User’s Guide for the amsmath Package(Version 2.1)
American Mathematical Society, LATEX Project
1999-12-13(revised 2002-02-25, 2016-11-14, 2018-04-05, 2019-10-14,
2020-02-18)

This file is maintained by the LATEX Project team.
Bug reports can be opened (category amslatex) at
https://latex-project.org/bugs/.

ii

CONTENTS

Contents
1 Introduction

1

2 Options for the amsmath package

2

3 Displayed equations
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 . . . . . . . . . . . . . . . . . . . . . . . . .

3
3
4
4
6
7
7
9
9
10
10
11

4 Miscellaneous mathematical features
12
4.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Math spacing commands . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.4 Nonbreaking dashes . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.5 Accents in math . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.6 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.7 Boxed formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.8 Over and under arrows . . . . . . . . . . . . . . . . . . . . . . . . 15
4.9 Extensible arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.10 Affixing symbols to other symbols . . . . . . . . . . . . . . . . . 16
4.11 Fractions and related constructions . . . . . . . . . . . . . . . . . 16
4.12 Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.13 Smash options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.14 Delimiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Operator names
19
5.1 Defining new operator names . . . . . . . . . . . . . . . . . . . . 19
5.2 \mod and its relatives . . . . . . . . . . . . . . . . . . . . . . . . . 20
6 The \text command

20

CONTENTS

iii

7 Integrals and sums
21
7.1 Multiline subscripts and superscripts . . . . . . . . . . . . . . . . 21
7.2 The \sideset command . . . . . . . . . . . . . . . . . . . . . . . 21
7.3 Placement of subscripts and limits . . . . . . . . . . . . . . . . . 22
7.4 Multiple integral signs . . . . . . . . . . . . . . . . . . . . . . . . 22
8 Commutative diagrams

23

9 Using math fonts
23
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
9.2 Recommended use of math font commands . . . . . . . . . . . . 24
9.3 Bold math symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 24
9.4 Italic Greek letters . . . . . . . . . . . . . . . . . . . . . . . . . . 25
10 Error messages and output problems
25
10.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
10.2 Error messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
10.3 Warning messages . . . . . . . . . . . . . . . . . . . . . . . . . . 32
10.4 Wrong output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
11 Additional information
33
11.1 Compatibility with other packages . . . . . . . . . . . . . . . . . 33
11.2 Converting existing documents . . . . . . . . . . . . . . . . . . . 33
11.3 Technical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
11.4 Getting help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
11.5 Of possible interest . . . . . . . . . . . . . . . . . . . . . . . . . . 34
References

35

Index

36

1

—1—
Introduction
The amsmath package is a LATEX package that provides miscellaneous enhancements for improving the information structure and printed output of documentsthat contain mathematical formulas. Readers unfamiliar with LATEX should refer to [3]. If you have an up-to-date version of LATEX, the amsmath package isnormally provided along with it. Upgrading when a newer version of the amsmath package is released can be done via http://mirror.ctan.org/macros/
latex/required/amsmath.zip.
This documentation describes the features of the amsmath package and discusses how they are intended to be used. It also covers some ancillary packages:
amsbsyamscd

amsopnamstext

amsxtra

These all have something to do with the contents of math formulas. For
information on extra math symbols and math fonts, see [8] and https://www.ams.org/tex/amsfonts.html. For documentation of the amsthm package or AMS document classes (amsart, amsbook, etc.) see [9] or [7] and https://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 listof 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 kindsof 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 summationor 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 packages:
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 \accentedsymbol, for compatibility with documents created using version
1.1.
The amsmath package incorporates amstext, amsopn, and amsbsy. The features of amscd and amsxtra, however, are available only by invoking thosepackages separately.
The independent mathtools package [10] provides some enhancements toamsmath; mathtools loads amsmath automatically, so there is no need to separately load amsmath if mathtools is used. Some mathtools facilities will be
noted below as appropriate.

—2—
Options for the amsmath package
The amsmath package has the following options:centertags (default) For an equation containing a split environment, place
equation numbers vertically centered on the total height of the equation.
tbtags ‘Top-or-bottom tags’: For an equation containing a split environment,
place equation numbers level with the last (resp. first) line, if numbers areon the right (resp. left).
sumlimits (default) Place the subscripts and superscripts of summation symbols above and below, in displayed
equations.
Q `N L This option also affects
other symbols of the same type— , , , , and so forth—but excluding 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 asdet, inf, lim, max, min, that traditionally have subscripts placed underneath when they occur in a displayed equation.
nonamelimits Opposite of namelimits.

3

3.1. INTRODUCTIONalignedleftspaceyesalignedleftspacenoalignedleftspaceyesifneg
To use one of these package options, put the option name in the optional argument of the \usepackage command—e.g., \usepackage[intlimits]{amsmath}.
For AMS document classes and any other classes that preload amsmath desired
options must be specified with the \documentclass—e.g.,\documentclass[intlimits,tbtags,reqno]{amsart}.
The amsmath package also recognizes the following options which are normally selected (implicitly or explicitly) through the \documentclass command,and thus need not be repeated in the option list of the \usepackage{amsmath}statement.
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 thancentered in the text column.
Three options have been added to control the space to the left of aligned
and gathered environments. Prior to the 2017 release a thin space was added
to the left but not the right of these constructs. This appears to have been an
accidental feature of the implementation and was typically corrected by prefixing
the environments with \!.
The new default behavior is aimed to ensure that the environments do nothave a thin space added in most cases, and that existing documents using\!\begin{aligned} continue to work as before.
alignedleftspaceyes Always add \, to the left of aligned and gathered.
alignedleftspaceno Never add \, to the left of aligned and gathered.
alignedleftspaceyesifneg Only add \, if the environment is prefixed bynegative space. (New default behavior.)—3—
Displayed equations
3.1

Introduction

The amsmath package provides a number of additional displayed equation structures beyond the ones provided in basic LATEX. The augmented set includes:
equationgathermultline
split

equation*gather*multline*alignalignatflalignalign*alignat*flalign*4

3. DISPLAYED EQUATIONS(Although the standard eqnarray environment remains available, it is better to
use align or equation+split instead. Within eqnarray, spacing around signsof relation is not the preferred mathematical spacing, and is inconsistent withthat spacing as it appears in other environments. Long lines in this environment
may result in misplaced or overprinted equation numbers. This environment also
does not support the use of \qed or \qedhere as provided by theorem packages.)
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 end of that line; \notag should not be used outside a display environment as it will mess up the numbering. You can also override a number with atag of your own using \tag{⟨label ⟩}, where ⟨label ⟩ means arbitrary text such as$*$ or ii used to “number” the equation. A tag can reference a different tagged
display by use of \tag{\ref{⟨label ⟩}⟨modifier ⟩} where ⟨modifier ⟩ is optional.
If you are using hyperref, use \ref*; use of the starred form of \ref prevents areference to a modified tag containing a nested link from linking to the original
display.
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 usedwithin 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 andsubeqn.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. split supports
only one alignment (&) column; if more are needed, aligned or alignedat
should be used. The width of a split structure is the full line width.
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.
In all multiline environments, lines are divided by \\. The \\ should not be
used to end the last line. Using it there will result in unwanted extra verticalspace following the display.
In all math environments (inline or display), blank lines (equivalent to \par)are not permitted, and will result in an error.
3.2

Single equations

The equation environment is for a single equation with an automatically generated number. The equation* environment is the same except for omitting
the number.1 The wrapper \[ ... \] is equivalent to equation*.
3.3

Split equations without alignment

The multline environment is a variation of the equation environment used forequations that don’t fit on a single line. The first line of a multline will be at
1 Basic L
AT X doesn’t provide an equation* environment, but rather a functionally equivE
alent environment named displaymath.

5

3.3. SPLIT EQUATIONS WITHOUT ALIGNMENT

Table 3.1. Comparison of displayed equation environments (vertical lines indicating 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}& = c_{11}&a_{12}& =b_{12}\\b_{21}& = c_{21}&a_{22}& =b_{22}+c_{22}\end{flalign*}

a=b

(1)

a=b

a=b+c−d
+e−f

(2)

=g+h
=i

(3) a + b + c + d + e + f+i+j+k+l+m+n(4)

a1 = b1 + c1

(5)

a2 = b2 + c2 − d2 + e2

(6)

a1 = b1 + c1

(7)

a2 = b2 + c2 − d2 + e2

(8)

a11 = b11a12 = b12(9)

a21 = b21

a22 = b22 + c22

a11 + b11 = c11

a12 = b12

b21 = c21

a22 = b22 + c22

6

3. DISPLAYED EQUATIONS

the left margin and the last line at the right margin, except for an indention on
both sides in the amount of \multlinegap. Any additional lines in between will
be centered independently within the display width (unless the fleqn option isin effect).
Like equation, multline has only a single equation number (thus, noneof the individual lines should be marked with \notag). The equation number
is placed on the last line (reqno option) or first line (leqno option); verticalcentering as for split is not supported by multline.
It’s possible to force one of the middle lines to the left or right with commands \shoveleft, \shoveright. These commands take the entire line as anargument, up to but not including the final \\; for example
A(3.10)

B
C
D\begin{multline}\framebox[.65\columnwidth]{A}\\\framebox[.5\columnwidth]{B}\\\shoveright{\framebox[.55\columnwidth]{C}}\\\framebox[.65\columnwidth]{D}\end{multline}
The value of \multlinegap can be changed with the usual LATEX commands\setlength or \addtolength.
3.4

Split equations with alignment

Like multline, the split environment is for single equations that are too longto fit on one line and hence must be split into multiple lines. Unlike multline,
however, the split environment provides for alignment among the split lines,
using & to mark alignment points. Unlike the other amsmath equation structures,the split environment provides no numbering, because it is intended to be used
only inside some other displayed equation structure, usually an equation, align,
or gather environment, which provides the numbering. For example:nHc =(3.11)

1 X(−1)l (n − l)p−2
2n
l=0

X

p  
Y
ni

l1 +···+lp =l i=1

li

p
h
i
X
· [(n − l) − (ni − li )]ni −li · (n − l)2 −(ni − li )2 .
j=1

\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}\\73.6. EQUATION GROUPS WITH MUTUAL ALIGNMENT&\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}
The split structure should constitute the entire body of the enclosing structure, apart from commands like \label that produce no visible material.
3.5

Equation groups without alignment

The 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 3.1). Equations inside gather are separated by a\\ command. Any equation in a gather may consist of a \begin{split} . . .
\end{split} structure—for example:\begin{gather}first equation\\\begin{split}
second & equation\\& on two lines\end{split}\\
third equation\end{gather}
3.6

Equation groups with mutual alignment

The align environment is used for two or more equations when vertical alignment is desired; usually binary relations such as equal signs are aligned (seeTable 3.1).
To have several equation columns side-by-side, use extra ampersands to separate the columns:(3.12)

x=y(3.13)

′

X=Y

x =y

(3.14)

x + x′ = y + y ′

′

′

X =Y

a=b+c
′

X + X′ = Y + Y ′a′ = ba′ b = c′ b\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}
Line-by-line annotations on an equation can be done by judicious application of\text inside an align environment:(3.15)

x = y1 − y2 + y3 − y5 + y8 − . . .
′∗(3.16)=y ◦y(3.17)= y(0)y ′by (3.24)
by (4.1)
by Axiom 1.

8

3. DISPLAYED EQUATIONS\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}
A variant environment alignat allows the horizontal space between equations
to be explicitly specified. This environment takes one argument, the numberof “equation columns” (the number of pairs of right-left aligned columns; theargument is the number of pairs): count the maximum number of &s in any row,
add 1 and divide by 2.
x = y1 − y2 + y3 − y5 + y8 − . . .

(3.18)′∗(3.19)=y ◦y(3.20)= y(0)y ′by (3.24)
by (4.1)
by Axiom 1.

\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}
The environment flalign (“full length alignment”) stretches the space between
the equation columns to the maximum possible width, leaving only enough spaceat the margin for the equation number, if present.
(3.21)

x=y(3.22)

′

x =y

X=Y
′

(3.23) x + x′ = y + y ′X′ = Y ′X + X′ = Y + Y ′\begin{flalign}x&=y& X&=Y\\x’&=y’& X’&=Y’\\
x+x’&=y+y’ & X+X’&=Y+Y’\end{flalign}
x=y

X=Y

x′ = y ′X′ = Y ′x + x′ = y + y ′X + X′ = Y + Y ′\begin{flalign*}
x&=y& X&=Y\\x’&=y’& X’&=Y’\\
x+x’&=y+y’ & X+X’&=Y+Y’\end{flalign*}

3.8. ADJUSTING TAG PLACEMENT

3.7

9

Alignment building blocks

Like equation, the multi-equation environments gather, align, and alignat
are designed to produce a structure whose width is the full line width. Thismeans, for example, that one cannot readily add parentheses around the entire
structure. But variants gathered, aligned, and alignedat are provided whose
total width is the actual width of the contents; thus they can be used as acomponent in a containing expression. E.g.,)
B ′ = −∂ × E,
Maxwell’s equations
E ′ = ∂ × B − 4πj,\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*}
Like the array environment, these -ed variants also take an optional [t], [b]or the default [c] argument to specify vertical positioning. For maximum interoperability, do not insert a space or line break before the option. See also
the note on page 10 regarding page breaking within the -ed environments.
“Cases” constructions like the following are common in mathematics:(
0
if r − j is odd,
(3.24)
Pr−j =(r−j)/2
r! (−1)
if r − j is even.
and in the amsmath package there is a cases environment to make them easy
to write:
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}
Notice the use of \text (cf. §6) and the nested math formulas. cases is set
in \textstyle. If \displaystyle is wanted, it must be requested explicitly;
mathtools provides a dcases environment for this purpose.
The -ed and cases environments must appear within an enclosing mathenvironment, which can be either in text, between $...$, or in any of thedisplay environments.
3.8

Adjusting tag placement

Placing equation numbers can be a rather complex problem in multiline displays.
The environments of the amsmath package try hard to avoid overprinting anequation number on the equation contents, if necessary moving the number

10

3. DISPLAYED EQUATIONS

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 lookright. A \raisetag command is provided to adjust the vertical position of thecurrent equation number, if it has been shifted away from its normal position. If
it is in its normal position the command has no effect whatsoever. In particular,
this means that one can’t use it to shift the tag of an equation environment.
To move a particular number up by six points, write \raisetag{6pt}. (Atthe end of a display, this also shifts up the text following the display.) Thiskind 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.
3.9

Vertical spacing and page breaks in multiline displays

You can use the \\[⟨dimension⟩] command to get extra vertical space betweenlines in all the amsmath displayed equation environments, as is usual in LATEX.
Do not type a space between the \\ and the following [; only for display environments defined by amsmath the space is interpreted to mean that the bracketedmaterial is part of the document content.
When the amsmath package is in use page breaks between equation lines arenormally disallowed; the philosophy is that page breaks in such material should
receive individual attention from the author. To get an individual page breakinside a particular displayed equation, a \displaybreak command is provided.
\displaybreak is best placed immediately before the \\ where it is to takeeffect. Like LATEX’s \pagebreak, \displaybreak takes an optional argument
between 0 and 4 denoting the desirability of the pagebreak. \displaybreak[0]means “it is permissible to break here” without encouraging a break; \displaybreak with no optional argument is the same as \displaybreak[4] and forcesa break.
If you prefer a strategy of letting page breaks fall where they may, even in the
middle of a multiline equation, then you might put \allowdisplaybreaks[1]in the preamble of your document. An optional argument 1–4 can be used forfiner control: [1] means allow page breaks, but avoid them as much as possible; values of 2,3,4 mean increasing permissiveness. When display breaks areenabled with \allowdisplaybreaks, the \\* command can be used to prohibita pagebreak after a given line, as usual.
Note: Certain equation environments wrap their contents in an
unbreakable box, with the consequence that neither \displaybreaknor \allowdisplaybreaks will have any effect on them. These include
split, aligned, gathered, and alignedat.
3.10

Interrupting a display

The command \intertext is used for a short interjection of one or two linesof text in the middle of a multiple-line display structure (see also the \textcommand in §6). Its salient feature is preservation of the alignment, which
would not happen if you simply ended the display and then started it up again

3.11. EQUATION NUMBERING

11

afterwards. \intertext may only appear right after a \\ or \\* command.
Notice the position of the word “and” in this example.
(3.25)

A1 = N0 (λ; Ω′ ) − ϕ(λ; Ω′ ),(3.26)

A2 = ϕ(λ; Ω′ ) − ϕ(λ; Ω),and(3.27)

A3 = N (λ; ω).

\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}
The mathtools package provides a command \shortintertext that is intended for use when the interjected text is only a few words; it uses less verticalspace than \intertext. This is most effective when equation numbers are on
the right.
3.11
3.11.1Equation numbering
Numbering hierarchy

In LATEX if you wanted to have equations numbered within sections—that is,have equation numbers (1.1), (1.2), . . . , (2.1), (2.2), . . . , in sections 1, 2, and soforth—you could redefine \theequation as suggested in the LATEX manual [3,§6.3, §C.8.4]:\renewcommand{\theequation}{\thesection.\arabic{equation}}
This works pretty well, except that the equation counter won’t be reset tozero at the beginning of a new section or chapter, unless you do it yourself using\setcounter. To make this a little more convenient, the amsmath package provides a command \numberwithin. To have equation numbering tied to section
numbering, with automatic reset of the equation counter, write\numberwithin{equation}{section}
As its name implies, the \numberwithin command can be applied to any counter,not just the equation counter.
3.11.2

Cross references to equation numbers

To make cross-references to equations easier, an \eqref command is provided.
This automatically supplies the parentheses around the equation number. I.e.,
if \ref{abc} produces 3.2 then \eqref{abc} produces (3.2). The parenthesesaround an \eqref equation number will be set in upright type regardless of typestyle of the context.

12

4. MISCELLANEOUS MATHEMATICAL FEATURES

3.11.3 Subordinate numbering sequences
The amsmath package provides also a wrapper environment, subequations, tomake it easy to number equations in a particular \align or similar group witha subordinate numbering scheme. For example\begin{subequations}...
\end{subequations}causes all numbered equations within that part of the document to be numbered(4.9a) (4.9b) (4.9c) . . . , if the preceding numbered equation was (4.8). A \labelcommand immediately after \begin{subequations} will produce a \ref of theparent number 4.9, not 4.9a. The counters used by the subequations environment are parentequation and equation and \addtocounter, \setcounter,\value, etc., can be applied as usual to those counter names. To get anythingother than lowercase letters for the subordinate numbers, use standard LATEX
methods for changing numbering style [3, §6.3, §C.8.4]. For example, redefining\theequation as follows will produce roman numerals.
\begin{subequations}\renewcommand{\theequation}{\theparentequation \roman{equation}}...
3.11.4 Numbering style
The default equation number is set in \normalfont. This means that in bold
section headings, bold is suppressed; a workaround for this is to use (\ref{. . . })rather than \eqref{. . . }.
If smaller type is specified for a numbered display, the size of the equation number will also be small. The default size can be ensured throughout adocument by applying this patch in the preamble:\makeatletter\renewcommand{\maketag@@@}[1]{\hbox{\m@th\normalsize\normalfont#1}}%\makeatother(This modification may be included in a future version of amsmath.)—4—
Miscellaneous mathematical features
4.1

Matrices

The amsmath package provides some environments for matrices beyond the basic
array environment of LATEX. The pmatrix, bmatrix, Bmatrix, vmatrix and
Vmatrix have (respectively) ( ), [ ], { }, | |, and ∥ ∥ delimiters built in. For
naming consistency there is a matrix environment sans delimiters. This is notentirely redundant with the array environment; the matrix environments alluse more economical horizontal spacing than the rather prodigal spacing of thearray environment. Also, unlike the array environment, you don’t have to13

4.2. MATH SPACING COMMANDS

give column specifications for any of the matrix environments; by default you
can have up to 10 centered columns.2 (If you need left or right alignment in acolumn or other special formats you may use array, or the mathtools package
which provides * variants of these environments with an optional argument tospecify left or right alignment.)
To produce a small matrix suitable for use in text, there is a smallmatrix
environment (e.g., ac db ) that comes closer to fitting within a single text line
than a normal matrix. Delimiters must be provided. (The mathtools package
provides p,b,B,v,V versions of smallmatrix, as well as * variants as describedabove.) The above example was produced by\bigl( \begin{smallmatrix}
a&b\\ c&d\end{smallmatrix} \bigr)\hdotsfor{⟨number ⟩} produces a row of dots in a matrix spanning the given
number of columns. For example,a b c d\begin{matrix} a&b&c&d\\e .......
e&\hdotsfor{3} \end{matrix}
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).

D1 t−a12 t2 . . . −a1n tn
 −a21 t1
D2 t. . . −a2n tn 


(4.1)
. . . . . . . . . . . . . . . . . . . . . . ,−an1 t1 −an2 t2 . . .
Dn t\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}
4.2

Math spacing commands

The amsmath package slightly extends the set of math spacing 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.
Abbrev.
\,\:\;Spelled out
no space\thinspace\medspace\thickspace\quad\qquadExample
⇒⇐
⇒⇐
⇒⇐
⇒⇐
⇒ ⇐
⇒
⇐

Abbrev.
\!

Spelled out
no space\negthinspace\negmedspace\negthickspace

Example
⇒⇐
⇒⇐
⇒⇐
⇒⇐

2 More precisely: The maximum number of columns in a matrix is determined by thecounter MaxMatrixCols (normal value = 10), which you can change if necessary using LATEX’s\setcounter or \addtocounter commands.

14

4. MISCELLANEOUS MATHEMATICAL FEATURES

For the greatest possible control over math spacing, use \mspace and ‘mathunits’. One math unit, or mu, is equal to 1/18 em. Thus to get a negative \quad
you could write \mspace{-18.0mu}.
4.3

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• \dotsc for “dots with commas”• \dotsb for “dots with binary operators/relations”• \dotsm for “multiplication dots”• \dotsi for “dots with integrals”• \dotso for “other dots” (none of the above)instead of \ldots and \cdots, you make it possible for your document to beadapted to different conventions on the fly, in case (for example) you have tosubmit it to a publisher who insists on following house tradition in this respect.
The default treatment for the various kinds follows American MathematicalSociety conventions:
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 $, andthe infinite integral\[\int_{A_1}\int_{A_2}\dotsi\].

Then we have the series A1 , A2 , . . . ,the regional sum A1 + A2 + · · · , theorthogonal product A1 A2 · · · , andthe infinite integral
Z Z
···.
A1

A2

For most situations, the undifferentiated \dots can be used, and amsmath will
output the most suitable form based on the immediate context; if an inappropriate form results, it can be corrected after examining the output.
4.4

Nonbreaking dashes

A command \nobreakdash is provided to suppress the possibility of a line breakafter the following hyphen or dash. For example, if you write ‘pages 1–9’ aspages 1\nobreakdash--9 then a line break will never occur between the dashand the 9. You can also use \nobreakdash to prevent undesirable hyphenations in combinations like $p$-adic. For frequent use, it’s advisable to makeabbreviations, e.g.,\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}}
The last example shows how to prohibit a line break after the hyphen but allow
normal hyphenation in the following word. (It suffices to add a zero-width space
after the hyphen.)15

4.9. EXTENSIBLE ARROWS

4.5

Accents in math

In ordinary LATEX the placement of the second accent in doubled math accentsis often poor. With the amsmath package you will get improved placement ofˆthe second accent: Â (\hat{\hat{A}}).
The commands \dddot and \ddddot are available to produce triple andquadruple dot accents in addition to the \dot and \ddot accents already available in LATEX.
To get a superscripted hat or tilde character, load the amsxtra package anduse \sphat or \sptilde. Usage is A\sphat (note the absence of the ^ character).
To place an arbitrary symbol in math accent position, or to get under accents,
see the accents package by Javier Bezos. (amsmath must be loaded before
accents.)
4.6

Roots

4.7

Boxed formulas√
In ordinary LATEX the placement of root indices is sometimes not so good: β k(\sqrt[\beta]{k}). In the amsmath package \leftroot and \uproot allow
you to adjust the position of the root:\sqrt[\leftroot{-2}\uproot{2}\beta]{k}√βwill move the beta up and to the right: k. The negative argument used with\leftroot moves the β to the right. The units are a small amount that is a
useful size for such adjustments.

The command \boxed puts a box around its argument, like \fbox except thatthe contents are in math mode:η ≤ C(δ(η) + ΛM (0, δ))(4.2)\boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))}
4.8Over and under arrows

Basic LATEX provides \overrightarrow and \overleftarrow commands. Some
additional over and under arrow commands are provided by the amsmath package to extend the set:\overleftarrow\overrightarrow\overleftrightarrow
4.9\underleftarrow\underrightarrow\underleftrightarrow

Extensible arrows\xleftarrow and \xrightarrow produce arrows that extend automatically toaccommodate unusually wide subscripts or superscripts. These commands takeone optional argument (the subscript) and one mandatory argument (the superscript, possibly empty):
(4.3)

n+µ−1

n±i−1

A ←−−−−− B −−−−→ CT16

4. MISCELLANEOUS MATHEMATICAL FEATURES\xleftarrow{n+\mu-1}\quad \xrightarrow[T]{n\pm i-1}
4.10

Affixing symbols to other symbols

LATEX provides \stackrel for placing a superscript above a binary relation. In
the amsmath package there are somewhat more general commands, \overset
and \underset, that can be used to place one symbol above or below anothersymbol, whether it’s a relation or something else.
The input \overset{*}{X}∗will place a superscript-size ∗ above the X : X ; \underset is the analog foradding a symbol underneath. The command \overunderset is a combinationof these, taking three arguments to place superscript sized expressions aboveand below the same base.
See also the description of \sideset in §7.2.
4.11

Fractions and related constructions

4.11.1 The \frac, \dfrac, and \tfrac commands
The \frac command, which is in the basic command set of LATEX, takes twoarguments—numerator and denominator—and typesets them in normal fraction
form. The amsmath package provides also \dfrac and \tfrac as convenient
abbreviations for {\displaystyle\frac ... } and {\textstyle\frac ... }.
r
r
1
1
1
log2 c(f ) k1 log2 c(f )
log2 c(f )
log2 c(f )
(4.4)
k
k
k
\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}
4.11.2 The \binom, \dbinom, and \tbinom commands
For binomial expressions such as nk amsmath has \binom, \dbinom and \tbinom:
(4.5)

 
 
k k−1
k k−2
2 −
2
+
2
1
2
k

2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}
4.11.3 The \genfrac command
The capabilities of \frac, \binom, and their variants are subsumed by a generalized fraction command \genfrac with six arguments. The last two correspondto \frac’s numerator and denominator; the first two are optional delimiters(as seen in \binom); the third is a line thickness override (\binom uses this to
set the fraction line thickness to 0—i.e., invisible); and the fourth argument
is a mathstyle override: integer values 0–3 select respectively \displaystyle,\textstyle, \scriptstyle, and \scriptscriptstyle. If the third argument
is left empty, the line thickness defaults to ‘normal’.
\genfrac{left-delim}{right-delim}{thickness}{mathstyle}{numerator}{denominator}

17

4.14. DELIMITERS

To illustrate, here is how \frac, \tfrac, and \binom might be defined.
\newcommand{\frac}[2]{\genfrac{}{}{}{}{#1}{#2}}\newcommand{\tfrac}[2]{\genfrac{}{}{}{1}{#1}{#2}}\newcommand{\binom}[2]{\genfrac{(}{)}{0pt}{}{#1}{#2}}
If you find yourself repeatedly using \genfrac throughout a document for aparticular notation, you will do yourself a favor (and your publisher) if you
define a meaningfully-named abbreviation for that notation, along the lines of\frac and \binom.
The primitive generalized fraction commands \over, \overwithdelims, \atop,\atopwithdelims, \above, \abovewithdelims produce warning messages if
used with the amsmath package, for reasons discussed in technote.tex.
4.12

Continued fractions

The continued fraction
1

(4.6)

√

2+

1
√

2+ √

1
2 + ···

can be obtained by typing\cfrac{1}{\sqrt{2}+\cfrac{1}{\sqrt{2}+\cfrac{1}{\sqrt{2}+\dotsb}}}
This produces better-looking results than straightforward use of \frac. Leftor right placement of any of the numerators is accomplished by using \cfrac[l]or \cfrac[r] instead of \cfrac.
4.13

Smash options

The command \smash is used to typeset a subformula with an effective heightand depth of zero, which is sometimes useful in adjusting the subformula’s
position with respect to adjacent symbols. With the amsmath package \smash
has optional arguments [t] and [b] because occasionally it is advantageous tobe able to “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 oftheir
to make them more consistent. Compare√ contents,√ \smash√ can√be employed√√x + y + z and x + y + z, where the latter was produced by $\sqrt{x}+ \sqrt{\smash[b]{y}} + \sqrt{z}$.
4.14

Delimiters

4.14.1 Delimiter sizes
The automatic delimiter sizing done by \left and \right has two limitations:First, it is applied mechanically to produce delimiters large enough to encompass

18

4. MISCELLANEOUS MATHEMATICAL FEATURES

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 mathfragment that is infinitesimally too large for a given delimiter size will get thenext 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 ofcommands that have ‘big’ in their names.
Delimiter
size

text
size

Resultc(b)( )d\left\right
c
(b)
d

\bigl
\bigr
b

 c
d

\Bigl
\Bigr
  c bd\biggl\biggr  
cbd\Biggl\Biggr!
!
cbd

The first kind of situation is a cumulative operator with limits above and below.
With \left and \right the delimiters usually turn out larger than necessary,and using the Big or bigg sizes instead gives better results:p 1/p
X
X

ai
xij 
i

j

versus

X
i

ai

X


p 1/p
xij

j

\biggl[\sum_i a_i\Bigl\lvert\sum_j x_{ij}\Bigr\rvert^p\biggr]^{1/p}The second kind of situation is clustered pairs of delimiters where \left and\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.
((a1 b1 ) − (a2 b2 )) ((a2 b1 ) + (a1 b2 )) versus (a1 b1 ) − (a2 b2 ) (a2 b1 ) + (a1 b2 )\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)
The third kind of situation is a slightly oversize object in running text, such asb′d′ where the delimiters produced by \left and \right cause too much line
spreading. In that case \bigl and \bigr can be used to produce delimiters thatare slightly larger than the base size but still able to fit within the normal line′
spacing: db ′ .
In ordinary LATEX \big, \bigg, \Big, and \Bigg delimiters aren’t scaled
properly over the full range of LATEX font sizes. With the amsmath package
they are.
4.14.2 Vertical bar notations
The amsmath package provides commands \lvert, \rvert, \lVert, \rVert(compare \langle, \rangle) to address the problem of overloading for thevert bar character |. This character is currently used in LATEX documents to5.1. DEFINING NEW OPERATOR NAMES

19

represent a wide variety of mathematical objects: the ‘divides’ relation in a
number-theory expression like p|q, or the absolute-value operation |z|, or the‘such that’ condition in set notation, or the ‘evaluated at’ notation fζ (t) t=0 .
The multiplicity of uses in itself is not so bad; what is bad, however, is that factthat not all of the uses take the same typographical treatment, and that thecomplex discriminatory powers of a knowledgeable reader cannot be replicatedin 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 | and a selected mathematical notation, andsimilarly for the double-bar command \|. This immediately rules out the useof | and \| 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:\providecommand{\abs}[1]{\lvert#1\rvert}\providecommand{\norm}[1]{\lVert#1\rVert}
whereupon the document would contain \abs{z} to produce |z| and \norm{v}
to produce ∥v∥. The mathtools provides the command \DeclarePairedDelimiter for defining \abs-like macros with scaling delimiters.

—5—
Operator names
5.1

Defining new operator names

Math functions such as log, sin, and lim are traditionally typeset in roman typeto make them visually more distinct from one-letter math variables, which are
set in math italic. The more common ones have predefined names, \log, \sin,\lim, and so forth, but new ones come up all the time in mathematical papers,
so the amsmath package provides a general mechanism for defining new ‘operator
names’. To define a math function \xxx to work like \sin, you write\DeclareMathOperator{\xxx}{xxx}whereupon ensuing uses of \xxx will produce xxx in the proper font and automatically add proper spacing on either side when necessary, so that you get
A xxx B instead of AxxxB. In the second argument of \DeclareMathOperator(the name text), a pseudo-text mode prevails: the hyphen character - will print
as a text hyphen rather than a minus sign and an asterisk * will print as a raised
text asterisk instead of a centered math star. (Compare a-b*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 * form of the\DeclareMathOperator command:\DeclareMathOperator*{\Lim}{Lim}20

6. THE \TEXT COMMAND

See also the discussion of subscript placement in Section 7.3.
The following operator names are predefined:\deg deg\lg lg\projlim proj lim\det det\lim lim\sec sec\dim dim\liminf lim inf\sin sin\exp exp\limsup lim sup\sinh sinh
\gcd gcd\ln ln\sup sup\hom hom\log log\tan tan\inf inf\max max\tanh tanh\injlim inj lim\min min\ker ker\Pr Pr\varinjlim lim \varliminf lim−→\varprojlim lim \varlimsup lim←−
There is also a command \operatorname such that using\operatorname{abc}in a math formula is equivalent to a use of \abc defined by \DeclareMathOperator. This may be occasionally useful for constructing more complex notationor other purposes. (Use the variant \operatorname* to get limits.)\arccos\arcsin\arctan
\arg
\cos
\cosh
\cot
\coth
\csc

5.2

arccosarcsinarctan
arg
cos
cosh
cot
coth
csc

\mod and its relatives

Commands \mod, \bmod, \pmod, \pod are provided to deal with the special
spacing conventions of “mod” notation. \bmod and \pmod are available in LATEX,
but with the amsmath package the spacing of \pmod will adjust to a smaller value
if it’s used in a non-display-mode formula. \mod and \pod are variants of \pmod
preferred by some authors; \mod omits the parentheses, whereas \pod omits the“mod” and retains the parentheses.
(5.1)

gcd(n, m mod n);x≡y(mod b);x≡ymod c;x≡y(d)\gcd(n,m\bmod n);\quad x\equiv y\pmod b;\quad x\equiv y\mod c;\quad x\equiv y\pod d—6—
The \text command
The main use of the command \text is for words or phrases in a display. It
is very similar to the LATEX command \mbox in its effects, but has a coupleof advantages. If you want a word or phrase of text in a subscript, you cantype ..._{\textrm{word or phrase}}, which is slightly easier than the \mbox
equivalent: ..._{\mbox{\rmfamily\scriptsize word or phrase}}. Note thatthe standard \textrm command will use the amsmath \text definition, but
ensure the \rmfamily font is used.
(6.1)

f[xi−1 ,xi ] is monotonic,i = 1, . . . , c + 1

21

7.2. THE \SIDESET COMMAND

f_{[x_{i-1},x_i]} \text{ is monotonic,}\quad i = 1,\dots,c+1
The font used for \text is the same as that of the surrounding environment;i.e., within a theorem, the contents of \text will be set in italic.
If a math expression is included in a \text string, it must be explicitly
marked as math ($...$).
∂s f (x) =

∂
f (x)
∂x0

for x = x0 + Ix1 .

\partial_s f(x) = \frac{\partial}{\partial x_0} f(x)\quad\text{for $x= x_0 + I x_1$.}
Function names should not be entered as \text. Instead, use \mathrm or\DeclareMathOperator as appropriate. These are fixed entities that should notchange depending on outside content (such as appearing within a theorem thatis set in italic), and in the case of declared operators, proper spacing is applied
automatically.

—7—
Integrals and sums
7.1

Multiline subscripts and superscripts

The \substack command can be used to produce a multiline subscript or superscript: for example\sum_{\substack{
0\le i\le m\\
0<j<n}}
P(i,j)X

P (i, j)

0≤i≤m
0<j<n

A slightly more generalized form is the subarray environment which allows you
to specify that each line should be left-aligned instead of centered, as here:\sum_{\begin{subarray}{l}i\in\Lambda\\ 0<j<n\end{subarray}}P(i,j)
7.2

X

P (i, j)

i∈Λ
0<j<n

The \sideset command

There’s also a command called \sideset, for a rather special purpose: putting
symbols
P at
Q the subscript and superscript corners of a large operator symbol suchasor . Note: this command is not designed to be applied to anything otherthan sum-class symbols. The prime example is the case when you want to put

22

7. INTEGRALS AND SUMSa prime on a sum symbol. If there are no limits above or below the sum, you
could just use \nolimits: here’s \sum\nolimits’ E_n in display mode:
X′

(7.1)

En

If, however, you want not only the prime but also something below or above thesum symbol, it’s not so easy—indeed, without \sideset, it would be downrightdifficult. With \sideset, you can write\sideset{}{’}\sum_{n<k,\;\text{$n$ odd}} nE_n

X′

nEn

n<k, n odd

The extra pair of empty braces is explained by the fact that \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\sideset{_*^*}{_*^*}\prod
7.3

∗ Y∗
∗

∗

Placement of subscripts and limits

The default positioning for subscripts depends on the base symbol involved. The
default for sum-class symbols is ‘displaylimits’ positioning: When a sum-classsymbol appears in a displayed formula, subscript and superscript are placed in‘limits’ position above and below, but in an inline formula, they are placed tothe 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 tothe side, even in displayed formulas. (See the discussion of the intlimits andrelated options in Section 2.)
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 \limits and \nolimits can be used to override the normal
behavior of a base symbol:
ZZ
X,,
limn→∞XA

To define a command whose subscripts follow the same ‘displaylimits’ behavior
as \sum, put \displaylimits at the tail end of the definition. When multiple
instances of \limits, \nolimits, or \displaylimits occur consecutively, thelast one takes precedence.
7.4

Multiple integral signs\iint, \iiint, and \iiiint give multiple integral signs with the spacing between them nicely adjusted, in both text and display style. \idotsint is an23

9.1. INTRODUCTION

extension of the same idea that gives two integral signs with dots between them.
ZZ
ZZZ
(7.2)
f (x, y) dx dyf (x, y, z) dx dy dz
A
ZZZZ Af (w, x, y, z) dw dx dy dz

(7.3)

A

Z

Z
···

f (x1 , . . . , xk )A

—8—Commutative diagrams
Some commutative diagram commands like the ones in AMS-TEX are available
as a separate package, amscd. For complex commutative diagrams authors willneed to turn to more comprehensive packages like Tik Z (in particular, tikzcd) or XY-pic, but for simple diagrams without diagonal arrows the amscd
commands may be more convenient. Here is one example.
j

S WΛ ⊗ T −−−−→y(S ⊗ T )/ITyEnd P(Z ⊗ T )/J\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}
In the 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 As or Vs ofvertical arrows will be typeset as left or right “sidescripts”. The commands @=and @| give horizontal and vertical double lines. A “null arrow” command @.
can be used instead of a visible arrow to fill out an array where needed.

—9—
Using math fonts
9.1

Introduction

For more comprehensive information on font use in LATEX, see the LATEX fontguide (fntguide.tex) or The LATEX Companion [4]. The basic set of math font

24

9. USING MATH FONTS

commands in LATEX includes \mathbf, \mathrm, \mathcal, \mathsf, \mathtt,\mathit. Additional math alphabet commands such as \mathbb for blackboard bold, \mathfrak for Fraktur, and \mathscr for Euler script are available
through the packages amsfonts and euscript (distributed separately).
9.2

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 \mb instead of \mathbf.
Of course, there is nothing to keep you from providing such abbreviations foryourself by suitable \newcommand statements. But for LATEX to provide shorter
names would actually be a disservice to authors, as that would obscure a muchbetter 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:\newcommand{\vect}[1]{\mathbf{#1}}
you can write \vect{a} + \vect{b} to produce a + 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 \vect; otherwise you would haveto replace all occurrences of \mathbf throughout your document, perhaps evenneeding 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 lettersof a particular font:\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}% or use amsfonts package\DeclareMathSymbol{\C}{\mathalpha}{AMSb}{"43}\DeclareMathSymbol{\R}{\mathalpha}{AMSb}{"52}These statements would define the commands \C and \R to produce blackboardbold letters from the ‘AMSb’ math symbols font. If you refer often to thecomplex numbers or real numbers in your document, you might find this method
more convenient than (let’s say) defining a \field command and writing\field{C}, \field{R}. But for maximum flexibility and control, define such a\field command and then define \C and \R in terms of that command:\usepackage{amsfonts}% to get the \mathbb alphabet\newcommand{\field}[1]{\mathbb{#1}}\newcommand{\C}{\field{C}}\newcommand{\R}{\field{R}}9.3

Bold math symbols

The \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\Delta \mathbf{\Delta}\mathbf{+}\delta \mathbf{\delta}
produces ∆∆+δδ; the \mathbf has no effect on the plus sign or the small delta.
The amsmath package therefore provides two additional commands, \boldsymbol and \pmb, that can be applied to other kinds of math symbols. \boldsymbol
can be used for a math symbol that remains unaffected by \mathbf if (and only

25

10.1. GENERAL REMARKSif) your current math font set includes a bold version of that symbol. \pmbcan be used as a last resort for any math symbols that do not have a true boldversion provided by your set of math fonts; “pmb” stands for “poor man’s bold”
and the command works by typesetting multiple copies of the symbol with slightoffsets. The quality of the output is inferior, especially for symbols that containany hairline strokes. When the standard default set of LATEX math fonts are inuse (Computer Modern), the only symbols that are likely to require \pmb arelarge operator symbols like \sum, extended delimiter symbols, or the extra mathsymbols provided by the amssymb package [8].
The following formula shows some of the results that are possible:
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}}A∞ + πA0 ∼ A∞ + πA0 ∼ A∞ + π A0
If you want to use only the \boldsymbol command without loading the wholeamsmath package, the bm package is recommended (this is a standard LATEX
package, not an AMS package; you probably have it already if you have a 1997or newer version of LATEX).
9.4

Italic Greek letters

For italic versions of the capital Greek letters, the following commands areprovided:\varGamma\varDelta\varTheta\varLambda\varXi\varPi

Γ
∆
Θ
Λ
Ξ
Π

\varSigma\varUpsilon\varPhi\varPsi\varOmega

Σ
Υ
Φ
Ψ
Ω

—10—
Error messages and output problems
10.1

General remarks

This is a supplement to Chapter 8 of the LATEX manual [3] (first edition: Chapter 6). Appendix B of the Companion [4] lists all LATEX errors including those
for amsmath with likely causes for each. For the reader’s convenience, the setof error messages discussed here overlaps somewhat with the set in those references, 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

26

10. ERROR MESSAGES AND OUTPUT PROBLEMS

such as \. Where examples are given, we show also the help messages thatappear on screen when you respond to an error message prompt by entering h.
Remember that the important error message is the first for a given line.
When running in nonstop mode, errors accumulate, and the first error may
create a further error condition from which it is impossible to recover. In such asituation processing will stop after 100 errors, so the reporting of genuine errors
may not be complete, and it may be impossible to determine which reported
errors are genuine and which are not.
A final section discusses some output errors, i.e., instances where the printed
document has something wrong but there was no LATEX error during typesetting.
10.2

Error messages\begin{split} won’t work here.
Example:! 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.
?
Explanation: The split environment does not construct a stand-alone displayed
equation; it needs to be used within some other environment such as equation
or gather.
Erroneous nesting of equation structures
Example:! 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*}
Explanation: The structures align, 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 align and most of its variants can beused inside the gather environment.

10.2. ERROR MESSAGES

27

Extra & on this line
Example:! 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.
?
Explanation: In an 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 anewline command \\ and the above error is issued.
Font OMX/cmex/m/n/7=cmex7 not loadable ...
Example:! Font OMX/cmex/m/n/7=cmex7 not loadable: Metric (TFM) file not found.
<to be read again>
relax
l.8 $ab+b^2$? 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>’.
?
Explanation: Certain extra sizes of some Computer Modern fonts that wereformerly available mainly through the AMSFonts distribution are consideredpart of standard LATEX (as of June 1994): cmex7–9, cmmib5–9, and cmbsy5–9.
If these extra sizes are missing on your system, you should try first to get themfrom the source where you obtained LATEX. If that fails, you could try getting
the fonts from CTAN (e.g., in the form of Metafont source files, directory /texarchive/fonts/latex/mf, or in PostScript Type 1 format, directory /texarchive/fonts/cm/ps-type1/bakoma).
If the font name begins with cmex, there is a special option cmex10 for
the amsmath package that provides a temporary workaround. I.e., change the\usepackage to\usepackage[cmex10]{amsmath}
This will force the use of the 10-point size of the cmex font in all cases. Depending on the contents of your document this may be adequate.

28

10. ERROR MESSAGES AND OUTPUT PROBLEMS

Improper argument for math accent
Example:! 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}}?
Explanation: Non-simple arguments for any LATEX command should be enclosed
in braces. In this example extra braces are needed as follows:... P_{\tilde{\mathcal{M}}}
Math formula deleted: Insufficient extension fonts
Example:! Math formula deleted: Insufficient extension fonts.
l.8 $ab+b^2$?
Explanation: This usually follows a previous error Font ... not loadable; see
the discussion of that error (above) for solutions.
Missing number, treated as zero
Example:! Missing number, treated as zero.
<to be read again>al.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.)?
Explanation: There are many possibilities that can lead to this error. However,one possibility that is relevant for the amsmath package is that you forgot togive the number argument of an alignat environment, as in:\begin{alignat}
a& =b&
c& =d\\
a’& =b’&c’& =d’\end{alignat}

10.2. ERROR MESSAGES

29

where the first line should read instead\begin{alignat}{2}Another possibility is that you have a left bracket character [ following aline break command \\ in a multiline construction such as array, tabular, or
eqnarray. This will be interpreted by LATEX as the beginning of an ‘additionalvertical space request [3, §C.1.6], even if it occurs on the following line and isintended to be part of the contents. For example,\begin{array}{c}a+b\\[f,g]\\
m+n\end{array}
To prevent the error message in such a case, you can add braces as discussed in
the LATEX manual [3, §C.1.1]:\begin{array}{c}a+b\\{[f,g]}\\
m+n\end{array}or precede the bracketed expression by \relax.
Missing \right. inserted
Example:! Missing \right. inserted.
<inserted text>\right .
l.10 \end{multline}? hI’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.
Explanation: This error typically arises when you try to insert a line breakor & between a \left-\right pair of delimiters in any multiline environment,
including split.
\begin{multline}
AAA\left(BBB\\
CCC\right)\end{multline}
There are two possible solutions: (1) instead of using \left and \right, use‘big’ delimiters of fixed size (\bigl \bigr \biggl \biggr . . . ; see §4.14.1); or30

10. ERROR MESSAGES AND OUTPUT PROBLEMS(2) use null delimiters to break up the \left-\right pair into parts for eachline (or cell):AAA\left(BBB\right.\\\left.CCC\right)
The latter solution may result in mismatched delimiter sizes; ensuring that they
match requires using \vphantom in the segment that has the smaller delimiter(or possibly \smash in the segment that has the larger delimiter). In the argument of \vphantom put a copy of the tallest element that occurs in the othersegment, e.g.,xxx \left(\int_t yyy\right.\\\left.\vphantom{\int_t} zzz ... \right)
Missing } inserted
Example:! Missing } inserted.
<inserted text>\right .
l.10 \end{multline}? hI’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.
Explanation: This error is often the result of using $ within a multiline display environment. Remove such $ signs (except when they appear within\text{...}).
Old form ‘\pmatrix’ should be \begin{pmatrix}.
Example:! 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.
?

10.2. ERROR MESSAGES

31

Explanation: When the amsmath package is used, the old forms of \pmatrix,\matrix, and \cases cannot be used any longer because of naming conflicts.
Their syntax did not conform with standard LATEX syntax in any case.
Paragraph ended before \xxx was complete
Example:
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 thiscontrol sequence to too much text. How can we recover?
My plan is to forget the whole thing and hope for the best.
?
Explanation: This might be produced by a blank line between the \begin and\end. Another possibility is a misspelling in the \end{multline} command,e.g.,\begin{multline}...
\end{multiline}
or by using abbreviations for certain environments, such as \bal and \eal for\begin{align} and \end{align}:
\bal
...
\eal
For technical reasons that kind of abbreviation does not work with the more
complex displayed equation environments of the amsmath package (gather,align, split, etc.; cf. technote.tex).
Runaway argument?
See the discussion for the error message Paragraph ended before \xxx
was complete.
Unknown option ‘xxx’ for package ‘yyy’
Example:! 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.
?
Explanation: 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.

32

10. ERROR MESSAGES AND OUTPUT PROBLEMS

10.3

Warning messages

Cannot use ‘split’ here
Example:Package amsmath Warning: Cannot use ‘split’ here;(amsmath)trying to recover with ‘aligned’
Explanation: The split environment is designed to serve as the entire body ofan equation, or an entire line of an align or gather environment. There cannot
be any printed material before or after it within the same enclosing structure:\begin{equation}\left\{ % <-- Not allowed\begin{split}...
\end{split}\right. % <-- Not allowed\end{equation}
Foreign command \over [or \atop or \above]
Example:Package amsmath Warning: Foreign command \over; \frac or \genfrac(amsmath)should be used instead.
Explanation: The primitive generalized fraction commands of TEX—\over, \atop,\above—are deprecated when the amsmath package is used because their syntax is foreign to LATEX and amsmath provides native LATEX equivalents. See
technote.tex for further information.
10.4

Wrong output

Section numbers 0.1, 5.1, 8.1 instead of 1, 2, 3This most likely means that you have the arguments for \numberwithin inreverse order:\numberwithin{section}{equation}
That means ‘print the section number as equation number.section number andreset to 1 every time an equation occurs’ when what you probably wanted was
the inverse\numberwithin{equation}{section}
The \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 the previous
item.
Bracketed material disappears at the beginning of aligned or gathered
For most multiline equation environments, amsmath disables the LATEX convention to recognize a [bracketed] expression as an option when it occurs at
the beginning of an environment or immediately following \\, provided a space11.2. CONVERTING EXISTING DOCUMENTS

33

intervenes. This fails at the beginning of aligned and gathered, and is a bug.
Insert \relax before the opening brace to restore the desired result.

—11—
Additional information
11.1

Compatibility with other packages

Several packages have already been mentioned as providing facilities lacking inamsmath.
• bm is recommended for specifying bold math symbols.
• mathtools provides enhancements to some amsmath environments as wellas additional compatible features, such as rcases (similar to cases butwith the brace on the right-hand side), multlined (a subsidiary environment comparable to multline), and \shortintertext (more compact
spacing than \intertext). If mathtools is used, amsmath is loaded automatically, so it is not necessary to load it separately.
• unicode-math provides commands for (most?) math symbols in Unicode.
These packages have known problems when used with amsmath.
• lineno may omit numbers before display environments.
• breqn redefines a number of commands from other packages, so is best
loaded after amsmath and other packages with math facilities.
breqn is not yet totally stable, and a number of bugs have been reported
via tex.stackexchange [13].
• wasysym, mathabx and perhaps other font packages also define multiple
integrals using the same names as amsmath.
11.2

Converting existing documents

11.2.1 Converting from plain LATEXA LATEX document will typically continue to work the same in most respects if\usepackage{amsmath} is added in the document preamble. By default, however, the amsmath package suppresses page breaks inside multiple-line displayed
equation structures such as eqnarray, align, and gather. To continue allowing
page breaks inside eqnarray after switching to amsmath, you will need to add
the following line in your document preamble:\allowdisplaybreaks[1]To ensure normal spacing around relation symbols, you might also want tochange eqnarray to align, multline, or equation/split as appropriate.
Most of the other differences in amsmath usage can be considered optional
refinements, e.g., using\DeclareMathOperator{\Hom}{Hom}instead of \newcommand{\Hom}{\mbox{Hom}}.

34

11. ADDITIONAL INFORMATION

11.2.2 Converting from AMS-LATEX 1.1
See diffs-m.txt.
11.3

Technical notes

The file technote.tex contains some remarks on miscellaneous technical questions that are less likely to be of general interest.
11.4

Getting help

The amsmath collection of packages is maintained by the LATEX team [11]. Bugs
should be reported using the instructions at https://www.latex-project.
org/bugs/. Please be careful not to confuse amsmath with the AMS document classes (amsart, amsbook, and amsproc), which automatically incorporate
amsmath; bugs in the classes should be reported directly to AMS, by email totech-support@ams.org.
The online forum tex.stackexchange [13] is a good source of answers tocommon questions on usage. Look first in the archive to see if your question
has already been asked and answered; if it has not, post your own question.
There is an online discussion group called comp.text.tex [12] that is afairly good source of information about LATEX and TEX in general. This group,
formerly a Unix newsgroup, predates tex.stackexchange by decades, and thearchives can be a useful resource for historical investigation.
11.5

Of possible interest

Information about obtaining AMSFonts or other TEX-related software from the
AMS web server at www.ams.org can be obtained by sending an email request
to tech-support@ams.org. This software is also posted on CTAN and included
in distributions based on TEX Live.
The TEX Users Group is a nonprofit organization that publishes a journal(TUGboat), holds meetings, and serves as a clearinghouse for general information about TEX and TEX-related software.
TEX Users GroupPO Box 2311
Portland, OR 97208-2311
USA
Phone: +1-503-223-9994
Email: office@tug.org
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 usergroups in other countries; information about contacting a local user group canbe gotten from the TEX Users Group.

REFERENCES

35

References
References in print[1] George Grätzer, More Math into LATEX, fifth ed., Springer, New York,2016.
[2] Donald E. Knuth, The TEXbook, Addison-Wesley, Reading, MA, 1984.
[3] Leslie Lamport, LATEX: A document preparation system, 2nd revised ed.,Addison-Wesley, Reading, MA, 1994.
[4] Frank Mittelbach, Michel Goossens, et al., The LATEX companion, second
ed., Addison-Wesley, Reading, MA, 2004. This is now also available as an
ebook, in both English and German; see
https://www.latex-project.org/help/books/. The front matter,
including the full Table of Contents, can be viewed online, from a link on
the same page.
[5] Frank Mittelbach and Rainer Schöpf, The new font family selection—user
interface to standard LATEX, TUGboat 11, no. 2 (June 1990), pp. 297–305.
[6] Michael Spivak, The joy of TEX, 2nd revised ed., Amer. Math. Soc.,
Providence, RI, 1990.
Package documentation[7] AMS author handbook, separate versions for journal articles, monographs
and proceedings articles, Amer. Math. Soc., Providence, RI, 2017;https://www.ams.org/publications/authors/tex/author-handbook.
[8] AMSFonts version 2.2d—user’s guide, Amer. Math. Soc., Providence, RI,
2002; distributed with the AMSFonts package http://mirror.ctan.org/tex-archive/fonts/amsfonts/doc/amsfndoc.pdf.
[9] Using the amsthm Package, version 2.20.3, Amer. Math. Soc., Providence,
RI, 2017; http://mirror.ctan.org/tex-archive/macros/latex/
required/amscls/doc/amsthdoc.pdf.
[10] Morten Høgholm, Lars Madsen, The mathtools package, 2018;
http://mirror.ctan.org/tex-archive/macros/latex/contrib/
mathtools/mathtools.pdf.
Online resources[11] The LATEX Project, https://www.latex-project.org.
[12] Online discussion group comp.text.tex,
https://groups.google.com/forum/#!forum/comp.text.tex.
[13] Online question and answer forum tex.stackexchange,
https://tex.stackexchange.com.

Index&, 29\!, 3, 13\,, 3, 13\:, 13\;, 13\[ ... \], 4\\, 4, 6, 7, 10, 11, 27, 29, 32\\*, 10, 11\|, 19|, 18, 19

amsthm package, 1, 35
amsxtra package, 1, 2, 15\arccos, 20\arcsin, 20\arctan, 20\arg, 20
array environment, 9, 12, 13, 29
arrowsextensible, 15
in commutative diagrams, 23\atop, 17, 32\atopwithdelims, 17\above, 17, 32\abovewithdelims, 17\accentedsymbol, 2
accents package, 15\addtocounter, 12, 13\addtolength, 6align environment, 4, 6, 7, 9, 26,
31–33\align, 12alignat environment, 8, 9, 26–28
aligned environment, 3, 4, 9, 10,32, 33alignedat environment, 4, 9, 10alignedleftspaceno option, 3alignedleftspaceyes option, 3alignedleftspaceyesifneg
option, 3\allowdisplaybreaks, 10
amsart class, 1, 34
amsbook class, 1, 34
amsbsy package, 1, 2
amscd package, 1, 2, 23
AMSFonts collection, 27
amsfonts package, 24
amsmath package, 1–34
amsopn package, 1, 2
amsproc class, 34
amssymb package, 25amstext package, 1, 2

b (bottom) option, 9, 17
BaKoMa fonts, 27\big, \Big, \bigg, . . . delimiters,
18\biggl, 29\biggr, 29\bigl, 29\bigr, 29\binom, 16, 17
binomials, 16
bm package, 2, 25, 33
Bmatrix environment, 12
bmatrix environment, 12\bmod, 20\boldsymbol, 24, 25\boxed, 15
breqn package, 33\C, 24c (center) option, 9cases environment, 9, 33\cases, 31
CD environment, 2, 23\cdots, 14centertags option, 2\cfrac, 17
cmbsy5, 27
36

37

INDEX

cmex, 27
cmex10 option, 27
cmex7, 27
cmmib5, 27
commutative diagrams, 23
comp.text.tex, 34
continued fractions, 17\cos, 20
\cosh, 20\cot, 20\coth, 20\csc, 20CTAN (Comprehensive TEX
Archive Network), 34\dbinom, 16
dcases environment, 9\ddddot, 15\dddot, 15\ddot, 15\DeclareMathOperator, 2, 19–21\DeclarePairedDelimiter, 19\deg, 20delimiters, 12, 13, 16–19, 25, 29
fixed size, 18, 29
mismatched sizes, 30
null, 30
\det, 20
\dfrac, 16diffs-m.txt, 34\dim, 20
\displaybreak, 10
displayed equations, 3centering, 3\displaylimits, 22
displaymath environment, 4\displaystyle, 9, 16\documentclass, 3\dot, 15
dots, see ellipsis dots
\dots, 14\dotsb, 14\dotsc, 14\dotsi, 14\dotsm, 14\dotso, 14-ed environments, 9, 32
ellipsis dots
in matrices, 13
eqnarray environment, 1, 4, 29, 33\eqref, 11
equation environment, 1, 4, 6, 9,
10, 26, 33
equation* environment, 4
equation numbers
cross-references, 11
hierarchy, 11left or right placement, 3, 11
overriding, 4subordinate numbering, 12vertical placement, 2
equations, see displayed equations
euscript package, 24\exp, 20
\fbox, 15flalign environment, 8
fleqn option, 3, 6
fntguide.tex, 23\frac, 16, 17
fractions, 16–17\fracwithdelims, 2function names, see operator
names
gather environment, 6, 7, 9, 26,
31–33
gathered environment, 3, 9, 10,32, 33\gcd, 20
\genfrac, 16, 17\hat, 15
\hdotsfor, 13\hom, 20
horizontal spacing, 4, 13
around operator names, 19, 21
hyperref package, 4\idotsint, 22\iiiint, 22\iiint, 22

38\iint, 22\inf, 20
\injlim, 20
integrals
multiple, 22, 33
placement of limits, 2\intertext, 10, 11, 33
intlimits option, 2, 22\ker, 20\label, 7, 12\langle, 18\ldots, 14\left, 17, 18, 29, 30\leftroot, 15
leqno option, 3, 6
\lg, 20\lim, 1, 2, 19, 20\liminf, 20
limits, see subscripts andsuperscripts\limits, 22\limsup, 20
line break, 9, 14, 29
lineno package, 33\ln, 20\log, 19, 20\lVert, 18\lvert, 18
math fonts, 24
math symbols, see math fonts
mathabx package, 33\mathbb, 24\mathbf, 24\mathcal, 24\mathfrak, 24\mathit, 24\mathrm, 21, 24\mathscr, 24\mathsf, 24
mathtools package, 2, 9, 11, 13, 19,
33, 35\mathtt, 24
matrices, 12

INDEX

ellipsis dots, 13
matrix environment, 12* variants, 13\matrix, 31\max, 20\mbox, 20\medspace, 13Metafont source files, 27\min, 20
\mod, 20\mspace, 14
multline environment, 4, 6, 33
multlined environment, 33\multlinegap, 6
namelimits option, 2\negmedspace, 13\negthickspace, 13\negthinspace, 13\newcommand, 24\nobreakdash, 14
nointlimits option, 2\nolimits, 22
nonamelimits option, 2\normalfont, 12
nosumlimits option, 2\notag, 4, 6
\numberwithin, 11, 32
online Q & A forum, 34
operator names, 19\operatorname, 20\operatorname*, 20
options
adjust positioning, 15, 17
amsmath package options, 2–3
behavior of particular options,6, 10, 13, 15, 17, 22, 27
extra vertical space after \\,10
positioning of -ed
environments, 9, 32
space before [, 9, 10, 32
unknown, 31\over, 17, 32\overleftarrow, 15

INDEX\overleftrightarrow, 15\overrightarrow, 15\overset, 16\overunderset, 16\overwithdelims, 17
page breaks, 10\pagebreak, 10\par, 4
pmatrix environment, 12\pmatrix, 31\pmb, 24, 25\pmod, 20\pod, 20
PostScript fonts, 27\Pr, 20\projlim, 20\qed, 4\qedhere, 4\qquad, 13\quad, 13, 14\R, 24\raisetag, 10\rangle, 18
rcases environment, 33\ref, 4, 12\ref*, 4\relax, 29, 33
reqno option, 3, 6
\right, 17, 18, 29, 30\rVert, 18\rvert, 18\scriptscriptstyle, 16\scriptstyle, 16\sec, 20
\setcounter, 11–13\setlength, 6\shortintertext, 11, 33\shoveleft, 6\shoveright, 6\sideset, 16, 21, 22\sin, 1, 2, 19, 20\sinh, 20

39
smallmatrix environment, 13* variants, 13
variant delimiters, 13\smash, 17, 30\sphat, 15
split environment, 2, 4, 6, 7, 10,26, 29, 31–33\sptilde, 15\stackrel, 16
subarray environment, 21
subeqn.tex, 4
subequations environment, 12
subscripts and superscripts, 16
multiline, 21
on sums, 21
placement, 2\substack, 21\sum, 22, 25
sumlimits option, 2\sup, 20superscripts, see subscripts andsuperscriptst (top) option, 9, 17
tabular environment, 29\tag, 4\tag*, 4\tan, 20
\tanh, 20\tbinom, 16
tbtags option, 2
technote.tex, 17, 31, 32, 34
testmath.tex, 4
TEX Live, 34
TEX Users Group, 34\text, 2, 7, 9, 10, 20, 21
text fragments inside math, 10, 20\textrm, 20\textstyle, 9, 16\tfrac, 16, 17\theequation, 11, 12\thickspace, 13\thinspace, 13
Tik Z package, 23
tikz-cd package, 23
TUGboat, 34

40\underleftarrow, 15\underleftrightarrow, 15\underrightarrow, 15\underset, 16
unicode-math package, 33\uproot, 15\usepackage, 3, 27\value, 12\varDelta, 25\varGamma, 25\varinjlim, 20\varLambda, 25\varliminf, 20\varlimsup, 20\varOmega, 25\varPhi, 25

INDEX\varPi, 25\varprojlim, 20\varPsi, 25\varSigma, 25\varTheta, 25\varUpsilon, 25\varXi, 25
Vmatrix environment, 12
vmatrix environment, 12\vphantom, 30
wasysym package, 33
www.ams.org, 34\xleftarrow, 15\xrightarrow, 15
XY-pic package, 23