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diff --git a/Master/texmf-dist/doc/latex/amslatex/amsldoc.tex b/Master/texmf-dist/doc/latex/amslatex/amsldoc.tex deleted file mode 100644 index abb532cccb8..00000000000 --- a/Master/texmf-dist/doc/latex/amslatex/amsldoc.tex +++ /dev/null @@ -1,2620 +0,0 @@ -%%% ==================================================================== -%%% @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} |