File: amsldoc.xml

package info (click to toggle)
tralics 2.14.4-2.1
links: PTS
area: main
in suites: forky, sid, trixie
size: 17,204 kB
sloc: cpp: 49,403; xml: 18,145; sh: 140; makefile: 118
file content (2357 lines) | stat: -rw-r--r-- 193,258 bytes
parent folder | download | duplicates (2)
<?xml version='1.0' encoding='iso-8859-1'?>
<!DOCTYPE std SYSTEM 'classes.dtd'>
<!-- Translated from latex by tralics 2.14.3, date: 2011/06/01-->
<std equation-number='left'>
<frontmatter><maketitle><title>User's Guide for the <EM r='pkg'>amsmath</EM> Package (Version2.0)</title><author>American Mathematical Society</author><date>1999-12-13 (revised 2002-02-25)</date></maketitle><p><pagestyle style='headings'/></p>
<tableofcontents depth='1'/><cleardoublepage/></frontmatter>
<mainmatter>
<div0 id-text='1' id='cid1'><head>Introduction</head>
<p>The <EM r='pkg'>amsmath</EM> package is a <LaTeX/> package that provides
miscellaneous enhancements for improving the information structure and
printed output of documents that contain mathematical formulas. Readers
unfamiliar with <LaTeX/> should refer to <cit><ref target='bid0'/></cit>. If you have an
up-to-date version of <LaTeX/>, the <EM r='pkg'>amsmath</EM> package is normally
provided along with it. Upgrading when a newer version of the
<EM r='pkg'>amsmath</EM> package is released can be done via
<xref url='http://www.ams.org/tex/amsmath.html'>http://<allowbreak/>www.<allowbreak/>ams.<allowbreak/>org/<allowbreak/>tex/<allowbreak/>amsmath.<allowbreak/>html</xref> or
<xref url='ftp://ftp.ams.org/pub/tex/'>ftp://<allowbreak/>ftp.<allowbreak/>ams.<allowbreak/>org/<allowbreak/>pub/<allowbreak/>tex/<allowbreak/></xref>.</p>
<p>This documentation describes the features of the <EM r='pkg'>amsmath</EM> package
and discusses how they are intended to be used. It also covers some
ancillary packages:</p>
<table rend='inline'><row><cell halign='left'><EM r='pkg'>amsbsy</EM></cell>
<cell halign='left'><EM r='pkg'>amstext</EM></cell>
</row><row><cell halign='left'><EM r='pkg'>amscd</EM></cell>
<cell halign='left'><EM r='pkg'>amsxtra</EM></cell>
</row><row><cell halign='left'><EM r='pkg'>amsopn</EM></cell>
</row></table><p>
These all have something to do with the contents of math formulas. For
information on extra math symbols and math fonts, see <cit><ref target='bid1'/></cit>
and <xref url='http://www.ams.org/tex/amsfonts.html'>http://<allowbreak/>www.<allowbreak/>ams.<allowbreak/>org/<allowbreak/>tex/<allowbreak/>amsfonts.<allowbreak/>html</xref>. For documentation of the
<EM r='pkg'>amsthm</EM> package or AMS document classes (<EM r='cls'>amsart</EM>,
<EM r='cls'>amsbook</EM>, etc.) see <cit><ref target='bid2'/></cit> or <cit><ref target='bid3'/></cit> and
<xref url='http://www.ams.org/tex/author-info.html'>http://<allowbreak/>www.<allowbreak/>ams.<allowbreak/>org/<allowbreak/>tex/<allowbreak/>author-info.<allowbreak/>html</xref>.</p>
<p>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 <EM r='pkg'>amsmath</EM> features:</p>
<list type='simple'>
<item id-text='1' id='uid1'><p noindent='true'>A convenient way to define new `operator name' commands analogous
to <EM r='cn'>sin</EM> and <EM r='cn'>lim</EM>, including proper side spacing and automatic
selection of the correct font style and size (even when used in
sub- or superscripts).</p>
</item>
<item id-text='2' id='uid2'><p noindent='true'>Multiple substitutes for the <EM r='cls'>eqnarray</EM> environment to make
various kinds of equation arrangements easier to write.</p>
</item>
<item id-text='3' id='uid3'><p noindent='true'>Equation numbers automatically adjust up or down to avoid
overprinting on the equation contents (unlike <EM r='cls'>eqnarray</EM>).</p>
</item>
<item id-text='4' id='uid4'><p noindent='true'>Spacing around equals signs matches the normal spacing in the
<EM r='cls'>equation</EM> environment (unlike <EM r='cls'>eqnarray</EM>).</p>
</item>
<item id-text='5' id='uid5'><p noindent='true'>A way to produce multiline subscripts as are often used with
summation or product symbols.</p>
</item>
<item id-text='6' id='uid6'><p noindent='true'>An easy way to substitute a variant equation number for a given
equation instead of the automatically supplied number.</p>
</item>
<item id-text='7' id='uid7'><p noindent='true'>An easy way to produce subordinate equation numbers of the form
(1.3a) (1.3b) (1.3c) for selected groups of equations.</p>
</item></list>
<p>The <EM r='pkg'>amsmath</EM> package is distributed together with some small
auxiliary packages:</p>
<list type='description'><label><EM r='pkg'>amsmath</EM></label>
<item id-text='1' id='uid8'><p noindent='true'>Primary package, provides various features for
displayed equations and other mathematical constructs.</p>
</item><label><EM r='pkg'>amstext</EM></label>
<item id-text='2' id='uid9'><p noindent='true'>Provides a <EM r='cn'>text</EM> command for
typesetting a fragment of text inside a display.</p>
</item><label><EM r='pkg'>amsopn</EM></label>
<item id-text='3' id='uid10'><p noindent='true'>Provides <EM r='cn'>DeclareMathOperator</EM> for defining new
`operator names' like <EM r='cn'>sin</EM> and <EM r='cn'>lim</EM>.</p>
</item><label><EM r='pkg'>amsbsy</EM></label>
<item id-text='4' id='uid11'><p noindent='true'>For backward compatibility this package continues
to exist but use of the newer <EM r='pkg'>bm</EM> package that comes with <LaTeX/>
is recommended instead.</p>
</item><label><EM r='pkg'>amscd</EM></label>
<item id-text='5' id='uid12'><p noindent='true'>Provides a <EM r='cls'>CD</EM> environment for simple
commutative diagrams (no support for diagonal arrows).</p>
</item><label><EM r='pkg'>amsxtra</EM></label>
<item id-text='6' id='uid13'><p noindent='true'>Provides certain odds and ends such as
<EM r='cn'>fracwithdelims</EM> and <EM r='cn'>accentedsymbol</EM>, for compatibility with
documents created using version 1.1.</p>
</item></list>
<p>The <EM r='pkg'>amsmath</EM> package incorporates <EM r='pkg'>amstext</EM>, <EM r='pkg'>amsopn</EM>, and
<EM r='pkg'>amsbsy</EM>. The features of <EM r='pkg'>amscd</EM> and <EM r='pkg'>amsxtra</EM>, however,
are available only by invoking those packages separately.</p>
</div0>
<div0 id-text='2' id='cid2'><head>Options for the <EM r='pkg'>amsmath</EM> package</head>
<p>The <EM r='pkg'>amsmath</EM> package has the following options:</p>
<list type='description'><label><EM r='opt'>centertags</EM></label>
<item id-text='1' id='uid14'><p noindent='true'>(default) For a split equation, place equation
numbers<anchor id-text='lid1' id='uid15'/> vertically centered
on the total height of the equation.</p>
</item><label><EM r='opt'>tbtags</EM></label>
<item id-text='2' id='uid16'><p noindent='true'>`Top-or-bottom tags': For a split equation, place
equation numbers<anchor id-text='lid2' id='uid17'/> level with
the last (resp. first) line, if numbers are on the right (resp.
left).</p>
</item><label><EM r='opt'>sumlimits</EM></label>
<item id-text='3' id='uid18'><p noindent='true'>(default) Place the subscripts and
superscripts<anchor id-text='lid3' id='uid19'/> of summation symbols
above and below, in displayed equations. This option also affects other
symbols of the same type&#x2014;<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&prod;</mo></math><texmath>\prod </texmath></formula>, <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&coprod;</mo></math><texmath>\coprod </texmath></formula>, <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&bigotimes;</mo></math><texmath>\bigotimes </texmath></formula>,
<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&bigoplus;</mo></math><texmath>\bigoplus </texmath></formula>, and so forth&#x2014;but excluding integrals (see below).</p>
</item><label><EM r='opt'>nosumlimits</EM></label>
<item id-text='4' id='uid20'><p noindent='true'>Always place the subscripts and superscripts of
summation-type symbols to the side, even in displayed equations.</p>
</item><label><EM r='opt'>intlimits</EM></label>
<item id-text='5' id='uid21'><p noindent='true'>Like <EM r='opt'>sumlimits</EM>, but for
integral<anchor id-text='lid5' id='uid22'/> symbols.</p>
</item><label><EM r='opt'>nointlimits</EM></label>
<item id-text='6' id='uid23'><p noindent='true'>(default) Opposite of <EM r='opt'>intlimits</EM>.</p>
</item><label><EM r='opt'>namelimits</EM></label>
<item id-text='7' id='uid24'><p noindent='true'>(default) Like <EM r='opt'>sumlimits</EM>, but for certain
`operator names' such as <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>det</mo></math><texmath>\det </texmath></formula>, <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>inf</mo></math><texmath>\inf </texmath></formula>, <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>lim</mo></math><texmath>\lim </texmath></formula>, <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>max</mo></math><texmath>\max </texmath></formula>, <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>min</mo></math><texmath>\min </texmath></formula>, that
traditionally have subscripts <anchor id-text='lid6' id='uid25'/> placed underneath when they occur in a displayed
equation.</p>
</item><label><EM r='opt'>nonamelimits</EM></label>
<item id-text='8' id='uid26'><p noindent='true'>Opposite of <EM r='opt'>namelimits</EM>.</p>
</item></list>
<p>To use one of these package options, put the option name in the optional
argument of the <EM r='cn'>usepackage</EM> command&#x2014;e.g.,
<hi rend='tt'>\usepackage[intlimits]{amsmath}</hi>.</p>
<p>The <EM r='pkg'>amsmath</EM> package also recognizes the following options which
are normally selected (implicitly or explicitly) through the
<EM r='cn'>documentclass</EM> command, and thus need not be repeated in the option
list of the <EM r='cn'>usepackage</EM><hi rend='tt'>{amsmath}</hi> statement.</p>
<list type='description'><label><EM r='opt'>leqno</EM></label>
<item id-text='1' id='uid27'><p noindent='true'>Place equation numbers on the left.<anchor id-text='lid7' id='uid28'/></p>
</item><label><EM r='opt'>reqno</EM></label>
<item id-text='2' id='uid29'><p noindent='true'>Place equation numbers on the right.</p>
</item><label><EM r='opt'>fleqn</EM></label>
<item id-text='3' id='uid30'><p noindent='true'>Position equations at a fixed indent from the left
margin rather than centered in the text column.<anchor id-text='lid8' id='uid31'/></p>
</item></list>
</div0>
<div0 id-text='3' id='cid3'><head>Displayed equations</head>
</div0>
<div0 id-text='1' id='cid4'><head>Introduction</head>
<p>The <EM r='pkg'>amsmath</EM> package provides a number of additional displayed
equation structures<anchor id-text='lid9' id='uid32'/> beyond the ones
provided in basic <LaTeX/>. The augmented set includes:</p>
<p noindent='true'><hi rend='tt'>equationequation*alignalign*</hi></p>
<p noindent='true'><hi rend='tt'>gathergather*flalignflalign*</hi></p>
<p noindent='true'><hi rend='tt'>multlinemultline*alignatalignat*</hi></p>
<p noindent='true'><hi rend='tt'>split</hi></p>
<p noindent='true'>(Although the standard <EM r='cls'>eqnarray</EM> environment remains available,
it is better to use <EM r='cls'>align</EM> or <EM r='cls'>equation</EM>+<EM r='cls'>split</EM> instead.)</p>
<p>Except for <EM r='cls'>split</EM>, each environment has both starred and unstarred
forms, where the unstarred forms have automatic numbering using
<LaTeX/>'s <EM r='cls'>equation</EM> counter. You can suppress the number on any
particular line by putting <EM r='cn'>notag</EM> before the <EM r='cn'>\\</EM>; you can also
override<anchor id-text='lid11' id='uid33'/> it with a tag of your own
using <EM r='cn'>tag</EM><hi rend='tt'>{</hi><hi rend='it'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&langle;</mo></math><texmath>\langle </texmath></formula>label<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&rangle;</mo></math><texmath>\rangle </texmath></formula></hi><hi rend='tt'>}</hi>, where <hi rend='it'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&langle;</mo></math><texmath>\langle </texmath></formula>label<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&rangle;</mo></math><texmath>\rangle </texmath></formula></hi> means arbitrary text such
as <hi rend='tt'>$*$</hi> or <hi rend='tt'>ii</hi> used to &#x201C;number&#x201D; the equation. There is also a
<EM r='cn'>tag*</EM> command that causes the text you supply to be typeset
literally, without adding parentheses around it. <EM r='cn'>tag</EM> and <EM r='cn'>tag*</EM>
can also be used within the unnumbered versions of all the <EM r='pkg'>amsmath</EM>
alignment structures. Some examples of the use of <EM r='cn'>tag</EM> may be found
in the sample files <EM r='fn'>testmath.tex</EM> and <EM r='fn'>subeqn.tex</EM> provided with
the <EM r='pkg'>amsmath</EM> package.</p>
<p>The <EM r='cls'>split</EM> environment is a special subordinate form that is used
only <hi rend='it'>inside</hi> one of the others. It cannot be used inside
<EM r='cls'>multline</EM>, however.</p>
<p>In the structures that do alignment (<EM r='cls'>split</EM>, <EM r='cls'>align</EM> and
variants), relation symbols have an <hi rend='tt'>&amp;</hi> before them but not
after&#x2014;unlike <EM r='cls'>eqnarray</EM>. Putting the <hi rend='tt'>&amp;</hi> after the
relation symbol will interfere with the normal spacing; it has to go
before.</p>
<table rend='display' id-text='1' id='uid34' place='p'><head>Comparison of displayed equation environments (vertical lines
indicating nominal margins)</head>
<unexpected><p rend='center'><minipage width='213.5pt'><p noindent='true'><hi rend='tt'>\begin{equation*}</hi></p>
<p noindent='true'><hi rend='tt'>a=b</hi></p>
<p noindent='true'><hi rend='tt'>\end{equation*}</hi></p>
</minipage><minipage width='213.5pt'><rule width='0.2pt'/>
<formula textype='equation*' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow></math><texmath>
a=b
</texmath></formula>
<rule width='0.0pt' depth='0.0pt' height='0.0pt'/><rule width='0.2pt'/></minipage></p><p rend='center'><minipage width='213.5pt'><p noindent='true'><hi rend='tt'>\begin{equation}</hi></p>
<p noindent='true'><hi rend='tt'>a=b</hi></p>
<p noindent='true'><hi rend='tt'>\end{equation}</hi></p>
</minipage><minipage width='213.5pt'><rule width='0.2pt'/>
<formula id-text='1' id='uid35' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow></math><texmath>
a=b
</texmath></formula>
<rule width='0.0pt' depth='0.0pt' height='0.0pt'/><rule width='0.2pt'/></minipage></p><p rend='center'><minipage width='213.5pt'><p noindent='true'><hi rend='tt'>\begin{equation}\label{xx}</hi></p>
<p noindent='true'><hi rend='tt'>\begin{split}</hi></p>
<p noindent='true'><hi rend='tt'>a&amp;=b+c-<zws/>d\\</hi></p>
<p noindent='true'><hi rend='tt'>&amp;\quad+e-<zws/>f\\</hi></p>
<p noindent='true'><hi rend='tt'>&amp;=g+h\\</hi></p>
<p noindent='true'><hi rend='tt'>&amp;=i</hi></p>
<p noindent='true'><hi rend='tt'>\end{split}</hi></p>
<p noindent='true'><hi rend='tt'>\end{equation}</hi></p>
</minipage><minipage width='213.5pt'><rule width='0.2pt'/>
<formula id-text='2' id='uid36' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd columnalign='right'><mi>a</mi></mtd><mtd columnalign='left'><mrow><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>-</mo><mi>d</mi></mrow></mtd></mtr><mtr><mtd/><mtd columnalign='left'><mrow><mspace width='1.em'/><mo>+</mo><mi>e</mi><mo>-</mo><mi>f</mi></mrow></mtd></mtr><mtr><mtd/><mtd columnalign='left'><mrow><mo>=</mo><mi>g</mi><mo>+</mo><mi>h</mi></mrow></mtd></mtr><mtr><mtd/><mtd columnalign='left'><mrow><mo>=</mo><mi>i</mi></mrow></mtd></mtr></mtable></math><texmath>
\begin{split}
a&amp; =b+c-d\\
&amp; \quad +e-f\\
&amp; =g+h\\
&amp; =i
\end{split}
</texmath></formula>
<rule width='0.0pt' depth='0.0pt' height='0.0pt'/><rule width='0.2pt'/></minipage></p><p rend='center'><minipage width='213.5pt'><p noindent='true'><hi rend='tt'>\begin{multline}</hi></p>
<p noindent='true'><hi rend='tt'>a+b+c+d+e+f\\</hi></p>
<p noindent='true'><hi rend='tt'>+i+j+k+l+m+n</hi></p>
<p noindent='true'><hi rend='tt'>\end{multline}</hi></p>
</minipage><minipage width='213.5pt'><rule width='0.2pt'/>
<formula id-text='1' id='uid37' textype='multline' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd columnalign='left'><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>e</mi><mo>+</mo><mi>f</mi></mrow></mtd></mtr><mtr><mtd columnalign='right'><mrow><mo>+</mo><mi>i</mi><mo>+</mo><mi>j</mi><mo>+</mo><mi>k</mi><mo>+</mo><mi>l</mi><mo>+</mo><mi>m</mi><mo>+</mo><mi>n</mi></mrow></mtd></mtr></mtable></math><texmath>
a+b+c+d+e+f\\
+i+j+k+l+m+n
</texmath></formula>
<rule width='0.0pt' depth='0.0pt' height='0.0pt'/><rule width='0.2pt'/></minipage></p><p rend='center'><minipage width='213.5pt'><p noindent='true'><hi rend='tt'>\begin{gather}</hi></p>
<p noindent='true'><hi rend='tt'>a_1=b_1+c_1\\</hi></p>
<p noindent='true'><hi rend='tt'>a_2=b_2+c_2-<zws/>d_2+e_2</hi></p>
<p noindent='true'><hi rend='tt'>\end{gather}</hi></p>
</minipage><minipage width='213.5pt'><rule width='0.2pt'/>
<formula id-text='1' id='uid38' textype='gather' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd><mrow><msub><mi>a</mi> <mn>1</mn> </msub><mo>=</mo><msub><mi>b</mi> <mn>1</mn> </msub><mo>+</mo><msub><mi>c</mi> <mn>1</mn> </msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>a</mi> <mn>2</mn> </msub><mo>=</mo><msub><mi>b</mi> <mn>2</mn> </msub><mo>+</mo><msub><mi>c</mi> <mn>2</mn> </msub><mo>-</mo><msub><mi>d</mi> <mn>2</mn> </msub><mo>+</mo><msub><mi>e</mi> <mn>2</mn> </msub></mrow></mtd></mtr></mtable></math><texmath>
a_1=b_1+c_1\\
a_2=b_2+c_2-d_2+e_2
</texmath></formula>
<rule width='0.0pt' depth='0.0pt' height='0.0pt'/><rule width='0.2pt'/></minipage></p><p rend='center'><minipage width='213.5pt'><p noindent='true'><hi rend='tt'>\begin{align}</hi></p>
<p noindent='true'><hi rend='tt'>a_1&amp;=b_1+c_1\\</hi></p>
<p noindent='true'><hi rend='tt'>a_2&amp;=b_2+c_2-<zws/>d_2+e_2</hi></p>
<p noindent='true'><hi rend='tt'>\end{align}</hi></p>
</minipage><minipage width='213.5pt'><rule width='0.2pt'/>
<formula id-text='1' id='uid39' textype='align' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd columnalign='right'><msub><mi>a</mi> <mn>1</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>b</mi> <mn>1</mn> </msub><mo>+</mo><msub><mi>c</mi> <mn>1</mn> </msub></mrow></mtd></mtr><mtr><mtd columnalign='right'><msub><mi>a</mi> <mn>2</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>b</mi> <mn>2</mn> </msub><mo>+</mo><msub><mi>c</mi> <mn>2</mn> </msub><mo>-</mo><msub><mi>d</mi> <mn>2</mn> </msub><mo>+</mo><msub><mi>e</mi> <mn>2</mn> </msub></mrow></mtd></mtr></mtable></math><texmath>
a_1&amp; =b_1+c_1\\
a_2&amp; =b_2+c_2-d_2+e_2
</texmath></formula>
<rule width='0.0pt' depth='0.0pt' height='0.0pt'/><rule width='0.2pt'/></minipage></p><p rend='center'><minipage width='213.5pt'><p noindent='true'><hi rend='tt'>\begin{align}</hi></p>
<p noindent='true'><hi rend='tt'>a_{11}&amp;=b_{11}&amp;</hi></p>
<p noindent='true'><hi rend='tt'>a_{12}&amp;=b_{12}\\</hi></p>
<p noindent='true'><hi rend='tt'>a_{21}&amp;=b_{21}&amp;</hi></p>
<p noindent='true'><hi rend='tt'>a_{22}&amp;=b_{22}+c_{22}</hi></p>
<p noindent='true'><hi rend='tt'>\end{align}</hi></p>
</minipage><minipage width='213.5pt'><rule width='0.2pt'/>
<formula id-text='1' id='uid40' textype='align' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd columnalign='right'><msub><mi>a</mi> <mn>11</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>b</mi> <mn>11</mn> </msub></mrow></mtd><mtd columnalign='right'><msub><mi>a</mi> <mn>12</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>b</mi> <mn>12</mn> </msub></mrow></mtd></mtr><mtr><mtd columnalign='right'><msub><mi>a</mi> <mn>21</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>b</mi> <mn>21</mn> </msub></mrow></mtd><mtd columnalign='right'><msub><mi>a</mi> <mn>22</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>b</mi> <mn>22</mn> </msub><mo>+</mo><msub><mi>c</mi> <mn>22</mn> </msub></mrow></mtd></mtr></mtable></math><texmath>
a_{11}&amp; =b_{11}&amp;
a_{12}&amp; =b_{12}\\
a_{21}&amp; =b_{21}&amp;
a_{22}&amp; =b_{22}+c_{22}
</texmath></formula>
<rule width='0.0pt' depth='0.0pt' height='0.0pt'/><rule width='0.2pt'/></minipage></p><p rend='center'><minipage width='213.5pt'><p noindent='true'><hi rend='tt'>\begin{flalign*}</hi></p>
<p noindent='true'><hi rend='tt'>a_{11}&amp;=b_{11}&amp;</hi></p>
<p noindent='true'><hi rend='tt'>a_{12}&amp;=b_{12}\\</hi></p>
<p noindent='true'><hi rend='tt'>a_{21}&amp;=b_{21}&amp;</hi></p>
<p noindent='true'><hi rend='tt'>a_{22}&amp;=b_{22}+c_{22}</hi></p>
<p noindent='true'><hi rend='tt'>\end{flalign*}</hi></p>
</minipage><minipage width='213.5pt'><rule width='0.2pt'/>
<formula textype='flalign*' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd columnalign='right'><msub><mi>a</mi> <mn>11</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>b</mi> <mn>11</mn> </msub></mrow></mtd><mtd columnalign='right'><msub><mi>a</mi> <mn>12</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>b</mi> <mn>12</mn> </msub></mrow></mtd></mtr><mtr><mtd columnalign='right'><msub><mi>a</mi> <mn>21</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>b</mi> <mn>21</mn> </msub></mrow></mtd><mtd columnalign='right'><msub><mi>a</mi> <mn>22</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>b</mi> <mn>22</mn> </msub><mo>+</mo><msub><mi>c</mi> <mn>22</mn> </msub></mrow></mtd></mtr></mtable></math><texmath>
a_{11}&amp; =b_{11}&amp;
a_{12}&amp; =b_{12}\\
a_{21}&amp; =b_{21}&amp;
a_{22}&amp; =b_{22}+c_{22}
</texmath></formula>
<rule width='0.0pt' depth='0.0pt' height='0.0pt'/><rule width='0.2pt'/></minipage></p></unexpected></table>
</div0>
<div0 id-text='2' id='cid5'><head>Single equations</head>
<p>The <EM r='cls'>equation</EM> environment is for a single equation with an
automatically generated number. The <EM r='cls'>equation*</EM> environment is the
same except for omitting the number.<note id-text='1' id='uid41' place='foot'>Basic <LaTeX/> doesn't provide an <EM r='cls'>equation*</EM> environment,
but rather a functionally equivalent environment named
<EM r='cls'>displaymath</EM>.</note></p>
</div0>
<div0 id-text='3' id='cid6'><head>Split equations without alignment</head>
<p>The <EM r='cls'>multline</EM> environment is a variation of the <EM r='cls'>equation</EM>
environment used for equations that don't fit on a single line. The
first line of a <EM r='cls'>multline</EM> will be at the left margin and the last
line at the right margin, except for an indention on both sides in the
amount of <EM r='cn'>multlinegap</EM>. Any additional lines in between will be
centered independently within the display width (unless the <EM r='opt'>fleqn</EM>
option is in effect).</p>
<p>Like <EM r='cls'>equation</EM>, <EM r='cls'>multline</EM> has only a single equation number
(thus, none of the individual lines should be marked with <EM r='cn'>notag</EM>).
The equation number is placed on the last line (<EM r='opt'>reqno</EM> option) or
first line (<EM r='opt'>leqno</EM> option); vertical centering as for <EM r='cls'>split</EM>
is not supported by <EM r='cls'>multline</EM>.</p>
<p>It's possible to force one of the middle lines to the left or right with
commands <EM r='cn'>shoveleft</EM>, <EM r='cn'>shoveright</EM>. These commands take the entire
line as an argument, up to but not including the final <EM r='cn'>\\</EM>; for
example</p>
<p>MISSING</p>
<p noindent='true'><hi rend='tt'>\begin{multline}</hi></p>
<p noindent='true'><hi rend='tt'>\framebox[.65\columnwidth]{A}\\</hi></p>
<p noindent='true'><hi rend='tt'>\framebox[.5\columnwidth]{B}\\</hi></p>
<p noindent='true'><hi rend='tt'>\shoveright{\framebox[.55\columnwidth]{C}}\\</hi></p>
<p noindent='true'><hi rend='tt'>\framebox[.65\columnwidth]{D}</hi></p>
<p noindent='true'><hi rend='tt'>\end{multline}</hi></p>
<p>The value of <EM r='cn'>multlinegap</EM> can be changed with the usual <LaTeX/>
commands <EM r='cn'>setlength</EM> or <EM r='cn'>addtolength</EM>.</p>
</div0>
<div0 id-text='4' id='cid7'><head>Split equations with alignment</head>
<p>Like <EM r='cls'>multline</EM>, the <EM r='cls'>split</EM> environment is for <hi rend='it'>single</hi>
equations that are too long to fit on one line and hence must be split
into multiple lines. Unlike <EM r='cls'>multline</EM>, however, the <EM r='cls'>split</EM>
environment provides for alignment among the split lines, using <hi rend='tt'>&amp;</hi> to
mark alignment points. Unlike the other <EM r='pkg'>amsmath</EM> equation
structures, the <EM r='cls'>split</EM> environment provides no numbering, because
it is intended to be used <hi rend='it'>only inside some other displayed
equation structure</hi>, usually an <EM r='cls'>equation</EM>, <EM r='cls'>align</EM>, or
<EM r='cls'>gather</EM> environment, which provides the numbering. For example:</p>
<formula id-text='7' id='uid42' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd columnalign='right'><msub><mi>H</mi> <mi>c</mi> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><mfrac><mn>1</mn> <mrow><mn>2</mn><mi>n</mi></mrow></mfrac><munderover><mo>&sum;</mo> <mrow><mi>l</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi> </munderover><msup><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow> <mi>l</mi> </msup><msup><mrow><mo>(</mo><mi>n</mi><mo>-</mo><mi>l</mi><mo>)</mo></mrow> <mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow> </msup><munder><mo>&sum;</mo> <mrow><msub><mi>l</mi> <mn>1</mn> </msub><mo>+</mo><mo>&ctdot;</mo><mo>+</mo><msub><mi>l</mi> <mi>p</mi> </msub><mo>=</mo><mi>l</mi></mrow> </munder><munderover><mo>&prod;</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>p</mi> </munderover><mfenced separators='' open='(' close=')'><mfrac linethickness='0pt'><msub><mi>n</mi> <mi>i</mi> </msub> <msub><mi>l</mi> <mi>i</mi> </msub></mfrac></mfenced></mrow></mtd></mtr><mtr><mtd/><mtd columnalign='left'><mrow><mspace width='1.em'/><mo>&middot;</mo><msup><mrow><mo>[</mo><mrow><mo>(</mo><mi>n</mi><mo>-</mo><mi>l</mi><mo>)</mo></mrow><mo>-</mo><mrow><mo>(</mo><msub><mi>n</mi> <mi>i</mi> </msub><mo>-</mo><msub><mi>l</mi> <mi>i</mi> </msub><mo>)</mo></mrow><mo>]</mo></mrow> <mrow><msub><mi>n</mi> <mi>i</mi> </msub><mo>-</mo><msub><mi>l</mi> <mi>i</mi> </msub></mrow> </msup><mo>&middot;</mo><mfenced separators='' open='[' close=']'><msup><mrow><mo>(</mo><mi>n</mi><mo>-</mo><mi>l</mi><mo>)</mo></mrow> <mn>2</mn> </msup> <mo>-</mo> <munderover><mo>&sum;</mo> <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow> <mi>p</mi> </munderover> <msup><mrow><mo>(</mo><msub><mi>n</mi> <mi>i</mi> </msub><mo>-</mo><msub><mi>l</mi> <mi>i</mi> </msub><mo>)</mo></mrow> <mn>2</mn> </msup></mfenced><mo>.</mo><mspace width='-20.0pt'/></mrow></mtd></mtr></mtable></math><texmath>\begin{split}
H_c&amp;=\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}\\
&amp;\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 ].
\hspace{-20.0pt}\end{split}</texmath></formula>
<p noindent='true'><hi rend='tt'>\begin{equation}\label{e:barwq}\begin{split}</hi></p>
<p noindent='true'><hi rend='tt'>H_c&amp;=\frac{1}{2n}\sum^n_{l=0}(-<zws/>1)^{l}(n-<zws/>{l})^{p-<zws/>2}</hi></p>
<p noindent='true'><hi rend='tt'>\sum_{l_1+\dots+l_p=l}\prod^p_{i=1}\binom{n_i}{l_i}\\</hi></p>
<p noindent='true'><hi rend='tt'>&amp;\quad\cdot[(n-<zws/>l)-<zws/>(n_i-<zws/>l_i)]^{n_i-<zws/>l_i}\cdot</hi></p>
<p noindent='true'><hi rend='tt'>\Bigl[(n-<zws/>l)^2-<zws/>\sum^p_{j=1}(n_i-<zws/>l_i)^2\Bigr].</hi></p>
<p noindent='true'><hi rend='tt'>\end{split}\end{equation}</hi></p>
<p>The <EM r='cls'>split</EM> structure should constitute the entire body of the
enclosing structure, apart from commands like <EM r='cn'>label</EM> that produce no
visible material.</p>
</div0>
<div0 id-text='5' id='cid8'><head>Equation groups without alignment</head>
<p>The <EM r='cls'>gather</EM> environment is used for a group of consecutive
equations when there is no alignment desired among them; each one is
centered separately within the text width (see Table<ref target='uid34'/>).
Equations inside <EM r='cls'>gather</EM> are separated by a <hi rend='tt'>\\</hi> command.
Any equation in a <EM r='cls'>gather</EM> may consist of a <hi rend='tt'>\begin{split}</hi>
... <hi rend='tt'>\end{split}</hi> structure&#x2014;for example:</p>
<p noindent='true'><hi rend='tt'>\begin{gather}</hi></p>
<p noindent='true'><hi rend='tt'>firstequation\\</hi></p>
<p noindent='true'><hi rend='tt'>\begin{split}</hi></p>
<p noindent='true'><hi rend='tt'>second&amp;equation\\</hi></p>
<p noindent='true'><hi rend='tt'>&amp;ontwolines</hi></p>
<p noindent='true'><hi rend='tt'>\end{split}</hi></p>
<p noindent='true'><hi rend='tt'>\\</hi></p>
<p noindent='true'><hi rend='tt'>thirdequation</hi></p>
<p noindent='true'><hi rend='tt'>\end{gather}</hi></p>
</div0>
<div0 id-text='6' id='cid9'><head>Equation groups with mutual alignment</head>
<p>The <EM r='cls'>align</EM> environment is used for two or more equations when
vertical alignment is desired; usually binary relations such as equal
signs are aligned (see Table<ref target='uid34'/>).</p>
<p>To have several equation columns side-by-side, use extra ampersands
to separate the columns:</p>
<formula id-text='6' id='uid43' textype='align' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd columnalign='right'><mi>x</mi></mtd><mtd columnalign='left'><mrow><mo>=</mo><mi>y</mi></mrow></mtd><mtd columnalign='right'><mi>X</mi></mtd><mtd columnalign='left'><mrow><mo>=</mo><mi>Y</mi></mrow></mtd><mtd columnalign='right'><mi>a</mi></mtd><mtd columnalign='left'><mrow><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mtd></mtr><mtr><mtd columnalign='right'><msup><mi>x</mi> <mo>&apos;</mo> </msup></mtd><mtd columnalign='left'><mrow><mo>=</mo><msup><mi>y</mi> <mo>&apos;</mo> </msup></mrow></mtd><mtd columnalign='right'><msup><mi>X</mi> <mo>&apos;</mo> </msup></mtd><mtd columnalign='left'><mrow><mo>=</mo><msup><mi>Y</mi> <mo>&apos;</mo> </msup></mrow></mtd><mtd columnalign='right'><msup><mi>a</mi> <mo>&apos;</mo> </msup></mtd><mtd columnalign='left'><mrow><mo>=</mo><mi>b</mi></mrow></mtd></mtr><mtr><mtd columnalign='right'><mrow><mi>x</mi><mo>+</mo><msup><mi>x</mi> <mo>&apos;</mo> </msup></mrow></mtd><mtd columnalign='left'><mrow><mo>=</mo><mi>y</mi><mo>+</mo><msup><mi>y</mi> <mo>&apos;</mo> </msup></mrow></mtd><mtd columnalign='right'><mrow><mi>X</mi><mo>+</mo><msup><mi>X</mi> <mo>&apos;</mo> </msup></mrow></mtd><mtd columnalign='left'><mrow><mo>=</mo><mi>Y</mi><mo>+</mo><msup><mi>Y</mi> <mo>&apos;</mo> </msup></mrow></mtd><mtd columnalign='right'><mrow><msup><mi>a</mi> <mo>&apos;</mo> </msup><mi>b</mi></mrow></mtd><mtd columnalign='left'><mrow><mo>=</mo><msup><mi>c</mi> <mo>&apos;</mo> </msup><mi>b</mi></mrow></mtd></mtr></mtable></math><texmath>
x&amp;=y &amp; X&amp;=Y &amp; a&amp;=b+c\\
x^{\prime }&amp;=y^{\prime } &amp; X^{\prime }&amp;=Y^{\prime } &amp; a^{\prime }&amp;=b\\
x+x^{\prime }&amp;=y+y^{\prime } &amp; X+X^{\prime }&amp;=Y+Y^{\prime } &amp; a^{\prime }b&amp;=c^{\prime }b
</texmath></formula>
<p noindent='true'><hi rend='tt'>\begin{align}</hi></p>
<p noindent='true'><hi rend='tt'>x&amp;=y&amp;X&amp;=Y&amp;a&amp;=b+c\\</hi></p>
<p noindent='true'><hi rend='tt'>x'<zws/>&amp;=y'<zws/>&amp;X'<zws/>&amp;=Y'<zws/>&amp;a'<zws/>&amp;=b\\</hi></p>
<p noindent='true'><hi rend='tt'>x+x'<zws/>&amp;=y+y'<zws/>&amp;X+X'<zws/>&amp;=Y+Y'<zws/>&amp;a'<zws/>b&amp;=c'<zws/>b</hi></p>
<p noindent='true'><hi rend='tt'>\end{align}</hi></p>
<p noindent='true'>Line-by-line annotations on an equation can be done by judicious
application of <EM r='cn'>text</EM> inside an <EM r='cls'>align</EM> environment:</p>
<formula id-text='6' id='uid44' textype='align' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd columnalign='right'><mi>x</mi></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>y</mi> <mn>1</mn> </msub><mo>-</mo><msub><mi>y</mi> <mn>2</mn> </msub><mo>+</mo><msub><mi>y</mi> <mn>3</mn> </msub><mo>-</mo><msub><mi>y</mi> <mn>5</mn> </msub><mo>+</mo><msub><mi>y</mi> <mn>8</mn> </msub><mo>-</mo><mo>&ctdot;</mo></mrow></mtd><mtd/><mtd columnalign='left'><mrow><mtext>by</mtext><mspace width='4.pt'/><mtext>(</mtext><mref target='uid46'/><mtext>)</mtext></mrow></mtd></mtr><mtr><mtd/><mtd columnalign='left'><mrow><mo>=</mo><msup><mi>y</mi> <mo>&apos;</mo> </msup><mo>&SmallCircle;</mo><msup><mi>y</mi> <mo>*</mo> </msup></mrow></mtd><mtd/><mtd columnalign='left'><mrow><mtext>by</mtext><mspace width='4.pt'/><mtext>(</mtext><mref target='uid59'/><mtext>)</mtext></mrow></mtd></mtr><mtr><mtd/><mtd columnalign='left'><mrow><mo>=</mo><mi>y</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><msup><mi>y</mi> <mo>&apos;</mo> </msup></mrow></mtd><mtd/><mtd columnalign='left'><mrow><mtext>by</mtext><mspace width='4.pt'/><mtext>Axiom</mtext><mspace width='4.pt'/><mtext>1.</mtext></mrow></mtd></mtr></mtable></math><texmath>
x&amp; = y_1-y_2+y_3-y_5+y_8-\dots &amp;&amp; \text{by (\ref {eq:C})}\\
&amp; = y^{\prime }\circ y^* &amp;&amp; \text{by (\ref {eq:D})}\\
&amp; = y(0) y^{\prime } &amp;&amp; \text{by Axiom 1.}
</texmath></formula>
<p noindent='true'><hi rend='tt'>\begin{align}</hi></p>
<p noindent='true'><hi rend='tt'>x&amp;=y_1-<zws/>y_2+y_3-<zws/>y_5+y_8-<zws/>\dots</hi></p>
<p noindent='true'><hi rend='tt'>&amp;&amp;\text{by\eqref{eq:C}}\\</hi></p>
<p noindent='true'><hi rend='tt'>&amp;=y'<zws/>\circy^*&amp;&amp;\text{by\eqref{eq:D}}\\</hi></p>
<p noindent='true'><hi rend='tt'>&amp;=y(0)y'<zws/>&amp;&amp;\text{byAxiom1.}</hi></p>
<p noindent='true'><hi rend='tt'>\end{align}</hi></p>
<p noindent='true'>A variant environment <EM r='cls'>alignat</EM> allows the horizontal space between
equations to be explicitly specified. This environment takes one argument,
the number of &#x201C;equation columns&#x201D;: count the maximum number of <hi rend='tt'>&amp;</hi>s
in any row, add 1 and divide by 2.</p>
<formula id-text='6' id='uid45' textype='alignat' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd columnalign='right'><mi>x</mi></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>y</mi> <mn>1</mn> </msub><mo>-</mo><msub><mi>y</mi> <mn>2</mn> </msub><mo>+</mo><msub><mi>y</mi> <mn>3</mn> </msub><mo>-</mo><msub><mi>y</mi> <mn>5</mn> </msub><mo>+</mo><msub><mi>y</mi> <mn>8</mn> </msub><mo>-</mo><mo>&ctdot;</mo></mrow></mtd><mtd columnalign='right'><mspace width='1.em'/></mtd><mtd columnalign='left'><mrow><mtext>by</mtext><mspace width='4.pt'/><mtext>(</mtext><mref target='uid46'/><mtext>)</mtext></mrow></mtd></mtr><mtr><mtd/><mtd columnalign='left'><mrow><mo>=</mo><msup><mi>y</mi> <mo>&apos;</mo> </msup><mo>&SmallCircle;</mo><msup><mi>y</mi> <mo>*</mo> </msup></mrow></mtd><mtd/><mtd columnalign='left'><mrow><mtext>by</mtext><mspace width='4.pt'/><mtext>(</mtext><mref target='uid59'/><mtext>)</mtext></mrow></mtd></mtr><mtr><mtd/><mtd columnalign='left'><mrow><mo>=</mo><mi>y</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><msup><mi>y</mi> <mo>&apos;</mo> </msup></mrow></mtd><mtd/><mtd columnalign='left'><mrow><mtext>by</mtext><mspace width='4.pt'/><mtext>Axiom</mtext><mspace width='4.pt'/><mtext>1.</mtext></mrow></mtd></mtr></mtable></math><texmath>
x&amp; = y_1-y_2+y_3-y_5+y_8-\dots &amp;\quad &amp; \text{by (\ref {eq:C})}\\
&amp; = y^{\prime }\circ y^* &amp;&amp; \text{by (\ref {eq:D})}\\
&amp; = y(0) y^{\prime } &amp;&amp; \text{by Axiom 1.}
</texmath></formula>
<p noindent='true'><hi rend='tt'>\begin{alignat}{2}</hi></p>
<p noindent='true'><hi rend='tt'>x&amp;=y_1-<zws/>y_2+y_3-<zws/>y_5+y_8-<zws/>\dots</hi></p>
<p noindent='true'><hi rend='tt'>&amp;\quad&amp;\text{by\eqref{eq:C}}\\</hi></p>
<p noindent='true'><hi rend='tt'>&amp;=y'<zws/>\circy^*&amp;&amp;\text{by\eqref{eq:D}}\\</hi></p>
<p noindent='true'><hi rend='tt'>&amp;=y(0)y'<zws/>&amp;&amp;\text{byAxiom1.}</hi></p>
<p noindent='true'><hi rend='tt'>\end{alignat}</hi></p>
</div0>
<div0 id-text='7' id='cid10'><head>Alignment building blocks</head>
<p>Like <EM r='cls'>equation</EM>, the multi-equation environments <EM r='cls'>gather</EM>,
<EM r='cls'>align</EM>, and <EM r='cls'>alignat</EM> are designed to produce a structure
whose width is the full line width. This means, for example, that one
cannot readily add parentheses around the entire structure. But variants
<EM r='cls'>gathered</EM>, <EM r='cls'>aligned</EM>, and <EM r='cls'>alignedat</EM> are provided whose
total width is the actual width of the contents; thus they can be used
as a component in a containing expression. E.g.,</p>
<formula textype='equation*' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced separators='' open='' close='&rbrace;'><mtable displaystyle='true'><mtr><mtd columnalign='right'><msup><mi>B</mi> <mo>&apos;</mo> </msup></mtd><mtd columnalign='left'><mrow><mo>=</mo><mo>-</mo><mi>&part;</mi><mo>&times;</mo><mi>E</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign='right'><msup><mi>E</mi> <mo>&apos;</mo> </msup></mtd><mtd columnalign='left'><mrow><mo>=</mo><mi>&part;</mi><mo>&times;</mo><mi>B</mi><mo>-</mo><mn>4</mn><mi>&pi;</mi><mi>j</mi><mo>,</mo></mrow></mtd></mtr></mtable></mfenced><mspace width='2.em'/><mtext>Maxwell's</mtext><mspace width='4.pt'/><mtext>equations</mtext></mrow></math><texmath>
\left.\begin{aligned}
B^{\prime }&amp;=-\partial \times E,\\
E^{\prime }&amp;=\partial \times B - 4\pi j,
\end{aligned}
\right\rbrace 
\qquad \text{Maxwell's equations}
</texmath></formula>
<p noindent='true'><hi rend='tt'>\begin{equation*}</hi></p>
<p noindent='true'><hi rend='tt'>\left.\begin{aligned}</hi></p>
<p noindent='true'><hi rend='tt'>B'<zws/>&amp;=-<zws/>\partial\timesE,\\</hi></p>
<p noindent='true'><hi rend='tt'>E'<zws/>&amp;=\partial\timesB-<zws/>4\pij,</hi></p>
<p noindent='true'><hi rend='tt'>\end{aligned}</hi></p>
<p noindent='true'><hi rend='tt'>\right\}</hi></p>
<p noindent='true'><hi rend='tt'>\qquad\text{Maxwell'<zws/>sequations}</hi></p>
<p noindent='true'><hi rend='tt'>\end{equation*}</hi></p>
<p noindent='true'>Like the <EM r='cls'>array</EM> environment, these <hi rend='tt'>-ed</hi> variants also take
an optional <hi rend='tt'>[t]</hi> or <hi rend='tt'>[b]</hi> argument to specify vertical
positioning.</p>
<p>&#x201C;Cases&#x201D; constructions like the following are common in
mathematics:</p>
<formula id-text='11' id='uid46' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>P</mi> <mrow><mi>r</mi><mo>-</mo><mi>j</mi></mrow> </msub><mo>=</mo><mfenced separators='' open='&lbrace;' close=''><mtable><mtr><mtd columnalign='left'><mn>0</mn></mtd><mtd columnalign='left'><mrow><mtext>if</mtext><mspace width='4.pt'/><mrow><mi>r</mi><mo>-</mo><mi>j</mi></mrow><mspace width='4.pt'/><mtext>is</mtext><mspace width='4.pt'/><mtext>odd</mtext><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign='left'><mrow><mi>r</mi><mo>!</mo><mspace width='0.166667em'/><msup><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow> <mrow><mo>(</mo><mi>r</mi><mo>-</mo><mi>j</mi><mo>)</mo><mo>/</mo><mn>2</mn></mrow> </msup></mrow></mtd><mtd columnalign='left'><mrow><mtext>if</mtext><mspace width='4.pt'/><mrow><mi>r</mi><mo>-</mo><mi>j</mi></mrow><mspace width='4.pt'/><mtext>is</mtext><mspace width='4.pt'/><mtext>even</mtext><mo>.</mo></mrow></mtd></mtr></mtable></mfenced></mrow></math><texmath>
P_{r-j}=
{\left\lbrace \begin{array}{ll}
0&amp; \text{if $r-j$ is odd},\\
r!\,(-1)^{(r-j)/2}&amp; \text{if $r-j$ is even}.
\end{array}\right.}
</texmath></formula>
<p noindent='true'>and in the <EM r='pkg'>amsmath</EM> package there is a <EM r='cls'>cases</EM> environment to
make them easy to write:</p>
<p noindent='true'><hi rend='tt'>P_{r-<zws/>j}=\begin{cases}</hi></p>
<p noindent='true'><hi rend='tt'>0&amp;\text{if$r-<zws/>j$isodd},\\</hi></p>
<p noindent='true'><hi rend='tt'>r!\,(-<zws/>1)^{(r-<zws/>j)/2}&amp;\text{if$r-<zws/>j$iseven}.</hi></p>
<p noindent='true'><hi rend='tt'>\end{cases}</hi></p>
<p noindent='true'>Notice the use of <EM r='cn'>text</EM> (cf.<ref target='cid33'/>) and the nested
math formulas.</p>
</div0>
<div0 id-text='8' id='cid11'><head>Adjusting tag placement</head>
<p>Placing equation numbers can be a rather complex problem in multiline
displays. The environments of the <EM r='pkg'>amsmath</EM> package try hard to
avoid overprinting an equation number on the equation contents, if
necessary moving the number down or up to a separate line. Difficulties
in accurately calculating the profile of an equation can occasionally
result in number movement that doesn't look right. There is a
<EM r='cn'>raisetag</EM> command provided to adjust the vertical position of the
current equation number, if it has been shifted away from its normal
position. To move a particular number up by six points, write
<hi rend='tt'>\raisetag{6pt}</hi>. This kind of adjustment is fine tuning like line
breaks and page breaks, and should therefore be left undone until your
document is nearly finalized, or you may end up redoing the fine tuning
several times to keep up with changing document contents.</p>
</div0>
<div0 id-text='9' id='cid12'><head>Vertical spacing and page breaks in multiline displays</head>
<p>You can use the <hi rend='tt'>\\[</hi><hi rend='it'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&langle;</mo></math><texmath>\langle </texmath></formula>dimension<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&rangle;</mo></math><texmath>\rangle </texmath></formula></hi><hi rend='tt'>]</hi> command to get extra vertical
space between lines in all the <EM r='pkg'>amsmath</EM> displayed equation
environments, as is usual in <LaTeX/>. When the <EM r='pkg'>amsmath</EM> package is
in use page breaks<anchor id-text='lid12' id='uid47'/> between equation lines are normally disallowed;
the philosophy is that page breaks in such material should receive
individual attention from the author. To get an individual page break
inside a particular displayed equation, a <EM r='cn'>displaybreak</EM> command is
provided. <EM r='cn'>displaybreak</EM> is best placed immediately before the
<hi rend='tt'>\\</hi> where it is to take effect. Like <LaTeX/>'s <EM r='cn'>pagebreak</EM>,
<EM r='cn'>displaybreak</EM> takes an optional argument between 0 and 4 denoting
the desirability of the pagebreak. <hi rend='tt'>\displaybreak[0]</hi> means &#x201C;it is
permissible to break here&#x201D; without encouraging a break;
<EM r='cn'>displaybreak</EM> with no optional argument is the same as
<hi rend='tt'>\displaybreak[4]</hi> and forces a break.</p>
<p>If you prefer a strategy of letting page breaks fall where they may,
even in the middle of a multi-line equation, then you might put
<EM r='cn'>allowdisplaybreaks</EM><hi rend='tt'>[1]</hi> in the preamble of your document. An
optional argument 1&#x2013;4 can be used for finer control: <hi rend='tt'>[1]</hi> means
allow page breaks, but avoid them as much as possible; values of 2,3,4
mean increasing permissiveness. When display breaks are enabled with
<EM r='cn'>allowdisplaybreaks</EM>, the <hi rend='tt'>\\*</hi> command can be used to prohibit a
pagebreak after a given line, as usual.</p>
<p><hi rend='bold'>Note: Certain equation environments wrap their contents in an
unbreakable box, with the consequence that neither <EM r='cn'>displaybreak</EM> nor
<EM r='cn'>allowdisplaybreaks</EM> will have any effect on them. These include
<EM r='cls'>split</EM>, <EM r='cls'>aligned</EM>, <EM r='cls'>gathered</EM>, and <EM r='cls'>alignedat</EM>.
</hi></p>
</div0>
<div0 id-text='10' id='cid13'><head>Interrupting a display</head>
<p>The command <EM r='cn'>intertext</EM> is used for a short interjection of one or
two lines of text<anchor id-text='lid13' id='uid48'/> in the middle of a
multiple-line display structure (see also the <EM r='cn'>text</EM> command in
<ref target='cid33'/>). Its salient feature is preservation of the alignment,
which would not happen if you simply ended the display and then started
it up again afterwards. <EM r='cn'>intertext</EM> may only appear right after a
<hi rend='tt'>\\</hi> or <hi rend='tt'>\\*</hi> command. Notice the position of the word &#x201C;and&#x201D; in
this example.</p>
<formula id-text='10' id='uid49' textype='align' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd columnalign='right'><msub><mi>A</mi> <mn>1</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><msub><mi>N</mi> <mn>0</mn> </msub><mrow><mo>(</mo><mi>&lambda;</mi><mo>;</mo><msup><mi>&Omega;</mi> <mo>&apos;</mo> </msup><mo>)</mo></mrow><mo>-</mo><mi>&phiv;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>;</mo><msup><mi>&Omega;</mi> <mo>&apos;</mo> </msup><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign='right'><msub><mi>A</mi> <mn>2</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><mi>&phiv;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>;</mo><msup><mi>&Omega;</mi> <mo>&apos;</mo> </msup><mo>)</mo></mrow><mo>-</mo><mi>&phiv;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>;</mo><mi>&Omega;</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign='left' columnspan='2'><mtext>and</mtext></mtd></mtr><mtr><mtd columnalign='right'><msub><mi>A</mi> <mn>3</mn> </msub></mtd><mtd columnalign='left'><mrow><mo>=</mo><mi>&Nscr;</mi><mo>(</mo><mi>&lambda;</mi><mo>;</mo><mi>&omega;</mi><mo>)</mo><mo>.</mo></mrow></mtd></mtr></mtable></math><texmath>
A_1&amp;=N_0(\lambda ;\Omega ^{\prime })-\phi (\lambda ;\Omega ^{\prime }),\\
A_2&amp;=\phi (\lambda ;\Omega ^{\prime })-\phi (\lambda ;\Omega ),\\
\multicolumn{2}{l}{\text{and}}\\
A_3&amp;=\mathcal {N}(\lambda ;\omega ).
</texmath></formula>
<p noindent='true'><hi rend='tt'>\begin{align}</hi></p>
<p noindent='true'><hi rend='tt'>A_1&amp;=N_0(\lambda;\Omega'<zws/>)-<zws/>\phi(\lambda;\Omega'<zws/>),\\</hi></p>
<p noindent='true'><hi rend='tt'>A_2&amp;=\phi(\lambda;\Omega'<zws/>)-<zws/>\phi(\lambda;\Omega),\\</hi></p>
<p noindent='true'><hi rend='tt'>\intertext{and}</hi></p>
<p noindent='true'><hi rend='tt'>A_3&amp;=\mathcal{N}(\lambda;\omega).</hi></p>
<p noindent='true'><hi rend='tt'>\end{align}</hi></p>
</div0>
<div0 id-text='11' id='cid14'><head>Equation numbering</head>
<div1 id-text='1' id='uid50'><head>Numbering hierarchy</head>
<p>In <LaTeX/> if you wanted to have equations numbered within
sections&#x2014;that is, have
equation numbers (1.1), (1.2), ..., (2.1), (2.2),
..., in sections 1, 2, and so forth&#x2014;you could redefine
<EM r='cn'>theequation</EM> as suggested in the <LaTeX/> manual <cit><ref target='bid0'>6.3, C.8.4</ref></cit>:</p>
<p noindent='true'><hi rend='tt'>\renewcommand{\theequation}{\thesection.\arabic{equation}}</hi></p>
<p>This works pretty well, except that the equation counter won't be reset
to zero at the beginning of a new section or chapter, unless you do it
yourself using <EM r='cn'>setcounter</EM>. To make this a little more convenient,
the <EM r='pkg'>amsmath</EM> package provides a command<anchor id-text='lid14' id='uid51'/> <EM r='cn'>numberwithin</EM>. To have equation numbering tied to
section numbering, with automatic reset of the equation counter, write</p>
<p noindent='true'><hi rend='tt'>\numberwithin{equation}{section}</hi></p>
<p noindent='true'>As its name implies, the <EM r='cn'>numberwithin</EM> command can be applied to
any counter, not just the <hi rend='tt'>equation</hi> counter.</p>
</div1>
<div1 id-text='2' id='uid52'><head>Cross references to equation numbers</head>
<p>To make cross-references to equations easier, an <EM r='cn'>eqref</EM>
command<anchor id-text='lid15' id='uid53'/> is provided. This
automatically supplies the parentheses around the equation number. I.e.,
if <hi rend='tt'>\ref{abc}</hi> produces 3.2 then <hi rend='tt'>\eqref{abc}</hi> produces
(3.2).</p>
</div1>
<div1 id-text='3' id='uid54'><head>Subordinate numbering sequences</head>
<p>The <EM r='pkg'>amsmath</EM> package provides also a <EM r='cls'>subequations</EM>
environment<anchor id-text='lid16' id='uid55'/> to make it
easy to number equations in a particular group with a subordinate
numbering scheme. For example</p>
<p noindent='true'><hi rend='tt'>\begin{subequations}</hi></p>
<p noindent='true'><hi rend='tt'>...</hi></p>
<p noindent='true'><hi rend='tt'>\end{subequations}</hi></p>
<p noindent='true'>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 <EM r='cn'>label</EM> command immediately after
<hi rend='tt'>\begin{subequations}</hi> will produce a <EM r='cn'>ref</EM> of the parent
number 4.9, not 4.9a. The counters used by the subequations
environment are <hi rend='tt'>parentequation</hi> and <hi rend='tt'>equation</hi> and
<EM r='cn'>addtocounter</EM>, <EM r='cn'>setcounter</EM>, <EM r='cn'>value</EM>, etc., can be applied
as usual to those counter names. To get anything other than lowercase
letters for the subordinate numbers, use standard <LaTeX/> methods for
changing numbering style <cit><ref target='bid0'>6.3, C.8.4</ref></cit>. For example,
redefining <EM r='cn'>theequation</EM> as follows will produce roman numerals.</p>
<p noindent='true'><hi rend='tt'>\begin{subequations}</hi></p>
<p noindent='true'><hi rend='tt'>\renewcommand{\theequation}{\theparentequation\roman{equation}}</hi></p>
<p noindent='true'><hi rend='tt'>...</hi></p>
</div1></div0>
<div0 id-text='4' id='cid15'><head>Miscellaneous mathematical features</head>
</div0>
<div0 id-text='1' id='cid16'><head>Matrices</head>
<p>The <EM r='pkg'>amsmath</EM> package provides some environments for
matrices<anchor id-text='lid17' id='uid56'/> beyond the basic <EM r='cls'>array</EM> environment of
<LaTeX/>. The <EM r='cls'>pmatrix</EM>, <EM r='cls'>bmatrix</EM>, <EM r='cls'>Bmatrix</EM>, <EM r='cls'>vmatrix</EM>
and <EM r='cls'>Vmatrix</EM> have (respectively) <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo>(</mo><mspace width='0.166667em'/><mo>)</mo></mrow></math><texmath>(\,)</texmath></formula>, <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo>[</mo><mspace width='0.166667em'/><mo>]</mo></mrow></math><texmath>[\,]</texmath></formula>, <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo>&lbrace;</mo><mspace width='0.166667em'/><mo>&rbrace;</mo></mrow></math><texmath>\lbrace \,\rbrace </texmath></formula>,
<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo>&vert;</mo><mspace width='0.166667em'/><mo>&vert;</mo></mrow></math><texmath>\vert \,\vert </texmath></formula>, and
<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo>&parallel;</mo><mspace width='0.166667em'/><mo>&parallel;</mo></mrow></math><texmath>\Vert \,\Vert </texmath></formula> delimiters built in. For naming consistency there is a
<EM r='cls'>matrix</EM> environment sans delimiters. This is not entirely redundant
with the <EM r='cls'>array</EM> environment; the matrix environments all use more
economical horizontal spacing than the rather prodigal spacing of the
<EM r='cls'>array</EM> environment. Also, unlike the <EM r='cls'>array</EM> environment, you
don't have to give column specifications for any of the matrix
environments; by default you can have up to 10 centered columns.<note id-text='2' id='uid57' place='foot'>More precisely: The maximum number of columns in a matrix is determined
by the counter <EM r='cn'>MaxMatrixCols</EM> (normal value = 10), which you can change
if necessary using <LaTeX/>'s <EM r='cn'>setcounter</EM> or <EM r='cn'>addtocounter</EM>
commands.</note> (If you need left or right alignment in a column or other special
formats you must resort to <EM r='cls'>array</EM>.)</p>
<p>To produce a small matrix suitable for use in text, there is a
<EM r='cls'>smallmatrix</EM> environment (e.g.,
<formula textype='math' type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mfenced open='(' close=')'><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced></math><texmath>
\bigl ( {\begin{matrix}
a&amp;b\\ c&amp;d
\end{matrix}} \bigr )
</texmath></formula>)
that comes closer to fitting within a single text line than a normal
matrix. Delimiters must be provided; there are no <hi rend='tt'>p</hi>,<hi rend='tt'>b</hi>,<hi rend='tt'>B</hi>,<hi rend='tt'>v</hi>,<hi rend='tt'>V</hi>
versions of <EM r='cls'>smallmatrix</EM>. The above example was produced by</p>
<p noindent='true'><hi rend='tt'>\bigl(\begin{smallmatrix}</hi></p>
<p noindent='true'><hi rend='tt'>a&amp;b\\c&amp;d</hi></p>
<p noindent='true'><hi rend='tt'>\end{smallmatrix}\bigr)</hi></p>
<EM r='cn'>hdotsfor</EM><p><hi rend='tt'>{</hi><hi rend='it'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&langle;</mo></math><texmath>\langle </texmath></formula>number<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&rangle;</mo></math><texmath>\rangle </texmath></formula></hi><hi rend='tt'>}</hi> produces a row of dots in a
matrix<anchor id-text='lid18' id='uid58'/> spanning the given number of
columns. For example,</p>
<p rend='center'><minipage width='128.1013pt'><p noindent='true'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr><mtr><mtd><mi>e</mi></mtd><mtd><mo>&ctdot;</mo></mtd><mtd><mo>&ctdot;</mo></mtd><mtd><mo>&ctdot;</mo></mtd></mtr></mtable></math><texmath>\begin{matrix} a&amp;b&amp;c&amp;d\\
e&amp;\dots &amp;\dots &amp;\dots  \end{matrix}</texmath></formula>
</p></minipage><minipage width='192.1487pt'><p noindent='true'><hi rend='tt'>\begin{matrix}a&amp;b&amp;c&amp;d\\</hi></p>
<p noindent='true'><hi rend='tt'>e&amp;\hdotsfor{3}\end{matrix}</hi></p>
<p noindent='true'/></minipage></p>
<p>The spacing of the dots can be varied through use of a square-bracket
option, for example, <hi rend='tt'>\hdotsfor[1.5]{3}</hi>. The number in square brackets
will be used as a multiplier (i.e., the normal value is 1.0).</p>
<formula id-text='13' id='uid59' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')'><mtable><mtr><mtd><mrow><msub><mi>D</mi> <mn>1</mn> </msub><mi>t</mi></mrow></mtd><mtd><mrow><mo>-</mo><msub><mi>a</mi> <mn>12</mn> </msub><msub><mi>t</mi> <mn>2</mn> </msub></mrow></mtd><mtd><mo>&ctdot;</mo></mtd><mtd><mrow><mo>-</mo><msub><mi>a</mi> <mrow><mn>1</mn><mi>n</mi></mrow> </msub><msub><mi>t</mi> <mi>n</mi> </msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>-</mo><msub><mi>a</mi> <mn>21</mn> </msub><msub><mi>t</mi> <mn>1</mn> </msub></mrow></mtd><mtd><mrow><msub><mi>D</mi> <mn>2</mn> </msub><mi>t</mi></mrow></mtd><mtd><mo>&ctdot;</mo></mtd><mtd><mrow><mo>-</mo><msub><mi>a</mi> <mrow><mn>2</mn><mi>n</mi></mrow> </msub><msub><mi>t</mi> <mi>n</mi> </msub></mrow></mtd></mtr><mtr><mtd><mo>&ctdot;</mo></mtd><mtd><mo>&ctdot;</mo></mtd><mtd><mo>&ctdot;</mo></mtd><mtd><mo>&ctdot;</mo></mtd></mtr><mtr><mtd><mrow><mo>-</mo><msub><mi>a</mi> <mrow><mi>n</mi><mn>1</mn></mrow> </msub><msub><mi>t</mi> <mn>1</mn> </msub></mrow></mtd><mtd><mrow><mo>-</mo><msub><mi>a</mi> <mrow><mi>n</mi><mn>2</mn></mrow> </msub><msub><mi>t</mi> <mn>2</mn> </msub></mrow></mtd><mtd><mo>&ctdot;</mo></mtd><mtd><mrow><msub><mi>D</mi> <mi>n</mi> </msub><mi>t</mi></mrow></mtd></mtr></mtable></mfenced><mo>,</mo></mrow></math><texmath>
\begin{pmatrix} D_1t&amp;-a_{12}t_2&amp;\dots &amp;-a_{1n}t_n\\
-a_{21}t_1&amp;D_2t&amp;\dots &amp;-a_{2n}t_n\\
\dots &amp;\dots &amp;\dots &amp;\dots \\
-a_{n1}t_1&amp;-a_{n2}t_2&amp;\dots &amp;D_nt\end{pmatrix},
</texmath></formula>
<p noindent='true'><hi rend='tt'>\begin{pmatrix}D_1t&amp;-<zws/>a_{12}t_2&amp;\dots&amp;-<zws/>a_{1n}t_n\\</hi></p>
<p noindent='true'><hi rend='tt'>-<zws/>a_{21}t_1&amp;D_2t&amp;\dots&amp;-<zws/>a_{2n}t_n\\</hi></p>
<p noindent='true'><hi rend='tt'>\hdotsfor[2]{4}\\</hi></p>
<p noindent='true'><hi rend='tt'>-<zws/>a_{n1}t_1&amp;-<zws/>a_{n2}t_2&amp;\dots&amp;D_nt\end{pmatrix}</hi></p>
</div0>
<div0 id-text='2' id='cid17'><head>Math spacing commands</head>
<p>The <EM r='pkg'>amsmath</EM> package slightly extends the set of math
spacing<anchor id-text='lid21' id='uid60'/> 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.</p>
<table rend='inline'><row bottom-border='true'><cell halign='left'>Abbrev.</cell>
<cell halign='left'>Spelled out</cell>
<cell right-border='true' halign='left'>Example</cell>
<cell halign='left'>Abbrev.</cell>
<cell halign='left'>Spelled out</cell>
<cell>Example</cell>
</row><row><cell halign='left'><rule width='0.0pt' height='10.625pt'/></cell>
<cell halign='left'>no space</cell>
<cell right-border='true' halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo>&Rightarrow;</mo><mspace width='-0.55542pt'/></mrow><mrow><mspace width='-0.55542pt'/><mo>&Leftarrow;</mo></mrow></mrow></math><texmath>\mathord {\Rightarrow \hspace{-0.55542pt}}\mathord {\hspace{-0.55542pt}\Leftarrow }</texmath></formula></cell>
<cell halign='left'/>
<cell halign='left'>no space</cell>
<cell><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo>&Rightarrow;</mo><mspace width='-0.55542pt'/></mrow><mrow><mspace width='-0.55542pt'/><mo>&Leftarrow;</mo></mrow></mrow></math><texmath>\mathord {\Rightarrow \hspace{-0.55542pt}}\mathord {\hspace{-0.55542pt}\Leftarrow }</texmath></formula></cell>
</row><row><cell halign='left'><EM r='cn'></EM></cell>
<cell halign='left'><EM r='cn'>thinspace</EM></cell>
<cell right-border='true' halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo>&Rightarrow;</mo><mspace width='-0.55542pt'/></mrow><mspace width='0.166667em'/><mrow><mspace width='-0.55542pt'/><mo>&Leftarrow;</mo></mrow></mrow></math><texmath>\mathord {\Rightarrow \hspace{-0.55542pt}}\,\mathord {\hspace{-0.55542pt}\Leftarrow }</texmath></formula></cell>
<cell halign='left'><EM r='cn'>!</EM></cell>
<cell halign='left'><EM r='cn'>negthinspace</EM></cell>
<cell><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo>&Rightarrow;</mo><mspace width='-0.55542pt'/></mrow><mspace width='-0.166667em'/><mrow><mspace width='-0.55542pt'/><mo>&Leftarrow;</mo></mrow></mrow></math><texmath>\mathord {\Rightarrow \hspace{-0.55542pt}}\!\mathord {\hspace{-0.55542pt}\Leftarrow }</texmath></formula></cell>
</row><row><cell halign='left'><EM r='cn'>:</EM></cell>
<cell halign='left'><EM r='cn'>medspace</EM></cell>
<cell right-border='true' halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo>&Rightarrow;</mo><mspace width='-0.55542pt'/></mrow><mspace width='0.222222em'/><mrow><mspace width='-0.55542pt'/><mo>&Leftarrow;</mo></mrow></mrow></math><texmath>\mathord {\Rightarrow \hspace{-0.55542pt}}\:\mathord {\hspace{-0.55542pt}\Leftarrow }</texmath></formula></cell>
<cell halign='left'/>
<cell halign='left'><EM r='cn'>negmedspace</EM></cell>
<cell><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo>&Rightarrow;</mo><mspace width='-0.55542pt'/></mrow><mspace width='0.0pt'/><mrow><mspace width='-0.55542pt'/><mo>&Leftarrow;</mo></mrow></mrow></math><texmath>\mathord {\Rightarrow \hspace{-0.55542pt}}\hspace{0.0pt}\mathord {\hspace{-0.55542pt}\Leftarrow }</texmath></formula></cell>
</row><row><cell halign='left'><EM r='cn'>;</EM></cell>
<cell halign='left'><EM r='cn'>thickspace</EM></cell>
<cell right-border='true' halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo>&Rightarrow;</mo><mspace width='-0.55542pt'/></mrow><mspace width='0.277778em'/><mrow><mspace width='-0.55542pt'/><mo>&Leftarrow;</mo></mrow></mrow></math><texmath>\mathord {\Rightarrow \hspace{-0.55542pt}}\;\mathord {\hspace{-0.55542pt}\Leftarrow }</texmath></formula></cell>
<cell halign='left'/>
<cell halign='left'><EM r='cn'>negthickspace</EM></cell>
<cell><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo>&Rightarrow;</mo><mspace width='-0.55542pt'/></mrow><mspace width='0.0pt'/><mrow><mspace width='-0.55542pt'/><mo>&Leftarrow;</mo></mrow></mrow></math><texmath>\mathord {\Rightarrow \hspace{-0.55542pt}}\hspace{0.0pt}\mathord {\hspace{-0.55542pt}\Leftarrow }</texmath></formula></cell>
</row><row><cell halign='left'/>
<cell halign='left'><EM r='cn'>quad</EM></cell>
<cell right-border='true' halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo>&Rightarrow;</mo><mspace width='-0.55542pt'/></mrow><mspace width='1.em'/><mrow><mspace width='-0.55542pt'/><mo>&Leftarrow;</mo></mrow></mrow></math><texmath>\mathord {\Rightarrow \hspace{-0.55542pt}}\quad \mathord {\hspace{-0.55542pt}\Leftarrow }</texmath></formula></cell>
</row><row><cell halign='left'/>
<cell halign='left'><EM r='cn'>qquad</EM></cell>
<cell right-border='true' halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo>&Rightarrow;</mo><mspace width='-0.55542pt'/></mrow><mspace width='2.em'/><mrow><mspace width='-0.55542pt'/><mo>&Leftarrow;</mo></mrow></mrow></math><texmath>\mathord {\Rightarrow \hspace{-0.55542pt}}\qquad \mathord {\hspace{-0.55542pt}\Leftarrow }</texmath></formula></cell>
</row></table><p>
For the greatest possible control over math spacing, use <EM r='cn'>mspace</EM>
and `math units'. One math unit, or <hi rend='tt'>mu</hi>, is equal to 1/18 em. Thus to
get a negative <EM r='cn'>quad</EM> you could write <hi rend='tt'>\mspace{-<zws/>18.0mu}</hi>.</p>
</div0>
<div0 id-text='3' id='cid18'><head>Dots</head>
<p>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</p>
<list type='simple'>
<item id-text='1' id='uid61'><p noindent='true'><EM r='cn'>dotsc</EM> for &#x201C;dots with commas&#x201D;</p>
</item>
<item id-text='2' id='uid62'><p noindent='true'><EM r='cn'>dotsb</EM> for &#x201C;dots with binary operators/relations&#x201D;</p>
</item>
<item id-text='3' id='uid63'><p noindent='true'><EM r='cn'>dotsm</EM> for &#x201C;multiplication dots&#x201D;</p>
</item>
<item id-text='4' id='uid64'><p noindent='true'><EM r='cn'>dotsi</EM> for &#x201C;dots with integrals&#x201D;</p>
</item>
<item id-text='5' id='uid65'><p noindent='true'><EM r='cn'>dotso</EM> for &#x201C;other dots&#x201D; (none of the above)</p>
</item></list>
<p>instead of <EM r='cn'>ldots</EM> and <EM r='cn'>cdots</EM>, 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:</p>
<table rend='inline'><row><cell><minipage pos='t' width='230.57713pt'><p noindent='true'><hi rend='tt'>Thenwehavetheseries$A_1,A_2,</hi></p>
<p noindent='true'><hi rend='tt'>\dotsc$,theregionalsum$A_1</hi></p>
<p noindent='true'><hi rend='tt'>+A_2+\dotsb$,theorthogonal</hi></p>
<p noindent='true'><hi rend='tt'>product$A_1A_2\dotsm$,and</hi></p>
<p noindent='true'><hi rend='tt'>theinfiniteintegral</hi></p>
<p noindent='true'><hi rend='tt'>\[\int_{A_1}\int_{A_2}\dotsi\].</hi></p>
<p noindent='true'/></minipage></cell>
<cell><minipage pos='t' width='192.1487pt'><p noindent='true'>Then we have the series <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>A</mi> <mn>1</mn> </msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn> </msub><mo>,</mo><mo>&ctdot;</mo></mrow></math><texmath>A_1,A_2,\cdots </texmath></formula>,
the regional sum <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>A</mi> <mn>1</mn> </msub><mo>+</mo><msub><mi>A</mi> <mn>2</mn> </msub><mo>+</mo><mo>&ctdot;</mo></mrow></math><texmath>A_1+A_2+\cdots </texmath></formula>,
the orthogonal product <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>A</mi> <mn>1</mn> </msub><msub><mi>A</mi> <mn>2</mn> </msub><mo>&ctdot;</mo></mrow></math><texmath>A_1A_2\cdots </texmath></formula>,
and the infinite integral</p>
<formula type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mo>&int;</mo> <msub><mi>A</mi> <mn>1</mn> </msub> </msub><msub><mo>&int;</mo> <msub><mi>A</mi> <mn>2</mn> </msub> </msub><mo>&ctdot;</mo><mo>.</mo></mrow></math><texmath>\int _{A_1}\int _{A_2}\cdots .</texmath></formula>
<p noindent='true'/></minipage></cell>
</row></table><p rend='center'/>
</div0>
<div0 id-text='4' id='cid19'><head>Nonbreaking dashes</head>
<p>A command <EM r='cn'>nobreakdash</EM> is provided to suppress the possibility
of a linebreak after the following hyphen or dash. For example, if you
write `pages 1&#x2013;9' as <hi rend='tt'>pages1\nobreakdash-<zws/>-<zws/>9</hi> then a linebreak will
never occur between the dash and the 9. You can also use
<EM r='cn'>nobreakdash</EM> to prevent undesirable hyphenations in combinations
like <hi rend='tt'>$p$-<zws/>adic</hi>. For frequent use, it's advisable to make abbreviations,
e.g.,</p>
<p noindent='true'><hi rend='tt'>\newcommand{\p}{$p$\nobreakdash}%for"\p-<zws/>adic"</hi></p>
<p noindent='true'><hi rend='tt'>\newcommand{\Ndash}{\nobreakdash-<zws/>-<zws/>}%for"pages1\Ndash9"</hi></p>
<p noindent='true'><hi rend='tt'>%For"\ndimensional"("n-<zws/>dimensional"):</hi></p>
<p noindent='true'><hi rend='tt'>\newcommand{\n}[1]{$n$\nobreakdash-<zws/>\hspace{0pt}}</hi></p>
<p noindent='true'>The last example shows how to prohibit a linebreak after the hyphen but
allow normal hyphenation in the following word. (It suffices to add a
zero-width space after the hyphen.)</p>
</div0>
<div0 id-text='5' id='cid20'><head>Accents in math</head>
<p>In ordinary <LaTeX/> the placement of the second accent in doubled math
accents is often poor. With the <EM r='pkg'>amsmath</EM> package you
will get improved placement of the second accent:
<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mover accent='true'><mover accent='true'><mi>A</mi> <mo>&Hat;</mo></mover> <mo>&Hat;</mo></mover></math><texmath>\hat{\hat{A}}</texmath></formula> (<EM r='cn'>hat</EM><hi rend='tt'>{\hat{A}}</hi>).</p>
<p>The commands <EM r='cn'>dddot</EM> and <EM r='cn'>ddddot</EM> are available to produce triple
and quadruple dot accents in addition to the <EM r='cn'>dot</EM> and <EM r='cn'>ddot</EM>
accents already available in <LaTeX/>.</p>
<p>To get a superscripted hat or tilde character, load the <EM r='pkg'>amsxtra</EM>
package and use <EM r='cn'>sphat</EM> or <EM r='cn'>sptilde</EM>. Usage is <hi rend='tt'>A\sphat</hi>
(note the absence of the <hi rend='tt'>^</hi> character).</p>
<p>To place an arbitrary symbol in math accent position, or to get under
accents, see the <EM r='pkg'>accents</EM> package by Javier Bezos.</p>
</div0>
<div0 id-text='6' id='cid21'><head>Roots</head>
<p>In ordinary <LaTeX/> the placement of root indices is sometimes not so
good: <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mroot><mi>k</mi> <mi>&beta;</mi></mroot></math><texmath>\@root \beta  \of {k}</texmath></formula> (<hi rend='tt'>\sqrt</hi><hi rend='tt'>[\beta]{k}</hi>). In the
<EM r='pkg'>amsmath</EM> package <EM r='cn'>leftroot</EM> and <EM r='cn'>uproot</EM> allow you to adjust
the position of the root:</p>
<p noindent='true'><hi rend='tt'>\sqrt[\leftroot{-<zws/>2}\uproot{2}\beta]{k}</hi></p>
<p noindent='true'>will move the beta up and to the right:
<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mroot><mi>k</mi> <mi>&beta;</mi></mroot></math><texmath>\@root \beta  \of {k}</texmath></formula>. The negative argument used
with <EM r='cn'>leftroot</EM> moves the <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&beta;</mi></math><texmath>\beta </texmath></formula> to the right. The units are a small
amount that is a useful size for such adjustments.</p>
</div0>
<div0 id-text='7' id='cid22'><head>Boxed formulas</head>
<p>The command <EM r='cn'>boxed</EM> puts a box around its
argument, like <EM r='cn'>fbox</EM> except that the contents are in math mode:</p>
<formula id-text='14' id='uid66' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable frame='solid'><mtr><mtd><mi>&eta;</mi><mo>&le;</mo><mi>C</mi><mo>(</mo><mi>&delta;</mi><mrow><mo>(</mo><mi>&eta;</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>&Lambda;</mi> <mi>M</mi> </msub><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>&delta;</mi><mo>)</mo></mrow><mo>)</mo></mtd></mtr></mtable></math><texmath>
\boxed{\eta \le C(\delta (\eta ) +\Lambda _M(0,\delta ))}
</texmath></formula>
<p noindent='true'><hi rend='tt'>\boxed{\eta\leqC(\delta(\eta)+\Lambda_M(0,\delta))}</hi></p>
</div0>
<div0 id-text='8' id='cid23'><head>Over and under arrows</head>
<p>Basic <LaTeX/> provides <EM r='cn'>overrightarrow</EM> and <EM r='cn'>overleftarrow</EM>
commands. Some additional over and under arrow commands are provided
by the <EM r='pkg'>amsmath</EM> package to extend the set:</p>
<table rend='inline'><row><cell halign='left'><EM r='cn'>overleftarrow</EM></cell>
<cell halign='center'><EM r='cn'>underleftarrow</EM></cell>
</row><row><cell halign='left'><EM r='cn'>overrightarrow</EM></cell>
<cell halign='center'><EM r='cn'>underrightarrow</EM></cell>
</row><row><cell halign='left'><EM r='cn'>overleftrightarrow</EM></cell>
<cell halign='center'><EM r='cn'>underleftrightarrow</EM></cell>
</row></table>
</div0>
<div0 id-text='9' id='cid24'><head>Extensible arrows</head>
<EM r='cn'>xleftarrow</EM><p>and <EM r='cn'>xrightarrow</EM> produce
arrows<anchor id-text='lid22' id='uid67'/> 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):</p>
<formula id-text='15' id='uid68' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>A</mi><mover><mo>&leftarrow;</mo> <mrow><mi>n</mi><mo>+</mo><mi>&mu;</mi><mo>-</mo><mn>1</mn></mrow></mover><mi>B</mi><munderover><mo>&rightarrow;</mo> <mi>T</mi> <mrow><mi>n</mi><mo>&pm;</mo><mi>i</mi><mo>-</mo><mn>1</mn></mrow></munderover><mi>C</mi></mrow></math><texmath>
A\xleftarrow{}B \xrightarrow[T]{n\pm i-1}C
</texmath></formula>
<p noindent='true'><hi rend='tt'>\xleftarrow{n+\mu-<zws/>1}\quad\xrightarrow[T]{n\pmi-<zws/>1}</hi></p>
</div0>
<div0 id-text='10' id='cid25'><head>Affixing symbols to other symbols</head>
<p><LaTeX/> provides <EM r='cn'>stackrel</EM> for placing a
superscript<anchor id-text='lid23' id='uid69'/> above a binary relation.
In the <EM r='pkg'>amsmath</EM> package there are somewhat more general commands,
<EM r='cn'>overset</EM> and <EM r='cn'>underset</EM>, that can be used to place one symbol
above or below another symbol, whether it's a relation or something
else. The input <hi rend='tt'>\overset{*}{X}</hi> will place a superscript-size <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>*</mo></math><texmath>*</texmath></formula> above
the <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>X</mi></math><texmath>X</texmath></formula>: <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mover><mi>X</mi> <mo>*</mo></mover></math><texmath>\overset{*}{X}</texmath></formula>; <EM r='cn'>underset</EM> is the analog for adding a
symbol underneath.</p>
<p>See also the description of <EM r='cn'>sideset</EM> in <ref target='cid36'/>.</p>
</div0>
<div0 id-text='11' id='cid26'><head>Fractions and related constructions</head>
<div1 id-text='1' id='uid70'><head>The <EM r='cn'>frac</EM>, <EM r='cn'>dfrac</EM>, and <EM r='cn'>tfrac</EM> commands</head>
<p>The <EM r='cn'>frac</EM> command, which is in the basic command set of
<LaTeX/>,<anchor id-text='lid24' id='uid71'/> takes two arguments&#x2014;numerator and
denominator&#x2014;and typesets them in normal fraction form. The
<EM r='pkg'>amsmath</EM> package provides also <EM r='cn'>dfrac</EM> and <EM r='cn'>tfrac</EM> as
convenient abbreviations for <hi rend='tt'>{\displaystyle\frac</hi> <hi rend='tt'>...</hi> <hi rend='tt'>}</hi>
and<EM r='indexcs'>textstyle</EM><EM r='indexcs'>displaystyle</EM> <hi rend='tt'>{\textstyle\frac</hi> <hi rend='tt'>...</hi> <hi rend='tt'>}</hi>.</p>
<formula id-text='16' id='uid72' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>1</mn> <mi>k</mi></mfrac><msub><mo form='prefix'>log</mo> <mn>2</mn> </msub><mi>c</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mspace width='1.em'/><mstyle scriptlevel='0' displaystyle='false'><mfrac><mn>1</mn> <mi>k</mi></mfrac></mstyle><msub><mo form='prefix'>log</mo> <mn>2</mn> </msub><mi>c</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mspace width='1.em'/><msqrt><mrow><mfrac><mn>1</mn> <mi>k</mi></mfrac><msub><mo form='prefix'>log</mo> <mn>2</mn> </msub><mi>c</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow></msqrt><mspace width='1.em'/><msqrt><mrow><mstyle scriptlevel='0' displaystyle='true'><mfrac><mn>1</mn> <mi>k</mi></mfrac></mstyle><msub><mo form='prefix'>log</mo> <mn>2</mn> </msub><mi>c</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow></msqrt></mrow></math><texmath>
\frac{1}{k}\log _2 c(f)\quad \tfrac{1}{k}\log _2 c(f)\quad \sqrt{\frac{1}{k}\log _2 c(f)}\quad \sqrt{\dfrac{1}{k}\log _2 c(f)}
</texmath></formula>
<p noindent='true'><hi rend='tt'>\begin{equation}</hi></p>
<p noindent='true'><hi rend='tt'>\frac{1}{k}\log_2c(f)\;\tfrac{1}{k}\log_2c(f)\;</hi></p>
<p noindent='true'><hi rend='tt'>\sqrt{\frac{1}{k}\log_2c(f)}\;\sqrt{\dfrac{1}{k}\log_2c(f)}</hi></p>
<p noindent='true'><hi rend='tt'>\end{equation}</hi></p>
</div1>
<div1 id-text='2' id='uid73'><head>The <EM r='cn'>binom</EM>, <EM r='cn'>dbinom</EM>, and <EM r='cn'>tbinom</EM> commands</head>
<p>For binomial expressions<anchor id-text='lid25' id='uid74'/> such as <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mfenced separators='' open='(' close=')'><mfrac linethickness='0pt'><mi>n</mi> <mi>k</mi></mfrac></mfenced></math><texmath>\binom{n}{k}</texmath></formula>
<EM r='pkg'>amsmath</EM> has <EM r='cn'>binom</EM>, <EM r='cn'>dbinom</EM> and <EM r='cn'>tbinom</EM>:</p>
<formula id-text='17' id='uid75' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mn>2</mn> <mi>k</mi> </msup><mo>-</mo><mfenced separators='' open='(' close=')'><mfrac linethickness='0pt'><mi>k</mi> <mn>1</mn></mfrac></mfenced><msup><mn>2</mn> <mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow> </msup><mo>+</mo><mfenced separators='' open='(' close=')'><mfrac linethickness='0pt'><mi>k</mi> <mn>2</mn></mfrac></mfenced><msup><mn>2</mn> <mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow> </msup></mrow></math><texmath>
2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}
</texmath></formula>
<p noindent='true'><hi rend='tt'>2^k-<zws/>\binom{k}{1}2^{k-<zws/>1}+\binom{k}{2}2^{k-<zws/>2}</hi></p>
</div1>
<div1 id-text='3' id='uid76'><head>The <EM r='cn'>genfrac</EM> command</head>
<p>The capabilities of <EM r='cn'>frac</EM>, <EM r='cn'>binom</EM>, and their variants are
subsumed by a generalized fraction command <EM r='cn'>genfrac</EM> with six
arguments. The last two correspond to <EM r='cn'>frac</EM>'s numerator and
denominator; the first two are optional delimiters (as seen in
<EM r='cn'>binom</EM>); the third is a line thickness override (<EM r='cn'>binom</EM> uses
this to set the fraction line thickness to 0&#x2014;i.e., invisible); and
the fourth argument is a mathstyle override: integer values 0&#x2013;3 select
respectively <EM r='cn'>displaystyle</EM>, <EM r='cn'>textstyle</EM>, <EM r='cn'>scriptstyle</EM>, and
<EM r='cn'>scriptscriptstyle</EM>. If the third argument is left empty, the line
thickness defaults to `normal'.</p>
<p rend='center'><minipage width='362.9526pt'><p rend='flushed-left'><hi rend='tt'>\genfrac{</hi><hi rend='it'>left-delim</hi><hi rend='tt'>}{</hi><hi rend='it'>right-delim</hi><hi rend='tt'>}{</hi><hi rend='it'>thickness</hi><hi rend='tt'>}{</hi><hi rend='it'>mathstyle</hi><hi rend='tt'>}{</hi><hi rend='it'>numerator</hi><hi rend='tt'>}{</hi><hi rend='it'>denominator</hi><hi rend='tt'>}</hi></p></minipage></p>
<p>To illustrate, here is how <EM r='cn'>frac</EM>, <EM r='cn'>tfrac</EM>, and
<EM r='cn'>binom</EM> might be defined.</p>
<p noindent='true'><hi rend='tt'>\newcommand{\frac}[2]{\genfrac{}{}{}{}{#1}{#2}}</hi></p>
<p noindent='true'><hi rend='tt'>\newcommand{\tfrac}[2]{\genfrac{}{}{}{1}{#1}{#2}}</hi></p>
<p noindent='true'><hi rend='tt'>\newcommand{\binom}[2]{\genfrac{(}{)}{0pt}{}{#1}{#2}}</hi></p>
<p noindent='true'>If you find yourself repeatedly using <EM r='cn'>genfrac</EM> throughout a document
for a particular notation, you will do yourself a favor (and your
publisher) if you define a meaningfully-named abbreviation for that
notation, along the lines of <EM r='cn'>frac</EM> and <EM r='cn'>binom</EM>.</p>
<p>The primitive generalized fraction commands <EM r='cs'>over</EM>, <EM r='cs'>overwithdelims</EM>,
<EM r='cs'>atop</EM>, <EM r='cs'>atopwithdelims</EM>, <EM r='cs'>above</EM>, <EM r='cs'>abovewithdelims</EM> produce
warning messages if used with the <EM r='pkg'>amsmath</EM> package, for reasons
discussed in <EM r='fn'>technote.tex</EM>.</p>
</div1></div0>
<div0 id-text='12' id='cid27'><head>Continued fractions</head>
<p>The continued fraction<anchor id-text='lid26' id='uid77'/></p>
<formula id-text='18' id='uid78' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mfrac><mn>1</mn> <mrow><msqrt><mn>2</mn></msqrt><mo>+</mo><mfrac><mn>1</mn> <mrow><msqrt><mn>2</mn></msqrt><mo>+</mo><mfrac><mn>1</mn> <mrow><msqrt><mn>2</mn></msqrt><mo>+</mo><mo>&ctdot;</mo></mrow></mfrac></mrow></mfrac></mrow></mfrac></math><texmath>
\cfrac{1}{\sqrt{2}+
\cfrac{1}{\sqrt{2}+
\cfrac{1}{\sqrt{2}+\cdots }}}
</texmath></formula>
<p noindent='true'>can be obtained by typing
<samepage/></p>
<p noindent='true'><hi rend='tt'>\cfrac{1}{\sqrt{2}+</hi></p>
<p noindent='true'><hi rend='tt'>\cfrac{1}{\sqrt{2}+</hi></p>
<p noindent='true'><hi rend='tt'>\cfrac{1}{\sqrt{2}+\dotsb</hi></p>
<p noindent='true'><hi rend='tt'>}}}</hi></p>
<p noindent='true'>This produces better-looking results than straightforward use of
<EM r='cn'>frac</EM>. Left or right placement of any of the numerators is
accomplished by using <EM r='cn'>cfrac</EM><hi rend='tt'>[l]</hi> or <EM r='cn'>cfrac</EM><hi rend='tt'>[r]</hi> instead of
<EM r='cn'>cfrac</EM>.</p>
</div0>
<div0 id-text='13' id='cid28'><head>Smash options</head>
<p>The command <EM r='cn'>smash</EM> is used to typeset a subformula with an effective height and depth of zero, which is sometimes
useful in adjusting the subformula's position with respect to adjacent
symbols. With the <EM r='pkg'>amsmath</EM> package <EM r='cn'>smash</EM> has optional
arguments <hi rend='tt'>t</hi> and <hi rend='tt'>b</hi>, because occasionally it is advantageous to be
able to &#x201C;smash&#x201D; only the top or only the bottom of something while
retaining the natural depth or height. For example, when adjacent
radical symbols are unevenly sized or positioned because of differences
in the height and depth of their contents, <EM r='cn'>smash</EM> can be employed to
make them more consistent. Compare
<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><msqrt><mi>y</mi></msqrt><mo>+</mo><msqrt><mi>z</mi></msqrt></mrow></math><texmath>\sqrt{x}+\sqrt{y}+\sqrt{z}</texmath></formula> and <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><msqrt><mpadded depth='0pt'><mi>y</mi></mpadded></msqrt><mo>+</mo><msqrt><mi>z</mi></msqrt></mrow></math><texmath>\sqrt{x}+\sqrt{\smash[b]{y}}+\sqrt{z}</texmath></formula>,
where the latter was produced by
<hi rend='tt'>$\sqrt{x}</hi> <hi rend='tt'>+</hi>
<hi rend='tt'>\sqrt{</hi>+<hi rend='tt'>\smash[b]{y}}</hi> <hi rend='tt'>+</hi> <hi rend='tt'>\sqrt{z}$</hi>.</p>
</div0>
<div0 id-text='14' id='cid29'><head>Delimiters</head>
<div1 id-text='1' id='uid79'><head>Delimiter sizes</head>
<p>The automatic delimiter sizing done by <EM r='cn'>left</EM> and <EM r='cn'>right</EM> has two
limitations: First, it is applied mechanically to produce delimiters
large enough to encompass the largest contained item, and second, the
range of sizes is not even approximately continuous but has fairly large
quantum jumps. This means that a math fragment that is infinitesimally
too large for a given delimiter size will get the next larger size, a
jump of 3pt or so in normal-sized text. There are two or three
situations where the delimiter size is commonly adjusted, using a set of
commands that have `big' in their names.</p>
<table rend='inline'><row><cell right-border='true' halign='left'>Delimiter</cell>
<cell halign='left'>text</cell>
<cell halign='left'><EM r='ncn'>left</EM></cell>
<cell halign='left'><EM r='ncn'>bigl</EM></cell>
<cell halign='left'><EM r='ncn'>Bigl</EM></cell>
<cell halign='left'><EM r='ncn'>biggl</EM></cell>
<cell><EM r='ncn'>Biggl</EM></cell>
</row><row bottom-border='true'><cell right-border='true' halign='left'>size</cell>
<cell halign='left'>size</cell>
<cell halign='left'><EM r='ncn'>right</EM></cell>
<cell halign='left'><EM r='ncn'>bigr</EM></cell>
<cell halign='left'><EM r='ncn'>Bigr</EM></cell>
<cell halign='left'><EM r='ncn'>biggr</EM></cell>
<cell><EM r='ncn'>Biggr</EM></cell>
</row><row><cell right-border='true' halign='left'><rule width='0.0pt' height='21.25pt'/>Result</cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><mrow><mrow><mo>(</mo><mi>b</mi><mo>)</mo></mrow><mrow><mo>(</mo><mfrac><mi>c</mi> <mi>d</mi></mfrac><mo>)</mo></mrow></mrow></mstyle></math><texmath>\displaystyle (b)(\frac{c}{d})</texmath></formula></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><mrow><mfenced open='(' close=')'><mi>b</mi></mfenced><mfenced separators='' open='(' close=')'><mfrac><mi>c</mi> <mi>d</mi></mfrac></mfenced></mrow></mstyle></math><texmath>\displaystyle \left(b\right)\left(\frac{c}{d}\right)</texmath></formula></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><mrow><mfenced open='(' close=')'><mi>b</mi></mfenced><mfenced open='(' close=')'><mfrac><mi>c</mi> <mi>d</mi></mfrac></mfenced></mrow></mstyle></math><texmath>\displaystyle \bigl (b\bigr )\bigl (\frac{c}{d}\bigr )</texmath></formula></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><mrow><mfenced open='(' close=')'><mi>b</mi></mfenced><mfenced open='(' close=')'><mfrac><mi>c</mi> <mi>d</mi></mfrac></mfenced></mrow></mstyle></math><texmath>\displaystyle \Bigl (b\Bigr )\Bigl (\frac{c}{d}\Bigr )</texmath></formula></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><mrow><mfenced open='(' close=')'><mi>b</mi></mfenced><mfenced open='(' close=')'><mfrac><mi>c</mi> <mi>d</mi></mfrac></mfenced></mrow></mstyle></math><texmath>\displaystyle \biggl (b\biggr )\biggl (\frac{c}{d}\biggr )</texmath></formula></cell>
<cell><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><mrow><mfenced open='(' close=')'><mi>b</mi></mfenced><mfenced open='(' close=')'><mfrac><mi>c</mi> <mi>d</mi></mfrac></mfenced></mrow></mstyle></math><texmath>\displaystyle \Biggl (b\Biggr )\Biggl (\frac{c}{d}\Biggr )</texmath></formula></cell>
</row></table><p>
The first kind of situation is a cumulative operator with limits above
and below. With <EM r='cn'>left</EM> and <EM r='cn'>right</EM> the delimiters usually turn out
larger than necessary, and using the <hi rend='tt'>Big</hi> or <hi rend='tt'>bigg</hi>
sizes<anchor id-text='lid27' id='uid80'/>
instead gives better results:</p>
<formula textype='equation*' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mfenced separators='' open='[' close=']'><munder><mo>&sum;</mo> <mi>i</mi> </munder><msub><mi>a</mi> <mi>i</mi> </msub><msup><mfenced separators='' open='&vert;' close='&vert;'><munder><mo>&sum;</mo> <mi>j</mi> </munder><msub><mi>x</mi> <mrow><mi>i</mi><mi>j</mi></mrow> </msub></mfenced> <mi>p</mi> </msup></mfenced> <mrow><mn>1</mn><mo>/</mo><mi>p</mi></mrow> </msup><mspace width='1.em'/><mtext>versus</mtext><mspace width='1.em'/><msup><mfenced separators='' open='[' close=']'><munder><mo>&sum;</mo> <mi>i</mi> </munder> <msub><mi>a</mi> <mi>i</mi> </msub> <msup><mfenced separators='' open='&vert;' close='&vert;'><munder><mo>&sum;</mo> <mi>j</mi> </munder> <msub><mi>x</mi> <mrow><mi>i</mi><mi>j</mi></mrow> </msub></mfenced> <mi>p</mi> </msup></mfenced> <mrow><mn>1</mn><mo>/</mo><mi>p</mi></mrow> </msup></mrow></math><texmath>
\left[\sum _i a_i\left|\sum _j x_{ij}\right|^p\right]^{1/p}
\quad \text{versus}\quad \biggl [\sum _i a_i\Bigl \vert \sum _j x_{ij}\Bigr \vert ^p\biggr ]^{1/p}
</texmath></formula>
<p noindent='true'><hi rend='tt'>\biggl[\sum_ia_i\Bigl\lvert\sum_jx_{ij}\Bigr\rvert^p\biggr]^{1/p}</hi></p>
<p noindent='true'>The second kind of situation is clustered pairs of delimiters where
<EM r='cn'>left</EM> and <EM r='cn'>right</EM> 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.</p>
<formula textype='equation*' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced separators='' open='(' close=')'><mrow><mo>(</mo><msub><mi>a</mi> <mn>1</mn> </msub><msub><mi>b</mi> <mn>1</mn> </msub><mo>)</mo></mrow><mo>-</mo><mrow><mo>(</mo><msub><mi>a</mi> <mn>2</mn> </msub><msub><mi>b</mi> <mn>2</mn> </msub><mo>)</mo></mrow></mfenced><mfenced separators='' open='(' close=')'><mrow><mo>(</mo><msub><mi>a</mi> <mn>2</mn> </msub><msub><mi>b</mi> <mn>1</mn> </msub><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><msub><mi>a</mi> <mn>1</mn> </msub><msub><mi>b</mi> <mn>2</mn> </msub><mo>)</mo></mrow></mfenced><mspace width='1.em'/><mtext>versus</mtext><mspace width='1.em'/><mfenced separators='' open='(' close=')'><mrow><mo>(</mo><msub><mi>a</mi> <mn>1</mn> </msub><msub><mi>b</mi> <mn>1</mn> </msub><mo>)</mo></mrow> <mo>-</mo> <mrow><mo>(</mo><msub><mi>a</mi> <mn>2</mn> </msub><msub><mi>b</mi> <mn>2</mn> </msub><mo>)</mo></mrow></mfenced><mfenced separators='' open='(' close=')'><mrow><mo>(</mo><msub><mi>a</mi> <mn>2</mn> </msub><msub><mi>b</mi> <mn>1</mn> </msub><mo>)</mo></mrow> <mo>+</mo> <mrow><mo>(</mo><msub><mi>a</mi> <mn>1</mn> </msub><msub><mi>b</mi> <mn>2</mn> </msub><mo>)</mo></mrow></mfenced></mrow></math><texmath>
\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 )
</texmath></formula>
<p noindent='true'><hi rend='tt'>\left((a_1b_1)-<zws/>(a_2b_2)\right)</hi></p>
<p noindent='true'><hi rend='tt'>\left((a_2b_1)+(a_1b_2)\right)</hi></p>
<p noindent='true'><hi rend='tt'>\quad\text{versus}\quad</hi></p>
<p noindent='true'><hi rend='tt'>\bigl((a_1b_1)-<zws/>(a_2b_2)\bigr)</hi></p>
<p noindent='true'><hi rend='tt'>\bigl((a_2b_1)+(a_1b_2)\bigr)</hi></p>
<p noindent='true'>The third kind of situation is a slightly oversize object in running
text, such as <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mfenced separators='' open='&vert;' close='&vert;'><mfrac><msup><mi>b</mi> <mo>&apos;</mo> </msup> <msup><mi>d</mi> <mo>&apos;</mo> </msup></mfrac></mfenced></math><texmath>\left|\frac{b^{\prime }}{d^{\prime }}\right|</texmath></formula> where the
delimiters produced by <EM r='cn'>left</EM> and <EM r='cn'>right</EM> cause too much line
spreading. In that case <EM r='ncn'>bigl</EM> and <EM r='ncn'>bigr</EM><anchor id-text='lid28' id='uid81'/> can be used to produce
delimiters that are slightly larger than the base size but still able to
fit within the normal line spacing:
<formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mfenced open='&vert;' close='&vert;'><mfrac><msup><mi>b</mi> <mo>&apos;</mo> </msup> <msup><mi>d</mi> <mo>&apos;</mo> </msup></mfrac></mfenced></math><texmath>\bigl \vert \frac{b^{\prime }}{d^{\prime }}\bigr \vert </texmath></formula>.</p>
<p>In ordinary <LaTeX/> <EM r='ncn'>big</EM>, <EM r='ncn'>bigg</EM>, <EM r='ncn'>Big</EM>, and <EM r='ncn'>Bigg</EM>
delimiters aren't scaled properly over the full range of <LaTeX/> font
sizes. With the <EM r='pkg'>amsmath</EM> package they are.</p>
</div1>
<div1 id-text='2' id='uid82'><head>Vertical bar notations</head>
<p>The <EM r='pkg'>amsmath</EM> package provides commands <EM r='cn'>lvert</EM>, <EM r='cn'>rvert</EM>,
<EM r='cn'>lVert</EM>, <EM r='cn'>rVert</EM> (compare <EM r='cn'>langle</EM>, <EM r='cn'>rangle</EM>) to address the
problem of overloading for the vert bar character <EM r='cn'>|</EM>. This
character is currently used in <LaTeX/> documents to represent a wide
variety of mathematical objects: the `divides' relation in a
number-theory expression like <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>p</mi><mo>&vert;</mo><mi>q</mi></mrow></math><texmath>p\vert q</texmath></formula>, or the absolute-value
operation <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo>&vert;</mo><mi>z</mi><mo>&vert;</mo></mrow></math><texmath>\vert z\vert </texmath></formula>, or the `such that' condition in set
notation, or the `evaluated at' notation <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>f</mi> <mi>&zeta;</mi> </msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msub><mo>&vert;</mo> <mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow> </msub></mrow></math><texmath>f_\zeta (t)\bigr \vert _{t=0}</texmath></formula>.
The multiplicity of uses in itself is not so bad; what is bad, however,
is that fact that not all of the uses take the same typographical
treatment, and that the complex discriminatory powers of a knowledgeable
reader cannot be replicated in computer processing of mathematical
documents. It is recommended therefore that there should be a one-to-one
correspondence in any given document between the vert bar character
<EM r='cn'>|</EM> and a selected mathematical notation, and similarly for the
double-bar command <EM r='ncn'>|</EM><anchor id-text='lid29' id='uid83'/>. This immediately
rules out the use of <EM r='cn'>|</EM>
and <EM r='ncn'>|</EM><anchor id-text='lid30' id='uid84'/> 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:</p>
<p noindent='true'><hi rend='tt'>\providecommand{\abs}[1]{\lvert#1\rvert}</hi></p>
<p noindent='true'><hi rend='tt'>\providecommand{\norm}[1]{\lVert#1\rVert}</hi></p>
<p noindent='true'>whereupon the document would contain <hi rend='tt'>\abs{z}</hi> to produce <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo>&vert;</mo><mi>z</mi><mo>&vert;</mo></mrow></math><texmath>\vert z\vert </texmath></formula> and <hi rend='tt'>\norm{v}</hi> to produce <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo>&parallel;</mo><mi>v</mi><mo>&parallel;</mo></mrow></math><texmath>\Vert v\Vert </texmath></formula>.</p>
</div1></div0>
<div0 id-text='5' id='cid30'><head>Operator names</head>
</div0>
<div0 id-text='1' id='cid31'><head>Defining new operator names</head>
<p>Math functions<anchor id-text='lid31' id='uid85'/> such as <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>log</mo></math><texmath>\log </texmath></formula>, <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>sin</mo></math><texmath>\sin </texmath></formula>, and <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>lim</mo></math><texmath>\lim </texmath></formula> are
traditionally typeset in roman type to make them visually more distinct
from one-letter math variables, which are set in math italic. The more
common ones have predefined names, <EM r='cn'>log</EM>, <EM r='cn'>sin</EM>, <EM r='cn'>lim</EM>, and so
forth, but new ones come up all the time in mathematical papers, so the
<EM r='pkg'>amsmath</EM> package provides a general mechanism for defining new
`operator names'. To define a math function <EM r='ncn'>xxx</EM> to work like
<EM r='cn'>sin</EM>, you write</p>
<p noindent='true'><hi rend='tt'>\DeclareMathOperator{\xxx}{xxx}</hi></p>
<p noindent='true'>whereupon ensuing uses of <EM r='ncn'>xxx</EM> will produce xxx in the
proper font and automatically add proper spacing<anchor id-text='lid33' id='uid86'/> on either side when necessary, so that you
get <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>A</mi><mo form='prefix'>xxx</mo><mi>B</mi></mrow></math><texmath>A\operatorname{xxx}B</texmath></formula> instead of <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>A</mi><mi> xxx </mi><mi>B</mi></mrow></math><texmath>A\mathrm {xxx}B</texmath></formula>. In the second argument of
<EM r='cn'>DeclareMathOperator</EM> (the name text), a pseudo-text mode prevails:
the hyphen character <EM r='cn'>-</EM> will print as a text hyphen rather than a
minus sign and an asterisk <EM r='cn'>*</EM> will print as a raised text asterisk
instead of a centered math star. (Compare
<hi rend='it'>a</hi>-<hi rend='it'>b</hi>*<hi rend='it'>c</hi> and <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo>-</mo><mi>b</mi><mo>*</mo><mi>c</mi></mrow></math><texmath>a-b*c</texmath></formula>.) But otherwise the name
text is printed in math mode, so that you can use, e.g., subscripts and
superscripts there.</p>
<p>If the new operator should have subscripts and superscripts placed in
`limits' position above and below as with <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>lim</mo></math><texmath>\lim </texmath></formula>, <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>sup</mo></math><texmath>\sup </texmath></formula>, or <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>max</mo></math><texmath>\max </texmath></formula>, use
the <EM r='cn'>*</EM> form of the <EM r='cn'>DeclareMathOperator</EM> command:</p>
<p noindent='true'><hi rend='tt'>\DeclareMathOperator*{\Lim}{Lim}</hi></p>
<p noindent='true'>See also the discussion of subscript placement in
Section<ref target='cid37'/>.</p>
<p>The following operator names are predefined:</p>
<table rend='inline'><row><cell halign='right'><EM r='cn'>arccos</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>arccos</mo></math><texmath>\arccos </texmath></formula></cell>
<cell halign='right'><EM r='cn'>deg</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>deg</mo></math><texmath>\deg </texmath></formula></cell>
<cell halign='right'><EM r='cn'>lg</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>lg</mo></math><texmath>\lg </texmath></formula></cell>
<cell halign='right'><EM r='cn'>projlim</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>proj lim</mo></math><texmath>\projlim </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>arcsin</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>arcsin</mo></math><texmath>\arcsin </texmath></formula></cell>
<cell halign='right'><EM r='cn'>det</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>det</mo></math><texmath>\det </texmath></formula></cell>
<cell halign='right'><EM r='cn'>lim</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>lim</mo></math><texmath>\lim </texmath></formula></cell>
<cell halign='right'><EM r='cn'>sec</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>sec</mo></math><texmath>\sec </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>arctan</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>arctan</mo></math><texmath>\arctan </texmath></formula></cell>
<cell halign='right'><EM r='cn'>dim</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>dim</mo></math><texmath>\dim </texmath></formula></cell>
<cell halign='right'><EM r='cn'>liminf</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>lim inf</mo></math><texmath>\liminf </texmath></formula></cell>
<cell halign='right'><EM r='cn'>sin</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>sin</mo></math><texmath>\sin </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>arg</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>arg</mo></math><texmath>\arg </texmath></formula></cell>
<cell halign='right'><EM r='cn'>exp</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>exp</mo></math><texmath>\exp </texmath></formula></cell>
<cell halign='right'><EM r='cn'>limsup</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>lim sup</mo></math><texmath>\limsup </texmath></formula></cell>
<cell halign='right'><EM r='cn'>sinh</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>sinh</mo></math><texmath>\sinh </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>cos</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>cos</mo></math><texmath>\cos </texmath></formula></cell>
<cell halign='right'><EM r='cn'>gcd</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>gcd</mo></math><texmath>\gcd </texmath></formula></cell>
<cell halign='right'><EM r='cn'>ln</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>ln</mo></math><texmath>\ln </texmath></formula></cell>
<cell halign='right'><EM r='cn'>sup</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>sup</mo></math><texmath>\sup </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>cosh</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>cosh</mo></math><texmath>\cosh </texmath></formula></cell>
<cell halign='right'><EM r='cn'>hom</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>hom</mo></math><texmath>\hom </texmath></formula></cell>
<cell halign='right'><EM r='cn'>log</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>log</mo></math><texmath>\log </texmath></formula></cell>
<cell halign='right'><EM r='cn'>tan</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>tan</mo></math><texmath>\tan </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>cot</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>cot</mo></math><texmath>\cot </texmath></formula></cell>
<cell halign='right'><EM r='cn'>inf</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>inf</mo></math><texmath>\inf </texmath></formula></cell>
<cell halign='right'><EM r='cn'>max</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>max</mo></math><texmath>\max </texmath></formula></cell>
<cell halign='right'><EM r='cn'>tanh</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>tanh</mo></math><texmath>\tanh </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>coth</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>coth</mo></math><texmath>\coth </texmath></formula></cell>
<cell halign='right'><EM r='cn'>injlim</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>inj lim</mo></math><texmath>\injlim </texmath></formula></cell>
<cell halign='right'><EM r='cn'>min</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>min</mo></math><texmath>\min </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>csc</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>csc</mo></math><texmath>\csc </texmath></formula></cell>
<cell halign='right'><EM r='cn'>ker</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>ker</mo></math><texmath>\ker </texmath></formula></cell>
<cell halign='right'><EM r='cn'>Pr</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>Pr</mo></math><texmath>\Pr </texmath></formula></cell>
</row></table>
<table rend='inline'><row><cell halign='right'><EM r='cn'>varlimsup</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><mover><mo movablelimits='false'>lim</mo> <mo>&OverBar;</mo></mover></mstyle></math><texmath>\displaystyle \varlimsup </texmath></formula></cell>
<cell halign='right'><EM r='cn'>varinjlim</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><munder accentunder='true'><mo movablelimits='false'>lim</mo> <mo>&rightarrow;</mo></munder></mstyle></math><texmath>\displaystyle \varinjlim </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>varliminf</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><munder><mo movablelimits='false'>lim</mo> <mo>&UnderBar;</mo></munder></mstyle></math><texmath>\displaystyle \varliminf </texmath></formula></cell>
<cell halign='right'><EM r='cn'>varprojlim</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><munder accentunder='true'><mo movablelimits='false'>lim</mo> <mo>&leftarrow;</mo></munder></mstyle></math><texmath>\displaystyle \varprojlim </texmath></formula></cell>
</row></table>
<p>There is also a command <EM r='cn'>operatorname</EM> such that using</p>
<p noindent='true'><hi rend='tt'>\operatorname{abc}</hi></p>
<p noindent='true'>in a math formula is equivalent to a use of <EM r='ncn'>abc</EM> defined by
<EM r='cn'>DeclareMathOperator</EM>. This may be occasionally useful for
constructing more complex notation or other purposes. (Use the variant
<EM r='cn'>operatorname*</EM> to get limits.)</p>
</div0>
<div0 id-text='2' id='cid32'><head><EM r='cn'>mod</EM> and its relatives</head>
<p>Commands <EM r='cn'>mod</EM>, <EM r='cn'>bmod</EM>, <EM r='cn'>pmod</EM>, <EM r='cn'>pod</EM> are provided to deal
with the special spacing conventions of &#x201C;mod&#x201D; notation. <EM r='cn'>bmod</EM> and
<EM r='cn'>pmod</EM> are available in <LaTeX/>, but with the <EM r='pkg'>amsmath</EM> package
the spacing of <EM r='cn'>pmod</EM> will adjust to a smaller value if it's used in
a non-display-mode formula. <EM r='cn'>mod</EM> and <EM r='cn'>pod</EM> are variants of
<EM r='cn'>pmod</EM> preferred by some authors; <EM r='cn'>mod</EM> omits the parentheses,
whereas <EM r='cn'>pod</EM> omits the &#x201C;mod&#x201D; and retains the parentheses.</p>
<formula id-text='19' id='uid87' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo movablelimits='true' form='prefix'>gcd</mo><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mspace width='0.277778em'/><mo form='prefix'>mod</mo><mspace width='0.277778em'/><mi>n</mi><mo>)</mo><mo>;</mo><mspace width='1.em'/><mi>x</mi><mo>&equiv;</mo><mi>y</mi><mspace width='10.0pt'/><mo>(</mo><mo form='prefix'>mod</mo><mspace width='0.277778em'/><mi>b</mi><mo>)</mo><mo>;</mo><mspace width='1.em'/><mi>x</mi><mo>&equiv;</mo><mi>y</mi><mspace width='3.33333pt'/><mo form='prefix'>mod</mo><mspace width='0.277778em'/><mi>c</mi><mo>;</mo><mspace width='1.em'/><mi>x</mi><mo>&equiv;</mo><mi>y</mi><mspace width='10.0pt'/><mo>(</mo><mi>d</mi><mo>)</mo></mrow></math><texmath>
\gcd (n,m\bmod n);\quad x\equiv y\pmod {b};
\quad x\equiv y\mod {c};\quad x\equiv y\hspace{10.0pt}(d)
</texmath></formula>
<p noindent='true'><hi rend='tt'>\gcd(n,m\bmodn);\quadx\equivy\pmodb;</hi></p>
<p noindent='true'><hi rend='tt'>\quadx\equivy\modc;\quadx\equivy\podd</hi></p>
</div0>
<div0 id-text='6' id='cid33'><head>The <EM r='cn'>text</EM> command</head>
<p>The main use of the command <EM r='cn'>text</EM> is for words or
phrases<anchor id-text='lid34' id='uid88'/> in a display. It is very
similar to the <LaTeX/> command <EM r='cn'>mbox</EM> in its effects, but has a
couple of advantages. If you want a word or phrase of text in a
subscript, you can type <hi rend='tt'>..._{\text{wordorphrase}}</hi>, which is slightly
easier than the <EM r='cn'>mbox</EM> equivalent: <hi rend='tt'>..._{\mbox{\scriptsize</hi> <hi rend='tt'>word</hi>
<hi rend='tt'>or</hi> <hi rend='tt'>phrase}}</hi>. The other advantage is the more descriptive name.</p>
<formula id-text='20' id='uid89' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>f</mi> <mrow><mo>[</mo><msub><mi>x</mi> <mrow><mi>i</mi><mo>-</mo><mn>1</mn></mrow> </msub><mo>,</mo><msub><mi>x</mi> <mi>i</mi> </msub><mo>]</mo></mrow> </msub><mspace width='4.pt'/><mtext>is</mtext><mspace width='4.pt'/><mtext>monotonic,</mtext><mspace width='1.em'/><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>&ctdot;</mo><mo>,</mo><mi>c</mi><mo>+</mo><mn>1</mn></mrow></math><texmath>
f_{[x_{i-1},x_i]} \text{ is monotonic,}
\quad i = 1,\dots ,c+1
</texmath></formula>
<p noindent='true'><hi rend='tt'>f_{[x_{i-<zws/>1},x_i]}\text{ismonotonic,}</hi></p>
<p noindent='true'><hi rend='tt'>\quadi=1,\dots,c+1</hi></p>
</div0>
<div0 id-text='7' id='cid34'><head>Integrals and sums</head>
</div0>
<div0 id-text='1' id='cid35'><head>Multiline subscripts and superscripts</head>
<p>The <EM r='cn'>substack</EM> command can be used to produce a multiline subscript
or superscript:<anchor id-text='lid35' id='uid90'/> for example</p>
<table rend='inline'><row><cell halign='left'><minipage pos='t' width='256.2026pt'><p noindent='true'><hi rend='tt'>\sum_{\substack{</hi></p>
<p noindent='true'><hi rend='tt'>0\lei\lem\\</hi></p>
<p noindent='true'><hi rend='tt'>0&lt;<zws/>j&lt;<zws/>n}}</hi></p>
<p noindent='true'><hi rend='tt'>P(i,j)</hi></p>
<p noindent='true'/></minipage></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><mrow><munder><mo>&sum;</mo> <mtable><mtr><mtd><mrow><mn>0</mn><mo>&le;</mo><mi>i</mi><mo>&le;</mo><mi>m</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn><mo>&lt;</mo><mi>j</mi><mo>&lt;</mo><mi>n</mi></mrow></mtd></mtr></mtable> </munder><mi>P</mi><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></mrow></mstyle></math><texmath>\displaystyle \sum _{\begin{array}{c}0\le i\le m\\ 0&lt;j&lt;n\end{array}} P(i,j)</texmath></formula></cell>
</row></table><p>
A slightly more generalized form is the <EM r='cls'>subarray</EM> environment which
allows you to specify that each line should be left-aligned instead of
centered, as here:</p>
<table rend='inline'><row><cell halign='left'><minipage pos='t' width='256.2026pt'><p noindent='true'><hi rend='tt'>\sum_{\begin{subarray}{l}</hi></p>
<p noindent='true'><hi rend='tt'>i\in\Lambda\\0&lt;<zws/>j&lt;<zws/>n</hi></p>
<p noindent='true'><hi rend='tt'>\end{subarray}}</hi></p>
<p noindent='true'><hi rend='tt'>P(i,j)</hi></p>
<p noindent='true'/></minipage></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><mrow><munder><mo>&sum;</mo> <mstyle scriptlevel='1' displaystyle='false'><mtable><mtr><mtd><mrow><mi>i</mi><mo>&Element;</mo><mi>&Lambda;</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn><mo>&lt;</mo><mi>j</mi><mo>&lt;</mo><mi>n</mi></mrow></mtd></mtr></mtable></mstyle> </munder><mi>P</mi><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></mrow></mstyle></math><texmath>\displaystyle \sum _{{{\scriptstyle \begin{matrix}
i\in \Lambda \\ 0&lt;j&lt;n
\end{matrix}}}}
P(i,j)</texmath></formula></cell>
</row></table>
</div0>
<div0 id-text='2' id='cid36'><head>The <EM r='cn'>sideset</EM> command</head>
<p>There's also a command called <EM r='cn'>sideset</EM>, for a rather special
purpose: putting symbols at the subscript and
superscript<anchor id-text='lid37' id='uid91'/> corners of a
large operator symbol such as <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&sum;</mo></math><texmath>\sum </texmath></formula> or <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo>&prod;</mo></math><texmath>\prod </texmath></formula>. <hi rend='it'>Note: this
command is not designed to be applied to anything other than sum-class symbols.</hi> The prime
example is the case when you want to put a prime on a sum symbol. If
there are no limits above or below the sum, you could just use
<EM r='cn'>nolimits</EM>: here's
<hi rend='tt'>\sum\nolimits'<zws/>E_n</hi> in display mode:</p>
<formula id-text='21' id='uid92' textype='equation' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mo>&sum;</mo> <mo>&apos;</mo> </msup><msub><mi>E</mi> <mi>n</mi> </msub></mrow></math><texmath>
\sum \nolimits ^{\prime } E_n
</texmath></formula>
<p noindent='true'>If, however, you want not only the prime but also something below or
above the sum symbol, it's not so easy&#x2014;indeed, without
<EM r='cn'>sideset</EM>, it would be downright difficult. With <EM r='cn'>sideset</EM>, you
can write</p>
<table rend='inline'><row><cell halign='left'><minipage pos='t' width='256.2026pt'><p noindent='true'><hi rend='tt'>\sideset{}{'<zws/>}</hi></p>
<p noindent='true'><hi rend='tt'>\sum_{n&lt;<zws/>k,\;\text{$n$odd}}nE_n</hi></p>
<p noindent='true'/></minipage></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><mrow><munder><mmultiscripts><mo>&sum;</mo><none/><mo>&apos;</mo></mmultiscripts> <mrow><mi>n</mi><mo>&lt;</mo><mi>k</mi><mo>,</mo><mspace width='0.277778em'/><mi>n</mi><mspace width='4.pt'/><mtext>odd</mtext></mrow> </munder><mi>n</mi><msub><mi>E</mi> <mi>n</mi> </msub></mrow></mstyle></math><texmath>\displaystyle \mathop {\mmlmultiscripts{\sum {\mmlnone }{\prime }}}\limits _{n&lt;k,\;\text{$n$ odd}} nE_n
</texmath></formula></cell>
</row></table><p>
The extra pair of empty braces is explained by the fact that
<EM r='cn'>sideset</EM> 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</p>
<table rend='inline'><row><cell halign='left'><minipage pos='t' width='256.2026pt'><p noindent='true'><hi rend='tt'>\sideset{_*^*}{_*^*}\prod</hi></p>
<p noindent='true'/></minipage></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle scriptlevel='0' displaystyle='true'><mmultiscripts><mo>&prod;</mo><mo>*</mo><mo>*</mo><mprescripts/><mo>*</mo><mo>*</mo></mmultiscripts></mstyle></math><texmath>\displaystyle \mathop {\mmlmultiscripts{\prod {*}{*}\mmlprescripts {*}{*}}}\limits </texmath></formula></cell>
</row></table>
</div0>
<div0 id-text='3' id='cid37'><head>Placement of subscripts and limits</head>
<p>The default positioning for subscripts depends on the
base symbol involved. The default for sum-class symbols is
`displaylimits' positioning: When a sum-class symbol appears
in a displayed formula, subscript and superscript are placed in `limits'
position above and below, but in an inline formula, they are placed to
the side, to avoid unsightly and wasteful spreading of the
surrounding text lines.
The default for integral-class symbols is to have sub- and
superscripts always to the side, even in displayed formulas.
(See the discussion of the <EM r='opt'>intlimits</EM> and related options in
Section<ref target='cid2'/>.)</p>
<p>Operator names such as <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo form='prefix'>sin</mo></math><texmath>\sin </texmath></formula> or <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mo movablelimits='true' form='prefix'>lim</mo></math><texmath>\lim </texmath></formula> may have either `displaylimits'
or `limits' positioning depending on how they were defined. The standard
operator names are defined according to normal mathematical usage.</p>
<p>The commands <EM r='cn'>limits</EM> and <EM r='cn'>nolimits</EM> can be used to override the
normal behavior of a base symbol:</p>
<formula textype='equation*' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mo>&sum;</mo> <mi>X</mi> </msub><mo>,</mo><mspace width='2.em'/><munder><mrow><mspace width='0.277778em'/><mpadded width='-3pt'><mo>&int;</mo></mpadded><mpadded width='-3pt'><mo>&int;</mo></mpadded><mspace width='0.277778em'/></mrow> <mi>A</mi> </munder><mo>,</mo><mspace width='2.em'/><msub><munder><mo movablelimits='false'>lim</mo> <mo>&UnderBar;</mo></munder> <mrow><mi>n</mi><mo>&rightarrow;</mo><mi>&infin;</mi></mrow> </msub></mrow></math><texmath>
\sum \nolimits _X,\qquad \iint \limits _{A},
\qquad \varliminf \nolimits _{n\rightarrow \infty }
</texmath></formula>
<p noindent='true'>To define a command whose subscripts follow the
same `displaylimits' behavior as <EM r='cn'>sum</EM>, put
<EM r='cn'>displaylimits</EM> at the tail end of the definition. When multiple
instances of <EM r='cn'>limits</EM>, <EM r='cn'>nolimits</EM>, or <EM r='cn'>displaylimits</EM> occur
consecutively, the last one takes precedence.</p>
</div0>
<div0 id-text='4' id='cid38'><head>Multiple integral signs</head>
<EM r='cn'>iint</EM><p>, <EM r='cn'>iiint</EM>, and <EM r='cn'>iiiint</EM> give multiple integral
signs<anchor id-text='lid38' id='uid93'/> with the spacing between them nicely
adjusted, in both text and display style. <EM r='cn'>idotsint</EM> is an extension
of the same idea that gives two integral signs with dots between them.</p>
<formula id-text='4' id='uid94' textype='gather' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable displaystyle='true'><mtr><mtd><mrow><munder><mrow><mspace width='0.277778em'/><mpadded width='-3pt'><mo>&int;</mo></mpadded><mpadded width='-3pt'><mo>&int;</mo></mpadded><mspace width='0.277778em'/></mrow> <mi>A</mi> </munder><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mspace width='0.166667em'/><mi>d</mi><mi>x</mi><mspace width='0.166667em'/><mi>d</mi><mi>y</mi><mspace width='2.em'/><munder><mrow><mspace width='0.277778em'/><mpadded width='-3pt'><mo>&int;</mo></mpadded><mpadded width='-3pt'><mo>&int;</mo></mpadded><mpadded width='-3pt'><mo>&int;</mo></mpadded><mspace width='0.277778em'/></mrow> <mi>A</mi> </munder><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mspace width='0.166667em'/><mi>d</mi><mi>x</mi><mspace width='0.166667em'/><mi>d</mi><mi>y</mi><mspace width='0.166667em'/><mi>d</mi><mi>z</mi></mrow></mtd></mtr><mtr><mtd><mrow><munder><mrow><mspace width='0.277778em'/><mpadded width='-3pt'><mo>&int;</mo></mpadded><mpadded width='-3pt'><mo>&int;</mo></mpadded><mpadded width='-3pt'><mo>&int;</mo></mpadded><mpadded width='-3pt'><mo>&int;</mo></mpadded><mspace width='0.277778em'/></mrow> <mi>A</mi> </munder><mi>f</mi><mrow><mo>(</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mspace width='0.166667em'/><mi>d</mi><mi>w</mi><mspace width='0.166667em'/><mi>d</mi><mi>x</mi><mspace width='0.166667em'/><mi>d</mi><mi>y</mi><mspace width='0.166667em'/><mi>d</mi><mi>z</mi><mspace width='2.em'/><munder><mrow><mo>&int;</mo><mo>...</mo><mo>&int;</mo></mrow> <mi>A</mi> </munder><mi>f</mi><mrow><mo>(</mo><msub><mi>x</mi> <mn>1</mn> </msub><mo>,</mo><mo>&ctdot;</mo><mo>,</mo><msub><mi>x</mi> <mi>k</mi> </msub><mo>)</mo></mrow></mrow></mtd></mtr></mtable></math><texmath>
\iint \limits _A f(x,y)\,dx\,dy\qquad \iiint \limits _A
f(x,y,z)\,dx\,dy\,dz\\
\iiiint \limits _A
f(w,x,y,z)\,dw\,dx\,dy\,dz\qquad \idotsint \limits _A f(x_1,\dots ,x_k)
</texmath></formula>
</div0>
<div0 id-text='8' id='cid39'><head>Commutative diagrams</head>
<p>Some commutative diagram commands like the ones in <amstex/> are
available as a separate package, <EM r='pkg'>amscd</EM>. For complex commutative
diagrams authors will need to turn to more comprehensive packages like
<EM r='pkg'>kuvio</EM> or <xypic/>, but for simple diagrams without diagonal
arrows<anchor id-text='lid39' id='uid95'/> the <EM r='pkg'>amscd</EM> commands
may be more convenient. Here is one example.</p>
<formula textype='equation*' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mtable><mtr><mtd><mrow><msup><mi>S</mi> <msub><mi>&Wscr;</mi> <mi>&Lambda;</mi> </msub> </msup><mo>&otimes;</mo><mi>T</mi></mrow></mtd><mtd><mover><mo>&rightarrow;</mo> <mi>j</mi></mover></mtd><mtd><mi>T</mi></mtd></mtr><mtr><mtd><mo>&downarrow;</mo></mtd><mtd/><mtd><mrow><mphantom><mstyle scriptlevel='1' displaystyle='false'><mo form='prefix'>End</mo><mi>P</mi></mstyle></mphantom><mo>&downarrow;</mo><mstyle scriptlevel='1' displaystyle='false'><mrow><mo form='prefix'>End</mo><mi>P</mi></mrow></mstyle></mrow></mtd><mtd/><mtd/></mtr><mtr><mtd><mrow><mo>(</mo><mi>S</mi><mo>&otimes;</mo><mi>T</mi><mo>)</mo><mo>/</mo><mi>I</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mrow><mo>(</mo><mi>Z</mi><mo>&otimes;</mo><mi>T</mi><mo>)</mo><mo>/</mo><mi>J</mi></mrow></mtd></mtr></mtable></math><texmath>
{\begin{matrix}
S^{{\mathcal {W}}_\Lambda }\otimes T &amp;\xrightarrow{}&amp; T\\

\downarrow 
&amp;&amp; 
\mathbox{mphantom}{\scriptstyle \operatorname{End}P}\downarrow {\scriptstyle \operatorname{End}P}
&amp;&amp;\\
(S\otimes T)/I &amp;=&amp; (Z\otimes T)/J
\end{matrix}}
</texmath></formula>
<p noindent='true'><hi rend='tt'>\begin{CD}</hi></p>
<p noindent='true'><hi rend='tt'>S^{{\mathcal{W}}_\Lambda}\otimesT@&gt;<zws/>j&gt;<zws/>&gt;<zws/>T\\</hi></p>
<p noindent='true'><hi rend='tt'>@VVV@VV{\EndP}V\\</hi></p>
<p noindent='true'><hi rend='tt'>(S\otimesT)/I@=(Z\otimesT)/J</hi></p>
<p noindent='true'><hi rend='tt'>\end{CD}</hi></p>
<p noindent='true'>In the <EM r='cls'>CD</EM> environment the commands <hi rend='tt'>@&gt;<zws/>&gt;<zws/>&gt;<zws/></hi>,
<hi rend='tt'>@&lt;<zws/>&lt;<zws/>&lt;<zws/></hi>, <hi rend='tt'>@VVV</hi>, and <hi rend='tt'>@AAA</hi> give respectively right, left, down, and up
arrows. For the horizontal arrows, material between the first and second
<hi rend='tt'>&gt;<zws/></hi> or <hi rend='tt'>&lt;<zws/></hi> 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 <hi rend='tt'>A</hi>s or <hi rend='tt'>V</hi>s
of vertical arrows will be typeset as left or right &#x201C;sidescripts&#x201D;.
The commands <hi rend='tt'>@=</hi> and <hi rend='tt'>@|</hi> give horizontal and vertical double lines.
A &#x201C;null arrow&#x201D; command <hi rend='tt'>@.</hi> can be used instead of a visible arrow
to fill out an array where needed.</p>
</div0>
<div0 id-text='9' id='cid40'><head>Using math fonts</head>
</div0>
<div0 id-text='1' id='cid41'><head>Introduction</head>
<p>For more comprehensive information on font use in <LaTeX/>, see the
<LaTeX/> font guide (<EM r='fn'>fntguide.tex</EM>) or <hi rend='it'>The <LaTeX/>
Companion</hi> <cit><ref target='bid4'/></cit>. The basic set of math font commands<anchor id-text='lid40' id='uid96'/> in <LaTeX/> includes
<EM r='cn'>mathbf</EM>, <EM r='cn'>mathrm</EM>, <EM r='cn'>mathcal</EM>, <EM r='cn'>mathsf</EM>, <EM r='cn'>mathtt</EM>,
<EM r='cn'>mathit</EM>. Additional math alphabet commands such as
<EM r='cn'>mathbb</EM> for blackboard bold, <EM r='cn'>mathfrak</EM> for Fraktur, and
<EM r='cn'>mathscr</EM> for Euler script are available through the packages
<EM r='pkg'>amsfonts</EM> and <EM r='pkg'>euscript</EM> (distributed separately).</p>
</div0>
<div0 id-text='2' id='cid42'><head>Recommended use of math font commands</head>
<p>If you find yourself employing math font commands frequently in your
document, you might wish that they had shorter names, such as <EM r='ncn'>mb</EM>
instead of <EM r='cn'>mathbf</EM>. Of course, there is nothing to keep you from
providing such abbreviations for yourself by suitable <EM r='cn'>newcommand</EM>
statements. But for <LaTeX/> to provide shorter names would actually be a
disservice to authors, as that would obscure a much better alternative:
defining custom command names derived from the names of the underlying
mathematical objects, rather than from the names of the fonts used to
distinguish the objects. For example, if you are using bold to indicate
vectors, then you will be better served in the long run if you define a
`vector' command instead of a `math-bold' command:</p>
<p noindent='true'><hi rend='tt'>\newcommand{\vect}[1]{\mathbf{#1}}</hi></p>
<p noindent='true'>you can write <hi rend='tt'>\vect{a}+\vect{b}</hi> to produce <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&#x1D41A;</mi><mo>+</mo><mi>&#x1D41B;</mi></mrow></math><texmath>\mathbf {a} +
\mathbf {b}</texmath></formula>.
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 <EM r='ncn'>vect</EM>; otherwise you would have to replace all
occurrences of <EM r='cn'>mathbf</EM> throughout your document, perhaps even
needing to inspect each one to see whether it is indeed
an instance of a vector.</p>
<p>It can also be useful to assign distinct
command names for different letters of a particular font:</p>
<p noindent='true'><hi rend='tt'>\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}%oruseamsfontspackage</hi></p>
<p noindent='true'><hi rend='tt'>\DeclareMathSymbol{\C}{\mathalpha}{AMSb}{"43}</hi></p>
<p noindent='true'><hi rend='tt'>\DeclareMathSymbol{\R}{\mathalpha}{AMSb}{"52}</hi></p>
<p noindent='true'>These statements would define the commands <EM r='cn'>C</EM> and <EM r='cn'>R</EM> to produce
blackboard-bold 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
<EM r='ncn'>field</EM> command and writing<hi rend='tt'>\field{C}</hi>, <hi rend='tt'>\field{R}</hi>. But for
maximum flexibility and control, define such a <EM r='ncn'>field</EM> command and
then define <EM r='ncn'>C</EM> and <EM r='ncn'>R</EM> in terms of that
command:<EM r='indexcs'>mathbb</EM></p>
<p noindent='true'><hi rend='tt'>\usepackage{amsfonts}%togetthe\mathbbalphabet</hi></p>
<p noindent='true'><hi rend='tt'>\newcommand{\field}[1]{\mathbb{#1}}</hi></p>
<p noindent='true'><hi rend='tt'>\newcommand{\C}{\field{C}}</hi></p>
<p noindent='true'><hi rend='tt'>\newcommand{\R}{\field{R}}</hi></p>
</div0>
<div0 id-text='3' id='cid43'><head>Bold math symbols</head>
<p>The <EM r='cn'>mathbf</EM> 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</p>
<p noindent='true'><hi rend='tt'>\Delta\mathbf{\Delta}\mathbf{+}\delta\mathbf{\delta}</hi></p>
<p noindent='true'>produces <formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&Delta;</mi><mi>&Delta;</mi><mo>+</mo><mi>&delta;</mi><mi>&delta;</mi></mrow></math><texmath>\Delta \mathbf {\Delta }\mathbf {+}\delta \mathbf {\delta }</texmath></formula>; the <EM r='cn'>mathbf</EM> has no effect on the plus
sign or the small delta.</p>
<p>The <EM r='pkg'>amsmath</EM> package therefore provides two additional commands,
<EM r='cn'>boldsymbol</EM> and <EM r='cn'>pmb</EM>, that can be applied to other kinds of math
symbols. <EM r='cn'>boldsymbol</EM> can be used for a math symbol that remains
unaffected by <EM r='cn'>mathbf</EM> if (and only if) your current math font set
includes a bold version of that symbol. <EM r='cn'>pmb</EM> can be used as a last
resort for any math symbols that do not have a true bold version
provided by your set of math fonts; &#x201C;pmb&#x201D; stands for &#x201C;poor man's
bold&#x201D; and the command works by typesetting multiple copies of the symbol
with slight offsets. The quality of the output is inferior, especially
for symbols that contain any hairline strokes. When the standard default set of
<LaTeX/> math fonts are in use (Computer Modern), the only symbols that
are likely to require <EM r='cn'>pmb</EM> are large operator symbols like <EM r='cn'>sum</EM>,
extended delimiter symbols, or the extra math symbols provided by
the <EM r='pkg'>amssymb</EM> package <cit><ref target='bid1'/></cit>.</p>
<p>The following formula shows some of the results that are possible:</p>
<p noindent='true'><hi rend='tt'>A_\infty+\piA_0</hi></p>
<p noindent='true'><hi rend='tt'>\sim\mathbf{A}_{\boldsymbol{\infty}}\boldsymbol{+}</hi></p>
<p noindent='true'><hi rend='tt'>\boldsymbol{\pi}\mathbf{A}_{\boldsymbol{0}}</hi></p>
<p noindent='true'><hi rend='tt'>\sim\pmb{A}_{\pmb{\infty}}\pmb{+}\pmb{\pi}\pmb{A}_{\pmb{0}}</hi></p>
<formula textype='equation*' type='display'><math mode='display' xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>A</mi> <mi>&infin;</mi> </msub><mo>+</mo><mi>&pi;</mi><msub><mi>A</mi> <mn>0</mn> </msub><mo>&sim;</mo><msub><mi>&#x1D400;</mi> <mi>&infin;</mi> </msub><mo>+</mo><mi>&pi;</mi><msub><mi>&#x1D400;</mi> <mn>0</mn> </msub><mo>&sim;</mo><msub><mi>A</mi> <mi>&infin;</mi> </msub><mo>+</mo><mi>&pi;</mi><msub><mi>A</mi> <mn>0</mn> </msub></mrow></math><texmath>
A_\infty + \pi A_0
\sim \mathbf {A}_{\infty } +
\pi  \mathbf {A}_{0}
\sim A_{\infty } +\pi  A_{0}
</texmath></formula>
<p noindent='true'>If you want to use only the <EM r='cn'>boldsymbol</EM> command without loading the
whole <EM r='pkg'>amsmath</EM> package, the <EM r='pkg'>bm</EM> package is recommended (this
is a standard <LaTeX/> package, not an AMS package; you probably have it
already if you have a 1997 or newer version of <LaTeX/>).</p>
</div0>
<div0 id-text='4' id='cid44'><head>Italic Greek letters</head>
<p>For italic versions of the capital Greek letters, the following commands
are provided:</p>
<table rend='inline'><row><cell halign='right'><EM r='cn'>varGamma</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&Gamma;</mi></math><texmath>\Gamma </texmath></formula></cell>
<cell halign='right'><EM r='cn'>varSigma</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&Sigma;</mi></math><texmath>\Sigma </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>varDelta</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&Delta;</mi></math><texmath>\Delta </texmath></formula></cell>
<cell halign='right'><EM r='cn'>varUpsilon</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&Upsi;</mi></math><texmath>\Upsilon </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>varTheta</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&Theta;</mi></math><texmath>\Theta </texmath></formula></cell>
<cell halign='right'><EM r='cn'>varPhi</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&Phi;</mi></math><texmath>\Phi </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>varLambda</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&Lambda;</mi></math><texmath>\Lambda </texmath></formula></cell>
<cell halign='right'><EM r='cn'>varPsi</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&Psi;</mi></math><texmath>\Psi </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>varXi</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&Xi;</mi></math><texmath>\Xi </texmath></formula></cell>
<cell halign='right'><EM r='cn'>varOmega</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&Omega;</mi></math><texmath>\Omega </texmath></formula></cell>
</row><row><cell halign='right'><EM r='cn'>varPi</EM></cell>
<cell halign='left'><formula type='inline'><math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&Pi;</mi></math><texmath>\Pi </texmath></formula></cell>
</row></table>
</div0>
<div0 id-text='10' id='cid45'><head>Error messages and output problems</head>
</div0>
<div0 id-text='1' id='cid46'><head>General remarks</head>
<p>This is a supplement to Chapter8 of the <LaTeX/> manual <cit><ref target='bid0'/></cit> (first
edition: Chapter6). For the reader's convenience, the set of error
messages discussed here overlaps somewhat with the set in that chapter,
but please be aware that we don't provide exhaustive coverage here.
The error messages are arranged in alphabetical order, disregarding
unimportant text such as <hi rend='tt'>!LaTeXError:</hi> at the beginning, and
nonalphabetical characters such as <hi rend='tt'>\\</hi>. Where examples are given, we
show also the help messages that appear on screen when you respond to an
error message prompt by entering <hi rend='tt'>h</hi>.</p>
<p>There is also a section discussing some output errors, i.e., instances
where the printed document has something wrong but there was no <LaTeX/>
error during typesetting.</p>
</div0>
<div0 id-text='2' id='cid47'><head>Error messages</head>
<error><msg>\beginsplit won't work here.</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>!PackageamsmathError:\begin{split}won'<zws/>tworkhere.</hi></p>
<p noindent='true'><hi rend='tt'>...</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>l.8\begin{split}</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>?h</hi></p>
<p noindent='true'><hi rend='tt'>\Didyouforgetapreceding\begin{equation}?</hi></p>
<p noindent='true'><hi rend='tt'>Ifnot,perhapsthe`<zws/>aligned'<zws/>environmentiswhatyouwant.</hi></p>
<p noindent='true'><hi rend='tt'>?</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: The <EM r='cls'>split</EM> environment does not construct a stand-alone displayed
equation; it needs to be used within some other environment such as
<EM r='cls'>equation</EM> or <EM r='cls'>gather</EM>.</p>
</error><error><msg>Extra &amp; on this line</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>!PackageamsmathError:Extra&amp;onthisline.</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>Seetheamsmathpackagedocumentationforexplanation.</hi></p>
<p noindent='true'><hi rend='tt'>TypeH&lt;<zws/>return&gt;<zws/>forimmediatehelp.</hi></p>
<p noindent='true'><hi rend='tt'>...</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>l.9\end{alignat}</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>?h</hi></p>
<p noindent='true'><hi rend='tt'>\Anextra&amp;hereissodisastrousthatyoushouldprobablyexit</hi></p>
<p noindent='true'><hi rend='tt'>andfixthingsup.</hi></p>
<p noindent='true'><hi rend='tt'>?</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: In an <EM r='cls'>alignat</EM> structure the number of alignment points per line
is dictated by the numeric argument given after <hi rend='tt'>\begin{alignat}</hi>.
If you use more alignment points in a line it is assumed that you
accidentally left out a newline command <hi rend='tt'>\\</hi> and the above error is
issued.</p>
</error><error><msg>Improper argument for math accent</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>!PackageamsmathError:Improperargumentformathaccent:</hi></p>
<p noindent='true'><hi rend='tt'>(amsmath)Extrabracesmustbeaddedto</hi></p>
<p noindent='true'><hi rend='tt'>(amsmath)preventwrongoutput.</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>Seetheamsmathpackagedocumentationforexplanation.</hi></p>
<p noindent='true'><hi rend='tt'>TypeH&lt;<zws/>return&gt;<zws/>forimmediatehelp.</hi></p>
<p noindent='true'><hi rend='tt'>...</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>l.415\tildek_{\lambda_j}=P_{\tilde\mathcal</hi></p>
<p noindent='true'><hi rend='tt'>{M}}</hi></p>
<p noindent='true'><hi rend='tt'>?</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: Non-simple arguments for any <LaTeX/> command should be enclosed in
braces. In this example extra braces are needed as follows:</p>
<p noindent='true'><hi rend='tt'>...P_{\tilde{\mathcal{M}}}</hi></p>
</error><error><msg>Font OMX/cmex/m/n/7=cmex7 not loadable ...</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>!FontOMX/cmex/m/n/7=cmex7notloadable:Metric(TFM)filenotfound.</hi></p>
<p noindent='true'><hi rend='tt'>&lt;<zws/>tobereadagain&gt;<zws/></hi></p>
<p noindent='true'><hi rend='tt'>relax</hi></p>
<p noindent='true'><hi rend='tt'>l.8$a</hi></p>
<p noindent='true'><hi rend='tt'>b+b^2$</hi></p>
<p noindent='true'><hi rend='tt'>?h</hi></p>
<p noindent='true'><hi rend='tt'>Iwasn'<zws/>tabletoreadthesizedataforthisfont,</hi></p>
<p noindent='true'><hi rend='tt'>soIwillignorethefontspecification.</hi></p>
<p noindent='true'><hi rend='tt'>[WizardscanfixTFMfilesusingTFtoPL/PLtoTF.]</hi></p>
<p noindent='true'><hi rend='tt'>Youmighttryinsertingadifferentfontspec;</hi></p>
<p noindent='true'><hi rend='tt'>e.g.,type`<zws/>I\font&lt;<zws/>samefontid&gt;<zws/>=&lt;<zws/>substitutefontname&gt;<zws/>'<zws/>.</hi></p>
<p noindent='true'><hi rend='tt'>?</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: Certain extra sizes of some Computer Modern fonts that were formerly
available mainly through the AMSFonts<anchor id-text='lid42' id='uid97'/>
distribution are considered part of standard <LaTeX/> (as of June 1994):
<EM r='fn'>cmex7</EM>&#x2013;<hi rend='tt'>9</hi>, <EM r='fn'>cmmib5</EM>&#x2013;<hi rend='tt'>9</hi>, and
<EM r='fn'>cmbsy5</EM>&#x2013;<hi rend='tt'>9</hi>. If these extra sizes are missing on your
system, you should try first to get them from the source where you
obtained <LaTeX/>. If that fails, you could try getting the fonts from
CTAN (e.g., in the form of Metafont<anchor id-text='lid43' id='uid98'/> source
files, directory <hi rend='tt'>/tex-archive/fonts/latex/mf</hi>, or in PostScript
Type 1 format, directory
<hi rend='tt'>/tex-archive/fonts/cm/ps-type1/bakoma</hi><anchor id-text='lid44' id='uid99'/>).</p>
<p>If the font name begins with <EM r='fn'>cmex</EM>, there is a special option
<EM r='fn'>cmex10</EM> for the <EM r='pkg'>amsmath</EM> package that provides a temporary
workaround. I.e., change the <EM r='cn'>usepackage</EM> to</p>
<p noindent='true'><hi rend='tt'>\usepackage[cmex10]{amsmath}</hi></p>
<p noindent='true'>This will force the use of the 10-point size of the <EM r='fn'>cmex</EM> font in
all cases. Depending on the contents of your document this may be
adequate.</p>
</error><error><msg>Math formula deleted: Insufficient extension fonts</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>!Mathformuladeleted:Insufficientextensionfonts.</hi></p>
<p noindent='true'><hi rend='tt'>l.8$ab+b^2$</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>?</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: This usually follows a previous error <hi rend='tt'>Font...notloadable</hi>; see the
discussion of that error (above) for solutions.</p>
</error><error><msg>Missing number, treated as zero</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>!Missingnumber,treatedaszero.</hi></p>
<p noindent='true'><hi rend='tt'>&lt;<zws/>tobereadagain&gt;<zws/></hi></p>
<p noindent='true'><hi rend='tt'>a</hi></p>
<p noindent='true'><hi rend='tt'>l.100\end{alignat}</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>?h</hi></p>
<p noindent='true'><hi rend='tt'>Anumbershouldhavebeenhere;Iinserted`<zws/>0'<zws/>.</hi></p>
<p noindent='true'><hi rend='tt'>(Ifyoucan'<zws/>tfigureoutwhyIneededtoseeanumber,</hi></p>
<p noindent='true'><hi rend='tt'>lookup`<zws/>weirderror'<zws/>intheindextoTheTeXbook.)</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>?</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: There are many possibilities that can lead to this error. However, one
possibility that is relevant for the <EM r='pkg'>amsmath</EM> package is that you
forgot to give the number argument of an <EM r='cls'>alignat</EM> environment, as
in:</p>
<p noindent='true'><hi rend='tt'>\begin{alignat}</hi></p>
<p noindent='true'><hi rend='tt'>a&amp;=b&amp;c&amp;=d\\</hi></p>
<p noindent='true'><hi rend='tt'>a'<zws/>&amp;=b'<zws/>&amp;c'<zws/>&amp;=d'<zws/></hi></p>
<p noindent='true'><hi rend='tt'>\end{alignat}</hi></p>
<p noindent='true'>where the first line should read instead</p>
<p noindent='true'><hi rend='tt'>\begin{alignat}{2}</hi></p>
<p>Another possibility is that you have a left bracket character <hi rend='tt'>[</hi>
following a linebreak command <hi rend='tt'>\\</hi> in a multiline construction such
as <EM r='cls'>array</EM>, <EM r='cls'>tabular</EM>, or <EM r='cls'>eqnarray</EM>. This will be
interpreted by <LaTeX/> as the beginning of an `additional vertical
space' request <cit><ref target='bid0'>C.1.6</ref></cit>, even if it occurs on the following
line and is intended to be part of the contents. For example</p>
<p noindent='true'><hi rend='tt'>\begin{array}</hi></p>
<p noindent='true'><hi rend='tt'>a+b\\</hi></p>
<p noindent='true'><hi rend='tt'>[f,g]\\</hi></p>
<p noindent='true'><hi rend='tt'>m+n</hi></p>
<p noindent='true'><hi rend='tt'>\end{array}</hi></p>
<p noindent='true'>To prevent the error message in such a case, you can
add braces as discussed in the <LaTeX/> manual <cit><ref target='bid0'>C.1.1</ref></cit>:</p>
<p noindent='true'><hi rend='tt'>\begin{array}</hi></p>
<p noindent='true'><hi rend='tt'>a+b\\</hi></p>
<p noindent='true'><hi rend='tt'>{[f,g]}\\</hi></p>
<p noindent='true'><hi rend='tt'>m+n</hi></p>
<p noindent='true'><hi rend='tt'>\end{array}</hi></p>
</error><error><msg>Missing \right. inserted</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>!Missing\right.inserted.</hi></p>
<p noindent='true'><hi rend='tt'>&lt;<zws/>insertedtext&gt;<zws/></hi></p>
<p noindent='true'><hi rend='tt'>\right.</hi></p>
<p noindent='true'><hi rend='tt'>l.10\end{multline}</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>?h</hi></p>
<p noindent='true'><hi rend='tt'>I'<zws/>veinsertedsomethingthatyoumayhaveforgotten.</hi></p>
<p noindent='true'><hi rend='tt'>(Seethe&lt;<zws/>insertedtext&gt;<zws/>above.)</hi></p>
<p noindent='true'><hi rend='tt'>Withluck,thiswillgetmeunwedged.Butifyou</hi></p>
<p noindent='true'><hi rend='tt'>reallydidn'<zws/>tforgetanything,trytyping`<zws/>2'<zws/>now;then</hi></p>
<p noindent='true'><hi rend='tt'>myinsertionandmycurrentdilemmawillbothdisappear.</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: This error typically arises when you try to insert a linebreak inside a
<EM r='cn'>left</EM>-<EM r='cn'>right</EM> pair of delimiters in a <EM r='cls'>multline</EM> or
<EM r='cls'>split</EM> environment:</p>
<p noindent='true'><hi rend='tt'>\begin{multline}</hi></p>
<p noindent='true'><hi rend='tt'>AAA\left(BBB\\</hi></p>
<p noindent='true'><hi rend='tt'>CCC\right)</hi></p>
<p noindent='true'><hi rend='tt'>\end{multline}</hi></p>
<p noindent='true'>There are two possible solutions: (1)instead of using <EM r='cn'>left</EM> and
<EM r='cn'>right</EM>, use `big' delimiters of fixed size (<EM r='cn'>bigl</EM> <EM r='cn'>bigr</EM>
<EM r='cn'>biggl</EM> <EM r='cn'>biggr</EM> ...; see <ref target='uid79'/>); or (2)use null
delimiters to break up the <EM r='cn'>left</EM>-<EM r='cn'>right</EM> pair into parts for each
line:</p>
<p noindent='true'><hi rend='tt'>AAA\left(BBB\right.\\</hi></p>
<p noindent='true'><hi rend='tt'>\left.CCC\right)</hi></p>
<p noindent='true'>The latter solution may result in mismatched delimiter sizes;
ensuring that they match requires using <EM r='cn'>vphantom</EM> in the line
that has the smaller delimiter (or possibly <EM r='cn'>smash</EM> in the line that
has the larger delimiter). In the argument of <EM r='cn'>vphantom</EM> put a copy
of the tallest element that occurs in the other line, e.g.,</p>
<p noindent='true'><hi rend='tt'>xxx\left(\int_tyyy\right.\\</hi></p>
<p noindent='true'><hi rend='tt'>\left.\vphantom{\int_t}zzz...\right)</hi></p>
</error><error><msg>Paragraph ended before \xxx was complete</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>Runawayargument?</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>!Paragraphendedbefore\multlinewascomplete.</hi></p>
<p noindent='true'><hi rend='tt'>&lt;<zws/>tobereadagain&gt;<zws/></hi></p>
<p noindent='true'><hi rend='tt'>\par</hi></p>
<p noindent='true'><hi rend='tt'>l.100</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>?h</hi></p>
<p noindent='true'><hi rend='tt'>Isuspectyou'<zws/>veforgottena`<zws/>}'<zws/>,causingmetoapplythis</hi></p>
<p noindent='true'><hi rend='tt'>controlsequencetotoomuchtext.Howcanwerecover?</hi></p>
<p noindent='true'><hi rend='tt'>Myplanistoforgetthewholethingandhopeforthebest.</hi></p>
<p noindent='true'><hi rend='tt'>?</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: This might be produced by a misspelling in the <hi rend='tt'>\end{multline}</hi> command,
e.g.,</p>
<p noindent='true'><hi rend='tt'>\begin{multline}</hi></p>
<p noindent='true'><hi rend='tt'>...</hi></p>
<p noindent='true'><hi rend='tt'>\end{multiline}</hi></p>
<p noindent='true'>or by using abbreviations for certain environments, such as <hi rend='tt'>\bal</hi> and
<hi rend='tt'>\eal</hi> for <hi rend='tt'>\begin{align}</hi> and <hi rend='tt'>\end{align}</hi>:</p>
<p noindent='true'><hi rend='tt'>\bal</hi></p>
<p noindent='true'><hi rend='tt'>...</hi></p>
<p noindent='true'><hi rend='tt'>\eal</hi></p>
<p noindent='true'>For technical reasons that kind of abbreviation does not work with
the more complex displayed equation environments of the <EM r='pkg'>amsmath</EM> package
(<EM r='cls'>gather</EM>, <EM r='cls'>align</EM>, <EM r='cls'>split</EM>, etc.; cf. <EM r='fn'>technote.tex</EM>).</p>
</error><error><msg>Runaway argument?</msg><p>See the discussion for the error message
<hi rend='tt'>Paragraph ended before <EM r='ncn'>xxx</EM> was complete</hi>.</p>
</error><error><msg>Unknown option `xxx' for package `yyy'</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>!LaTeXError:Unknownoption`<zws/>intlim'<zws/>forpackage`<zws/>amsmath'<zws/>.</hi></p>
<p noindent='true'><hi rend='tt'>...</hi></p>
<p noindent='true'><hi rend='tt'>?h</hi></p>
<p noindent='true'><hi rend='tt'>Theoption`<zws/>intlim'<zws/>wasnotdeclaredinpackage`<zws/>amsmath'<zws/>,perhapsyou</hi></p>
<p noindent='true'><hi rend='tt'>misspelleditsname.Trytyping&lt;<zws/>return&gt;<zws/>toproceed.</hi></p>
<p noindent='true'><hi rend='tt'>?</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: 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.</p>
</error><error><msg>Old form `\pmatrix' should be \beginpmatrix.</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>!PackageamsmathError:Oldform`<zws/>\pmatrix'<zws/>shouldbe</hi></p>
<p noindent='true'><hi rend='tt'>\begin{pmatrix}.</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>Seetheamsmathpackagedocumentationforexplanation.</hi></p>
<p noindent='true'><hi rend='tt'>TypeH&lt;<zws/>return&gt;<zws/>forimmediatehelp.</hi></p>
<p noindent='true'><hi rend='tt'>...</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>\pmatrix-<zws/>&gt;<zws/>\left(\matrix@check\pmatrix</hi></p>
<p noindent='true'><hi rend='tt'>\env@matrix</hi></p>
<p noindent='true'><hi rend='tt'>l.16\pmatrix</hi></p>
<p noindent='true'><hi rend='tt'>{a&amp;b\crc&amp;d\cr}</hi></p>
<p noindent='true'><hi rend='tt'>?h</hi></p>
<p noindent='true'><hi rend='tt'>`<zws/>\pmatrix{...}'<zws/>isoldPlain-<zws/>TeXsyntaxwhoseuseis</hi></p>
<p noindent='true'><hi rend='tt'>ill-<zws/>advisedinLaTeX.</hi></p>
<p noindent='true'><hi rend='tt'>?</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: When the <EM r='pkg'>amsmath</EM> package is used, the old forms of <EM r='cn'>pmatrix</EM>,
<EM r='cn'>matrix</EM>, and <EM r='cn'>cases</EM> cannot be used any longer because of naming
conflicts. Their syntax did not conform with standard <LaTeX/> syntax
in any case.</p>
</error><error><msg>Erroneous nesting of equation structures</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>!PackageamsmathError:Erroneousnestingofequationstructures;</hi></p>
<p noindent='true'><hi rend='tt'>(amsmath)tryingtorecoverwith`<zws/>aligned'<zws/>.</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>Seetheamsmathpackagedocumentationforexplanation.</hi></p>
<p noindent='true'><hi rend='tt'>TypeH&lt;<zws/>return&gt;<zws/>forimmediatehelp.</hi></p>
<p noindent='true'><hi rend='tt'>...</hi></p>
<p noindent='true'><hi rend='tt'></hi></p>
<p noindent='true'><hi rend='tt'>l.260\end{alignat*}</hi></p>
<p noindent='true'><hi rend='tt'>\end{equation*}</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: The structures <EM r='cls'>align</EM>, <EM r='cls'>alignat</EM>, 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
<EM r='cls'>align</EM> and most of its variants can be used inside the
<EM r='cls'>gather</EM> environment.</p>
</error></div0>
<div0 id-text='3' id='cid48'><head>Warning messages</head>
<error><msg>Foreign command \over [or \atop or \above]</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>PackageamsmathWarning:Foreigncommand\over;\fracor\genfrac</hi></p>
<p noindent='true'><hi rend='tt'>(amsmath)shouldbeusedinstead.</hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: The primitive generalized fraction commands of <TeX/>&#x2014;<EM r='cs'>over</EM>, <EM r='cs'>atop</EM>, <EM r='cs'>above</EM>&#x2014;are deprecated when the
<EM r='pkg'>amsmath</EM> package is used because their syntax is foreign to <LaTeX/>
and <EM r='pkg'>amsmath</EM> provides native <LaTeX/> equivalents. See
<EM r='fn'>technote.tex</EM> for further information.</p>
</error><error><msg>Cannot use `split' here</msg><p noindent='true'><hi rend='it'>Example</hi>:</p>
<p noindent='true'><hi rend='tt'>PackageamsmathWarning:Cannotuse`<zws/>split'<zws/>here;</hi></p>
<p noindent='true'><hi rend='tt'>(amsmath)tryingtorecoverwith`<zws/>aligned'<zws/></hi></p>
<p noindent='true'><hi rend='it'>Explanation</hi>: The <EM r='cls'>split</EM> environment is designed to serve as the entire
body of an equation, or an entire line of an <EM r='cls'>align</EM> or <EM r='cls'>gather</EM>
environment. There cannot be any printed material before or
after it within the same enclosing structure:</p>
<p noindent='true'><hi rend='tt'>\begin{equation}</hi></p>
<p noindent='true'><hi rend='tt'>\left\{%&lt;<zws/>-<zws/>-<zws/>Notallowed</hi></p>
<p noindent='true'><hi rend='tt'>\begin{split}</hi></p>
<p noindent='true'><hi rend='tt'>...</hi></p>
<p noindent='true'><hi rend='tt'>\end{split}</hi></p>
<p noindent='true'><hi rend='tt'>\right.%&lt;<zws/>-<zws/>-<zws/>Notallowed</hi></p>
<p noindent='true'><hi rend='tt'>\end{equation}</hi></p>
</error></div0>
<div0 id-text='4' id='cid49'><head>Wrong output</head>
<div1 id-text='1' id='uid100'><head>Section numbers 0.1, 5.1, 8.1 instead of 1, 2, 3</head>
<p>This most likely means that you have the arguments for <EM r='cn'>numberwithin</EM>
in reverse order:</p>
<p noindent='true'><hi rend='tt'>\numberwithin{section}{equation}</hi></p>
<p noindent='true'>That means `print the section number as <hi rend='it'>equation
number</hi>.<hi rend='it'>section number</hi> and reset to 1 every time an equation
occurs' when what you probably wanted was the inverse</p>
<p noindent='true'><hi rend='tt'>\numberwithin{equation}{section}</hi></p>
</div1>
<div1 id-text='2' id='uid101'><head>The <EM r='cn'>numberwithin</EM> command had no effect on equation
numbers</head>
<p>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
<ref target='uid100'/>.</p>
</div1></div0>
<div0 id-text='11' id='cid50'><head>Additional information</head>
</div0>
<div0 id-text='1' id='cid51'><head>Converting existing documents</head>
<div1 id-text='1' id='uid102'><head>Converting from plain <LaTeX/></head>
<p>A <LaTeX/> document will typically continue to work the same in most
respects if <hi rend='tt'>\usepackage{amsmath}</hi> is added in the document
preamble. By default, however, the <EM r='pkg'>amsmath</EM> package suppresses page
breaks inside multiple-line displayed equation structures such as
<EM r='cls'>eqnarray</EM>, <EM r='cls'>align</EM>, and <EM r='cls'>gather</EM>. To continue allowing page
breaks inside <EM r='cls'>eqnarray</EM> after switching to <EM r='pkg'>amsmath</EM>, you will
need to add the following line in your document preamble:</p>
<p noindent='true'><hi rend='tt'>\allowdisplaybreaks[1]</hi></p>
<p noindent='true'>To ensure normal spacing around relation symbols, you might also want to
change <EM r='cls'>eqnarray</EM> to <EM r='cls'>align</EM>, <EM r='cls'>multline</EM>, or
<EM r='cls'>equation</EM>/<EM r='cls'>split</EM> as appropriate.</p>
<p>Most of the other differences in <EM r='pkg'>amsmath</EM> usage can be considered
optional refinements, e.g., using</p>
<p noindent='true'><hi rend='tt'>\DeclareMathOperator{\Hom}{Hom}</hi></p>
<p noindent='true'>instead of <hi rend='tt'>\newcommand{\Hom}{\mbox{Hom}}</hi>.</p>
</div1>
<div1 id-text='2' id='uid103'><head>Converting from <amslatex/> 1.1</head>
<p>See <EM r='fn'>diffs-m.txt</EM>.</p>
</div1></div0>
<div0 id-text='2' id='cid52'><head>Technical notes</head>
<p>The file <EM r='fn'>technote.tex</EM> contains some remarks on miscellaneous
technical questions that are less likely to be of general interest.</p>
</div0>
<div0 id-text='3' id='cid53'><head>Getting help</head>
<p>Questions or comments regarding <EM r='pkg'>amsmath</EM> and related packages
should be sent to:</p>
<label/>
<item id-text='3' id='uid104'><p rend='center'><minipage width='298.8987pt'><p rend='flushed-left'>American Mathematical Society</p>
<p noindent='true' rend='flushed-left'>Technical Support</p>
<p noindent='true' rend='flushed-left'>Electronic Products and Services</p>
<p noindent='true' rend='flushed-left'>P. O. Box 6248</p>
<p noindent='true' rend='flushed-left'>Providence, RI 02940</p>
<p noindent='true' spacebefore='3.0pt' rend='flushed-left'>Phone: 800-321-4AMS (321-4267) or 401-455-4080</p>
<p noindent='true' rend='flushed-left'>Internet: <hi rend='tt'>tech-support@ams.org</hi>
</p></minipage>
If you are reporting a problem you should include
the following information to make proper investigation possible:</p>
<list type='ordered'>
<item id-text='1' id='uid105' label='(1)'><p noindent='true'>The source file where the problem occurred, preferably reduced
to minimum size by removing any material that can be removed without
affecting the observed problem.</p>
</item>
<item id-text='2' id='uid106' label='(2)'><p noindent='true'>A <LaTeX/> log file showing the error message (if applicable) and
the version numbers of the document class and option files being used.</p>
</item></list>
<div0 id-text='4' id='cid54'><head>Of possible interest</head>
<p>Information about obtaining AMSFonts or other <TeX/>-related
software from the AMS Internet archive <EM r='fn'>e-math.ams.org</EM>
can be obtained by sending a request through electronic mail to:
<hi rend='tt'>webmaster@ams.org</hi>.</p>
<p>Information about obtaining the <EM r='pkg'>amsmath</EM> distribution on diskette
from the AMS is available from:</p>
<label/>
<item id-text='4' id='uid107'><p rend='center'><minipage width='298.8987pt'><p rend='flushed-left'>American Mathematical Society</p>
<p noindent='true' rend='flushed-left'>Customer Services</p>
<p noindent='true' rend='flushed-left'>P. O. Box 6248</p>
<p noindent='true' rend='flushed-left'>Providence, RI 02940</p>
<p noindent='true' spacebefore='3.0pt' rend='flushed-left'>Phone: 800-321-4AMS (321-4267) or 401-455-4000</p>
<p noindent='true' rend='flushed-left'>Internet: <hi rend='tt'>cust-serv@ams.org</hi>
</p></minipage></p>
<p>The <TeX/> Users Group<anchor id-text='lid46' id='uid108'/> is a nonprofit
organization that publishes a journal
(<hi rend='it'>TUGboat</hi><anchor id-text='lid47' id='uid109'/>), holds
meetings, and serves as a clearing-house of general information about
<TeX/> and <TeX/>-related software.</p>
</item><label/>
<item id-text='4' id='uid110'><p rend='center'><minipage width='298.8987pt'><p rend='flushed-left'><TeX/> Users Group</p>
<p noindent='true' rend='flushed-left'>PO Box 2311</p>
<p noindent='true' rend='flushed-left'>Portland, OR 97208-2311</p>
<p noindent='true' rend='flushed-left'>USA</p>
<p noindent='true' spacebefore='3.0pt' rend='flushed-left'>Phone: +1-503-223-9994</p>
<p noindent='true' rend='flushed-left'>Email: <hi rend='tt'>office@tug.org</hi>
</p></minipage>
Membership in the <TeX/> Users Group is a good way to support continued
development of free <TeX/>-related software. There are also many local
<TeX/> user groups in other countries; information about contacting a
local user group can be gotten from the <TeX/> Users Group.</p>
<p>There is a Usenet newsgroup called <EM r='fn'>comp.text.tex</EM> that is a fairly
good source of information about <LaTeX/> and <TeX/> in general. If you
don't know about reading newsgroups, check with your local system
administrator to see if newsgroup service is available at your site.</p>
<Bibliography><p>tocchapterBibliography</p>
<p noindent='true'><bibitem id='bid1'/><hi rend='it'>AMSFonts version </hi>2.2<hi rend='it'>&#x2014;user's guide</hi>,
Amer. Math. Soc., Providence, RI, 1994; distributed
with the AMSFonts package.</p>
<p noindent='true'><bibitem id='bid3'/><hi rend='it'>Instructions for preparation of
papers and monographs&#x2014;<amslatex/></hi>,
Amer. Math. Soc., Providence, RI, 1996, 1999.</p>
<p noindent='true'><bibitem id='bid2'/><hi rend='it'>Using the <EM r='pkg'>amsthm</EM> Package</hi>,
Amer. Math. Soc., Providence, RI, 1999.</p>
<p noindent='true'><bibitem id='bid4'/>Michel Goossens, Frank Mittelbach, and Alexander Samarin,
<hi rend='it'>The <LaTeX/> companion</hi>, Addison-Wesley, Reading, MA, 1994.
[<hi rend='it'>Note: The 1994 edition is not a reliable guide for the
<EM r='pkg'>amsmath</EM> package unless you refer to the errata for Chapter
8&#x2014;file <EM r='fn'>compan.err</EM>, distributed with <LaTeX/>.</hi>]</p>
<p noindent='true' rend='flushed-left'><bibitem id='bid5'/>G. Grtzer,
<hi rend='it'>Math into <LaTeX/>: An Introduction to <LaTeX/> and AMS-<LaTeX/></hi>
<xref url='http://www.ams.org/cgi-bin/bookstore/bookpromo?fn=91&amp;arg1=bookvideo&amp;itmc=MLTEX'>http://<allowbreak/>www.<allowbreak/>ams.<allowbreak/>org/<allowbreak/>cgi-bin/<allowbreak/>bookstore/<allowbreak/>bookpromo?fn=91&amp;arg1=bookvideo&amp;itmc=MLTEX</xref>,
Birkhuser, Boston, 1995.</p>
<p noindent='true'><bibitem id='bid6'/>Donald E. Knuth, <hi rend='it'>The <TeX/>book</hi>,
Addison-Wesley, Reading, MA, 1984.</p>
<p noindent='true'><bibitem id='bid0'/>Leslie Lamport, <hi rend='it'><LaTeX/>: A document preparation
system</hi>, 2nd revised ed., Addison-Wesley, Reading, MA, 1994.</p>
<p noindent='true'><bibitem id='bid7'/>Frank Mittelbach and Rainer Schpf,
<hi rend='it'>The new font family selection&#x2014;user
interface to standard <LaTeX/></hi>, <hi rend='it'>TUGboat</hi> <hi rend='bold'>11</hi>,
no.2 (June 1990), pp.297&#x2013;305.</p>
<p noindent='true'><bibitem id='bid8'/>Michael Spivak, <hi rend='it'>The joy of <TeX/></hi>, 2nd revised ed.,
Amer. Math. Soc., Providence, RI, 1990.</p>
</Bibliography><list type='simple'>
<item id-text='1' id='uid111'><p noindent='true'><hi rend='tt'>\! </hi>, 11</p>
</item>
<item id-text='2' id='uid112'><p noindent='true'><hi rend='tt'>\,</hi>, 11</p>
</item>
<item id-text='3' id='uid113'><p noindent='true'><hi rend='tt'>\:</hi>, 11</p>
</item>
<item id-text='4' id='uid114'><p noindent='true'><hi rend='tt'>\;</hi>, 11</p>
</item>
<item id-text='5' id='uid115'><p noindent='true'><hi rend='tt'>\\</hi>, 3, 5, 6, 8, 9, 24, 26</p>
</item>
<item id-text='6' id='uid116'><p noindent='true'><hi rend='tt'>\\*</hi>, 9</p>
</item>
<item id-text='7' id='uid117'><p noindent='true'><hi rend='tt'>\|</hi>, 16</p>
</item>
<item id-text='8' id='uid118'><p noindent='true'><hi rend='tt'>\above</hi>, 14, 29</p>
</item>
<item id-text='9' id='uid119'><p noindent='true'><hi rend='tt'>\abovewithdelims</hi>, 14</p>
</item>
<item id-text='10' id='uid120'><p noindent='true'><hi rend='tt'>\accentedsymbol</hi>, 2</p>
</item>
<item id-text='11' id='uid121'><p noindent='true'><hi rend='tt'>accents</hi> package, 12</p>
</item>
<item id-text='12' id='uid122'><p noindent='true'><hi rend='tt'>\addtocounter</hi>, 10</p>
</item>
<item id-text='13' id='uid123'><p noindent='true'><hi rend='tt'>\addtolength</hi>, 5</p>
</item>
<item id-text='14' id='uid124'><p noindent='true'><hi rend='tt'>align</hi> environment, 3, 5&#x2013;7, 28&#x2013;30</p>
</item>
<item id-text='15' id='uid125'><p noindent='true'><hi rend='tt'>alignat</hi> environment, 7, 24, 26, 29</p>
</item>
<item id-text='16' id='uid126'><p noindent='true'><hi rend='tt'>aligned</hi> environment, 7, 9</p>
</item>
<item id-text='17' id='uid127'><p noindent='true'><hi rend='tt'>alignedat</hi> environment, 7, 9</p>
</item>
<item id-text='18' id='uid128'><p noindent='true'><hi rend='tt'>\allowdisplaybreaks</hi>, 8, 9</p>
</item>
<item id-text='19' id='uid129'><p noindent='true'><hi rend='tt'>amsart</hi> class, 1</p>
</item>
<item id-text='20' id='uid130'><p noindent='true'><hi rend='tt'>amsbook</hi> class, 1</p>
</item>
<item id-text='21' id='uid131'><p noindent='true'><hi rend='tt'>amsbsy</hi> package, 1, 2</p>
</item>
<item id-text='22' id='uid132'><p noindent='true'><hi rend='tt'>amscd</hi> package, 1, 2, 20</p>
</item>
<item id-text='23' id='uid133'><p noindent='true'><hi rend='tt'>amsfonts</hi> package, 21</p>
</item>
<item id-text='24' id='uid134'><p noindent='true'>AMSFonts collection, 25</p>
</item>
<item id-text='25' id='uid135'><p noindent='true'><hi rend='tt'>amsmath</hi> package, i, 1&#x2013;3, 5, 8&#x2013;18, 22, 23, 25, 26,
28&#x2013;32</p>
</item>
<item id-text='26' id='uid136'><p noindent='true'><hi rend='tt'>amsopn</hi> package, 1, 2</p>
</item>
<item id-text='27' id='uid137'><p noindent='true'><hi rend='tt'>amssymb</hi> package, 22</p>
</item>
<item id-text='28' id='uid138'><p noindent='true'><hi rend='tt'>amstext</hi> package, 1, 2</p>
</item>
<item id-text='29' id='uid139'><p noindent='true'><hi rend='tt'>amsthm</hi> package, 1, 32</p>
</item>
<item id-text='30' id='uid140'><p noindent='true'><hi rend='tt'>amsxtra</hi> package, 1, 2, 12</p>
</item>
<item id-text='31' id='uid141'><p noindent='true'><hi rend='tt'>\arccos</hi>, 18</p>
</item>
<item id-text='32' id='uid142'><p noindent='true'><hi rend='tt'>\arcsin</hi>, 18</p>
</item>
<item id-text='33' id='uid143'><p noindent='true'><hi rend='tt'>\arctan</hi>, 18</p>
</item>
<item id-text='34' id='uid144'><p noindent='true'><hi rend='tt'>\arg</hi>, 18</p>
</item>
<item id-text='35' id='uid145'><p noindent='true'><hi rend='tt'>array</hi> environment, 8, 10, 26</p>
</item>
<item id-text='36' id='uid146'><p noindent='true'>arrows
extensible, 13
in commutative diagrams, 20</p>
</item>
<item id-text='37' id='uid147'><p noindent='true'><hi rend='tt'>\atop</hi>, 14, 29</p>
</item>
<item id-text='38' id='uid148'><p noindent='true'><hi rend='tt'>\atopwithdelims</hi>, 14</p>
</item>
<item id-text='39' id='uid149'><p noindent='true'>BaKoMa fonts, 25</p>
</item>
<item id-text='40' id='uid150'><p noindent='true'><EM r='cn'>big</EM>, <EM r='cn'>Big</EM>, <EM r='cn'>bigg</EM>, ... delimiters, 16</p>
</item>
<item id-text='41' id='uid151'><p noindent='true'><hi rend='tt'>\biggl</hi>, 27</p>
</item>
<item id-text='42' id='uid152'><p noindent='true'><hi rend='tt'>\biggr</hi>, 27</p>
</item>
<item id-text='43' id='uid153'><p noindent='true'><hi rend='tt'>\bigl</hi>, 27</p>
</item>
<item id-text='44' id='uid154'><p noindent='true'><hi rend='tt'>\bigr</hi>, 27</p>
</item>
<item id-text='45' id='uid155'><p noindent='true'><hi rend='tt'>\binom</hi>, 14</p>
</item>
<item id-text='46' id='uid156'><p noindent='true'>binomials, 14</p>
</item>
<item id-text='47' id='uid157'><p noindent='true'><hi rend='tt'>bm</hi> package, 2, 23</p>
</item>
<item id-text='48' id='uid158'><p noindent='true'><hi rend='tt'>Bmatrix</hi> environment, 10</p>
</item>
<item id-text='49' id='uid159'><p noindent='true'><hi rend='tt'>bmatrix</hi> environment, 10</p>
</item>
<item id-text='50' id='uid160'><p noindent='true'><hi rend='tt'>\bmod</hi>, 18</p>
</item>
<item id-text='51' id='uid161'><p noindent='true'><hi rend='tt'>\boldsymbol</hi>, 22, 23</p>
</item>
<item id-text='52' id='uid162'><p noindent='true'><hi rend='tt'>\boxed</hi>, 13</p>
</item>
<item id-text='53' id='uid163'><p noindent='true'><hi rend='tt'>\C</hi>, 22</p>
</item>
<item id-text='54' id='uid164'><p noindent='true'><hi rend='tt'>cases</hi> environment, 8</p>
</item>
<item id-text='55' id='uid165'><p noindent='true'><hi rend='tt'>\cases</hi>, 28</p>
</item>
<item id-text='56' id='uid166'><p noindent='true'><hi rend='tt'>CD</hi> environment, 2, 21</p>
</item>
<item id-text='57' id='uid167'><p noindent='true'><hi rend='tt'>\cdots</hi>, 12</p>
</item>
<item id-text='58' id='uid168'><p noindent='true'><hi rend='tt'>centertags</hi> option, 2</p>
</item>
<item id-text='59' id='uid169'><p noindent='true'><hi rend='tt'>\cfrac</hi>, 15</p>
</item>
<item id-text='60' id='uid170'><p noindent='true'><hi rend='tt'>cmbsy5</hi>, 25</p>
</item>
<item id-text='61' id='uid171'><p noindent='true'><hi rend='tt'>cmex</hi>, 25</p>
</item>
<item id-text='62' id='uid172'><p noindent='true'><hi rend='tt'>cmex10</hi>, 25</p>
</item>
<item id-text='63' id='uid173'><p noindent='true'><hi rend='tt'>cmex7</hi>, 25</p>
</item>
<item id-text='64' id='uid174'><p noindent='true'><hi rend='tt'>cmmib5</hi>, 25</p>
</item>
<item id-text='65' id='uid175'><p noindent='true'><hi rend='tt'>comp.text.tex</hi>, 31</p>
</item>
<item id-text='66' id='uid176'><p noindent='true'><hi rend='tt'>compan.err</hi>, 32</p>
</item>
<item id-text='67' id='uid177'><p noindent='true'>continued fractions, 15</p>
</item>
<item id-text='68' id='uid178'><p noindent='true'><hi rend='tt'>\cos</hi>, 18</p>
</item>
<item id-text='69' id='uid179'><p noindent='true'><hi rend='tt'>\cosh</hi>, 18</p>
</item>
<item id-text='70' id='uid180'><p noindent='true'><hi rend='tt'>\cot</hi>, 18</p>
</item>
<item id-text='71' id='uid181'><p noindent='true'><hi rend='tt'>\coth</hi>, 18</p>
</item>
<item id-text='72' id='uid182'><p noindent='true'><hi rend='tt'>\csc</hi>, 18</p>
</item>
<item id-text='73' id='uid183'><p noindent='true'><hi rend='tt'>\dbinom</hi>, 14</p>
</item>
<item id-text='74' id='uid184'><p noindent='true'><hi rend='tt'>\ddddot</hi>, 12</p>
</item>
<item id-text='75' id='uid185'><p noindent='true'><hi rend='tt'>\dddot</hi>, 12</p>
</item>
<item id-text='76' id='uid186'><p noindent='true'><hi rend='tt'>\ddot</hi>, 12</p>
</item>
<item id-text='77' id='uid187'><p noindent='true'><hi rend='tt'>\DeclareMathOperator</hi>, 2, 17, 18</p>
</item>
<item id-text='78' id='uid188'><p noindent='true'><hi rend='tt'>\deg</hi>, 18</p>
</item>
<item id-text='79' id='uid189'><p noindent='true'><hi rend='tt'>\det</hi>, 18</p>
</item>
<item id-text='80' id='uid190'><p noindent='true'><hi rend='tt'>\dfrac</hi>, 14</p>
</item>
<item id-text='81' id='uid191'><p noindent='true'><hi rend='tt'>diffs-m.txt</hi>, 30</p>
</item>
<item id-text='82' id='uid192'><p noindent='true'><hi rend='tt'>\dim</hi>, 18</p>
</item>
<item id-text='83' id='uid193'><p noindent='true'><hi rend='tt'>\displaybreak</hi>, 8, 9</p>
</item>
<item id-text='84' id='uid194'><p noindent='true'>displayed equations, 3</p>
</item>
<item id-text='85' id='uid195'><p noindent='true'>displayed equations
centering, 3</p>
</item>
<item id-text='86' id='uid196'><p noindent='true'><hi rend='tt'>\displaylimits</hi>, 20</p>
</item>
<item id-text='87' id='uid197'><p noindent='true'><hi rend='tt'>displaymath</hi> environment, 3</p>
</item>
<item id-text='88' id='uid198'><p noindent='true'><hi rend='tt'>\displaystyle</hi>, 14</p>
</item>
<item id-text='89' id='uid199'><p noindent='true'><hi rend='tt'>\documentclass</hi>, 2</p>
</item>
<item id-text='90' id='uid200'><p noindent='true'><hi rend='tt'>\dot</hi>, 12</p>
</item>
<item id-text='91' id='uid201'><p noindent='true'>dots, <hi rend='it'>see</hi> ellipsis dots</p>
</item>
<item id-text='92' id='uid202'><p noindent='true'><hi rend='tt'>\dotsb</hi>, 12</p>
</item>
<item id-text='93' id='uid203'><p noindent='true'><hi rend='tt'>\dotsc</hi>, 12</p>
</item>
<item id-text='94' id='uid204'><p noindent='true'><hi rend='tt'>\dotsi</hi>, 12</p>
</item>
<item id-text='95' id='uid205'><p noindent='true'><hi rend='tt'>\dotsm</hi>, 12</p>
</item>
<item id-text='96' id='uid206'><p noindent='true'><hi rend='tt'>\dotso</hi>, 12</p>
</item>
<item id-text='97' id='uid207'><p noindent='true'><hi rend='tt'>e-math.ams.org</hi>, 31</p>
</item>
<item id-text='98' id='uid208'><p noindent='true'>ellipsis dots
in matrices, 11</p>
</item>
<item id-text='99' id='uid209'><p noindent='true'><hi rend='tt'>eqnarray</hi> environment, 1, 3, 26, 30</p>
</item>
<item id-text='100' id='uid210'><p noindent='true'><hi rend='tt'>\eqref</hi>, 10</p>
</item>
<item id-text='101' id='uid211'><p noindent='true'><hi rend='tt'>equation</hi> environment, 1, 3, 5, 7, 24, 30</p>
</item>
<item id-text='102' id='uid212'><p noindent='true'>equation numbers
cross-references, 10
hierarchy, 9
left or right placement, 3
overriding, 3
subordinate numbering, 10
vertical placement, 2</p>
</item>
<item id-text='103' id='uid213'><p noindent='true'><hi rend='tt'>equation*</hi> environment, 3</p>
</item>
<item id-text='104' id='uid214'><p noindent='true'>equations, <hi rend='it'>see</hi> displayed equations</p>
</item>
<item id-text='105' id='uid215'><p noindent='true'><hi rend='tt'>euscript</hi> package, 21</p>
</item>
<item id-text='106' id='uid216'><p noindent='true'><hi rend='tt'>\exp</hi>, 18</p>
</item>
<item id-text='107' id='uid217'><p noindent='true'><hi rend='tt'>\fbox</hi>, 13</p>
</item>
<item id-text='108' id='uid218'><p noindent='true'><hi rend='tt'>fleqn</hi> option, 3, 5</p>
</item>
<item id-text='109' id='uid219'><p noindent='true'><hi rend='tt'>fntguide.tex</hi>, 21</p>
</item>
<item id-text='110' id='uid220'><p noindent='true'><hi rend='tt'>\frac</hi>, 14, 15</p>
</item>
<item id-text='111' id='uid221'><p noindent='true'>fractions, 14</p>
</item>
<item id-text='112' id='uid222'><p noindent='true'><hi rend='tt'>\fracwithdelims</hi>, 2</p>
</item>
<item id-text='113' id='uid223'><p noindent='true'>function names, <hi rend='it'>see</hi> operator names</p>
</item>
<item id-text='114' id='uid224'><p noindent='true'><hi rend='tt'>gather</hi> environment, 5&#x2013;7, 24, 28&#x2013;30</p>
</item>
<item id-text='115' id='uid225'><p noindent='true'><hi rend='tt'>gathered</hi> environment, 7, 9</p>
</item>
<item id-text='116' id='uid226'><p noindent='true'><hi rend='tt'>\gcd</hi>, 18</p>
</item>
<item id-text='117' id='uid227'><p noindent='true'><hi rend='tt'>\genfrac</hi>, 14</p>
</item>
<item id-text='118' id='uid228'><p noindent='true'><hi rend='tt'>\hat</hi>, 12</p>
</item>
<item id-text='119' id='uid229'><p noindent='true'><hi rend='tt'>\hdotsfor</hi>, 11</p>
</item>
<item id-text='120' id='uid230'><p noindent='true'><hi rend='tt'>\hom</hi>, 18</p>
</item>
<item id-text='121' id='uid231'><p noindent='true'>horizontal space
around operator names, 17
in math mode, 11</p>
</item>
<item id-text='122' id='uid232'><p noindent='true'><hi rend='tt'>\idotsint</hi>, 20</p>
</item>
<item id-text='123' id='uid233'><p noindent='true'><hi rend='tt'>\iiiint</hi>, 20</p>
</item>
<item id-text='124' id='uid234'><p noindent='true'><hi rend='tt'>\iiint</hi>, 20</p>
</item>
<item id-text='125' id='uid235'><p noindent='true'><hi rend='tt'>\iint</hi>, 20</p>
</item>
<item id-text='126' id='uid236'><p noindent='true'><hi rend='tt'>\inf</hi>, 18</p>
</item>
<item id-text='127' id='uid237'><p noindent='true'><hi rend='tt'>\injlim</hi>, 18</p>
</item>
<item id-text='128' id='uid238'><p noindent='true'>integrals
multiple, 20
placement of limits, 2</p>
</item>
<item id-text='129' id='uid239'><p noindent='true'><hi rend='tt'>\intertext</hi>, 9</p>
</item>
<item id-text='130' id='uid240'><p noindent='true'><hi rend='tt'>intlimits</hi> option, 2, 20</p>
</item>
<item id-text='131' id='uid241'><p noindent='true'><hi rend='tt'>\ker</hi>, 18</p>
</item>
<item id-text='132' id='uid242'><p noindent='true'><hi rend='tt'>kuvio</hi> package, 20</p>
</item>
<item id-text='133' id='uid243'><p noindent='true'><hi rend='tt'>\label</hi>, 6, 10</p>
</item>
<item id-text='134' id='uid244'><p noindent='true'><hi rend='tt'>\langle</hi>, 16</p>
</item>
<item id-text='135' id='uid245'><p noindent='true'><hi rend='tt'>\ldots</hi>, 12</p>
</item>
<item id-text='136' id='uid246'><p noindent='true'><hi rend='tt'>\left</hi>, 15, 16, 27</p>
</item>
<item id-text='137' id='uid247'><p noindent='true'><hi rend='tt'>\leftroot</hi>, 13</p>
</item>
<item id-text='138' id='uid248'><p noindent='true'><hi rend='tt'>leqno</hi> option, 3, 5</p>
</item>
<item id-text='139' id='uid249'><p noindent='true'><hi rend='tt'>\lg</hi>, 18</p>
</item>
<item id-text='140' id='uid250'><p noindent='true'><hi rend='tt'>\lim</hi>, 1, 2, 17, 18</p>
</item>
<item id-text='141' id='uid251'><p noindent='true'><hi rend='tt'>\liminf</hi>, 18</p>
</item>
<item id-text='142' id='uid252'><p noindent='true'>limits, <hi rend='it'>see</hi> subscripts and superscripts</p>
</item>
<item id-text='143' id='uid253'><p noindent='true'><hi rend='tt'>\limits</hi>, 20</p>
</item>
<item id-text='144' id='uid254'><p noindent='true'><hi rend='tt'>\limsup</hi>, 18</p>
</item>
<item id-text='145' id='uid255'><p noindent='true'><hi rend='tt'>\ln</hi>, 18</p>
</item>
<item id-text='146' id='uid256'><p noindent='true'><hi rend='tt'>\log</hi>, 17, 18</p>
</item>
<item id-text='147' id='uid257'><p noindent='true'><hi rend='tt'>\lVert</hi>, 16</p>
</item>
<item id-text='148' id='uid258'><p noindent='true'><hi rend='tt'>\lvert</hi>, 16</p>
</item>
<item id-text='149' id='uid259'><p noindent='true'>math fonts, 21</p>
</item>
<item id-text='150' id='uid260'><p noindent='true'>math symbols, <hi rend='it'>see</hi> math fonts</p>
</item>
<item id-text='151' id='uid261'><p noindent='true'><hi rend='tt'>\mathbb</hi>, 21, 22</p>
</item>
<item id-text='152' id='uid262'><p noindent='true'><hi rend='tt'>\mathbf</hi>, 21, 22</p>
</item>
<item id-text='153' id='uid263'><p noindent='true'><hi rend='tt'>\mathcal</hi>, 21</p>
</item>
<item id-text='154' id='uid264'><p noindent='true'><hi rend='tt'>\mathfrak</hi>, 21</p>
</item>
<item id-text='155' id='uid265'><p noindent='true'><hi rend='tt'>\mathit</hi>, 21</p>
</item>
<item id-text='156' id='uid266'><p noindent='true'><hi rend='tt'>\mathrm</hi>, 21</p>
</item>
<item id-text='157' id='uid267'><p noindent='true'><hi rend='tt'>\mathscr</hi>, 21</p>
</item>
<item id-text='158' id='uid268'><p noindent='true'><hi rend='tt'>\mathsf</hi>, 21</p>
</item>
<item id-text='159' id='uid269'><p noindent='true'><hi rend='tt'>\mathtt</hi>, 21</p>
</item>
<item id-text='160' id='uid270'><p noindent='true'>matrices, 10
ellipsis dots, 11</p>
</item>
<item id-text='161' id='uid271'><p noindent='true'><hi rend='tt'>matrix</hi> environment, 10</p>
</item>
<item id-text='162' id='uid272'><p noindent='true'><hi rend='tt'>\matrix</hi>, 28</p>
</item>
<item id-text='163' id='uid273'><p noindent='true'><hi rend='tt'>\max</hi>, 18</p>
</item>
<item id-text='164' id='uid274'><p noindent='true'><hi rend='tt'>\mbox</hi>, 18</p>
</item>
<item id-text='165' id='uid275'><p noindent='true'><hi rend='tt'>\medspace</hi>, 11</p>
</item>
<item id-text='166' id='uid276'><p noindent='true'>Metafont source files, 25</p>
</item>
<item id-text='167' id='uid277'><p noindent='true'><hi rend='tt'>\min</hi>, 18</p>
</item>
<item id-text='168' id='uid278'><p noindent='true'><hi rend='tt'>\mod</hi>, 18</p>
</item>
<item id-text='169' id='uid279'><p noindent='true'><hi rend='tt'>\mspace</hi>, 11</p>
</item>
<item id-text='170' id='uid280'><p noindent='true'><hi rend='tt'>multline</hi> environment, 3, 5, 27, 30</p>
</item>
<item id-text='171' id='uid281'><p noindent='true'><hi rend='tt'>\multlinegap</hi>, 5</p>
</item>
<item id-text='172' id='uid282'><p noindent='true'><hi rend='tt'>namelimits</hi> option, 2</p>
</item>
<item id-text='173' id='uid283'><p noindent='true'><hi rend='tt'>\negmedspace</hi>, 11</p>
</item>
<item id-text='174' id='uid284'><p noindent='true'><hi rend='tt'>\negthickspace</hi>, 11</p>
</item>
<item id-text='175' id='uid285'><p noindent='true'><hi rend='tt'>\negthinspace</hi>, 11</p>
</item>
<item id-text='176' id='uid286'><p noindent='true'><hi rend='tt'>\newcommand</hi>, 21</p>
</item>
<item id-text='177' id='uid287'><p noindent='true'><hi rend='tt'>\nobreakdash</hi>, 12</p>
</item>
<item id-text='178' id='uid288'><p noindent='true'><hi rend='tt'>nointlimits</hi> option, 2</p>
</item>
<item id-text='179' id='uid289'><p noindent='true'><hi rend='tt'>\nolimits</hi>, 19, 20</p>
</item>
<item id-text='180' id='uid290'><p noindent='true'><hi rend='tt'>nonamelimits</hi> option, 2</p>
</item>
<item id-text='181' id='uid291'><p noindent='true'><hi rend='tt'>nosumlimits</hi> option, 2</p>
</item>
<item id-text='182' id='uid292'><p noindent='true'><hi rend='tt'>\notag</hi>, 3, 5</p>
</item>
<item id-text='183' id='uid293'><p noindent='true'><hi rend='tt'>\numberwithin</hi>, 9, 29, 30</p>
</item>
<item id-text='184' id='uid294'><p noindent='true'>operator names, 17</p>
</item>
<item id-text='185' id='uid295'><p noindent='true'><hi rend='tt'>\operatorname</hi>, 18</p>
</item>
<item id-text='186' id='uid296'><p noindent='true'><hi rend='tt'>\operatorname*</hi>, 18</p>
</item>
<item id-text='187' id='uid297'><p noindent='true'><hi rend='tt'>\over</hi>, 14, 29</p>
</item>
<item id-text='188' id='uid298'><p noindent='true'><hi rend='tt'>\overleftarrow</hi>, 13</p>
</item>
<item id-text='189' id='uid299'><p noindent='true'><hi rend='tt'>\overleftrightarrow</hi>, 13</p>
</item>
<item id-text='190' id='uid300'><p noindent='true'><hi rend='tt'>\overrightarrow</hi>, 13</p>
</item>
<item id-text='191' id='uid301'><p noindent='true'><hi rend='tt'>\overset</hi>, 13</p>
</item>
<item id-text='192' id='uid302'><p noindent='true'><hi rend='tt'>\overwithdelims</hi>, 14</p>
</item>
<item id-text='193' id='uid303'><p noindent='true'>page breaks, 8</p>
</item>
<item id-text='194' id='uid304'><p noindent='true'><hi rend='tt'>\pagebreak</hi>, 8</p>
</item>
<item id-text='195' id='uid305'><p noindent='true'><hi rend='tt'>pmatrix</hi> environment, 10</p>
</item>
<item id-text='196' id='uid306'><p noindent='true'><hi rend='tt'>\pmatrix</hi>, 28</p>
</item>
<item id-text='197' id='uid307'><p noindent='true'><hi rend='tt'>\pmb</hi>, 22</p>
</item>
<item id-text='198' id='uid308'><p noindent='true'><hi rend='tt'>\pmod</hi>, 18</p>
</item>
<item id-text='199' id='uid309'><p noindent='true'><hi rend='tt'>\pod</hi>, 18</p>
</item>
<item id-text='200' id='uid310'><p noindent='true'>PostScript fonts, 25</p>
</item>
<item id-text='201' id='uid311'><p noindent='true'><hi rend='tt'>\Pr</hi>, 18</p>
</item>
<item id-text='202' id='uid312'><p noindent='true'><hi rend='tt'>\projlim</hi>, 18</p>
</item>
<item id-text='203' id='uid313'><p noindent='true'><hi rend='tt'>\qquad</hi>, 11</p>
</item>
<item id-text='204' id='uid314'><p noindent='true'><hi rend='tt'>\quad</hi>, 11</p>
</item>
<item id-text='205' id='uid315'><p noindent='true'><hi rend='tt'>\R</hi>, 22</p>
</item>
<item id-text='206' id='uid316'><p noindent='true'><hi rend='tt'>\raisetag</hi>, 8</p>
</item>
<item id-text='207' id='uid317'><p noindent='true'><hi rend='tt'>\rangle</hi>, 16</p>
</item>
<item id-text='208' id='uid318'><p noindent='true'><hi rend='tt'>\ref</hi>, 10</p>
</item>
<item id-text='209' id='uid319'><p noindent='true'><hi rend='tt'>reqno</hi> option, 3, 5</p>
</item>
<item id-text='210' id='uid320'><p noindent='true'><hi rend='tt'>\right</hi>, 15, 16, 27</p>
</item>
<item id-text='211' id='uid321'><p noindent='true'><hi rend='tt'>\rVert</hi>, 16</p>
</item>
<item id-text='212' id='uid322'><p noindent='true'><hi rend='tt'>\rvert</hi>, 16</p>
</item>
<item id-text='213' id='uid323'><p noindent='true'><hi rend='tt'>\scriptscriptstyle</hi>, 14</p>
</item>
<item id-text='214' id='uid324'><p noindent='true'><hi rend='tt'>\scriptstyle</hi>, 14</p>
</item>
<item id-text='215' id='uid325'><p noindent='true'><hi rend='tt'>\sec</hi>, 18</p>
</item>
<item id-text='216' id='uid326'><p noindent='true'><hi rend='tt'>\setcounter</hi>, 9, 10</p>
</item>
<item id-text='217' id='uid327'><p noindent='true'><hi rend='tt'>\setlength</hi>, 5</p>
</item>
<item id-text='218' id='uid328'><p noindent='true'><hi rend='tt'>\shoveleft</hi>, 5</p>
</item>
<item id-text='219' id='uid329'><p noindent='true'><hi rend='tt'>\shoveright</hi>, 5</p>
</item>
<item id-text='220' id='uid330'><p noindent='true'><hi rend='tt'>\sideset</hi>, 13, 19</p>
</item>
<item id-text='221' id='uid331'><p noindent='true'><hi rend='tt'>\sin</hi>, 1, 2, 17, 18</p>
</item>
<item id-text='222' id='uid332'><p noindent='true'><hi rend='tt'>\sinh</hi>, 18</p>
</item>
<item id-text='223' id='uid333'><p noindent='true'><hi rend='tt'>smallmatrix</hi> environment, 11</p>
</item>
<item id-text='224' id='uid334'><p noindent='true'><hi rend='tt'>\smash</hi>, 15, 27</p>
</item>
<item id-text='225' id='uid335'><p noindent='true'><hi rend='tt'>\sphat</hi>, 12</p>
</item>
<item id-text='226' id='uid336'><p noindent='true'><hi rend='tt'>split</hi> environment, 3, 5, 6, 9, 24, 27&#x2013;30</p>
</item>
<item id-text='227' id='uid337'><p noindent='true'><hi rend='tt'>\sptilde</hi>, 12</p>
</item>
<item id-text='228' id='uid338'><p noindent='true'><hi rend='tt'>\stackrel</hi>, 13</p>
</item>
<item id-text='229' id='uid339'><p noindent='true'><hi rend='tt'>subarray</hi> environment, 19</p>
</item>
<item id-text='230' id='uid340'><p noindent='true'><hi rend='tt'>subeqn.tex</hi>, 3</p>
</item>
<item id-text='231' id='uid341'><p noindent='true'><hi rend='tt'>subequations</hi> environment, 10</p>
</item>
<item id-text='232' id='uid342'><p noindent='true'>subscripts and superscripts, 13
multi-line, 19
on sums, 19
placement, 2</p>
</item>
<item id-text='233' id='uid343'><p noindent='true'><hi rend='tt'>\substack</hi>, 19</p>
</item>
<item id-text='234' id='uid344'><p noindent='true'><hi rend='tt'>\sum</hi>, 20, 22</p>
</item>
<item id-text='235' id='uid345'><p noindent='true'><hi rend='tt'>sumlimits</hi> option, 2</p>
</item>
<item id-text='236' id='uid346'><p noindent='true'><hi rend='tt'>\sup</hi>, 18</p>
</item>
<item id-text='237' id='uid347'><p noindent='true'>superscripts, <hi rend='it'>see</hi> subscripts and superscripts</p>
</item>
<item id-text='238' id='uid348'><p noindent='true'><hi rend='tt'>tabular</hi> environment, 26</p>
</item>
<item id-text='239' id='uid349'><p noindent='true'><hi rend='tt'>\tag</hi>, 3</p>
</item>
<item id-text='240' id='uid350'><p noindent='true'><hi rend='tt'>\tag*</hi>, 3</p>
</item>
<item id-text='241' id='uid351'><p noindent='true'><hi rend='tt'>\tan</hi>, 18</p>
</item>
<item id-text='242' id='uid352'><p noindent='true'><hi rend='tt'>\tanh</hi>, 18</p>
</item>
<item id-text='243' id='uid353'><p noindent='true'><hi rend='tt'>\tbinom</hi>, 14</p>
</item>
<item id-text='244' id='uid354'><p noindent='true'><hi rend='tt'>tbtags</hi> option, 2</p>
</item>
<item id-text='245' id='uid355'><p noindent='true'><hi rend='tt'>technote.tex</hi>, 14, 28&#x2013;30</p>
</item>
<item id-text='246' id='uid356'><p noindent='true'><hi rend='tt'>testmath.tex</hi>, 3</p>
</item>
<item id-text='247' id='uid357'><p noindent='true'><TeX/> Users Group, 31</p>
</item>
<item id-text='248' id='uid358'><p noindent='true'><hi rend='tt'>\text</hi>, 2, 7&#x2013;9, 18</p>
</item>
<item id-text='249' id='uid359'><p noindent='true'>text fragments inside math, 9, 18</p>
</item>
<item id-text='250' id='uid360'><p noindent='true'><hi rend='tt'>\textstyle</hi>, 14</p>
</item>
<item id-text='251' id='uid361'><p noindent='true'><hi rend='tt'>\tfrac</hi>, 14</p>
</item>
<item id-text='252' id='uid362'><p noindent='true'><hi rend='tt'>\theequation</hi>, 9, 10</p>
</item>
<item id-text='253' id='uid363'><p noindent='true'><hi rend='tt'>\thickspace</hi>, 11</p>
</item>
<item id-text='254' id='uid364'><p noindent='true'><hi rend='tt'>\thinspace</hi>, 11</p>
</item>
<item id-text='255' id='uid365'><p noindent='true'><hi rend='it'>TUGboat</hi>, 31</p>
</item>
<item id-text='256' id='uid366'><p noindent='true'><hi rend='tt'>\underleftarrow</hi>, 13</p>
</item>
<item id-text='257' id='uid367'><p noindent='true'><hi rend='tt'>\underleftrightarrow</hi>, 13</p>
</item>
<item id-text='258' id='uid368'><p noindent='true'><hi rend='tt'>\underrightarrow</hi>, 13</p>
</item>
<item id-text='259' id='uid369'><p noindent='true'><hi rend='tt'>\underset</hi>, 13</p>
</item>
<item id-text='260' id='uid370'><p noindent='true'><hi rend='tt'>\uproot</hi>, 13</p>
</item>
<item id-text='261' id='uid371'><p noindent='true'><hi rend='tt'>\usepackage</hi>, 2, 25</p>
</item>
<item id-text='262' id='uid372'><p noindent='true'><hi rend='tt'>\value</hi>, 10</p>
</item>
<item id-text='263' id='uid373'><p noindent='true'><hi rend='tt'>\varDelta</hi>, 23</p>
</item>
<item id-text='264' id='uid374'><p noindent='true'><hi rend='tt'>\varGamma</hi>, 23</p>
</item>
<item id-text='265' id='uid375'><p noindent='true'><hi rend='tt'>\varinjlim</hi>, 18</p>
</item>
<item id-text='266' id='uid376'><p noindent='true'><hi rend='tt'>\varLambda</hi>, 23</p>
</item>
<item id-text='267' id='uid377'><p noindent='true'><hi rend='tt'>\varliminf</hi>, 18</p>
</item>
<item id-text='268' id='uid378'><p noindent='true'><hi rend='tt'>\varlimsup</hi>, 18</p>
</item>
<item id-text='269' id='uid379'><p noindent='true'><hi rend='tt'>\varOmega</hi>, 23</p>
</item>
<item id-text='270' id='uid380'><p noindent='true'><hi rend='tt'>\varPhi</hi>, 23</p>
</item>
<item id-text='271' id='uid381'><p noindent='true'><hi rend='tt'>\varPi</hi>, 23</p>
</item>
<item id-text='272' id='uid382'><p noindent='true'><hi rend='tt'>\varprojlim</hi>, 18</p>
</item>
<item id-text='273' id='uid383'><p noindent='true'><hi rend='tt'>\varPsi</hi>, 23</p>
</item>
<item id-text='274' id='uid384'><p noindent='true'><hi rend='tt'>\varSigma</hi>, 23</p>
</item>
<item id-text='275' id='uid385'><p noindent='true'><hi rend='tt'>\varTheta</hi>, 23</p>
</item>
<item id-text='276' id='uid386'><p noindent='true'><hi rend='tt'>\varUpsilon</hi>, 23</p>
</item>
<item id-text='277' id='uid387'><p noindent='true'><hi rend='tt'>\varXi</hi>, 23</p>
</item>
<item id-text='278' id='uid388'><p noindent='true'><hi rend='tt'>Vmatrix</hi> environment, 10</p>
</item>
<item id-text='279' id='uid389'><p noindent='true'><hi rend='tt'>vmatrix</hi> environment, 10</p>
</item>
<item id-text='280' id='uid390'><p noindent='true'><hi rend='tt'>\vphantom</hi>, 27</p>
</item>
<item id-text='281' id='uid391'><p noindent='true'><hi rend='tt'>\xleftarrow</hi>, 13</p>
</item>
<item id-text='282' id='uid392'><p noindent='true'><hi rend='tt'>\xrightarrow</hi>, 13</p>
</item></list>
</item></div0></item></div0></mainmatter><theindex title='Index'><index target='uid97' level='1'>AMSFonts collection</index>
<index target='' level='1'>arrows</index>
<index target='uid67' level='2'>extensible</index>
<index target='uid95' level='2'>in commutative diagrams</index>
<index target='uid99' level='1'>BaKoMa fonts</index>
<index target='uid80 uid81' level='1'><EM r='cn'>big</EM>, <EM r='cn'>Big</EM>, <EM r='cn'>bigg</EM>, ... delimiters</index>
<index target='uid74' level='1'>binomials</index>
<index target='uid77' level='1'>continued fractions</index>
<index target='uid32' level='1'>displayed equations</index>
<index target='uid31' level='2'>centering</index>
<index target='uid58' level='1' encap='see{ellipsis dots}'>dots</index>
<index target='' level='1'>ellipsis dots</index>
<index target='uid58' level='2'>in matrices</index>
<index target='' level='1'>equation numbers</index>
<index target='uid53' level='2'>cross-references</index>
<index target='uid51' level='2'>hierarchy</index>
<index target='uid28' level='2'>left or right placement</index>
<index target='uid33' level='2'>overriding</index>
<index target='uid55' level='2'>subordinate numbering</index>
<index target='uid15 uid17' level='2'>vertical placement</index>
<index target='uid32' level='1' encap='see{displayed equations}'>equations</index>
<index target='uid71' level='1'>fractions</index>
<index target='uid85' level='1' encap='see{operator names}'>function names</index>
<index target='' level='1'>horizontal space</index>
<index target='uid86' level='2'>around operator names</index>
<index target='uid60' level='2'>in math mode</index>
<index target='' level='1'>integrals</index>
<index target='uid93' level='2'>multiple</index>
<index target='uid22' level='2'>placement of limits</index>
<index target='uid19' level='1' encap='see{subscripts and superscripts}'>limits</index>
<index target='uid96' level='1'>math fonts</index>
<index target='uid96' level='1' encap='see{math fonts}'>math symbols</index>
<index target='uid56' level='1'>matrices</index>
<index target='uid58' level='2'>ellipsis dots</index>
<index target='uid98' level='1'>Metafont source files</index>
<index target='uid85' level='1'>operator names</index>
<index target='uid47' level='1'>page breaks</index>
<index target='uid99' level='1'>PostScript fonts</index>
<index target='uid69' level='1'>subscripts and superscripts</index>
<index target='uid90' level='2'>multi-line</index>
<index target='uid91' level='2'>on sums</index>
<index target='uid19 uid25' level='2'>placement</index>
<index target='uid90' level='1' encap='see{subscripts and superscripts}'>superscripts</index>
<index target='uid108' level='1'><TeX/> Users Group</index>
<index target='uid48 uid88' level='1'>text fragments inside math</index>
<index target='uid83 uid84' level='1'>this-is-wrong</index>
<index target='uid109' level='1'><hi rend='it'>TUGboat</hi></index>
</theindex></std>