% This is file `short-math-guide.tex'. % % Copyright 1995-2017 % American Mathematical Society % % American Mathematical Society % Technical Support % Publications Technical Group % 201 Charles Street % Providence, RI 02904 % USA % tel: (401) 455-4080 % (800) 321-4267 (USA and Canada only) % fax: (401) 331-3842 % email: tech-support@ams.org % % This work may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either % version 1.3c of this license or (at your option) any % later version. The latest version of this license is in % http://www.latex-project.org/lppl.txt % and version 1.3c or later is part of all distributions of LaTeX % version 2005/12/01 or later. % % This file has the LPPL maintenance status "maintained". % % The Current Maintainer of this work is the American Mathematical % Society. % % Note: When updating, don't forget to update \smgversion, below \begin{filecontents}{mathdoc.sty} \NeedsTeXFormat{LaTeX2e}[1995/12/01] \ProvidesPackage{mathdoc}[2017/12/22 v2.0] \providecommand{\lat}[1]{\protect\LaTeX{}} \providecommand{\mdash}{\textemdash} \providecommand{\ndash}{\textendash} \newcommand{\ntt}{\fontseries{m}\fontshape{n}\ttfamily} \newcommand{\mntt}[1]{\mbox{\ntt #1}} \providecommand{\pkg}[1]{\mntt{#1}} \providecommand{\cls}[1]{\mntt{#1}} \providecommand{\opt}[1]{\mntt{#1}} \providecommand{\fn}[1]{\mntt{#1}} \providecommand{\env}[1]{\mntt{#1}} \chardef\bslash=`\\ \let\charhack=\char \providecommand{\cn}[1]{\mntt{\bslash\charhack`#1}} 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[mjd,1999/11/05] \newenvironment{tex2html_preform}{}{} \providecommand{\strong}{\textbf} \@ifundefined{ht@url}{ \newcommand\htlink{\href} \newenvironment{makeimage}{}{} }{ \renewcommand\htlink[2]{#1\htlinkfootnote{\ht@url{#2}}} \newcommand{\htlinkfootnote}[1]{} } \providecommand{\url}{\texttt} \endinput \end{filecontents} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \errorcontextlines=99 \documentclass{article} \pagestyle{myheadings} %\usepackage{hthtml} \usepackage{amsmath} %\usepackage[cmex10]{amsmath} \usepackage{amssymb} \usepackage{mathrsfs} %\usepackage[mathcal]{euscript} \usepackage{euscript} %\usepackage{mathtime} %\usepackage{stmaryrd} \numberwithin{equation}{section} %% at the moment, using either url or hyperref crashes %% get the content correct before debugging that problem %\usepackage{url} \let\url\texttt \usepackage[breaklinks,colorlinks]{hyperref} \usepackage{xcolor} \definecolor{hycitecolor}{rgb}{0,0.65,0} \usepackage{mathdoc} \hoffset=-1 true in \voffset=-1 true in \topmargin=0.75 true in %% \oddsidemargin\topmargin %\textwidth=210 true mm % A4 \textwidth=139 true mm \textheight=11 true in \advance\textheight-2\topmargin \headheight=7pt \advance\textheight-\headheight \headsep=11pt \advance\textheight-\headsep \oddsidemargin=\paperwidth \advance\oddsidemargin-\textwidth \oddsidemargin=.5\oddsidemargin \evensidemargin=\oddsidemargin \hfuzz=14pt % suppress uninteresting warnings \providecommand{\abs}[1]{\lvert#1\rvert} \newcommand{\colhead}[1]{% \textbf{\begin{tabular}[b]{@{}l@{}}#1\end{tabular}}% } \newcommand{\secref}[1]{Section~\ref{#1}} \newcommand{\tabref}[1]{Table~\ref{#1}} \newcommand{\begend}[1]{% \cn{begin}\texttt{\symbol{123}#1\symbol{125}}% \ \dots\ \cn{end}\texttt{\symbol{123}#1\symbol{125}}% } \newcommand{\dbldollars}{\texttt{\$\$} \dots\ \texttt{\$\$}} \newenvironment{lstack}[1][t]{% \begin{tabular}[#1]{@{}l@{}}% }{% \end{tabular} } \newenvironment{cstack}[1][t]{% \begin{tabular}[#1]{@{}c@{}}% }{% \end{tabular} } \newenvironment{llstack}[1][t]{% \begin{tabular}[#1]{@{}ll@{}}% }{% \end{tabular} } %%\newcommand{\lspx}{\mbox{\rule{5pt}{.6pt}\rule{.6pt}{6pt}}} %%\newcommand{\rspx}{\mbox{\rule[-1pt]{.6pt}{7pt}% %% \rule[-1pt]{5pt}{.6pt}}} %%\newcommand{\lspx}{\mathord{\otimes}} %%\newcommand{\rspx}{\mathord{\odot}} \newcommand{\lspx}{3} \newcommand{\rspx}{4} \newcommand{\spx}[1]{$\lspx #1\rspx$} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\esssup}{ess\,sup} \newcommand{\dotsref}{\leavevmode\unskip\space (see Section~\ref{dots})} \newcommand{\vertref}{\leavevmode\unskip\space (see Section~\ref{verts})} \providecommand{\pdfinfo}[1]{} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{Short Math Guide for \LaTeX{}} \newcommand{\smgversion}{\textup{2.0 (2017/12/22)}} \markboth{Short Math Guide for \LaTeX{}, version \protect\smgversion} {Short Math Guide for \LaTeX{}, version \protect\smgversion} \author{Michael Downes, updated by Barbara Beeton} \date{American Mathematical Society} \pdfinfo{ /Title (Short Math Guide for LaTeX) /Author (Michael Downes, updated by Barbara Beeton) /Subject (Short Math Guide for LaTeX) /Keywords (LaTeX,amsmath,amsfonts,amssymb,equation,math,formula) } \maketitle \begin{center} Version \smgversion, currently available from a link at \mbox{\texttt{https://www.ams.org/tex/amslatex}} \end{center} %%\begin{rawhtml} %%

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%%\end{rawhtml} \setcounter{tocdepth}{3} \tableofcontents \vspace{\fill} \section*{Acknowledgments and plans for future work} Thanks to all who contributed suggestions, assistance and encouragement. Special thanks to David Carlisle for repairing unruly macros and to Jennifer Wright Sharp for applying consistent editing in AMS style. \smallskip\noindent Plans for a future edition include addition of an index. \smallskip\noindent Reports concerning errors and suggestions for improvement should be sent to\\[2pt] \hspace*{\fill} \href{mailto:tech-support@ams.org}{\texttt{tech-support@ams.org}}\,. \hspace{\fill}\null \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} This is a concise summary of recommended features in \LaTeX{} and a couple of extension packages for \textbf{writing math formulas}. Readers needing greater depth of detail are referred to the sources listed in the bibliography, especially \cite{lamport}, \cite{amsldoc}, and \cite{fntguide}. A certain amount of familiarity with standard \LaTeX{} terminology is assumed; if your memory needs refreshing on the \LaTeX{} meaning of \emph{command}, \emph{optional argument}, \emph{environment}, \emph{package}, and so forth, see \cite{lamport}. Most of the features described here are available to you if you use \LaTeX{} with two extension packages published by the American Mathematical Society: \pkg{amssymb} and \pkg{amsmath}. Thus, the source file for this document begins with \begin{verbatim} \documentclass{article} \usepackage{amssymb,amsmath} \end{verbatim} The \pkg{amssymb} package might be omissible for documents whose math symbol usage is relatively modest; in \secref{mathsymbols}, the symbols that require \pkg{amssymb} are marked with \textsuperscript{a} or \textsuperscript{b} (font \fn{msam} or \fn{msbm}). In \secref{alpha-digit}, a few additional fonts are included; the necessary packages are identified there. Many noteworthy features found in other packages are not covered here; see \secref{other-packages}. Regarding math symbols, please note especially that the list given here is not intended to be comprehensive, but to illustrate such symbols as users will normally find already present in their \lat/ system and usable without installing any additional fonts or doing other setup work. If you have a need for a symbol not shown here, you will probably want to consult \emph{The Comprehensive \LaTeX{} Symbol List}~\cite{comprehensive}. %\[\url{http://www.ctan.org/tex-archive/info/symbols/comprehensive/}\] If your \lat/ installation is based on \TeX\,Live, and includes documentation, the list can also be accessed by typing \texttt{texdoc comprehensive} at a system prompt. \begin{table}[p] \caption[]{Multiline equations and equation groups\\ \phantom{Table 1:} (vertical lines indicate nominal margins).} \label{displays} \bigskip \begin{makeimage} \begin{minipage}{\textwidth} \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\\ +o+p+q+r+s \end{multline} \end{verbatim} \producing \begin{multline} a+b+c+d+e+f\\ +i+j+k+l+m+n\\ +o+p+q+r+s \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{alignat}{2} a_1& =b_1+c_1& &+e_1-f_1\\ a_2& =b_2+c_2&{}-d_2&+e_2 \end{alignat} \end{verbatim} \producing \begin{alignat}{2} a_1& =b_1+c_1& &+e_1-f_1\\ a_2& =b_2+c_2&{}-d_2&+e_2 \end{alignat} \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} \def\containsMSABM{TF} \begin{notes} \item Applying \env{*} to any primary environment will suppress the assignment of equation numbers. However, \cn{tag} may be used to apply a visible label, and \cn{eqref} can be used to reference such manually tagged lines. Use of either \env{*} or a \cn{tag} on a subordinate environment is an error. \item The \env{split} environment is something of a special case. It is a subordinate environment that can be used as the contents of an \env{equation} environment or the contents of one \qq{line} in a multiple-equation structure such as \env{align} or \env{gather}. \item The primary environments \env{gather}, \env{align} and \env{alignat} have subordinate ``\env{-ed}'' counterparts (\env{gathered}, \env{aligned} and \env{alignedat}) that can be used as components of more complicated displays, or within in-line math. These ``\env{-ed}'' environments can be positioned vertically using an optional argument \verb+[t]+, \verb+[c]+ or~\verb+[b]+. \item The name \env{flalign} is meant as ``full length'', not ``flush left'' as often mistakenly reported. However, since a display occupying the full width will often begin at the left margin, this confusion is understandable. The indent applied to \env{flalign} from both margins is set with \cn{multlinegap}. \end{notes} \end{minipage} \end{makeimage} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Inline math formulas and displayed equations}\label{first-step} \subsection{The fundamentals} Entering and leaving math mode in \LaTeX{} is normally done with the following commands and environments.% \begin{center} \begin{tabular}{ccc} \colhead{inline formulas}&& \colhead{displayed equations}\\[3pt] \cline{1-1}\cline{3-3}\noalign{\medskip} \begin{cstack} \verb'$' \dots\ \verb'$'\\ \verb'\(' \dots\ \verb'\)' \end{cstack}% && \begin{llstack} \begin{lstack}\verb'\[...\]'\\[6pt]\end{lstack}& unnumbered\\ \begin{lstack} \verb'\begin{equation*}'\\ \dots\\ \verb'\end{equation*}'\\[6pt] \end{lstack}& unnumbered\\ \begin{lstack} \verb'\begin{equation}'\\ \dots\\ \verb'\end{equation}' \end{lstack}& \begin{lstack}automatically\\numbered\end{lstack} \end{llstack} \end{tabular} \begin{notes} % \singlenote \item Do not leave a blank line between text and a displayed equation. This allows a page break at that location, which is bad style. It also causes the spacing between text and display to be incorrect, usually larger than it should be. If a visual break is desired in the input, insert a line containing only a \verb+%+ at the beginning. Leave a blank line between a display and following text only if a new paragraph is intended. \item Do not group multiple display structures in the input (\verb+\[...\]+, \env{equation}, etc.). Instead, use a multiline structure with substructures (\env{split}, \env{aligned}, etc.)\ as appropriate. \item The alternative environments \begend{math} and\\ \begend{displaymath} are seldom needed in practice. Using the plain \TeX{} notation \dbldollars\ for displayed equations is strongly discouraged. Although it is not expressly forbidden in \LaTeX{}, it is not documented anywhere in the \LaTeX{} book as being part of the \LaTeX{} command set, and it interferes with the proper operation of various features such as the \opt{fleqn} option. \item The \env{eqnarray} and \env{eqnarray*} environments described in \cite{lamport} are strongly discouraged because they produce inconsistent spacing of the equal signs and make no attempt to prevent overprinting of the equation body by the equation number. \end{notes} \end{center} Environments for handling equation groups and multiline equations are shown in \tabref{displays}. \newpage \subsection{Automatic numbering and cross-referencing} To get an auto-numbered equation, use the \env{equation} environment; to assign a label for cross-referencing, use the \cn{label} command: \begin{verbatim} \begin{equation}\label{reio} ... \end{equation} \end{verbatim} To get a cross-reference to an auto-numbered equation, use the \cn{eqref} command: \begin{verbatim} ... using equations~\eqref{ax1} and~\eqref{bz2}, we can derive ... \end{verbatim} The above example would produce something like \begin{quote} using equations (3.2) and (3.5), we can derive \end{quote} In other words, \verb'\eqref{ax1}' is equivalent to \verb'(\ref{ax1})', but the parentheses produced by \cn{eqref} are always upright. To give your equation numbers the form \textit{m.n} (\textit{section-number.equation-number}), use the \cn{numberwithin} command in the preamble of your document: \begin{verbatim} \numberwithin{equation}{section} \end{verbatim} For more details on custom numbering schemes see \cite[\S 6.3, \S C.8.4]{lamport}. The \env{subequations} environment provides a convenient way to number equations in a group with a subordinate numbering scheme. For example, supposing that the current equation number is \theequation, write \begin{verbatim} \begin{equation}\label{first} a=b+c \end{equation} some intervening text \begin{subequations}\label{grp} \begin{align} a&=b+c\label{second}\\ d&=e+f+g\label{third}\\ h&=i+j\label{fourth} \end{align} \end{subequations} \end{verbatim} to get \begin{equation}\label{first} a=b+c \end{equation} some intervening text \begin{subequations}\label{grp} \begin{align} a&=b+c\label{second}\\ d&=e+f+g\label{third}\\ h&=i+j\label{fourth} \end{align} \end{subequations} By putting a \cn{label} command immediately after \verb'\begin{subequations}' you can get a reference to the parent number; \verb'\eqref{grp}' from the above example would produce \eqref{grp} while \verb'\eqref{second}' would produce \eqref{second}. An example at \url{https://tex.stackexchange.com/questions/220001/} shows a variant of the above example, with numbering like (2.1), (2.1a), \dots, rather than (2.1), (2.2a), \dots. This is accomplished by using \cn{tag} with a cross-reference to the principal component of the subequation number. \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \providecommand{\dotsref}{\leavevmode\unskip\ignorespaces} \providecommand{\vertref}{\leavevmode\unskip\ignorespaces} \section{Math symbols and math fonts}\label{mathsymbols} \subsection{Classes of math symbols} The symbols in a math formula fall into different classes that correspond more or less to the part of speech each symbol would have if the formula were expressed in words. Certain spacing and positioning cues are traditionally used for the different symbol classes to increase the readability of formulas. \begin{center} \begin{tabular}{clll} \colhead{Class\\number}& \colhead{Mnemonic}& \colhead{Description\\(part of speech)}& \colhead{Examples}\\\hline\noalign{\smallskip} 0& Ord& simple/ordinary (\qq{noun})& $A\;0\;\Phi\;\infty$\\ 1& Op& prefix operator& $\sum\;\prod\;\int$\\ 2& Bin& binary operator (conjunction)& ${+}\;{\cup}\;{\wedge}$\\ 3& Rel& relation/comparison (verb)& ${=}\;{<}\;{\subset}$\\ 4& Open& left/opening delimiter& $(\;{[}\;{\lbrace}\;{\langle}$\\ 5& Close& right/closing delimiter& $)\;{]}\;{\rbrace}\;{\rangle}$\\ 6& Punct& postfix/punctuation& ${.}\;{,}\;{;}\;{!}$\\ \end{tabular} \end{center} \begin{notes} \item The distinction in \TeX{} between class 0 and an additional class 7 has to do only with font selection issues, and it is immaterial here. \item Symbols of class 2 (Bin), notably the minus sign $-$, are automatically printed by \LaTeX{} as class~0 (no space) if they do not have a suitable left operand\mdash e.g., at the beginning of a math formula or after an opening delimiter. \end{notes} The spacing for a few symbols follows tradition instead of the general rule: although $/$ is (semantically speaking) of class~2, we write $k/2$ with no space around the slash rather than $k\mathbin{/}2$. And compare \verb'p|q' $p\vert q$ (no space) with \verb'p\mid q' $p\mid q$ (class-3 spacing). The proper way to define a new math symbol is discussed in \emph{\LaTeXe{} font selection} \cite{fntguide}. It is not really possible to give a useful synopsis here because one needs first to understand the ramifications of font specifications. But supposing one knows that a Cyrillic font named \fn{wncyr10} is available, here is a minimal example showing how to define a \LaTeX{} command to print one letter from that font as a math symbol: \begin{verbatim} % Declare that the combination of font attributes OT2/wncyr/m/n % should select the wncyr font. \DeclareFontShape{OT2}{wncyr}{m}{n}{<->wncyr10}{} % Declare that the symbolic math font name "cyr" should resolve to % OT2/wncyr/m/n. \DeclareSymbolFont{cyr}{OT2}{wncyr}{m}{n} % Declare that the command \Sh should print symbol 88 from the math font % "cyr", and that the symbol class is 0 (= alphabetic = Ord). \DeclareMathSymbol{\Sh}{\mathalpha}{cyr}{88} \end{verbatim} \subsection{Some symbols intentionally omitted here} The following math symbols that are mentioned in the \LaTeX{} book \cite{lamport} are intentionally omitted from this discussion because they are superseded by equivalent symbols when the \pkg{amssymb} package is loaded. If you are using the \pkg{amssymb} package anyway, the only thing that you are likely to gain by using the alternate name is an unnecessary increase in the number of fonts used by your document. \begin{center} \def\jdo#1{\cn{#1} \ $\csname #1\endcsname$} \begin{tabular}{r@{\,, see \ }l} \cn{Box}&\jdo{square}\\ \cn{Diamond}&\jdo{lozenge}\\ \cn{leadsto}&\jdo{rightsquigarrow}\\ \cn{Join}&\jdo{bowtie}\\ \cn{lhd}&\jdo{vartriangleleft}\\ \cn{unlhd}&\jdo{trianglelefteq}\\ \cn{rhd}&\jdo{vartriangleright}\\ \cn{unrhd}&\jdo{trianglerighteq} \end{tabular} \end{center} Furthermore, there are \strong{many, many additional symbols} available for \lat/ use above and beyond the ones included here. This list is not intended to be comprehensive. For a much more comprehensive list of symbols, including nonmathematically oriented ones, such as phonetic alphabetic or dingbats, see \emph{The Comprehensive \LaTeX{} Symbol List}~\cite{comprehensive}. (Full font tables, ordered by font name, for all the fonts covered by the comprehensive list are included in the documentation provided by \TeX~Live: \texttt{texdoc rawtables}. These tables do not include symbol names.) Another source of symbol information is the \pkg{unicode-math} package; see~\cite{uc-math}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\newpage \subsection{Alphabets and digits\nopunct}\label{alpha-digit} \subsubsection{Latin letters and Arabic numerals} The Latin letters are simple symbols, class~0. The default font for them in math formulas is italic. \begin{center} \begin{tabular}{c} $A\,B\,C\,D\,E\,F\,G\,H\,I\,J\,K\,L\,M% \,N\,O\,P\,Q\,R\,S\,T\,U\,V\,W\,X\,Y\,Z$\\ $a\,b\,c\,d\,e\,f\,g\,h\,i\,j\,k\,l\,m% \,n\,o\,p\,q\,r\,s\,t\,u\,v\,w\,x\,y\,z$ \end{tabular} \end{center} When adding an accent to an $i$ or $j$ in math, dotless variants can be obtained with \cn{imath} and \cn{jmath}: \begin{symlist} \dosymbol{Var}{imath}{kernel} \dosymbol{Var}{jmath}{kernel} \symbox{\hat{\jmath}}{\string\hat\string{\string\jmath\string}} \end{symlist} Arabic numerals 0\ndash 9 are also of class~0. Their default font is upright/roman. \[0\,1\,2\,3\,4\,5\,6\,7\,8\,9\] \subsubsection{Greek letters} Like the Latin letters, the Greek letters are simple symbols, class~0. For obscure historical reasons, the default font for lowercase Greek letters in math formulas is italic while the default font for capital Greek letters is upright/roman. (In other fields such as physics and chemistry, however, the typographical traditions are somewhat different.) The capital Greek letters not present in this list are the letters that have the same appearance as some Latin letter: A for Alpha, B for Beta, and so on. In the list of lowercase letters there is no omicron because it would be identical in appearance to Latin $o$. In practice, the Greek letters that have Latin look-alikes are seldom used in math formulas, to avoid confusion. \begin{symlist}[adjustheight=12pt] \dosymbol{Var}{Gamma}{kernel} \dosymbol{Var}{Delta}{kernel} \dosymbol{Var}{Lambda}{kernel} \dosymbol{Var}{Phi}{kernel} \dosymbol{Var}{Pi}{kernel} \dosymbol{Var}{Psi}{kernel} \dosymbol{Var}{Sigma}{kernel} \dosymbol{Var}{Theta}{kernel} \dosymbol{Var}{Upsilon}{kernel} \dosymbol{Var}{Xi}{kernel} \dosymbol{Var}{Omega}{kernel} \newcolumn \dosymbol{Var}{alpha}{kernel} \dosymbol{Var}{beta}{kernel} \dosymbol{Var}{gamma}{kernel} \dosymbol{Var}{delta}{kernel} \dosymbol{Var}{epsilon}{kernel} \dosymbol{Var}{zeta}{kernel} \dosymbol{Var}{eta}{kernel} \dosymbol{Var}{theta}{kernel} \dosymbol{Var}{iota}{kernel} \dosymbol{Var}{kappa}{kernel} \dosymbol{Var}{lambda}{kernel} \dosymbol{Var}{mu}{kernel} \newcolumn \dosymbol{Var}{nu}{kernel} \dosymbol{Var}{xi}{kernel} %\dosymbol{Var}{omicron}{??} \dosymbol{Var}{pi}{kernel} \dosymbol{Var}{rho}{kernel} \dosymbol{Var}{sigma}{kernel} \dosymbol{Var}{tau}{kernel} \dosymbol{Var}{upsilon}{kernel} \dosymbol{Var}{phi}{kernel} \dosymbol{Var}{chi}{kernel} \dosymbol{Var}{psi}{kernel} \dosymbol{Var}{omega}{kernel} \newcolumn \dosymbol{Ord}{digamma}{amssymb} \dosymbol{Var}{varepsilon}{kernel} \dosymbol{Ord}{varkappa}{amssymb} \dosymbol{Var}{varphi}{kernel} \dosymbol{Var}{varpi}{kernel} \dosymbol{Var}{varrho}{kernel} \dosymbol{Var}{varsigma}{kernel} \dosymbol{Var}{vartheta}{kernel} \end{symlist} \subsubsection{Other ``basic'' alphabetic symbols} These are also class~0. \begin{symlist} \dosymbol{Ord}{aleph}{kernel} \dosymbol{Ord}{beth}{amssymb} \dosymbol{Ord}{daleth}{amssymb} \dosymbol{Ord}{gimel}{amssymb} \dosymbol{Ord}{complement}{amssymb} \dosymbol{Ord}{ell}{kernel} \dosymbol{Ord}{eth}{amssymb} \dosymbol{Ord}{hbar}{amssymb} \dosymbol{Ord}{hslash}{amssymb} \dosymbol{Ord}{mho}{amssymb} \dosymbol{Ord}{partial}{kernel} \dosymbol{Ord}{wp}{kernel} \dosymbol{Ord}{circledS}{amssymb} \dosymbol{Ord}{Bbbk}{amssymb} \dosymbol{Ord}{Finv}{amssymb} \dosymbol{Ord}{Game}{amssymb} \dosymbol{Ord}{Im}{kernel} \dosymbol{Ord}{Re}{kernel} \vfil \end{symlist} \begin{notes} \end{notes} \subsubsection{Math font switches}\label{mathfonts} Not all of the fonts necessary to support comprehensive math font switching are commonly available in a typical \LaTeX{} setup. Here are the results of applying various font switches to a wide range of math symbols when the standard set of Computer Modern fonts is in use. It can be seen that the only symbols that respond correctly to all of the font switches are the uppercase Latin letters. In fact, \emph{nearly all} math symbols apart from Latin letters remain unaffected by font switches; and although the lowercase Latin letters, capital Greek letters, and numerals do respond properly to some font switches, they produce bizarre results for other font switches. (Use of alternative math font sets such as Lucida New Math may ameliorate the situation somewhat.) \[\renewcommand{\arraystretch}{1.3} \begin{array}{cccccccc} %\text{default}& \cn{mathbf}& \cn{mathsf}& \cn{mathit}& \cn{mathcal}& % \cn{mathscr}& \cn{mathbb}& \cn{mathfrak}\\ \text{default}& \cn{mathbf}& \cn{mathrm}& \cn{mathsf}& \cn{mathit}& \cn{mathcal}& \cn{mathbb}& \cn{mathfrak}\\ \hline \symrow{X} \\ \symrow{x} \\ \symrow{0} \\ \symrow{[\,]} \\ \symrow{+} \\ \symrow{-} \\ \symrow{=} \\ \symrow{\Xi} \\ \symrow{\xi} \\ \symrow{\infty} \\ \symrow{\aleph} \\ \symrow{\sum}\\ \symrow{\amalg} \\ \symrow{\Re} \\ \end{array}\] A common desire is to get a bold version of a particular math symbol. For those symbols where \cn{mathbf} is not applicable, the \cn{boldsymbol} or \cn{pmb} commands can be used. \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} \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} The \cn{boldsymbol} command is obtained preferably by using the \pkg{bm} package, which provides a newer, more powerful version than the one provided by the \pkg{amsmath} package. It is usually ill-advised to apply \cn{boldsymbol} to more than one symbol at a time; if such a need seems to arise, it more likely means that there is another, better way of going about it. \subsubsection{Blackboard Bold letters (\fn{msbm}; no lowercase)} Usage: \verb'\mathbb{R}'. Requires \pkg{amsfonts}. \[ \mathbb{A}\,\mathbb{B}\,\mathbb{C}\,\mathbb{D}\,\mathbb{E}\,\mathbb{F} \,\mathbb{G}\,\mathbb{H}\,\mathbb{I}\,\mathbb{J}\,\mathbb{K}\,\mathbb{L} \,\mathbb{M}\,\mathbb{N}\,\mathbb{O}\,\mathbb{P}\,\mathbb{Q}\,\mathbb{R} \,\mathbb{S}\,\mathbb{T}\,\mathbb{U}\,\mathbb{V}\,\mathbb{W}\,\mathbb{X} \,\mathbb{Y}\,\mathbb{Z} \] One lowercase letter is available with a distinct name:\qquad $\Bbbk$\quad \cn{Bbbk} %\newpage \subsubsection{Calligraphic letters (\fn{cmsy}; no lowercase)} Usage: \verb'\mathcal{M}'. \[ \mathcal{A}\,\mathcal{B}\,\mathcal{C}\,\mathcal{D}\,\mathcal{E} \,\mathcal{F}\,\mathcal{G}\,\mathcal{H}\,\mathcal{I}\,\mathcal{J} \,\mathcal{K}\,\mathcal{L}\,\mathcal{M}\,\mathcal{N}\,\mathcal{O} \,\mathcal{P}\,\mathcal{Q}\,\mathcal{R}\,\mathcal{S}\,\mathcal{T} \,\mathcal{U}\,\mathcal{V}\,\mathcal{W}\,\mathcal{X}\,\mathcal{Y} \,\mathcal{Z} \] \subsubsection{Non-CM calligraphic and script letters} (\fn{rsfs}; no lowercase) Usage: \verb'\usepackage{mathrsfs}' \verb'\mathscr{B}'. \[ \mathscr{A}\,\mathscr{B}\,\mathscr{C}\,\mathscr{D}\,\mathscr{E} \,\mathscr{F}\,\mathscr{G}\,\mathscr{H}\,\mathscr{I}\,\mathscr{J} \,\mathscr{K}\,\mathscr{L}\,\mathscr{M}\,\mathscr{N}\,\mathscr{O} \,\mathscr{P}\,\mathscr{Q}\,\mathscr{R}\,\mathscr{S}\,\mathscr{T} \,\mathscr{U}\,\mathscr{V}\,\mathscr{W}\,\mathscr{X}\,\mathscr{Y} \,\mathscr{Z} \] \begingroup \noindent (\fn{eusm}; no lowercase) Usage: \verb'\usepackage{euscript}' \verb'\mathscr{E}'. \renewcommand{\mathscr}{\EuScript} \[ \mathscr{A}\,\mathscr{B}\,\mathscr{C}\,\mathscr{D}\,\mathscr{E} \,\mathscr{F}\,\mathscr{G}\,\mathscr{H}\,\mathscr{I}\,\mathscr{J} \,\mathscr{K}\,\mathscr{L}\,\mathscr{M}\,\mathscr{N}\,\mathscr{O} \,\mathscr{P}\,\mathscr{Q}\,\mathscr{R}\,\mathscr{S}\,\mathscr{T} \,\mathscr{U}\,\mathscr{V}\,\mathscr{W}\,\mathscr{X}\,\mathscr{Y} \,\mathscr{Z} \] \endgroup \subsubsection{Fraktur letters (\fn{eufm})} Usage: \verb'\mathfrak{S}'. Requires \pkg{amsfonts}. \[ \mathfrak{A}\,\mathfrak{B}\,\mathfrak{C}\,\mathfrak{D}\,\mathfrak{E} \,\mathfrak{F}\,\mathfrak{G}\,\mathfrak{H}\,\mathfrak{I}\,\mathfrak{J} \,\mathfrak{K}\,\mathfrak{L}\,\mathfrak{M}\,\mathfrak{N}\,\mathfrak{O} \,\mathfrak{P}\,\mathfrak{Q}\,\mathfrak{R}\,\mathfrak{S}\,\mathfrak{T} \,\mathfrak{U}\,\mathfrak{V}\,\mathfrak{W}\,\mathfrak{X}\,\mathfrak{Y} \,\mathfrak{Z} \] \[ \mathfrak{a}\,\mathfrak{b}\,\mathfrak{c}\,\mathfrak{d}\,\mathfrak{e} \,\mathfrak{f}\,\mathfrak{g}\,\mathfrak{h}\,\mathfrak{i}\,\mathfrak{j} \,\mathfrak{k}\,\mathfrak{l}\,\mathfrak{m}\,\mathfrak{n}\,\mathfrak{o} \,\mathfrak{p}\,\mathfrak{q}\,\mathfrak{r}\,\mathfrak{s}\,\mathfrak{t} \,\mathfrak{u}\,\mathfrak{v}\,\mathfrak{w}\,\mathfrak{x}\,\mathfrak{y} \,\mathfrak{z} \] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Miscellaneous simple symbols} These symbols are also of class~0 (ordinary) which means they do not have any built-in spacing.% \begin{symlist} \dosymbol{Ordx}{\#}{kernel} \dosymbol{Ordx}{\&}{kernel} \dosymbol{Ord}{angle}{amssymb} \dosymbol{Ord}{backprime}{amssymb} \dosymbol{Ord}{bigstar}{amssymb} \dosymbol{Ord}{blacklozenge}{amssymb} \dosymbol{Ord}{blacksquare}{amssymb} \dosymbol{Ord}{blacktriangle}{amssymb} \dosymbol{Ord}{blacktriangledown}{amssymb} \dosymbol{Ord}{bot}{kernel} \dosymbol{Ord}{clubsuit}{kernel} \dosymbol{Ord}{diagdown}{amssymb} \dosymbol{Ord}{diagup}{amssymb} \dosymbol{Ord}{diamondsuit}{kernel} \dosymbol{Ord}{emptyset}{kernel} \dosymbol{Ord}{exists}{kernel} \dosymbol{Ord}{flat}{kernel} \dosymbol{Ord}{forall}{kernel} \dosymbol{Ord}{heartsuit}{kernel} \dosymbol{Ord}{infty}{kernel} \dosymbol{Ord}{lozenge}{amssymb} \dosymbol{Ord}{measuredangle}{amssymb} \dosymbol{Ord}{nabla}{kernel} \dosymbol{Ord}{natural}{kernel} \dosymbol{Ord}{neg}{kernel} \dosymbol{Ord}{nexists}{amssymb} \dosymbol{Ord}{prime}{kernel} \dosymbol{Ord}{sharp}{kernel} \dosymbol{Ord}{spadesuit}{kernel} \dosymbol{Ord}{sphericalangle}{amssymb} \dosymbol{Ord}{square}{amssymb} \dosymbol{Ord}{surd}{kernel} \dosymbol{Ord}{top}{kernel} \dosymbol{Ord}{triangle}{kernel} \dosymbol{Ord}{triangledown}{amssymb} \dosymbol{Ord}{varnothing}{amssymb} \end{symlist} \begin{notes} \item A common mistake in the use of the symbols $\square$ and $\#$ is to try to make them serve as binary operators or relation symbols without using a properly defined math symbol command. If you merely use the existing commands \cn{square} or \cn{\#} the intersymbol spacing will be incorrect because those commands produce a class-0 symbol. \item Synonyms: \alias{lnot} \end{notes} \subsection{Binary operator symbols\nopunct} \begin{symlist} \dosymbol{Binc}{*}{kernel} \dosymbol{Binc}{+}{kernel} \dosymbol{Binc}{-}{kernel} \dosymbol{Bin}{amalg}{kernel} \dosymbol{Bin}{ast}{kernel} \dosymbol{Bin}{barwedge}{amssymb} \dosymbol{Bin}{bigcirc}{kernel} \dosymbol{Bin}{bigtriangledown}{kernel} \dosymbol{Bin}{bigtriangleup}{kernel} \dosymbol{Bin}{boxdot}{amssymb} \dosymbol{Bin}{boxminus}{amssymb} \dosymbol{Bin}{boxplus}{amssymb} \dosymbol{Bin}{boxtimes}{amssymb} \dosymbol{Bin}{bullet}{kernel} \dosymbol{Bin}{cap}{kernel} \dosymbol{Bin}{Cap}{amssymb} \dosymbol{Bin}{cdot}{kernel} \dosymbol{Bin}{centerdot}{amssymb} \dosymbol{Bin}{circ}{kernel} \dosymbol{Bin}{circledast}{amssymb} \dosymbol{Bin}{circledcirc}{amssymb} \dosymbol{Bin}{circleddash}{amssymb} \dosymbol{Bin}{cup}{kernel} \dosymbol{Bin}{Cup}{amssymb} \dosymbol{Bin}{curlyvee}{amssymb} \dosymbol{Bin}{curlywedge}{amssymb} \dosymbol{Bin}{dagger}{kernel} \dosymbol{Bin}{ddagger}{kernel} \dosymbol{Bin}{diamond}{kernel} \dosymbol{Bin}{div}{kernel} \dosymbol{Bin}{divideontimes}{amssymb} \dosymbol{Bin}{dotplus}{amssymb} \dosymbol{Bin}{doublebarwedge}{amssymb} \dosymbol{Bin}{gtrdot}{amssymb} \dosymbol{Bin}{intercal}{amssymb} \dosymbol{Bin}{leftthreetimes}{amssymb} \dosymbol{Bin}{lessdot}{amssymb} \dosymbol{Bin}{ltimes}{amssymb} \dosymbol{Bin}{mp}{kernel} \dosymbol{Bin}{odot}{kernel} \dosymbol{Bin}{ominus}{kernel} \dosymbol{Bin}{oplus}{kernel} \dosymbol{Bin}{oslash}{kernel} \dosymbol{Bin}{otimes}{kernel} \dosymbol{Bin}{pm}{kernel} \dosymbol{Bin}{rightthreetimes}{amssymb} \dosymbol{Bin}{rtimes}{amssymb} \dosymbol{Bin}{setminus}{kernel} \dosymbol{Bin}{smallsetminus}{amssymb} \dosymbol{Bin}{sqcap}{kernel} \dosymbol{Bin}{sqcup}{kernel} \dosymbol{Bin}{star}{kernel} \dosymbol{Bin}{times}{kernel} \dosymbol{Bin}{triangleleft}{kernel} \dosymbol{Bin}{triangleright}{kernel} \dosymbol{Bin}{uplus}{kernel} \dosymbol{Bin}{vee}{kernel} \dosymbol{Bin}{veebar}{amssymb} \dosymbol{Bin}{wedge}{kernel} \dosymbol{Bin}{wr}{kernel} \end{symlist} \begin{notes} \synonyms \alias{land}, \alias{lor}, \alias{doublecup}, \alias{doublecap} \end{notes} \subsection{Relation symbols: \texorpdfstring{$<$ $=$ $>$ $\succ$ $\sim$}{< + > succeed ~} and variants\nopunct} \begin{symlist}[adjustheight=10pt] \dosymbol{Relc}{<}{kernel} \dosymbol{Relc}{=}{kernel} \dosymbol{Relc}{>}{kernel} \dosymbol{Rel}{approx}{kernel} \dosymbol{Rel}{approxeq}{amssymb} \dosymbol{Rel}{asymp}{kernel} \dosymbol{Rel}{backsim}{amssymb} \dosymbol{Rel}{backsimeq}{amssymb} \dosymbol{Rel}{bumpeq}{amssymb} \dosymbol{Rel}{Bumpeq}{amssymb} \dosymbol{Rel}{circeq}{amssymb} \dosymbol{Rel}{cong}{kernel} \dosymbol{Rel}{curlyeqprec}{amssymb} \dosymbol{Rel}{curlyeqsucc}{amssymb} \dosymbol{Rel}{doteq}{kernel} \dosymbol{Rel}{doteqdot}{amssymb} \dosymbol{Rel}{eqcirc}{amssymb} \dosymbol{Rel}{eqsim}{amssymb} \dosymbol{Rel}{eqslantgtr}{amssymb} \dosymbol{Rel}{eqslantless}{amssymb} \dosymbol{Rel}{equiv}{kernel} \dosymbol{Rel}{fallingdotseq}{amssymb} \dosymbol{Rel}{geq}{kernel} \dosymbol{Rel}{geqq}{amssymb} \dosymbol{Rel}{geqslant}{amssymb} \dosymbol{Rel}{gg}{kernel} \dosymbol{Rel}{ggg}{amssymb} \dosymbol{Rel}{gnapprox}{amssymb} \dosymbol{Rel}{gneq}{amssymb} \dosymbol{Rel}{gneqq}{amssymb} \dosymbol{Rel}{gnsim}{amssymb} \dosymbol{Rel}{gtrapprox}{amssymb} \dosymbol{Rel}{gtreqless}{amssymb} \dosymbol{Rel}{gtreqqless}{amssymb} \dosymbol{Rel}{gtrless}{amssymb} \dosymbol{Rel}{gtrsim}{amssymb} \dosymbol{Rel}{gvertneqq}{amssymb} \dosymbol{Rel}{leq}{kernel} \dosymbol{Rel}{leqq}{amssymb} \dosymbol{Rel}{leqslant}{amssymb} \dosymbol{Rel}{lessapprox}{amssymb} \dosymbol{Rel}{lesseqgtr}{amssymb} \dosymbol{Rel}{lesseqqgtr}{amssymb} \dosymbol{Rel}{lessgtr}{amssymb} \dosymbol{Rel}{lesssim}{amssymb} \dosymbol{Rel}{ll}{kernel} \dosymbol{Rel}{lll}{amssymb} \dosymbol{Rel}{lnapprox}{amssymb} \dosymbol{Rel}{lneq}{amssymb} \dosymbol{Rel}{lneqq}{amssymb} \dosymbol{Rel}{lnsim}{amssymb} \dosymbol{Rel}{lvertneqq}{amssymb} \dosymbol{Rel}{ncong}{amssymb} \dosymbol{Rel}{neq}{kernel} \dosymbol{Rel}{ngeq}{amssymb} \dosymbol{Rel}{ngeqq}{amssymb} \dosymbol{Rel}{ngeqslant}{amssymb} \dosymbol{Rel}{ngtr}{amssymb} \dosymbol{Rel}{nleq}{amssymb} \dosymbol{Rel}{nleqq}{amssymb} \dosymbol{Rel}{nleqslant}{amssymb} \dosymbol{Rel}{nless}{amssymb} \dosymbol{Rel}{nprec}{amssymb} \dosymbol{Rel}{npreceq}{amssymb} \dosymbol{Rel}{nsim}{amssymb} \dosymbol{Rel}{nsucc}{amssymb} \dosymbol{Rel}{nsucceq}{amssymb} \dosymbol{Rel}{prec}{kernel} \dosymbol{Rel}{precapprox}{amssymb} \dosymbol{Rel}{preccurlyeq}{amssymb} \dosymbol{Rel}{preceq}{kernel} \dosymbol{Rel}{precnapprox}{amssymb} \dosymbol{Rel}{precneqq}{amssymb} \dosymbol{Rel}{precnsim}{amssymb} \dosymbol{Rel}{precsim}{amssymb} \dosymbol{Rel}{risingdotseq}{amssymb} \dosymbol{Rel}{sim}{kernel} \dosymbol{Rel}{simeq}{kernel} \dosymbol{Rel}{succ}{kernel} \dosymbol{Rel}{succapprox}{amssymb} \dosymbol{Rel}{succcurlyeq}{amssymb} \dosymbol{Rel}{succeq}{kernel} \dosymbol{Rel}{succnapprox}{amssymb} \dosymbol{Rel}{succneqq}{amssymb} \dosymbol{Rel}{succnsim}{amssymb} \dosymbol{Rel}{succsim}{amssymb} \dosymbol{Rel}{thickapprox}{amssymb} \dosymbol{Rel}{thicksim}{amssymb} \dosymbol{Rel}{triangleq}{amssymb} \end{symlist} \begin{notes} \synonyms \alias{ne}, \alias{le}, \alias{ge}, \alias{Doteq}, \alias{llless}, \alias{gggtr} \end{notes} \subsection{Relation symbols: arrows} See also \secref{notations}. \begin{symlist}[adjustheight=10pt] \dosymbol{Rel}{circlearrowleft}{amssymb} \dosymbol{Rel}{circlearrowright}{amssymb} \dosymbol{Rel}{curvearrowleft}{amssymb} \dosymbol{Rel}{curvearrowright}{amssymb} \dosymbol{Rel}{downdownarrows}{amssymb} \dosymbol{Rel}{downharpoonleft}{amssymb} \dosymbol{Rel}{downharpoonright}{amssymb} \dosymbol{Rel}{hookleftarrow}{kernel} \dosymbol{Rel}{hookrightarrow}{kernel} \dosymbol{Rel}{leftarrow}{kernel} \dosymbol{Rel}{Leftarrow}{kernel} \dosymbol{Rel}{leftarrowtail}{amssymb} \dosymbol{Rel}{leftharpoondown}{kernel} \dosymbol{Rel}{leftharpoonup}{kernel} \dosymbol{Rel}{leftleftarrows}{amssymb} \dosymbol{Rel}{leftrightarrow}{kernel} \dosymbol{Rel}{Leftrightarrow}{kernel} \dosymbol{Rel}{leftrightarrows}{amssymb} \dosymbol{Rel}{leftrightharpoons}{amssymb} \dosymbol{Rel}{leftrightsquigarrow}{amssymb} \dosymbol{Rel}{Lleftarrow}{amssymb} \dosymbol{Rel}{longleftarrow}{kernel} \dosymbol{Rel}{Longleftarrow}{kernel} \dosymbol{Rel}{longleftrightarrow}{kernel} \dosymbol{Rel}{Longleftrightarrow}{kernel} \dosymbol{Rel}{longmapsto}{kernel} \dosymbol{Rel}{longrightarrow}{kernel} \dosymbol{Rel}{Longrightarrow}{kernel} \dosymbol{Rel}{looparrowleft}{amssymb} \dosymbol{Rel}{looparrowright}{amssymb} \dosymbol{Rel}{Lsh}{amssymb} \dosymbol{Rel}{mapsto}{kernel} \dosymbol{Rel}{multimap}{amssymb} \dosymbol{Rel}{nLeftarrow}{amssymb} \dosymbol{Rel}{nLeftrightarrow}{amssymb} \dosymbol{Rel}{nRightarrow}{amssymb} \dosymbol{Rel}{nearrow}{kernel} \dosymbol{Rel}{nleftarrow}{amssymb} \dosymbol{Rel}{nleftrightarrow}{amssymb} \dosymbol{Rel}{nrightarrow}{amssymb} \dosymbol{Rel}{nwarrow}{kernel} \dosymbol{Rel}{rightarrow}{kernel} \dosymbol{Rel}{Rightarrow}{kernel} \dosymbol{Rel}{rightarrowtail}{amssymb} \dosymbol{Rel}{rightharpoondown}{kernel} \dosymbol{Rel}{rightharpoonup}{kernel} \dosymbol{Rel}{rightleftarrows}{amssymb} \dosymbol{Rel}{rightleftharpoons}{amssymb} \dosymbol{Rel}{rightrightarrows}{amssymb} \dosymbol{Rel}{rightsquigarrow}{amssymb} \dosymbol{Rel}{Rrightarrow}{amssymb} \dosymbol{Rel}{Rsh}{amssymb} \dosymbol{Rel}{searrow}{kernel} \dosymbol{Rel}{swarrow}{kernel} \dosymbol{Rel}{twoheadleftarrow}{amssymb} \dosymbol{Rel}{twoheadrightarrow}{amssymb} \dosymbol{Rel}{upharpoonleft}{amssymb} \dosymbol{Rel}{upharpoonright}{amssymb} \dosymbol{Rel}{upuparrows}{amssymb} \end{symlist} \begin{notes} \synonyms \alias{gets}, \alias{to}, \alias{restriction} \end{notes} \subsection{Relation symbols: miscellaneous\nopunct} \begin{symlist}[adjustheight=10pt] \dosymbol{Rel}{backepsilon}{amssymb} \dosymbol{Rel}{because}{amssymb} \dosymbol{Rel}{between}{amssymb} \dosymbol{Rel}{blacktriangleleft}{amssymb} \dosymbol{Rel}{blacktriangleright}{amssymb} \dosymbol{Rel}{bowtie}{kernel} \dosymbol{Rel}{dashv}{kernel} \dosymbol{Rel}{frown}{kernel} \dosymbol{Rel}{in}{kernel} \dosymbol{Rel}{mid}{kernel} \dosymbol{Rel}{models}{kernel} \dosymbol{Rel}{ni}{kernel} \dosymbol{Rel}{nmid}{amssymb} \dosymbol{Rel}{notin}{kernel} \dosymbol{Rel}{nparallel}{amssymb} \dosymbol{Rel}{nshortmid}{amssymb} \dosymbol{Rel}{nshortparallel}{amssymb} \dosymbol{Rel}{nsubseteq}{amssymb} \dosymbol{Rel}{nsubseteqq}{amssymb} \dosymbol{Rel}{nsupseteq}{amssymb} \dosymbol{Rel}{nsupseteqq}{amssymb} \dosymbol{Rel}{ntriangleleft}{amssymb} \dosymbol{Rel}{ntrianglelefteq}{amssymb} \dosymbol{Rel}{ntriangleright}{amssymb} \dosymbol{Rel}{ntrianglerighteq}{amssymb} \dosymbol{Rel}{nvdash}{amssymb} \dosymbol{Rel}{nVdash}{amssymb} \dosymbol{Rel}{nvDash}{amssymb} \dosymbol{Rel}{nVDash}{amssymb} \dosymbol{Rel}{parallel}{kernel} \dosymbol{Rel}{perp}{kernel} \dosymbol{Rel}{pitchfork}{amssymb} \dosymbol{Rel}{propto}{kernel} \dosymbol{Rel}{shortmid}{amssymb} \dosymbol{Rel}{shortparallel}{amssymb} \dosymbol{Rel}{smallfrown}{amssymb} \dosymbol{Rel}{smallsmile}{amssymb} \dosymbol{Rel}{smile}{kernel} \dosymbol{Rel}{sqsubset}{amssymb} \dosymbol{Rel}{sqsubseteq}{kernel} \dosymbol{Rel}{sqsupset}{amssymb} \dosymbol{Rel}{sqsupseteq}{kernel} \dosymbol{Rel}{subset}{kernel} \dosymbol{Rel}{Subset}{amssymb} \dosymbol{Rel}{subseteq}{kernel} \dosymbol{Rel}{subseteqq}{amssymb} \dosymbol{Rel}{subsetneq}{amssymb} \dosymbol{Rel}{subsetneqq}{amssymb} \dosymbol{Rel}{supset}{kernel} \dosymbol{Rel}{Supset}{amssymb} \dosymbol{Rel}{supseteq}{kernel} \dosymbol{Rel}{supseteqq}{amssymb} \dosymbol{Rel}{supsetneq}{amssymb} \dosymbol{Rel}{supsetneqq}{amssymb} \dosymbol{Rel}{therefore}{amssymb} \dosymbol{Rel}{trianglelefteq}{amssymb} \dosymbol{Rel}{trianglerighteq}{amssymb} \dosymbol{Rel}{varpropto}{amssymb} \dosymbol{Rel}{varsubsetneq}{amssymb} \dosymbol{Rel}{varsubsetneqq}{amssymb} \dosymbol{Rel}{varsupsetneq}{amssymb} \dosymbol{Rel}{varsupsetneqq}{amssymb} \dosymbol{Rel}{vartriangle}{amssymb} \dosymbol{Rel}{vartriangleleft}{amssymb} \dosymbol{Rel}{vartriangleright}{amssymb} \dosymbol{Rel}{vdash}{kernel} \dosymbol{Rel}{Vdash}{amssymb} \dosymbol{Rel}{vDash}{amssymb} \dosymbol{Rel}{Vvdash}{amssymb} \end{symlist} \begin{notes} \synonyms \alias{owns} \end{notes} \subsection{Cumulative (variable-size) operators\nopunct} \begin{symlist}[adjustcols=-1] \openup3pt \dosymbol{COi}{int}{kernel} \dosymbol{COi}{oint}{kernel} \dosymbol{COs}{bigcap}{kernel} \dosymbol{COs}{bigcup}{kernel} \dosymbol{COs}{bigodot}{kernel} \dosymbol{COs}{bigoplus}{kernel} \dosymbol{COs}{bigotimes}{kernel} \dosymbol{COs}{bigsqcup}{kernel} \dosymbol{COs}{biguplus}{kernel} \dosymbol{COs}{bigvee}{kernel} \dosymbol{COs}{bigwedge}{kernel} \dosymbol{COs}{coprod}{kernel} \dosymbol{COs}{prod}{kernel} \dosymbol{COs}{smallint}{kernel} \dosymbol{COs}{sum}{kernel} \end{symlist} \subsection{Punctuation\nopunct} \begin{symlist}[adjustcols=-4] \openup2pt \dosymbol{Ordc}{.}{kernel} \dosymbol{Ordc}{/}{kernel} \dosymbol{Ordc}{|}{kernel} \dosymbol{Punc}{,}{kernel} \dosymbol{Punc}{;}{kernel} \dosymbol{Pun}{colon}{kernel} \dosymbol{Relc}{:}{kernel} %\dosymbol{DeR}{!}{kernel} %\dosymbol{DeR}{?}{kernel} \dosymbol{Punc}{!}{kernel} \dosymbol{Punc}{?}{kernel} \dosymbol{Inn}{dotsb}{kernel} \dosymbol{Inn}{dotsc}{kernel} \dosymbol{Inn}{dotsi}{kernel} \dosymbol{Inn}{dotsm}{kernel} \dosymbol{Inn}{dotso}{kernel} \dosymbol{Inn}{ddots}{kernel} \dosymbol{Ord}{vdots}{kernel} \end{symlist} \begin{notes} \item The \verb':' by itself produces a colon with class-3 (relation) spacing. The command \cn{colon} produces special spacing for use in constructions such as \verb'f\colon A\to B' $f\colon A\to B$. \item Although the commands \cn{cdots} and \cn{ldots} are frequently used, we recommend the more semantically oriented commands \cn{dotsb} \cn{dotsc} \cn{dotsi} \cn{dotsm} \cn{dotso} for most purposes\dotsref. \end{notes} \subsection{Pairing delimiters (extensible)}\label{pair-delims} See Section~\ref{delim} for more information. \begin{symlist} \openup7pt \dosymbol{DeLRc}{(}{)}{kernel} \dosymbol{DeLRc}{[}{]}{kernel} \dosymbol{DeLR}{lbrace}{rbrace}{kernel} \dosymbol{DeLR}{lvert}{rvert}{kernel} \dosymbol{DeLR}{lVert}{rVert}{kernel} \dosymbol{DeLR}{langle}{rangle}{kernel} \dosymbol{DeLR}{lceil}{rceil}{kernel} \dosymbol{DeLR}{lfloor}{rfloor}{kernel} \dosymbol{DeLR}{lgroup}{rgroup}{kernel} \dosymbol{DeLR}{lmoustache}{rmoustache}{kernel} \end{symlist} \subsection{Nonpairing extensible symbols\nopunct} \begin{symlist} \dosymbol{DeB}{vert}{kernel} \dosymbol{DeB}{Vert}{kernel} \dosymbol{DeBc}{/}{kernel} \dosymbol{DeB}{backslash}{kernel} \dosymbol{DeB}{arrowvert}{kernel} \dosymbol{DeB}{Arrowvert}{kernel} \dosymbol{DeB}{bracevert}{kernel} \end{symlist} \begin{notes} \item Using \cn{vert}, \verb'|', \cn{Vert}, or \cn{|} for paired delimiters is not recommended\vertref. Instead, use delimiters from the list in Section~\ref{pair-delims}. \synonyms \alias{|} \end{notes} \subsection{Extensible vertical arrows\nopunct} \begin{symlist} \dosymbol{DeA}{uparrow}{kernel} \dosymbol{DeA}{Uparrow}{kernel} \dosymbol{DeA}{downarrow}{kernel} \dosymbol{DeA}{Downarrow}{kernel} \dosymbol{DeA}{updownarrow}{kernel} \dosymbol{DeA}{Updownarrow}{kernel} \end{symlist} \subsection{Math accents\nopunct}\label{accents} \begin{symlist}[adjustcols=1] \dosymbol{Acc}{acute}{kernel} \dosymbol{Acc}{grave}{kernel} \dosymbol{Acc}{ddot}{kernel} \dosymbol{Acc}{tilde}{kernel} \dosymbol{Acc}{bar}{kernel} \dosymbol{Acc}{breve}{kernel} \dosymbol{Acc}{check}{kernel} \dosymbol{Acc}{hat}{kernel} \dosymbol{Acc}{vec}{kernel} \dosymbol{Acc}{dot}{kernel} \dosymbol{Acc}{ddot}{amsmath} \dosymbol{Acc}{dddot}{amsmath} \dosymbol{Acc}{mathring}{amsmath} \dosymbol{Accw}{widetilde}{kernel} \dosymbol{Accw}{widehat}{kernel} \end{symlist} \subsection{Named operators} These operators are represented by a multiletter abbreviation. \shiftlistright40pt \begin{symlist}[adjustcols=-1] \dosymbol{Opn}{arccos}{kernel} \dosymbol{Opn}{arcsin}{kernel} \dosymbol{Opn}{arctan}{kernel} \dosymbol{Opn}{arg}{kernel} \dosymbol{Opn}{cos}{kernel} \dosymbol{Opn}{cosh}{kernel} \dosymbol{Opn}{cot}{kernel} \dosymbol{Opn}{coth}{kernel} \dosymbol{Opn}{csc}{kernel} \dosymbol{Opn}{deg}{kernel} \dosymbol{Opn}{det}{kernel} \dosymbol{Opn}{dim}{kernel} \dosymbol{Opn}{exp}{kernel} \dosymbol{Opn}{gcd}{kernel} \dosymbol{Opn}{hom}{kernel} \dosymbol{Opn}{inf}{kernel} \dosymbol{Opn}{injlim}{kernel} \dosymbol{Opn}{ker}{kernel} \dosymbol{Opn}{lg}{kernel} \dosymbol{Opn}{lim}{kernel} \dosymbol{Opn}{liminf}{kernel} \dosymbol{Opn}{limsup}{kernel} \dosymbol{Opn}{ln}{kernel} \dosymbol{Opn}{log}{kernel} \dosymbol{Opn}{max}{kernel} \dosymbol{Opn}{min}{kernel} \dosymbol{Opn}{Pr}{kernel} \dosymbol{Opn}{projlim}{kernel} \dosymbol{Opn}{sec}{kernel} \dosymbol{Opn}{sin}{kernel} \dosymbol{Opn}{sinh}{kernel} \dosymbol{Opn}{sup}{kernel} \dosymbol{Opn}{tan}{kernel} \dosymbol{Opn}{tanh}{kernel} \dosymbol{Opn}{varinjlim}{kernel} \dosymbol{Opn}{varprojlim}{kernel} \dosymbol{Opn}{varliminf}{kernel} \dosymbol{Opn}{varlimsup}{kernel} \end{symlist} To define additional named operators outside the above list, use the \cn{DeclareMathOperator} command; for example, after \begin{verbatim} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\esssup}{ess\,sup} \end{verbatim} one could write \begin{center} \begin{tabular}{rl} \verb'\rank(x)'& $\rank(x)$\\ \verb'\esssup(y,z)'& $\esssup(y,z)$ \end{tabular} \end{center} The star form \cn{DeclareMathOperator*} creates an operator that takes limits in a displayed formula, such as $\sup$ or $\max$. When predefining such a named operator is problematic (e.g., when using one in the title or abstract of an article), there is an alternative form that can be used directly: \[\verb'\operatorname{rank}(x)'\quad \rightarrow\quad\operatorname{rank}(x)\] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Notations} \label{notations} \subsection{Top and bottom embellishments} These are visually similar to accents but generally span multiple symbols rather than being applied to a single base symbol. For ease of reference, \cn{widetilde} and \cn{widehat} are redundantly included here and in the table of math accents. \begin{symlist} \dosymbol{Accw}{widetilde}{kernel} \dosymbol{Accw}{widehat}{kernel} \dosymbol{Accw}{overline}{kernel} \dosymbol{Accw}{underline}{kernel} \dosymbol{Accw}{overbrace}{kernel} \dosymbol{Accw}{underbrace}{kernel} \dosymbol{Accw}{overleftarrow}{kernel} \dosymbol{Accw}{underleftarrow}{amsmath} \dosymbol{Accw}{overrightarrow}{kernel} \dosymbol{Accw}{underrightarrow}{amsmath} \dosymbol{Accw}{overleftrightarrow}{amsmath} \dosymbol{Accw}{underleftrightarrow}{amsmath} \end{symlist} \subsection{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} \subsection{Affixing symbols to other symbols} In addition to the standard accents (Section~\ref{accents}), other symbols can be placed above or below a base symbol with the \cn{overset} and \cn{underset} commands. For example, writing \verb|\overset{*}{X}| will place a superscript-size $*$ above the $X$, thus: $\overset{*}{X}$. See also the description of \cn{sideset} in \secref{sideset}. \subsection{Matrices}\label{ss:matrix} The environments \env{pmatrix}, \env{bmatrix}, \env{Bmatrix}, \env{vmatrix}, and \env{Vmatrix} have (respectively) $(\,)$, $[\,]$, $\lbrace\,\rbrace$, $\lvert\,\rvert$, and $\lVert\,\rVert$ delimiters built in. There is also a \env{matrix} environment without delimiters and an \env{array} environment that can be used to obtain left alignment or other variations in the column specs. \begin{center} \begin{minipage}{.4\columnwidth} \begin{verbatim} \begin{pmatrix} \alpha& \beta^{*}\\ \gamma^{*}& \delta \end{pmatrix} \end{verbatim} \end{minipage} \qquad \begin{minipage}{.4\columnwidth} \[ \begin{pmatrix} \alpha& \beta^{*}\\ \gamma^{*}& \delta \end{pmatrix} \] \end{minipage} \end{center} 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. This example was produced by \begin{verbatim} \bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr) \end{verbatim} By default, all elements in a matrix are centered horizontally. The \pkg{mathtools} package provides starred versions of all the matrix environments that facilitate other alignments. That package also provides fenced versions of \env{smallmatrix} with parallel names in both starred and nonstarred versions. To produce a row of dots in a matrix\index{matrices!ellipsis dots}\index{ellipsis dots!in matrices}\index{dots|see{ellipsis dots}} spanning a given number of columns, use \cn{hdotsfor}. For example, \verb'\hdotsfor{3}' in the second column of a four-column matrix will print a row of dots across the final three columns. For piecewise function definitions there is a \env{cases} environment: \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} and the embedded math. \begin{notes} \singlenote The plain \TeX{} form \verb'\matrix{...\cr...\cr}' and the related commands \cn{pmatrix}, \cn{cases} should be avoided in \LaTeX{} (and when the \pkg{amsmath} package is loaded they are disabled). \end{notes} \subsection{Math spacing commands} When the \pkg{amsmath} package is used, all of these math spacing commands can be used both in and out of math mode. \begin{center}\begin{tabular}{llllll} Abbrev.& Spelled out& Example & Abbrev.& Spelled out& Example\\ \hline \strut & no space& \spx{}& & no space& \spx{}\\ \cn{\,}& \cn{thinspace}& \spx{\,}& \cn{!}& \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{tabular}\end{center} For finer control over math spacing, use \cn{mspace} and `math units'. One math unit, or \verb|mu|, is equal to 1/18 em. Thus to get a negative half \cn{quad} write \verb|\mspace{-9.0mu}|. There are also three commands that leave a space equal to the height and/or width of a given fragment of \lat/ material: \begin{center}\begin{tabular}{ll} \colhead{Example}& \colhead{Result}\\ \hline \verb'\phantom{XXX}'& space as wide and high as three X's\strut \\ \verb'\hphantom{XXX}'& space as wide as three X's; height 0\\ \verb'\vphantom{X}'& space of width 0, height = height of X \end{tabular}\end{center} \subsection{Dots}\label{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. In most situations, the generic \cn{dots} can be used, and \pkg{amsmath} will interpret it in the manner preferred by the AMS, namely low dots (\cn{ldots}) between commas or raised dots (\cn{cdots}) between binary operators and relations, etc. If what follows the dots is ambiguous as to the choice, the specific form of the command can be used. However, by using the semantically oriented commands \begin{itemize} \setlength{\itemsep}{0pt} \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} \vspace{-\topsep} \begin{tabular}{@{}ll@{}} \begin{minipage}[t]{.50\textwidth} \small \begin{verbatim} 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{verbatim} \end{minipage} & \begin{minipage}[t]{.48\textwidth} \noindent 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} \subsection{Nonbreaking dashes} The command \cn{nobreakdash} suppresses the possibility of a linebreak after the following hyphen or dash. For example, if you write `pages 1\ndash 9' as \verb|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 \verb|$p$-adic|. For frequent use, it's advisable to make abbreviations, e.g., \begin{verbatim} \newcommand{\p}{$p$\nobreakdash}% for "\p adic" ("p-adic") \newcommand{\Ndash}{\nobreakdash\textendash}% for "pages 1\Ndash 9" % For "\n dimensional" ("n-dimensional"): \newcommand{\n}{$n$\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. (Add a zero-width space after the hyphen.) \subsection{Roots} The command \cn{sqrt} produces a square root. To specify an explicit radix, give it as an optional argument. \[ \verb'\sqrt{\frac{n}{n-1} S}'\quad\sqrt{\frac{n}{n-1} S}, \qquad \verb'\sqrt[3]{2}'\quad \sqrt[3]{2} \] \subsection{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} If you need to box an equation including the equation number, it may be difficult, depending on the context; there are some suggestions in the AMS author FAQ; see the entry outlined in red on the page \url{https://www.ams.org/faq?faq_id=290}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Fractions and related constructions} \subsection{The \hycn{frac}, \hycn{dfrac}, and \hycn{tfrac} commands} The \cn{frac} command\index{fractions} takes two arguments\mdash numerator and denominator\mdash and typesets them in normal fraction form. Use \cn{dfrac} or \cn{tfrac} to overrule \LaTeX{}'s guess about the proper size to use for the fraction's contents (t = text style, d = display style). \begin{equation} \frac{1}{k}\log_2 c(f),\quad\dfrac{1}{k}\log_2 c(f),\quad\tfrac{1}{k}\log_2 c(f) \end{equation} \begin{verbatim} \begin{equation} \frac{1}{k}\log_2 c(f),\quad\dfrac{1}{k}\log_2 c(f), \quad\tfrac{1}{k}\log_2 c(f) \end{equation} \end{verbatim} \begin{equation} \Re{z} =\frac{n\pi \dfrac{\theta +\psi}{2}}{ \left(\dfrac{\theta +\psi}{2}\right)^2 + \left( \dfrac{1}{2} \log \left\lvert\dfrac{B}{A}\right\rvert\right)^2}. \end{equation} \begin{verbatim} \begin{equation} \Re{z} =\frac{n\pi \dfrac{\theta +\psi}{2}}{ \left(\dfrac{\theta +\psi}{2}\right)^2 + \left( \dfrac{1}{2} \log \left\lvert\dfrac{B}{A}\right\rvert\right)^2}. \end{equation} \end{verbatim} \subsection{The \hycn{binom}, \hycn{dbinom}, and \hycn{tbinom} commands} For binomial expressions\index{binomials} such as $\binom{n}{k}$ there are \cn{binom}, \cn{dbinom} and \cn{tbinom} commands: \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 \hycn{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 pt\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{cmdspec}[25em] \string\genfrac \ma{left-delim} \ma{right-delim} \ma{thickness} \ma{mathstyle} \ma{numerator} \ma{denominator} \end{cmdspec} 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} \begin{notes} \singlenote For technical reasons, using the primitive fraction commands \cn{over}, \cn{atop}, \cn{above} in a \LaTeX{} document is not recommended (see, e.g., \url{https://www.ams.org/faq?faq\_id=288}, the entry outlined in red). \end{notes} \subsection{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}\verb|[l]| or \cn{cfrac}\verb|[r]| instead of \cn{cfrac}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Delimiters}\label{delim} \subsection{Delimiter sizes}\label{bigdel} Unless you indicate otherwise, delimiters in math formulas will remain at the standard size regardless of the height of the enclosed material. To get larger sizes, you can either select a particular size using a \cn{big...} prefix (see below), or you can use \cn{left} and \cn{right} prefixes for autosizing. 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 has fairly large quantum jumps. This means that an expression that is infinitesimally too large for a given delimiter size will get the next larger size, a jump of 6pt or so (3pt top and bottom) in normal-sized text. There are two or three situations where the delimiter size is commonly adjusted. These adjustments are done using the following commands: \begin{center}\begin{tabular}{l|llllll} Delimiter& no size& \ncn{left}& \ncn{bigl}& \ncn{Bigl}& \ncn{biggl}& \ncn{Biggl}\\ size& specified& \ncn{right}& \ncn{bigr}& \ncn{Bigr}& \ncn{biggr}& \ncn{Biggr}\\[4pt] %\hline\omit\rule{0pt}{1ex}\\ \hline\omit\rule{0pt}{1ex}\\[-1ex] Result $\vphantom{\Bigg|^{\frac{1}{2}}}$ & % force height to avoid gap in vertical $\displaystyle(b)(\frac{c}{d})$& $\displaystyle\left(b\right)\left(\frac{c}{d}\right)$& $\displaystyle\bigl(b\bigr)\bigl(\frac{c}{d}\bigr)$& $\displaystyle\Bigl(b\Bigr)\Bigl(\frac{c}{d}\Bigr)$& $\displaystyle\biggl(b\biggr)\biggl(\frac{c}{d}\biggr)$& $\displaystyle\Biggl(b\Biggr)\Biggl(\frac{c}{d}\Biggr)$ \end{tabular}\end{center} The first kind of adjustment is done for cumulative operators with limits, such as summation signs. With \cn{left} and \cn{right} the delimiters usually turn out larger than necessary, and using the \verb|Big| or \verb|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 larger than the base size but still able to fit within the normal line spacing: $\bigl\lvert\frac{b'}{d'}\bigr\rvert$. The \pkg{mathtools} package provides a feature \cn{DeclarePairedDelimiter} that can simplify sizing; see the package documentation for details. \subsection{Vertical bar notations}\label{verts} The use of the \verb'|' character to produce paired delimiters is not recommended. There is an ambiguity about the directionality of the symbol that will in rare cases produce incorrect spacing\mdash e.g., \verb'|k|=|-k|' produces $|k|=|-k|$, and \verb'|\sin x|' produces $|\sin x|$ instead of the correct $\lvert\sin x\rvert$. Using \cn{lvert} for a \qq{left vert bar} and \cn{rvert} for a \qq{right vert bar} whenever they are used in pairs will prevent this problem; compare $\lvert -k\rvert$, produced by \verb'\lvert -k\rvert'. For double bars there are analogous \cn{lVert}, \cn{rVert} commands. Recommended practice is 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 \verb|\abs{z}| would produce $\lvert z\rvert$ and \verb|\norm{v}| would produce $\lVert v\rVert$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The \hycn{text} command} The main use of the command \cn{text} is for words or phrases\index{text fragments inside math} in a display. It is similar to \cn{mbox} in its effects but, unlike \cn{mbox}, automatically produces subscript-size text if used in a subscript. \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} \subsection{\hycn{mod} and its relatives} Commands \cn{mod}, \cn{bmod}, \cn{pmod}, \cn{pod} deal with the special spacing conventions of \qq{mod} notation. \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Integrals and sums} \subsection{Altering the placement of limits} The limits on integrals, sums, and similar symbols are placed either to the side of or above and below the base symbol, depending on convention and context. \lat/ has rules for automatically choosing one or the other, and most of the time the results are satisfactory. In the event they are not, there are three \lat/ commands that can be used to influence the placement of the limits: \cn{limits}, \cn{nolimits}, \cn{displaylimits}. Compare \begin{center} \begin{minipage}{.4\columnwidth} \[\int_{\abs{x-x_z(t)} 0} z^n}{\prod_{1\leq k\leq n} (1-q^k)} \qquad\begin{minipage}{.5\columnwidth} \begin{verbatim} \frac{\sum_{n > 0} z^n} {\prod_{1\leq k\leq n} (1-q^k)} \end{verbatim} \end{minipage} \] Using \cn{dfrac} instead of \cn{frac} wouldn't change anything in this case; if you want the sum and product symbols to appear full size, you need the \cn{displaystyle} command: \[ \frac{{\displaystyle\sum_{n > 0} z^n}} {{\displaystyle\prod_{1\leq k\leq n} (1-q^k)}} \qquad\begin{minipage}{.7\columnwidth} \begin{verbatim} \frac{{\displaystyle\sum_{n > 0} z^n}} {{\displaystyle\prod_{1\leq k\leq n} (1-q^k)}} \end{verbatim} \end{minipage} \] And if you want full-size symbols but with limits on the side, use the \cn{nolimits} command also: \[ \frac{{\displaystyle\sum\nolimits_{n> 0} z^n}} {{\displaystyle\prod\nolimits_{1\leq k\leq n} (1-q^k)}} \qquad\begin{minipage}{.76\columnwidth} \begin{verbatim} \frac{{\displaystyle\sum\nolimits_{n> 0} z^n}} {{\displaystyle\prod\nolimits_{1\leq k\leq n} (1-q^k)}} \end{verbatim} \end{minipage} \] There are similar commands \cn{textstyle}, \cn{scriptstyle}, and \cn{scriptscriptstyle}, to force \lat/ to use the symbol size and spacing that would be applied in (respectively) inline math, first-order subscript, or second-order subscript, even when the current context would normally yield some other size. \textbf{Note:} These commands belong to a special class of commands referred to in the \lat/ book as ``declarations''. In particular, notice where the braces fall that delimit the effect of the command: \begin{center} \textbf{Right: } \verb'{\displaystyle ...}' \qquad\qquad\textbf{Wrong: } \verb'\displaystyle{...}' \end{center} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Other packages of interest} \label{other-packages} Many other \LaTeX{} packages that address some aspect of mathematical formulas are available from CTAN (the Comprehensive \TeX{} Archive Network). To recommend a few examples: \begin{description} \raggedright \item[mathtools] Additional features extending \pkg{amsmath}; loads \pkg{amsmath}. \item[amsthm] General theorem and proof setup. \item[amsfonts] Defines \cn{mathbb} and \cn{mathfrak}, and provides access to many additional symbols (without names; \pkg{amssymb} provides the names). \item[accents] Under accents and accents using arbitrary symbols. \item[bm] Bold math package, provides a more general and more robust implementation of \cn{boldsymbol}. \item[mathrsfs] Ralph Smith's Formal Script, font setup. \item[cases] Apply a large brace to two or more equations without losing the individual equation numbers. \item[delarray] Delimiters spanning multiple rows of an array. %%\item[kuvio] Commutative diagrams and other diagrams. % not in TeX Live \item[xypic] Commutative diagrams and other diagrams. \item[TikZ] Comprehensive graphical facilities, including features for drawing diagrams. \end{description} The \TeX{} Catalogue,\\ \null\hspace{2\parindent} \url{http://mirror.ctan.org/help/Catalogue/alpha.html},\\ is a good place to look if you know a package's name. \medskip Questions and answers on specific \TeX-related topics are the \emph{raison d'\^etre} of this forum:\\ \null\hspace{2\parindent} \url{https://tex.stackexchange.com/questions}\\ Check the archives for existing answers; pointers to selected topics may expedite your search:\\ \null\hspace{2\parindent} \url{https://tex.meta.stackexchange.com/a/2425#2425}\\ If nothing useful turns up, ask your own question. %\newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Other documentation of interest} \begin{thebibliography}{AMUG} \raggedright \bibitem[AMUG]{amsldoc} American Mathematical Society and the \LaTeX3 Project: \emph{User's Guide for the \textnormal{\ttfamily amsmath} package}, Version~2.$+$, \url{http://mirror.ctan.org/macros/latex/required/amsmath/amsldoc.tex} and \url{http://mirror.ctan.org/macros/latex/required/amsmath/amsldoc.pdf}, 2017. \bibitem[AFUG]{amsfndoc} American Mathematical Society: \emph{User's Guide, AMSFonts}, \url{http://mirror.ctan.org/fonts/amsfonts/amsfndoc.pdf}, 2002. \bibitem[CLSL]{comprehensive} Scott Pakin: \emph{The Comprehensive \LaTeX{} Symbol List}, \url{http://mirror.ctan.org/tex-archive/info/symbols/comprehensive/}, January 2017. Raw font tables, without symbol names, are shown alphabetically by font name in the \fn{rawtables*.pdf} files in the same area of CTAN and from \TeX\,Live with \texttt{texdoc rawtables}. \bibitem[Lam]{lamport} Leslie Lamport: \emph{\LaTeX{}: A document preparation system}, 2nd edition, Addison-Wesley, 1994. \bibitem[LC]{companion} Frank Mittelbach and Michel Goossens, with Johannes Braams, David Carlisle, and Chris Rowley: \emph{The \LaTeX{} Companion}, 2nd edition, Addison-Wesley, 2004. \bibitem[LFG]{fntguide} \LaTeX3 Project Team: \emph{\LaTeXe{} font selection}, % \fn{fntguide.tex}, November 2005. \url{http://mirror.ctan.org/macros/latex/doc/fntguide.pdf}, 2005. \bibitem[LGC]{graphics-companion} Michel Goossens, Frank Mittelbach, Sebastian Rahtz, Denis Roegel, and Herbert~Vo\ss: \emph{The \LaTeX{} Graphics Companion}, 2nd edition, Addison-Wesley, 2008. \bibitem[LGG]{grfguide} D.~P. Carlisle, \LaTeX3 Project: \emph{Packages in the `graphics' bundle}, %\fn{grfguide.tex}, 2017. \url{http://mirror.ctan.org/macros/latex/required/graphics/grfguide.pdf}, 2017. \bibitem[LUG]{usrguide} \LaTeX3 Project Team: \emph{\LaTeXe~for authors}, % \fn{usrguide.tex}, 2015. \url{http://mirror.ctan.org/macros/latex/doc/usrguide.pdf}, 2015. \bibitem[MML]{gratzer} George Gr\"atzer: \textit{More Math into \LaTeX}, 5th edition, Springer, New York, 2016. \bibitem[UCM]{uc-math} Will Robertson: \emph{Every symbol \textup{(}most symbols\textup{)} defined by \pkg{unicode-math}}, \url{http://mirror.ctan.org/macros/latex/contrib/unicode-math/unimath-symbols.pdf}, 2017; and\\ Will Robertson, Philipp Stephani, Joseph Wright, and Khaled Hosny: \emph{Experimental Unicode mathematical typesetting: The \pkg{unicode-math} package}, \url{http://mirror.ctan.org/macros/latex/contrib/unicode-math/unicode-math.pdf}, 2017. \end{thebibliography} \end{document}