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author | Karl Berry <karl@freefriends.org> | 2006-01-09 00:49:07 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2006-01-09 00:49:07 +0000 |
commit | 007f67a693e4d031fd3d792df8e4d5f43e2cb2e7 (patch) | |
tree | 90d17e00e572ecb1e24764b6f29c80e098b08d29 /Master/texmf-dist/doc/latex/amslatex/amsldoc.tex | |
parent | 950209b26f70aa87ed07c54f82a95b6f03b7c3a0 (diff) |
doc/latex
git-svn-id: svn://tug.org/texlive/trunk@84 c570f23f-e606-0410-a88d-b1316a301751
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diff --git a/Master/texmf-dist/doc/latex/amslatex/amsldoc.tex b/Master/texmf-dist/doc/latex/amslatex/amsldoc.tex new file mode 100644 index 00000000000..abb532cccb8 --- /dev/null +++ b/Master/texmf-dist/doc/latex/amslatex/amsldoc.tex @@ -0,0 +1,2620 @@ +%%% ==================================================================== +%%% @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} |