 1
- \item \texttt{amsbook} class, 1
- \item \texttt{amsbsy} package, 1, 2
- \item \texttt{amscd} package, 1, 2, 20
- \item \texttt{amsfonts} package, 21
- \item AMSFonts collection, 25
- \item \texttt{amsmath} package, i, 1--3, 5, 8--18, 22, 23, 25, 26,
- 28--32
- \item \texttt{amsopn} package, 1, 2
- \item \texttt{amssymb} package, 22
- \item \texttt{amstext} package, 1, 2
- \item \texttt{amsthm} package, 1, 32
- \item \texttt{amsxtra} package, 1, 2, 12
- \item \verb*+\arccos+, 18
- \item \verb*+\arcsin+, 18
- \item \verb*+\arctan+, 18
- \item \verb*+\arg+, 18
- \item \texttt{array} environment, 8, 10, 26
- \item arrows
- \subitem extensible, 13
- \subitem in commutative diagrams, 20
- \item \verb*+\atop+, 14, 29
- \item \verb*+\atopwithdelims+, 14
-
- \indexspace
-
- \item BaKoMa fonts, 25
- \item \cn{big}, \cn{Big}, \cn{bigg}, \dots\ delimiters, 16
- \item \verb*+\biggl+, 27
- \item \verb*+\biggr+, 27
- \item \verb*+\bigl+, 27
- \item \verb*+\bigr+, 27
- \item \verb*+\binom+, 14
- \item binomials, 14
- \item \texttt{bm} package, 2, 23
- \item \texttt{Bmatrix} environment, 10
- \item \texttt{bmatrix} environment, 10
- \item \verb*+\bmod+, 18
- \item \verb*+\boldsymbol+, 22, 23
- \item \verb*+\boxed+, 13
-
- \indexspace
-
- \item \verb*+\C+, 22
- \item \texttt{cases} environment, 8
- \item \verb*+\cases+, 28
- \item \texttt{CD} environment, 2, 21
- \item \verb*+\cdots+, 12
- \item \texttt{centertags} option, 2
- \item \verb*+\cfrac+, 15
- \item \texttt{cmbsy5}, 25
- \item \texttt{cmex}, 25
- \item \texttt{cmex10}, 25
- \item \texttt{cmex7}, 25
- \item \texttt{cmmib5}, 25
- \item \texttt{comp.text.tex}, 31
- \item \texttt{compan.err}, 32
- \item continued fractions, 15
- \item \verb*+\cos+, 18
- \item \verb*+\cosh+, 18
- \item \verb*+\cot+, 18
- \item \verb*+\coth+, 18
- \item \verb*+\csc+, 18
-
- \indexspace
-
- \item \verb*+\dbinom+, 14
- \item \verb*+\ddddot+, 12
- \item \verb*+\dddot+, 12
- \item \verb*+\ddot+, 12
- \item \verb*+\DeclareMathOperator+, 2, 17, 18
- \item \verb*+\deg+, 18
- \item \verb*+\det+, 18
- \item \verb*+\dfrac+, 14
- \item \texttt{diffs-m.txt}, 30
- \item \verb*+\dim+, 18
- \item \verb*+\displaybreak+, 8, 9
- \item displayed equations, 3
- \item displayed equations
- \subitem centering, 3
- \item \verb*+\displaylimits+, 20
- \item \texttt{displaymath} environment, 3
- \item \verb*+\displaystyle+, 14
- \item \verb*+\documentclass+, 2
- \item \verb*+\dot+, 12
- \item dots, \see{ellipsis dots}{11}
- \item \verb*+\dotsb+, 12
- \item \verb*+\dotsc+, 12
- \item \verb*+\dotsi+, 12
- \item \verb*+\dotsm+, 12
- \item \verb*+\dotso+, 12
-
- \indexspace
-
- \item \texttt{e-math.ams.org}, 31
- \item ellipsis dots
- \subitem in matrices, 11
- \item \texttt{eqnarray} environment, 1, 3, 26, 30
- \item \verb*+\eqref+, 10
- \item \texttt{equation} environment, 1, 3, 5, 7, 24, 30
- \item equation numbers
- \subitem cross-references, 10
- \subitem hierarchy, 9
- \subitem left or right placement, 3
- \subitem overriding, 3
- \subitem subordinate numbering, 10
- \subitem vertical placement, 2
- \item \texttt{equation*} environment, 3
- \item equations, \see{displayed equations}{3}
- \item \texttt{euscript} package, 21
- \item \verb*+\exp+, 18
-
- \indexspace
-
- \item \verb*+\fbox+, 13
- \item \texttt{fleqn} option, 3, 5
- \item \texttt{fntguide.tex}, 21
- \item \verb*+\frac+, 14, 15
- \item fractions, 14
- \item \verb*+\fracwithdelims+, 2
- \item function names, \see{operator names}{17}
-
- \indexspace
-
- \item \texttt{gather} environment, 5--7, 24, 28--30
- \item \texttt{gathered} environment, 7, 9
- \item \verb*+\gcd+, 18
- \item \verb*+\genfrac+, 14
-
- \indexspace
-
- \item \verb*+\hat+, 12
- \item \verb*+\hdotsfor+, 11
- \item \verb*+\hom+, 18
- \item horizontal space
- \subitem around operator names, 17
- \subitem in math mode, 11
-
- \indexspace
-
- \item \verb*+\idotsint+, 20
- \item \verb*+\iiiint+, 20
- \item \verb*+\iiint+, 20
- \item \verb*+\iint+, 20
- \item \verb*+\inf+, 18
- \item \verb*+\injlim+, 18
- \item integrals
- \subitem multiple, 20
- \subitem placement of limits, 2
- \item \verb*+\intertext+, 9
- \item \texttt{intlimits} option, 2, 20
-
- \indexspace
-
- \item \verb*+\ker+, 18
- \item \texttt{kuvio} package, 20
-
- \indexspace
-
- \item \verb*+\label+, 6, 10
- \item \verb*+\langle+, 16
- \item \verb*+\ldots+, 12
- \item \verb*+\left+, 15, 16, 27
- \item \verb*+\leftroot+, 13
- \item \texttt{leqno} option, 3, 5
- \item \verb*+\lg+, 18
- \item \verb*+\lim+, 1, 2, 17, 18
- \item \verb*+\liminf+, 18
- \item limits, \see{subscripts and superscripts}{2}
- \item \verb*+\limits+, 20
- \item \verb*+\limsup+, 18
- \item \verb*+\ln+, 18
- \item \verb*+\log+, 17, 18
- \item \verb*+\lVert+, 16
- \item \verb*+\lvert+, 16
-
- \indexspace
-
- \item math fonts, 21
- \item math symbols, \see{math fonts}{21}
- \item \verb*+\mathbb+, 21, 22
- \item \verb*+\mathbf+, 21, 22
- \item \verb*+\mathcal+, 21
- \item \verb*+\mathfrak+, 21
- \item \verb*+\mathit+, 21
- \item \verb*+\mathrm+, 21
- \item \verb*+\mathscr+, 21
- \item \verb*+\mathsf+, 21
- \item \verb*+\mathtt+, 21
- \item matrices, 10
- \subitem ellipsis dots, 11
- \item \texttt{matrix} environment, 10
- \item \verb*+\matrix+, 28
- \item \verb*+\max+, 18
- \item \verb*+\mbox+, 18
- \item \verb*+\medspace+, 11
- \item Metafont source files, 25
- \item \verb*+\min+, 18
- \item \verb*+\mod+, 18
- \item \verb*+\mspace+, 11
- \item \texttt{multline} environment, 3, 5, 27, 30
- \item \verb*+\multlinegap+, 5
-
- \indexspace
-
- \item \texttt{namelimits} option, 2
- \item \verb*+\negmedspace+, 11
- \item \verb*+\negthickspace+, 11
- \item \verb*+\negthinspace+, 11
- \item \verb*+\newcommand+, 21
- \item \verb*+\nobreakdash+, 12
- \item \texttt{nointlimits} option, 2
- \item \verb*+\nolimits+, 19, 20
- \item \texttt{nonamelimits} option, 2
- \item \texttt{nosumlimits} option, 2
- \item \verb*+\notag+, 3, 5
- \item \verb*+\numberwithin+, 9, 29, 30
-
- \indexspace
-
- \item operator names, 17
- \item \verb*+\operatorname+, 18
- \item \verb*+\operatorname*+, 18
- \item \verb*+\over+, 14, 29
- \item \verb*+\overleftarrow+, 13
- \item \verb*+\overleftrightarrow+, 13
- \item \verb*+\overrightarrow+, 13
- \item \verb*+\overset+, 13
- \item \verb*+\overwithdelims+, 14
-
- \indexspace
-
- \item page breaks, 8
- \item \verb*+\pagebreak+, 8
- \item \texttt{pmatrix} environment, 10
- \item \verb*+\pmatrix+, 28
- \item \verb*+\pmb+, 22
- \item \verb*+\pmod+, 18
- \item \verb*+\pod+, 18
- \item PostScript fonts, 25
- \item \verb*+\Pr+, 18
- \item \verb*+\projlim+, 18
-
- \indexspace
-
- \item \verb*+\qquad+, 11
- \item \verb*+\quad+, 11
-
- \indexspace
-
- \item \verb*+\R+, 22
- \item \verb*+\raisetag+, 8
- \item \verb*+\rangle+, 16
- \item \verb*+\ref+, 10
- \item \texttt{reqno} option, 3, 5
- \item \verb*+\right+, 15, 16, 27
- \item \verb*+\rVert+, 16
- \item \verb*+\rvert+, 16
-
- \indexspace
-
- \item \verb*+\scriptscriptstyle+, 14
- \item \verb*+\scriptstyle+, 14
- \item \verb*+\sec+, 18
- \item \verb*+\setcounter+, 9, 10
- \item \verb*+\setlength+, 5
- \item \verb*+\shoveleft+, 5
- \item \verb*+\shoveright+, 5
- \item \verb*+\sideset+, 13, 19
- \item \verb*+\sin+, 1, 2, 17, 18
- \item \verb*+\sinh+, 18
- \item \texttt{smallmatrix} environment, 11
- \item \verb*+\smash+, 15, 27
- \item \verb*+\sphat+, 12
- \item \texttt{split} environment, 3, 5, 6, 9, 24, 27--30
- \item \verb*+\sptilde+, 12
- \item \verb*+\stackrel+, 13
- \item \texttt{subarray} environment, 19
- \item \texttt{subeqn.tex}, 3
- \item \texttt{subequations} environment, 10
- \item subscripts and superscripts, 13
- \subitem multi-line, 19
- \subitem on sums, 19
- \subitem placement, 2
- \item \verb*+\substack+, 19
- \item \verb*+\sum+, 20, 22
- \item \texttt{sumlimits} option, 2
- \item \verb*+\sup+, 18
- \item superscripts, \see{subscripts and superscripts}{19}
-
- \indexspace
-
- \item \texttt{tabular} environment, 26
- \item \verb*+\tag+, 3
- \item \verb*+\tag*+, 3
- \item \verb*+\tan+, 18
- \item \verb*+\tanh+, 18
- \item \verb*+\tbinom+, 14
- \item \texttt{tbtags} option, 2
- \item \texttt{technote.tex}, 14, 28--30
- \item \texttt{testmath.tex}, 3
- \item \tex/ Users Group, 31
- \item \verb*+\text+, 2, 7--9, 18
- \item text fragments inside math, 9, 18
- \item \verb*+\textstyle+, 14
- \item \verb*+\tfrac+, 14
- \item \verb*+\theequation+, 9, 10
- \item \verb*+\thickspace+, 11
- \item \verb*+\thinspace+, 11
- \item \journalname{TUGboat}, 31
-
- \indexspace
-
- \item \verb*+\underleftarrow+, 13
- \item \verb*+\underleftrightarrow+, 13
- \item \verb*+\underrightarrow+, 13
- \item \verb*+\underset+, 13
- \item \verb*+\uproot+, 13
- \item \verb*+\usepackage+, 2, 25
-
- \indexspace
-
- \item \verb*+\value+, 10
- \item \verb*+\varDelta+, 23
- \item \verb*+\varGamma+, 23
- \item \verb*+\varinjlim+, 18
- \item \verb*+\varLambda+, 23
- \item \verb*+\varliminf+, 18
- \item \verb*+\varlimsup+, 18
- \item \verb*+\varOmega+, 23
- \item \verb*+\varPhi+, 23
- \item \verb*+\varPi+, 23
- \item \verb*+\varprojlim+, 18
- \item \verb*+\varPsi+, 23
- \item \verb*+\varSigma+, 23
- \item \verb*+\varTheta+, 23
- \item \verb*+\varUpsilon+, 23
- \item \verb*+\varXi+, 23
- \item \texttt{Vmatrix} environment, 10
- \item \texttt{vmatrix} environment, 10
- \item \verb*+\vphantom+, 27
-
- \indexspace
-
- \item \verb*+\xleftarrow+, 13
- \item \verb*+\xrightarrow+, 13
-
-\end{theindex}
-
-\end{document}