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User’s Guide for the amsmath Package (Version 2.0) American Mathematical Society 1999/12/13

ii CONTENTS Contents 1 Introduction 2 Options for the amsmath package 3 Displayed equations Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Single equations . . . . . . . . . . . . . . . . . 3.3 Split equations without alignment 3.4 Split equations with alignment . . . . . . . . . . . . . . . . . . . 3.5 Equation groups without alignment . . . . . . . . . . . . . . . . . 3.6 Equation groups with mutual alignment . . . . . . . . . . . . . . 3.7 Alignment building blocks . . . . . . . . . . . . . . . . . . . . . . 3.8 Adjusting tag placement . . . . . . . . . . . . . . . . . . . . . . . 3.9 Vertical spacing and page breaks in multiline displays . . . . . . 3.10 Interrupting a display . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Equation numbering . . . . . . . . . . . . . . . . . . . . . . . . . 4 Miscellaneous mathematical features 4.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Math spacing commands . . . . . . . . . . . . . . . . . . . . . . . 4.3 Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Nonbreaking dashes . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Accents in math . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Boxed formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Over and under arrows . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Extensible arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Aﬃxing symbols to other symbols . . . . . . . . . . . . . . . . . 4.11 Fractions and related constructions . . . . . . . . . . . . . . . . . 4.12 Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Smash options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Delimiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Operator names 5.1 Deﬁning new operator names . . . . . . . . . . . . . . . . . . . . 5.2 \mod and its relatives . . . . . . . . . . . . . . . . . . . . . . . . . 6 The \text command 1 2 3 3 3 5 5 6 6 7 8 8 9 9 10 10 11 11 12 12 13 13 13 13 13 14 15 15 15 17 17 18 18

CONTENTS 7 Integrals and sums 7.1 Multiline subscripts and superscripts . . . . . . . . . . . . . . . . 7.2 The \sideset command . . . . . . . . . . . . . . . . . . . . . . . 7.3 Placement of subscripts and limits . . . . . . . . . . . . . . . . . 7.4 Multiple integral signs . . . . . . . . . . . . . . . . . . . . . . . . 8 Commutative diagrams 9 Using math fonts Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 9.2 Recommended use of math font commands . . . . . . . . . . . . 9.3 Bold math symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Italic Greek letters . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Error messages and output problems 10.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Error messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Warning messages . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Wrong output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Additional information 11.1 Converting existing documents . . . . . . . . . . . . . . . . . . . 11.2 Technical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Getting help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Of possible interest . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography Index iii 19 19 19 20 20 20 21 21 21 22 23 23 23 23 29 29 30 30 30 30 31 31 33

iv CONTENTS

1 —1— Introduction The amsmath package is a LATEX package that provides miscellaneous enhance- ments for improving the information structure and printed output of documents that contain mathematical formulas. Readers unfamiliar with LATEX should refer to [7]. If you have an up-to-date version of LATEX, the amsmath package is nor- mally provided along with it. Upgrading when a newer version of the amsmath package is released can be done via http://www.ams.org/tex/amsmath.html or ftp://ftp.ams.org/pub/tex/. This documentation describes the features of the amsmath package and dis- cusses how they are intended to be used. It also covers some ancillary packages: amsbsy amscd amsopn amstext amsxtra These all have something to do with the contents of math formulas. For infor- mation on extra math symbols and math fonts, see [1] and http://www.ams. org/tex/amsfonts.html. For documentation of the amsthm package or AMS document classes (amsart, amsbook, etc.) see [3] or [2] and 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 amsmath features: • A convenient way to deﬁne new ‘operator name’ commands analogous to \sin and \lim, including proper side spacing and automatic selection of the correct font style and size (even when used in sub- or superscripts). • Multiple substitutes for the eqnarray environment to make various kinds of equation arrangements easier to write. • Equation numbers automatically adjust up or down to avoid overprinting on the equation contents (unlike eqnarray). • Spacing around equals signs matches the normal spacing in the equation environment (unlike eqnarray). • A way to produce multiline subscripts as are often used with summation or product symbols. • An easy way to substitute a variant equation number for a given equation instead of the automatically supplied number. • An easy way to produce subordinate equation numbers of the form (1.3a) (1.3b) (1.3c) for selected groups of equations. The amsmath package is distributed together with some small auxiliary pack- ages: amsmath Primary package, provides various features for displayed equations and other mathematical constructs.

2 2. OPTIONS FOR THE AMSMATH PACKAGE amstext Provides a \text command for typesetting a fragment of text inside a display. amsopn Provides \DeclareMathOperator for deﬁning new ‘operator names’ like \sin and \lim. amsbsy For backward compatibility this package continues to exist but use of the newer bm package that comes with LATEX is recommended instead. amscd Provides a CD environment for simple commutative diagrams (no support for diagonal arrows). amsxtra Provides certain odds and ends such as \fracwithdelims and \ac- centedsymbol, for compatibility with documents created using version 1.1. The amsmath package incorporates amstext, amsopn, and amsbsy. The fea- tures of amscd and amsxtra, however, are available only by invoking those packages separately. —2— Options for the amsmath package The amsmath package has the following options: centertags (default) For a split equation, place equation numbers vertically centered on the total height of the equation. tbtags ‘Top-or-bottom tags’: For a split equation, place equation numbers level with the last (resp. ﬁrst) line, if numbers are on the right (resp. left). sumlimits (default) Place the subscripts and superscripts of summation sym- bols above and below, in displayed equations. This option also aﬀects other symbols of the same type—,,,, and so forth—but exclud- ing integrals (see below). nosumlimits Always place the subscripts and superscripts of summation-type symbols to the side, even in displayed equations. intlimits Like sumlimits, but for integral symbols. nointlimits (default) Opposite of intlimits. namelimits (default) Like sumlimits, but for certain ‘operator names’ such as det, inf, lim, max, min, that traditionally have subscripts placed under- neath when they occur in a displayed equation. nonamelimits Opposite of namelimits. To use one of these package options, put the option name in the optional ar- gument of the \usepackage command—e.g., \usepackage[intlimits]{amsmath}. The amsmath package also recognizes the following options which are nor- mally selected (implicitly or explicitly) through the \documentclass command, and thus need not be repeated in the option list of the \usepackage{amsmath} statement.

3.2. SINGLE EQUATIONS 3 leqno Place equation numbers on the left. reqno Place equation numbers on the right. fleqn Position equations at a ﬁxed indent from the left margin rather than centered in the text column. —3— Displayed equations Introduction 3.1 The amsmath package provides a number of additional displayed equation struc- tures beyond the ones provided in basic LATEX. The augmented set includes: equation gather multline split equation* gather* multline* align flalign alignat align* flalign* alignat* (Although the standard eqnarray environment remains available, it is better to use align or equation+split instead.) Except for split, each environment has both starred and unstarred forms, where the unstarred forms have automatic numbering using LATEX’s equation counter. You can suppress the number on any particular line by putting \notag before the \\; you can also override it with a tag of your own using \tag{label}, where label means arbitrary text such as $*$ or ii used to “number” the equation. There is also a \tag* command that causes the text you supply to be typeset literally, without adding parentheses around it. \tag and \tag* can also be used within the unnumbered versions of all the amsmath alignment structures. Some examples of the use of \tag may be found in the sample ﬁles testmath.tex and subeqn.tex provided with the amsmath package. The split environment is a special subordinate form that is used only inside one of the others. It cannot be used inside multline, however. In the structures that do alignment (split, align and variants), relation symbols have an & before them but not after—unlike eqnarray. Putting the & after the relation symbol will interfere with the normal spacing; it has to go before. 3.2 Single equations The equation environment is for a single equation with an automatically gen- erated number. The equation* environment is the same except for omitting the number.1 1Basic LATEX doesn’t provide an equation* environment, but rather a functionally equiv- alent environment named displaymath.

4 3. DISPLAYED EQUATIONS Table 3.1. Comparison of displayed equation environments (vertical lines in- dicating nominal margins) \begin{equation*} a=b \end{equation*} \begin{equation} a=b \end{equation} \begin{equation}\label{xx} \begin{split} a& =b+c-d\\ & \quad +e-f\\ & =g+h\\ & =i \end{split} \end{equation} \begin{multline} a+b+c+d+e+f\\ +i+j+k+l+m+n \end{multline} \begin{gather} a_1=b_1+c_1\\ a_2=b_2+c_2-d_2+e_2 \end{gather} \begin{align} a_1& =b_1+c_1\\ a_2& =b_2+c_2-d_2+e_2 \end{align} \begin{align} a_{11}& =b_{11}& a_{12}& =b_{12}\\ a_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \end{align} \begin{flalign*} a_{11}& =b_{11}& a_{12}& =b_{12}\\ a_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \end{flalign*} (1) (2) a = b a = b a = b + c − d + e − f = g + h = i (3) a + b + c + d + e + f + i + j + k + l + m + n (4) (5) (6) (7) (8) (9) a1 = b1 + c1 a2 = b2 + c2 − d2 + e2 a1 = b1 + c1 a2 = b2 + c2 − d2 + e2 a11 = b11 a21 = b21 a12 = b12 a22 = b22 + c22 a11 = b11 a21 = b21 a12 = b12 a22 = b22 + c22

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