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\input amsman.hdr
\document
\topmatter
\TITLE  Summary of \amstex  \endTITLE
\author Michael Spivak \endauthor
\date{ September 13, 1983}
\endtopmatter

This is a brief summary of \amstex, as it is now written for the new \tex.
  The old ``Joy of \tex'' manual
is woefully out of date, but I will not be able even
to begin on the new version until November, so this summary has to be
used as a supplement.  It probably won't be of much help for learning
\amstex\ from scratch, but
people who are already familiar with the old \amstex\ can use it to
see what new things are available, and what changes have been made. (Some
of the more esoteric features of \amstex\ aren't mentioned here, and will
have to wait for the manual.)

\heading Basics\endheading
\subheading{ Control Sequences} Control sequences are of two types.
The first, like *\par*, consists of
*\* followed by any number of {\sl letters}. The second, like *\$*,
consists of *\* followed by just one {\sl non-letter}.  A control sequence
of the first kind is ended by any non-letter, including a space, and
a space after such a control sequence is {\sl not\/} considered as a
space within the text.  When an interword space does have to be indicated
after such a control sequence, you can use *\"space*, where *"space*
indicates a typed space; this is a control sequence of the second type.
 A control sequence of the second kind needs
no special ending, since it is only one character long.  A space
after such a control sequence {\sl does} count as a space in the text.
Multiple spaces  count as one space,
so *\$"space"space* gives only one space after the printed \$.
A special exception is made for the control
sequence *\"space*\,;
 here a space after the control sequence
{\sl is\/} ignored, so that *\"space"space*
still gives one space (this of course complies with the general rule that
multiple spaces count as one space).  Also, spaces in math mode still don't
count, so it doesn't matter about spaces after control sequences of this sort.

\subheading{ Document Formatting}   Your file
should begin
**
\input amstex
\documentstyle{amsppt}
...
\document
**
with the line
**
\enddocument"hskip10em
**
at the very end of the file.
(If you are using a program *amstex*, which consists of *tex* with the file
*amstex.tex* preloaded, then you shouldn't have the initial line *\input
amstex*.)


The control sequence *\documentstyle* is supposed to allow you to select
the particular format, but
at the moment the only style available is the *amsppt* (AMS
preprint) style.
When you use this style you will get the question
**
Do you want output? (y or n, follow answer by return)
**
on your terminal.
If you type anything other than *y* or *n* (or *Y* or *N*), followed by
\CR, the
question will be repeated.

The other things that go at the beginning of the paper, between
*\documentstyle* and *\document*,
 will be covered
later.

\heading Defining Control Sequences\endheading

To define your own control sequence *\cs* you type
**
\define\cs{...}
**
*\define* will
check for you whether *\cs* already has a meaning,
 and if so it will issue
an error message.  It's actually rather improbable that you will stumble
upon one of \tex's control sequences, since most of them have long
and unlikely
names. Most of the control sequences consisting of *\* followed by a
single non-letter do have meanings, but *\0*, *\1*, \dots,*\9* and *\.*
don't, and they are convenient ones to choose, especially for
temporary definitions in a complicated math formula (definitions
made inside *$* signs disappear at the closing *$* sign). If
you want to redefine one of your own control sequences, you can use
*\redefine*.  You might even want to *\redefine* some of \tex's control
sequences, provided you can be reasonably sure that they are not used
in some hidden way somewhere else.  For example, we'll see later that
*\b*  and *\d* are used for certain accents under letters, and
you might want to use them for something else, say for *\beta* ($\beta$),
 and *\delta* ($\delta$).  So you would say
**
\redefine\b{\beta}
\redefine\d{\delta}
**
(using *\define* would give an error message).  If it turns out that
you also need the original
*\d* (which  gives an under-dot accent),
 you can use  *\predefine*:
**
\predefine\underdot{\d}
\redefine\d{\delta}
**
Then you can type `*\underdot x*' to get `\d x' if you ever need it.
The only important thing is that the *\predefine* comes before the *\redefine*
for *\d*.


When you are defining control sequences with arguments,
**
\define\cs#1#2{...}
**
you must  be careful not to have spaces in the expression *#1#2{*.
But you {\sl don't\/} have to worry about spaces when using *\cs*.
 \tex\ will ignore spaces between arguments.

\heading Ordinary Text\endheading
\subheading{Control Sequences for Special Symbols} If your keyboard is
missing certain symbols you might be able to make do with
$$\example
*\plus*&*+*\\
*\equal*&*=*\\
*\less*&*<*\\
*\more*&*>*\\
*\lq*&*`*\\
*\rq*&*'*\\
*\vert*&*|*
\endexample$$
(If you need *\vert* for the *|*, then you will also need *\Vert*, for
*\|*.)

More important, the following symbols are special in \amstex:
**
\"qquad{"qquad}"qquad$"qquad&"qquad#"qquad%"qquad"""qquad@"qquad~"qquad^"qquad_
**
Some of these have printed versions that can appear as symbols in ordinary
text, and you get them by typing the obvious things:
$$\example\openup.5\jot
*\{*&\{\\
*\}*&\}\\
*\$*&\$\\
*\&*&\&\\
*\#*&\#\\
*\%*&\%\\
*\@*&\@\endexample$$
(You can also use *\_* to get a printed \_, for things like
 {\it first\_letter}, which computer scientists like to use.)
\eject

\subheading{Paragraphing and special text symbols}
New paragraphs are indicated by either a blank line, or the specific
control sequence *\par*.  Double quotes `` and '' are produced simply
by typing two single quotes in a row: *``* and *''*.  If you don't have
*`* or *'* you can use *\lq* and *\rq*. These work even in pairs---*\lq \lq*
gives ``.  Hyphens, en-dashes and em-dashes should be distinguished as
follows
$$\example
hyphen ( - )& *-*\\
en-dash ( -- )& *--*\\
em-dash ( --- )& *---*\endexample$$
A hyphen in math mode (between *$* signs) becomes a minus sign $-$.

{\it Italic\/} type is indicated by *\it* and {\bf boldface} type by *\bf*,
while *\sl* gives type that is {\sl slanted},
and *\rm* indicates a return to ordinary
(roman) type. You should
add the ``italic
correction'' *\/* after something in italic or slanted
type unless it is followed by
a period or a comma:
**
{\it Italic\/} type ... type that is {\sl slanted}, and ...
**


\subheading{ Accents} Here are the common accents:
$$
\example
*\`o*&\`o\\
*\'o*&\'o\\
*\^o*&\^o\\
*\""o*&\"o\\
*\~o*&\~o\\
*\=o*&\=o\\
*\.o*&\.o\\
*\v o*&\v o\\
*\u o*&\u o\\
*\H o*&\H o (long Hungarian umlaut)\\
*\t oo*&\t oo (tie-after accent)\\
*\c o*&\c o (cedilla)\\
*\d o*&\d o\\
*\b o*&\b o\endexample
$$
You can also type
*\`{o}* or *\v{o}*, etc., but you should accent only single letters.

And here are the special letters:
$$
\example
*\oe*, *\OE*& \oe, \OE\\
*\ae*, *\AE*& \ae, \AE\\
*\aa*, *\AA*& \aa, \AA\\
*\o*, *\O*& \o, \O\\
*\l*, *\L*& \l, \L\\
*\ss*& \ss\endexample$$
Also, for accents over `i' and `j' one needs the dotless
 `\i' and `\j', which you obtain by typing *\i* and *\j*.


\subheading{Miscellaneous considerations for ordinary type}

\noindent 1)  \tex\ puts more space after the ends of sentences than
between words, so it needs to know which periods represent ends of
sentences.  Periods after upper-case letters are not considered to
be ends of sentences, since they are usually initials.  But periods
after abbreviations like ``Mr.''\ cause problems.  So you can use *\"space*
to indicate an ordinary interword space: *\Mr.\ Jones*.  Actually,
in this case it would be better to type *\Mr.@Jones*, using
\amstex's ``tie'' *~*, which
indicates an ordinary interword space and also tells \tex\ to try not
to break the line at that point.

 On those rare occasions when  you need a period after
a capital letter, and it really is the end of a sentence, use *\period*:
**
Supported by the NSF \period And numerous others.
**


\noindent 3)
To get a footnote at some point,
\footnote {Here is a footnote}
you just type
**
... at some point,
\footnote {Here is a footnote}
you just type
**
and sure enough it will appear, at the bottom of the page.  The particular
style you are using will determine what symbols to use and whether numbering
is consecutive or begins anew on each page.  In the *amsppt* style, superscript
numbers are used, and they begin anew on each page.  On rare occasions the
numbering won't be right, because \tex\ will have decided that a footnote
should
have number 3, say, before realizing that it will actually wind up on the next
page and should
therefore have the number 1.  On such occasions you can say
**
\adjustfootnote{-2}
**
right before the footnote, to get the numbering decreased by 2 as needed.

\adjustfootnote{1}
Notice that the space or \CR\ after the final *}* of the *\footnote*
is what provides the space before the next word ``you''.  If you don't want
the space \footnote {When \tex\ set the  footnote at the top of this page it
was still working on the previous page (since it didn't know yet just where
the best page break would be).  Luckily, that footnote still got the
number 1, because there were no footnotes on the previous page.  But
after \tex\ had completed the previous page, it started numbering footnotes
over again, so {\sl this\/} footnote originally got the number 1 also!
I had to put {\eighttt \bs adjustfootnote\char'173 1\char'175 } right
before it.},
you just omit it:
**
the space \footnote {...},
you just omit it:
**

\noindent 4) Use *\dots* for the three dots \dots that indicate an ellipsis:
**
the three dots \dots that indicate an ellipsis:
**

\par\noindent 5) A *%* on a line of input causes \tex\ to ignore
 everything on the line from that *%* to the end, {\sl including\/} the
\CR\ at the end. So
**
... by Schwartz's Lemma%   Is this how you spell Schwartz?
we have
**
prints out
$$\text{by Schwartz's Lemmawe have}$$
which isn't quite what you had in mind when you wrote that reminder to
yourself; be sure to type
**
... by Schwartz's Lemma %Is this how you spell Schwartz?
we have
**
to get the space after ``Lemma'', since the \CR\ that normally gives it
has been obliterated by the *%*.  You can take advantage of this behavior
of *%* to break a long word near the end of a line right in the middle.

If you want to ``comment out'' a large portion of text, instead of
putting *%* in front of each line, you can type
**
\comment
...
...
\endcomment
**
and everything between *\comment* and *\endcomment* will be ignored. But
the *\endcomment* must be on a line by itself (although it can be
preceded by blank spaces).
\eject
\par\noindent 6) For the proper spacing between quotes within quotes,
 use *\qspace*:
**
``They call this `typesetting'\qspace'' he sneered.
**
This will work for 'qspace'' or ''\qspace', and `\qspace`` or ``\qspace`.

\smallvspace
\par\noindent 7) Use *\dash* for an em-dash ( --- ) that can go at the
beginning of a line, if you really need it to avoid a bad line break (normally,
\tex\ will only put an em-dash at the end of a line, not at the beginning
of one).  Similarly, use *\slash* instead of */* in a word like *and/or* if
you want to allow the ``and/'' to be at the end of one line, with the
``or'' at the beginning of the next.
\medvspace
\par\noindent 8) To get an paragraph to be unindented, use *\noindent* at
the beginning of it.
\bigvspace
\par\noindent 9) Between paragraphs you can use *\smallvspace*, *\medvspace*
and *\bigvspace*  to get a
small, medium or big amount of vertical space; there is a *\bigvspace* before
this *\noindent*'ed paragraph, a *\medvspace* before the previous one, and
a *\smallvspace* before that one.

\par\noindent 10) Between
paragraphs if you type *\midspace{"<dim">}* for some ``dimension''
*"<dim">*  you will get a space of that dimension, if it will fit on
the page; otherwise it will be on the top of the next page. A ``dimension''
is a number followed by a unit like *in* (inches), *cm* (centimeters),
*pt* (point) or *pc* (pica), to name the ones most likely to be used.
If you type *\topspace* instead of *\midspace* the space will be at the
top of the current page if it fits, or at the top of the next page.


\par\noindent 11)
 *\newline* breaks a line right where it is typed, without spreading the
line out. *\linebreak* spreads the line out.

*\pagebreak* breaks the page immediately after the present
line is completed.  *\newpage* does the same, but fills up the remainder
 of the page with blank space, instead of spreading out the lines if the
page is short.

\par\noindent 12) Use *\TeX* to get the \tex\  logo, and *\AmSTeX*
to get the \amstex\ logo.



\heading More About\\ Document Formatting\endheading

\subheading{Topmatter}
At the beginning of your file, right after the *\documentstyle* line,
there will normally be
**
\topmatter
...
...
\endtopmatter
**
for the ``topmatter'', like the title, author, etc. (some of which may
actually end up at the bottom of the title page, or even at the end of
the paper).
Here are the things you can have (the order is irrelevant):
**
\title...\endtitle
\author...\endauthor
\affil...\endaffil
\address{...}
\date{...}
\thanks{...}
\keywords{...}
\subjclass{...}
\abstract{...}
**

The first three use a different syntax
 because each of them can be made multi-line, by inserting *\\* between
the lines.  *\address* can be used several times in a row, for different
authors, say.  In the *amsppt* style, the addresses appear at the end,
along with the *\keywords*, prefaced by ``{\it Keywords.}''
 and the *\subjclass*, prefaced
by ``{1980 \it Mathematics subject classifications.}''
  The abstract begins with the word ``{\smc Abstract.}'', so this
shouldn't be typed in.

Sometimes acknowledgements and affiliations are handled as footnotes
on the title and/or author, instead of using *\thanks* and
*\affil...\endaffil*. That's  no problem---*\footnote* can  be used
in *\title...\endtitle* and *\author...\endauthor* (and most other
places, as well, although the footnote may disappear if you bury it really
deeply within some complicated construction---don't worry about it until
it happens).

There's also *\TITLE...\endTITLE* and *\AUTHOR...\endAUTHOR* to automatically
print the title and author in capital letters.  Normally, these will
be used by various journal styles, when it is desirable to have the
title and author appearing in the file the same way that it may appear
in some index, say.  In fact, various journal styles will probably just have
*\title* mean *\TITLE*, so that there really isn't any need to use *\TITLE*
yourself.

There are a couple of other neat little things that you can do near the
beginning of the file.  First of all, you can say *\guidelines* before
*\document* to get horizontal lines at the top and bottom of the page---%
these can be useful when your output come out on a roll of paper, and you
can't figure out just where to cut it.  The guidelines are normally about
two lines above and below the actual page. You can say
*\guidelinegap"<dim">* to specify some other dimension *"<dim">* for
the gap.

In a few places the *amsppt* style assumes that your paper is in English,
for it begins the abstract with the word ``{\smc Abstract.}'', puts the
logo ``Typeset by \amstex'' at the bottom of the first page, labels
the bibliography (which we'll get to later) as ``{\smc References}'', and
prefaces the *\keywords* and *\subjclass* with appropriate English
words.  If you put *\german* before the *\document*, however, all of
these will come out in German.  Actually, at the moment only
``{\smc Abstract.}''
will be changed, to ``{\smc Zusammenfassung.}'', since I don't know what the
proper German for the other things is.  But this will be changed as soon
as someone tells me, and other languages will be added as requested.

Incidentally, if you can't stand having the ``Typeset by \amstex'' logo
at the bottom, no matter what language it is in, put *\nologo* at the
beginning.


\subheading{The paper proper} The heading for this section was made by
typing
**
\heading More About\\Document Formatting\endheading
**
with *\\* indicating linebreaks, as usual.  *\subheading{...}*
provides
the boldface flush-left headings like `{\bf The paper proper.}'
(the period is supplied
automatically); this can't be multi-line, so the syntax is different.

To state theorems, lemmas, and other proclamations, you just say
**
\proclaim{Theorem 4} Blah, blah, blah.
\finishproclaim
**
and you will get
\proclaim{Theorem 4} Blah, blah, blah. \finishproclaim
Notice that the punctuation after the label of the theorem comes for free.
All *\finishproclaim* really does is switch back to regular type, and
perhaps provide a little extra spacing,
so if you
forget it the paper will look strange, but it won't be catastrophic.
We use *\finishproclaim* instead of *\endproclaim* because the
*\end...*\ syntax is reserved for things with *\\* in them; but
\amstex\ recognizes
*\endproclaim* as a synonym for *\finishproclaim*, in case you forget.

For the proofs, use
**
\demo{Proof} Here is the proof.
\finishdemo
**
Once again, omitting *\finishdemo* will not be catastrophic---all it will
lose is the special spacing after the end of the proof.



Mathematicians frequently list things like this:
\conditions
(a) : The first thing on the list.  We will make this very long so that you
can see what happens when the condition is more than one line long,\\
(b): The second thing,\\
(c): The final thing.\endconditions
\noindent You get this by typing
**
\conditions
(a) : The first thing on the list.  We will make this very long so that you
can see what happens when the condition is more than one line long,\\
(b): The second thing,\\
(c): The final thing.\endconditions
**
As usual, *\\* indicates the line breaks in this construction: the colon
separates the label for the condition from the condition itself. In the
*amsppt* style the
label gets printed in *\rm*, unless some other font is explicitly given, while
the condition is printed in whatever font is being used at the time (so that
*\conditions* in a *\proclaim* will get printed in *\sl*).  When you need
``runin'' conditions like
\runin\conditions (a): The first,\\
(b): The second,\\
(c): The last.\endconditions
\noindent just type *\runin* right before the *\conditions*.

By the way, sometimes you may not want to start a paragraph after the
statement of a theorem, but you may find that *\finishproclaim* does it for
you, even though you didn't leave a blank line.  The same problem
will arise with *\finishdemo* and *\endconditions*.
You will then need to
use *\noindent*; in later versions of \amstex\ this annoyance
will go away.


There are
 *\proclaimnp* and *\demonp*, if you need to have the *\proclaim*
or *\demo* supply No Punctuation.

Finally, there's  *\refto*.  *\refto{10}* gives \refto{10}, while
*\refto{10, Theorem 4}* gives \refto{10, Theorem 4}, with only the
stuff before the comma in boldface.




\heading Bibliography\endheading

When you are ready to type the bibliography, you first type
*\Refs*, which produces the heading {\smc References}, or whatever
the particular style uses, and sets things up for typing individual
references.  A typical individual reference would be
\def\setref#1{{\eightpoint\sfcode`.=1000 \par\vskip 2pt #1}\par}%
**
\ref \no 9 \by S. S. Chern \pages 947--055
\paper Integral formulas for hypersurfaces in Euclidean space and
  their applications to uniqueness theorems
\yr 1959 \vol 8
\jour J. Math. Mech.
\endref
**
which in the *amsppt* style will produce
\setref{\ref \no 9 \by S. S. Chern \pages 947--055
\paper Integral formulas for hypersurfaces in Euclidean space and
  their applications to uniqueness theorems
\yr 1959 \vol 8
\jour J. Math. Mech.
\endref}
\noindent Notice that there is no need to specify the various pieces
of the reference in the order that they will be printed, nor do you
have to worry about the proper font or details like enclosing
parentheses; all this is done by the style
file.
Notice also that we don't have *\"space* after *Math.*\ and *Mech.*;
within *\Refs* all spaces will count as usual interword spaces, even those
after periods (if you do need the usual space after a period, for example if a
title consists of two sentences, you can always use *\period*, which
has already been mentioned in another context).

There is no specific amount of information that you have to include
in each *\ref...\endref*.  The style file will do the reasonable thing
with the information you give.  For example,
if you leave out the *\vol*, then it just won't
get printed, but if you leave out *\jour*, then *\vol* and *\yr* will
be ignored, even if you put them in.

Other things you can have within *\ref...\endref* are *\toappear*,
which typesets ``(to appear)'' and  *\issue*, which might be needed for
some sort of special issue. If the next reference is by the same author,
you can type *\bysame* instead of *\by ...*; in the *amsppt* style this
will produce a line just the length of the author's name in the previous
reference instead of repeating it.

Instead of *\no* for a number you can use *\key* for some other sort of
``key'', like *\key[{\bf C1}]* to get ``[{\bf C1}]'' instead of a number
in front of the reference.  Both *\key* and *\no* can be omitted.

If the paper is just one page long, you might use *\page* instead of
*\pages*; this will make sure that the single page gets ``p. '' printed
in front of it to avoid confusion.

Finally, there is indeed a use for *\\* in *\ref...\endref*, though it's
for special things, where two references by the same author are combined
because they are closely related. For example,
**
\ref\no4 \by L. Auslander \paper On the Euler characteristic of compact
locally affine spaces \jour Comment. Math. Helv. \vol 35 \yr 1961
\pages 25--27\\ \paper II \jour Bull. Amer. Math. Soc.
\vol 67 \yr1961\pages 405--406 \endref
**
will give
\setref{\ref\no4 \by L. Auslander \paper On the Euler characteristic of compact
locally affine spaces \jour Comment. Math. Helv. \vol 35 \yr 1961
\pages 25--27\\ \paper II \jour Bull. Amer. Math. Soc.
\vol 67 \yr1961\pages 405--406 \endref}

For book references there are *\book*, *\publ* and *\publaddr*.  For
example,
**
\ref\no7\by H. Bass\book Algebraic $K$-theory\publ W. A. Benjamin
\publaddr New York\yr 1968\pages 15--19\endref
**
gives
\setref{\ref\no7\by H. Bass\book Algebraic $K$-theory\publ W. A. Benjamin
\publaddr New York\yr 1968\pages 15--19\endref}
\noindent  Notices that in this cases *\pages* gave ``pp. '' before
the pages.

For a paper that appears in a book, rather than in a journal, just use
*\paper* together with *\inbook*.

Finally, there are *\paperinfo*, *\bookinfo* and *\finalinfo* to put
in extra information after the paper, after the
 book, or at the very end.  Punctuation
around *\paperinfo* and *\bookinfo* is handled automatically.  Punctuation
for *\finalinfo* has to be supplied.







\heading Mathematics\endheading
\subheading{Getting Into Mathematics}
Math formulas in text are indicated by *$...$*; displayed math formulas
by *$$...$$*.  Within a math formula, spaces are completely irrelevant
(except, of course, the spaces needed at the end of control sequences).

\subheading{Special Symbols}
Aside from the symbols $<$, $>$, $+$, $=$ and $|$, which
are probably on your keyboard, you will need control sequences for most
of the other special symbols of mathematics.  They are listed in
Appendix F of The \tex book.  Some important ones are
$$\example
*\le* or *\leq*&$\le$ (``less than or equal'')\\
*\ge* or *\geq*&$\ge$ (``greater than or equal'')\\
*\ne* or *\neq*&$\ne$ (``not equal'')\\
*\in*&$\in$\\
*\notin*&$\notin$\\
*\infty*&$\infty$
\endexample
$$
All the Greek letters have special control sequences to name them,
like *\alpha* ($\alpha$), *\beta* ($\beta$), *\gamma* ($\gamma$) and
*\Gamma* ($\Gamma$).  Several characters have variants:
**
$$(\epsilon,\phi, \theta,      \pi)$$
**
yields
$$(\epsilon,\phi, \theta,      \pi)$$
while
**
$$(\varepsilon, \varphi, \vartheta,\varpi)$$
**
yields
$$(\varepsilon, \varphi, \vartheta,\varpi)$$


\subheading{Switching Fonts}
You can also switch to other fonts within a math formula, exactly as in
text. If you type
**
$${\rm a + \bf b - c}/d$$
**
you will get
$${\rm a + \bf b - c}/d$$
Notice that, just as in text, font changes work for the whole formula, not
just for the next letter.  Only the ``variables'' change fonts; symbols like
$+$ and $-$ stay the same.

There's a font called *\cal* that gives ``calligraphic'' characters:
**
$${\cal A} + B$$
**
gives
$${\cal A} + B$$
This font  should only be used for upper case letters.  However, there's also
a character $\ell$, which you get with the control sequence *\ell*.

One other font change that's useful for math is *\mit*, the ``math italic''
font.  This should be used only for capital Greek letters, to give
things like $\mit \Gamma$ (*$\mit \Gamma$*), as compared to the ordinary
$\Gamma$, which you get simply by typing *$\Gamma$* (or even *$\rm \Gamma$*).



\subheading{Bigger Parentheses, etc} In addition to the ordinary
size parentheses that you get in math formulas, like $f(x+y)$,
you can get slightly larger ones with *\bigl* and *\bigr*, which also
apply to things like *[* and~*]* and *\{* and *\}*:
$$\example
*\bigl( X \bigr)*&$\bigl(X\bigr)$\\
*\bigl[ X \bigr]*&$\bigl[X\bigr]$\\
*\bigl\{X \bigr\}*&$\bigl\{X\bigr\}$\\
*\bigl|X\bigr|*&$\bigl|X\bigr|$\\
*\bigl\|X\bigr\|*&$\bigl\|X\bigr\|$\endexample$$
These are useful for formulas like
$$\bigl(x-f(y)\bigr)\bigl(x+f(y)\bigr)$$
There are also *\biggl* and *\biggr* sizes, which fit nicely around
two-line fractions:
$$\hbox{*\biggl(\biggr)*}\qquad\biggl(\biggr)$$



\subheading{Superscripts and Subscripts}

Superscripts are indicated by *^*, subscripts by *_* (that's the ``underscore''
key, not the hyphen).  You can also use *\sp* and *\sb*.  The
symbols *^* and *_* apply
only to the next character, so you need braces if you want them to apply
to more:
$$\example
*$x^\alpha$*&$x^\alpha$\\
*$x_{x+y}$*&$x_{x+y}$\\
*$x^{(x+y)}$*&$x^{(x+y)}$\\
*$x^{12}$*&$x^{12}$\endexample$$
Notice that *\alpha* is just one thing in \tex's mind, so no braces are
needed there, but the (typeset) number 12 is two things.

For things like
$$a^{b^c}$$
just think of the formula as a mathematician does: the $a$ has a
superscript, and this superscript is---not $b$, but the whole formula
$b^c$.  So you type
**
$$a^{b^c}$$
**

You can have simultaneous superscripts and subscripts:
$$\example
*$A_a^b$*&$A_a^b$\endexample$$

Remember that *\prime* ($\prime$) is special, because you need to superscript
it to get it the right size:
$$\example
*$f^\prime$*&$f^\prime$\\
*$f^{\prime\prime}$*&$f^{\prime\prime}$\endexample$$
But  any number of single quote marks
(*'*) in a row
in a math formula will  translate into
*^{\prime\prime...}* with the appropriate number of *\prime*'s:
$$\example
*$f''_2$*&$f''_2$\endexample
$$
(You'll still need *\prime* for the occasional formula like
$$\example
*$f^{\prime2}$*&$f^{\prime2}$\endexample
$$
which is the common way to typeset such things.)


\subheading{Fractions, etc}
To get a fraction like
$$\frac ab$$
you type
**
$$\frac ab$$
**
and to get
$$1+\frac {a+b}{c+d}$$
you type
\footnote{ If you are enamored of \amstex's old way of doing things, just use
\tex's primitive {\eighttt \bs over}; in the old \amstex\ {\eighttt
\bs frac} was just a
new name for {\eighttt \bs over}, but {\eighttt \bs over} seems to cause
 so much confusion
that it's been replaced by a control sequence that works like all the others.
Sometimes {\eighttt \bs
over} will save a pair of braces (though sometimes the
new {\eighttt \bs frac} will also),
 but you often have to think ahead and put in certain
braces before their time has come, and it seems easier to do things the new
way.}
**
$$1+\frac {a+b}{c+d}$$
**

Notice that
**
$$\frac {\frac ab}{c+d}$$
**
gives
$$\frac {\frac ab}{c+d}$$
where the numerator is now in the size normally used for text.  \tex\ uses
four sizes for mathematics formulas: *\dsize* is the size used for display,
*\tsize* the size for text, *\ssize* the size for superscripts, and
*\sssize* the size for superscripts to superscripts (the letter $\sssize c$
in $a^{b^c}$). You can get the fraction $\frac ab$ in the above formula
bigger by typing
**
$$\frac {\dsize\frac ab}{c+d}$$
**
to get
$$\frac {\dsize\frac ab}{c+d}$$
The size change macros *\dsize*, etc., work just like font changes---they
change sizes permanently within a formula, or in some subformula delimited
by braces.  These are the only macros that work this way.


An easier way to get the above formula is to use *\Frac*, which automatically
makes the *\frac* into *\dsize*:
**
$$\frac {\Frac ab}{c+d}$$
**

In addition to *\frac*, there's *\stack* and *\binom*:
$$\openup \jot\example
*$$\stack ab$$*&$\dsize\stack ab$\\
*$$\binom {n+1}k$$*&$\Binom {n+1}k$\endexample$$
*\binom* is common; *\stack* occurs in only a few special situations, that
we'll see later.  There's also *\Binom* to get the *\binom* in *\dsize* (but
there's no special *\Stack*, since that's almost never wanted).  Finally,
there's *\thickfrac* for a fraction with a thick line (rarely used):
$$\example
*$$\thickfrac {\Frac ab +\Frac cd}{C+D}$$*&$\dsize\thickfrac
   {\Frac ab+\Frac cd}{C+D}$\endexample
$$
^^L
\subheading{Variable Size Symbols}
You can underline and overline formulas:
$$\openup 1.5\jot\example
*$$\underline{x+y+z}$$*&$\dsize\underline{x+y+z}$\\
*$$\overline{x+y+2^{\underline 2}}$$*&$\dsize\overline{x+y+2
    ^{\underline 2}}$\endexample
$$

There are also *\overrightarrow*, a.k.a. *\overarrow*, *\overleftarrow*,
*\overleftrightarrow*, *\underrightarrow*, *\underleftarrow* and
*\underleftrightarrow*, all of which do the obvious thing.

*\sqrt* gives square roots automatically of the right size:
$$\openup\jot\example
*$$\sqrt 2$$*&$\sqrt2$\\
*$$\sqrt{\frac ab+1}$$*&$\dsize\sqrt{\frac ab+1}$\endexample
$$

Most important, you can get variable size parentheses, etc., using
*\left* and *\right*:
$$\openup\jot\example
*$$\left(\frac 1{1-x^2}\right)$$*&$\dsize\left(\frac 1{1-x^2}\right)$\\
*$$\left[\frac 1{1-x^2}\right)$$*&$\dsize\left[\frac 1{1-x^2}\right)$
\endexample$$
Note that the *\left* and *\right* things don't have to match; you can
even do things like *\left]*.  But every *\left* needs a matching
*\right*, so for special things like
$$
\left.\frac {dy}{dx}\right|_{x=a}
$$
there are *\left.* and *\right.* to give a *\left* or *\right* empty
symbol:
**
$$\left.\frac {dy}{dx}\right|_{x=a}$$
**

Other  possible delimiters are:
$$\example
*|* (or *\vert*) &$|$\\
*\|* (or *\Vert*) &$\|$\\
*\{* &$\{$\\
*\}* &$\}$\\
*\lfloor* &$\lfloor$\\
*\rfloor* &$\rfloor$\\
*\langle* &$\langle$\\
*\rangle* &$\rangle$\\
*\slash* &$\slash$\endexample
$$

The last three are special, because they come in only a limited number
of sizes; the others are, in the largest sizes, built up of extensible
pieces:
$$\left(\quad\vbox to 30pt{}\quad\right)$$

\subheading{Large Operators}
The next thing you want to know about are ``large operators'', which
include things like $\int$ (*\int*) and $\sum$ (*\sum*), as well as
related things to be found in Appendix F of The \tex book.
  The important thing to
remember about *\sum* is that when you type *\sum_{i=1}^na_i* between
dollar signs
it comes out as $\sum_{i=1}^na_i$, as expected, but when you type the
very same thing as a displayed equation it comes out as
$$
\sum_{i=1}^na_i
$$
which is the way such formulas are usually displayed: Not only does the
$\sum$ sign change size, but the sub- and
super-script are set as ``limits''.


*\int*s are similar, except that they merely change sign; normally
**
$$\int_a^b f(x)dx$$
**
gives
$$\int_a^b f(x)dx$$

Some printers set things differently, so the format you are using may use
``limits'' for *\int*s, and not use them for *\sum*s.  You can change
things yourself in the *amsspt* style by typing
**
\LimitsOnInts
**
to get limits on *\int*s
and
**
\NoLimitsOnSums
**
to avoid having limits on *\sum*s.  (And, *\NoLimitsOnInts* and
*\LimitsOnSums* get you back again.)  But these should be used only for
changing the *amsppt* style;
most journals insist on their own styles,
 and their style files will then simply ignore
these instructions

Occasionally, no matter what style is being used,
you need to change the way an individual $\sum$ or $\int$ is set. This
is easily done
by following
it with *\limits* or *\nolimits* to tell \tex\ that limits should
or should not be used.  This works even in text.  For example,
*$\sum\limits_{i=1}^Na_i$* gives $\sum\limits_{i=1}^Na_i$, as opposed
to\newline
$\dsize\sum_{i=1}^Na_i$, which we got with *$\dsize\sum_{i=1}^Na_i$*.

Whatever convention is being used for *\sum* will also hold for most
of the other large operators, like $\prod$ (*\prod*).  But the
symbol $\oint$ (*\oint*) will be treated like $\int$.  The same is
true for
$$\openup 1.5\jot\example
*$$\intii$$*&$\dsize\intii$\\
*$$\intiii$$*&$\dsize\intiii$\\
*$$\intiv$$*&$\dsize\intiv$\\
*$$\intdotsint$$*&$\dsize\intdotsint$\endexample$$

 One  of the things to note about *\sum* is that you usually {\sl don't\/}
want to type
**
$$\left(\sum_{i=1}^N a_i\right)$$
**
because it gives
$$\left(\sum_{i=1}^N a_i\right)$$
instead of the preferred
$$\biggl(\sum_{i=1}^N a_i\biggr)$$
This formula was typed using *\biggl* and *\biggr*.

One other thing you sometimes see is something like
$$\sum_{\stack{\ssize1\le i\le N}{\ssize1\le j \le M}}a_{ij}$$
Here we use *\stack*:
**
$$\sum_{\stack{\ssize1\le i\le N}{\ssize1\le j \le M}}a_{ij}$$
**
Notice that *\ssize* was needed for both *1\le i\le N* and *1\le j \le M*,
since they would normally be in *\ssize*, since the whole stack is in
*\ssize*.


\subheading{Spacing}
You seldom have to tell \tex\ to put in spacing in math formulas.  But
there is one case where you always should. The formula
$$\int_a^b f(x)dx$$
given above really isn't right because it's conventional to leave
a certain amount of space before the $dx$.  This space, which printers
call a ``thin space'', is called *\,* by \tex.  So you should type
$$\example
*$$\int_a^bf (x) \, dx$$*&$\dsize\int_a^b f(x)\,dx$\endexample
$$

The other case where you need to specify spacing is in a formula like
$$f(x,y)=0\qquad x,y>0$$
The space before the ``side-condition'' $x,y>0$ is traditionally two
``quads'', where a quad is another standard printer's space.  \amstex\ has
*\quad* for this amount of space and *\qquad* for the more usual
*\quad\quad*, so the above formula was typed as
**
$$f(x,y)=0\qquad x,y>0$$
**


\subheading{Ordinary type}
Sometimes you have a formula like
$$f(x,y)=0\qquad\text{for all }x, y >0$$
that includes ordinary type.  One way to get this is to switch to *\rm*:
**
$$f(x,y)=0\qquad{\rm for all\ }x, y > 0$$
**
Notice that we had to sequester the *\rm* so that it didn't affect
the $x$ and $y$.  Notice also that we had to use *\"space*, since
spaces are ignored in math.  For such situations it is almost always
better to use \amstex's *\text{...}*, which puts you back into ordinary
text:
**
$$f(x,y)=0\qquad\text{for all }x,y>0$$
**
Now all we need to remember is that space after the *all* (it doesn't have
to be *\"space* since we're out of math when we're in *\text*).  Even
easier is to go into *\text* and then within *\text* switch back to
math!
**
$$f(x,y)=0\qquad\text{for all $x,y>0$}$$
**
Just be sure you get the order of *$* signs and *}*s correct.

Sometimes roman type is used in a different way, as in the formula
$$\sin  (x+y) = \sin x \cos y + \cos x\sin y$$
Here $\sin$ stands for a special mathematical operator, and the
spacing follows certain important rules; note, for example, that there
is no space between $\sin$ and $(x+y)$, but there is space between
$\sin$ and $x$.  You don't have to worry about any of this, just use
*\sin*:

**
$$\sin  (x+y) = \sin x \cos y + \cos x\sin y$$
**
Other familiar operators are
$$\vcenter{\halign{#&&\quad#\cr
*\arccos*&*\cos*&*\csc*&*\exp*&*\ker*&*\limsup*&*\min*&*\sinh*\cr
*\arcsin*&*\cosh*&*\deg*&*\gcd*&*\lg*&*\ln*&*\Pr*&*\sup*\cr
*\arctan*&*\cot*&*\det*&*\hom*&*\lim*&*\log*&*\sec*&*\tan*\cr
*\arg*&*\coth*&*\dim*&*\inf*&*\liminf*&*\max*&*\sin*&*\tanh*\cr}}$$
Some of these have ``limits'' just like $\sum$:
$$
\example
*$$\max_{1\le n\le m}\log_2P_n$$*&$\dsize\max_{1\le n\le m}\log_2P_n$
\endexample$$

The control sequence *\operatorname* produces new gadgets like *\sin*,
and *\operatornamewithlimits* produces new gadgets like *\max*.  If you
say
\define\gadget{\operatorname{gadget}}%
\define\Gadget{\operatornamewithlimits{Gadget}}%
**
\define\gadget{\operatorname{gadget}}
\define\Gadget{\operatornamewithlimits{Gadget}}
**
then you can type
**
$$\gadget^2(x+y)-\gadget x + \Gadget_{i<j}a_{ij}$$
**
to get
$$\gadget^2(x+y)-\gadget x + \Gadget_{i<j}a_{ij}$$


\subheading{ Numbered equations} If you type
**
$$x=y\tag 3--1$$
**
you will get
$$x=y\tag 3--1$$
or \TagsOnRight
$$x=y\tag 3--1$$
depending on the format style. \TagsOnLeft%
  The *amsppt* style puts tags on
the left.
Typing *\TagsOnRight* puts them on the right
 (and *\TagsOnLeft* get them back to the left again).
  Like the control sequences *\LimitsOnInts*, etc.,
*\TagsOnRight* and *\TagsOnLeft* will be ignored by the style files for
journals.

Note that the
parentheses, or whatever,
around the tags are put in automatically by the style file.
  Note also that the tag was processed as text, rather than
as mathematics---so the *--* gave an en-dash rather than two hyphens.
For a tag like ``$2'$'' you will
need to type
**
$$x=y\tag {$2'$}$$
**
(making sure to have the math part *$2'$* inside braces). However
*\TagsAsMath* automatically treats all tags as math, so that
*...\tag 2'$$*
will give the tag $(2')$ [while *...\tag \text{2--3}$$* will give the
tag (2--3)]. *\TagsAsText* gets you back to tags treated as text.  Journal
formats will not ignore the control sequences *\TagsAsMath* and *\TagsAsText*,
since these do not actually effect the form of the output, merely the
way that it is specified.


Finally, we ought to mention that a tag might end up on the line
just before an equation (or on the line just after it, in the case of
right-tagging), if the tag and the equation won't fit nicely on one line.
But \tex\ does all this for you automatically, so you don't have to worry.

\subheading{Aligned Equations} You get
$$
\align x&=y=z\\
 x^2 &=y^2\\
 x^3 &\le y^3
  \endalign
$$
by typing
**
$$
\align x&=y=z\\
 x^2 &=y^2\\
 x^3 &\le y^3
  \endalign
$$
**
The *&*'s come right before the symbols which get lined up, and
the *\\*'s separate individual lines.

An *\align*'ed equation is a unit, so that you can get
$$
\define\1{\align a&=b\\a^2&=b^2\endalign}
\define\2{\align c&=d\\c^2&=d^2\\c^3&=d^3\endalign}
\left\{ \1 \right.\qquad\left\{ \2 \right.
$$
by typing the following, which incidentally illustrates how a couple
of on-the-spot *\define*'s can make life easier:
**
$$
\define\1{\align a&=b\\a^2&=b^2\endalign}
\define\2{\align c&=d\\c^2&=d^2\\c^3&=d^3\endalign}
\left\{ \1 \right.\qquad\left\{ \2 \right.
$$
**

Similarly, you can tag an *\align*.  For example,
**
$$\align a&=b\\a^2&=b^2\\a^3&=b^3\endalign \tag 3-1$$
**
comes out as
$$\align a&=b\\a^2&=b^2\\a^3&=b^3\endalign \tag 3-1$$
with the tag centered with respect to the *\align*, since it's the
whole *\align* that's being tagged.

A long broken equation like
$$
\broken (a+b)(a-b) &=(a+b)a - (a + b)b\\
   & = a^2 +ab-ab-b^2\\
  &=a^2-b^2
\endbroken
$$
can be gotten by typing
**
$$
\align (a+b)(a-b) &=(a+b)a - (a + b)b\\
   & = a^2 +ab-ab-b^2\\
  &=a^2-b^2
\endalign
$$
**


You will get the same result if you use *\broken...\endbroken*:
**
$$
\broken (a+b)(a-b) &=(a+b)a - (a + b)b\\
   & = a^2 +ab-ab-b^2\\
  &=a^2-b^2
\endbroken
$$
**
But things work out quite differently when you
 put a *\tag* on a *\broken* equation, like
**
$$\broken a&=b\\&=c\\&=d\endbroken \tag 1--2$$
**
Instead of having the tag centered on the equation you will get either
$$\broken a&=b\\&=c\\&=d\endbroken \tag 1--2$$
or \TagsOnRight
$$\broken a&=b\\&=c\\&=d\endbroken \tag 1--2$$
depending on the format.
\TagsOnLeft

[Some formats center tags on broken equations,
just as if they were an ordinary *\align*.  Typing *\CenteredTagsOnBrokens*
will cause the *amsppt* style to do this, and *\TopOrBottomTagsOnBrokens*
will get the original behavior.  These control sequences are, of course,
ignored by a journal format style.  You can get an individual *\broken*
to be tagged in the center (to save some space, perhaps) by using
the construction
*\cbroken...\endcbroken*.]

*\align* or *\broken* can also be used for broken equations like
$$\align Z=(X+Y)&(A+B+C+D+E+F\\
    &\qquad +G+H+K+L+M+N)\endalign
$$
which we got by typing
**
$$\align Z=(X+Y)&(A+B+C+D+E+F\\
    &\qquad +G+H+K+L+M+N)\endalign
$$
**

  Sometimes an equation is broken as follows:
$$
\multline A+B+C+D+E+F+G+H+I+J+K+L+M+N+P+Q+R+S+T+U\\
   =A'+B'+C'+D'+E'+F'+G'\endmultline$$
with the first line beginning near the left margin, and the second line
ending near the right margin.  A long equation might even require three
or more lines:
$$
\multline A+B+C+D+E+F+G+H+I+J+K+L+M+N\\
  + A'+B'+C'+D'+E'+F'+G'+H'+I'+J'+K'+L'+M'+N'+P'\\
     =P+Q+R+S
\endmultline
$$
You get such displays by using *\multline*:
**
$$
\multline A+B+C+D+E+F+G+H+I+J+K+L+M+N\\
  + A'+B'+C'+D'+E'+F'+G'+H'+I'+J'+K'+L'+M'+N'+P'\\
     =P+Q+R+S
\endmultline
$$
**
The lines between the first and last are centered, but you can shove any
of these to the left or right by typing
$$
\hbox{*\shoveleft{*second line*}*, *\shoveright{*third line*}*, etc.}
$$

When a *\multline* is tagged the tag will either be at the beginning
of the first line or the end of the last line.  One caution here:
If the tag and the first line together won't fit, *\multline* {\sl
won't\/} put the tag on a separate line, you'll have to do it by hand
(by having an empty first equation).  In version 2 of \amstex\ we hope
to fix this problem.

Finally, every once in a while you will need a display consisting of a bunch
of equations that are individually centered, instead of being aligned:
$$\bunch a=b+c\\
  d=e\\
   f+g=h\endbunch$$
You shouldn't type
**
$$a=b+c$$ $$d=e$$ $$f+g=h$$
**
because you'll get too much space between the equations.  Instead
use *\bunch*:
**
$$\bunch a=b+c\\
  d=e\\
   f+g=h\endbunch$$
**
Notice that there {\sl must not\/} be
any *&*'s here, since nothing is being aligned.
Of course, you might well have
 *&*'s within
some *\align* appearing within the *\bunch*.
For example,
**
$$
\bunch a=b+c\\
\align d&=e\\&=f\endalign\\
 f+g=h\endbunch
$$
**
gives
$$
\bunch a=b+c\\
\align d&=e\\&=f\endalign\\
 f+g=h\endbunch
$$

\subheading{Aligned and tagged equations} For something like
$$
\aligntag a&=b=c\tag 1\\
a^2&=b^2\\
a^3&=c^3\tag 2\endaligntag$$
you use *\aligntag*:
**
$$
\aligntag a&=b=c\tag 1\\
a^2&=b^2\\
a^3&=c^3\tag 2\endaligntag$$
**
Like *\multline*, *\aligntag* won't automatically put a tag on a separate
line if necessary---you may get a tag overlapping an equation, without
getting an *Overfull box* message!  In version 2 this should be fixed.

You can have a *\broken* inside an *\aligntag*, with or without a tag,
and {\sl everything} will line up nicely.
For example,
**
$$
\aligntag (a+b)(a+b)&=a^2+2ab+b^2,\tag 1\\
\broken (a+b)(a-b) &=(a+b)a - (a + b)b\\
   & = a^2 +ab-ab-b^2\\
  &=a^2-b^2
\endbroken\tag2\endaligntag$$
**
gives
$$
\aligntag (a+b)(a+b)&=a^2+2ab+b^2,\tag1\\
\broken (a+b)(a-b) &=(a+b)a - (a + b)b\\
   & = a^2 +ab-ab-b^2\\
  &=a^2-b^2
\endbroken\tag2\endaligntag$$
with the $=$ signs of the *\broken* lining up with the ones from the
*\align*.  (*\broken*  wasn't designed to
 behave this way inside a bare *\align* since there's
no point using *\broken* instead of *\align* if you don't have tags to
worry about.)

Between lines of an *\aligntag* you can type *\vspace{"<dim">}* to get
extra vertical space of dimension *"<dim">* between them.  If you
type *\xtext{...}* the extra text *...* will appear on a separate line,
starting at the left margin, without destroying the alignment of the
other lines.

Although an *\aligntag* is usually treated as a unit, like an *\align*,
you can put *\allowbreak* between lines to allow a page break, *\goodbreak*
to indicate an especially nice place to break, and even *\break* to
force a break.

There's also *\bunchtag*, which bears the same relation to *\aligntag*
as *\bunch* bears to *\align*.

\subheading{Matrices} A ``matrix'' is a square array like
$$\matrix
  a&b&c\\a'&b'&c+c'\\a''&b''&c+c'+c''\endmatrix$$
You get this by typing
**
$$\matrix
  a&b&c\\a'&b'&c+c'\\a''&b''&c+c'+c''\endmatrix$$
**
with *\\* separating rows, as usual, and *&* separating elements of
the various columns.
  The elements within each column are centered, and there is a quad of
space between the columns. The number of columns will be the largest
number specified in all the rows;
 rows with fewer specified columns will simply
have a blank formula at the corresponding place.  Thus,
**
$$\matrix
  a\\a'&b'\\a''&b''&c+c'+c''\endmatrix$$
**
gives
$$\matrix
  a\\a'&b'\\a''&b''&c+c'+c''\endmatrix$$

 You can get a different format for a *\matrix* by using
**
\format ...\\
**
right after *\matrix*.  For example, if you want the matrix
$$\matrix \format \c & \quad\r & \quad\l\\
x&1&.1\\
x+y&11&.11\\
x+y+z&111&.111\endmatrix$$
with the first column centered, the second set flush right and the third
set flush left, you type
**
$$\matrix \format \c & \quad\r & \quad\l\\
x&1&.1\\
x+y&11&.11\\
x+y+z&111&.111\endmatrix$$
**
In the format line *\format \c & \quad\r & \quad\l\\* the specifications
for the columns are separated by *&*, just as the elements of each row
are later; the *\c* indicates that the elements of the first column
are centered, the *\quad\r* indicates that the elements of the second
column are set flush right, with a *\quad* of space before the column,
etc.  Any other spacing, like *\,* or *\;* or *\qquad* can be used
instead.

There is also a special way of abbreviating formats with many columns that are
all treated the
same.  Suppose, for example, that we want all centered columns, but
separated by a thin space *\,* instead of by a *\quad*.  We can just use
**
\format \c && \,\c \\
**
The double *&&* means that the remaining specifications are repeated
over and over, for as many columns as is necessary.  Similarly,
**
\format \c && \quad\l & \quad\r \\
**
will center the first column, and then produce columns that are alternately
set flush left and flush right, with all columns separated by a *\quad* of
space.  You mustn't have *&&* twice in a *\format...\\*.


If you have a special format that you
use many times you unfortunately {\sl can't\/} just say
**
\define\specialformat{\format ... \\}
**
and  then save work by typing
**
\matrix\specialformat
  a & b & ...
   ...\endmatrix
**
*\matrix* has to see the explicit sequence *\format ... \\* if it is
used.  (However, you can say, for example,
**
\define\specialformat{\c&&\quad\l&\quad\r}
**
and then
type
**
\matrix\format \specialformat \\
  ...
**
which is almost as good.)

In a *\matrix* the elements are set as math formulas in *\tsize*; use
*\dsize* if you explicitly need it.

Usually a matrix will occur with parentheses or something else around it,
like
$$\matrixp
  a&b&c\\a'&b'&c+c'\\a''&b''&c+c'+c''\endmatrixp$$
You can always put in the parentheses with *\left* and *\right*, but
there's also *\matrixp* and *\endmatrixp* to do it:
**
$$\matrixp
  a&b&c\\a'&b'&c+c'\\a''&b''&c+c'+c''\endmatrixp$$
**
There's also *\matrixv* to give *\left|*, *\matrixvv* for *\left\|* and
*\matrixb* for *\left[*, with corresponding *\endmatrix...*\ constructions.
You can even mix them, to get one thing on the left and another on the
right. But be sure not to begin with *\matrixp*, say, and end simply
with *\endmatrix*, since this gives a *\left(* unbalanced by a *\right* (if
you really want nothing on the right use *\endmatrix\right.*).

For a matrix like \let\hdots=\ldots
$$
\matrixp \format \c&&\;\c\\
a_{11}&a_{12}&\hdots&a_{1n}\\
a_{21}&a_{22}&\hdots&a_{2n}\\
\vdots&\vdots& &\vdots\\
a_{m1}&a_{m2}&\hdots&a_{mn}
\endmatrixp
$$
you use *\hdots* for the horizontal dots, and *\vdots* for the vertical
dots:
**
$$
\matrixp \format \c&&\;\c\\
a_{11}&a_{12}&\hdots&a_{1n}\\
a_{21}&a_{22}&\hdots&a_{2n}\\
\vdots&\vdots& &\vdots\\
a_{m1}&a_{m2}&\hdots&a_{mn}
\endmatrixp
$$
**
(Actually, at the moment you'll have to use *\ldots* (low dots), because
I forgot to add the alternative name *\hdots*.)

You might want to get something like
$$
\matrixp \format \c&&\;\c\\
a_{11}&a_{12}&\hdots&a_{1n}\\
a_{21}&a_{22}&\hdots&a_{2n}\\
\vdots&\vdots&\ddots &\vdots\\
a_{m1}&a_{m2}&\hdots&a_{mn}
\endmatrixp
$$
To get the diagonal dots just use *\ddots*:
**
$$
\matrixp \format \c&&\;\c\\
a_{11}&a_{12}&\hdots&a_{1n}\\
a_{21}&a_{22}&\hdots&a_{2n}\\
\vdots&\vdots&\ddots &\vdots\\
a_{m1}&a_{m2}&\hdots&a_{mn}
\endmatrix
$$
**
If instead you want
$$
\matrixp \format \c&&\;\c\\
a_{11}&a_{12}&\hdots&a_{1n}\\
a_{21}&a_{22}&\hdots&a_{2n}\\
\dotsfor 4\\
\dotsfor 4\\
a_{m1}&a_{m2}&\hdots&a_{mn}
\endmatrixp
$$
you type
**
$$
\matrixp \format \c&&\;\c\\
a_{11}&a_{12}&\hdots&a_{1n}\\
a_{21}&a_{22}&\hdots&a_{2n}\\
\dotsfor 4\\
\dotsfor 4\\
a_{m1}&a_{m2}&\hdots&a_{mn}
\endmatrixp
$$
**
You have to tell *\dotsfor* how many columns it should put dots in for (since
*\matrix* doesn't even know for sure how many columns there are going to
be until it has read the final row).

A construction like
$$
f(x)=\cases x+1, & \text{for $x>0$}\\
           x-1, & \text{for $x\le0$}\endcases
$$
can be constructed from *\matrix* using *\left\{* and *\right.* but there's
also *\cases* to do it all for you:
**
$$
f(x)=\cases x+1, & \text{for $x>0$}\\
           x-1, & \text{for $x\le0$}\endcases
$$
**


A *\matrix* can be used inside another *\matrix* or anything similar.  For
example, you can get the following matrix of parenthesized matrices
$$
\def\1{\matrixp 1&0\\0&1\endmatrixp}
\def\2{\matrixp 0&1\\1&0\endmatrixp}
\def\3{\matrixp 1&1\\1&0\endmatrixp}
\def\4{\matrixp 1&0\\1&1\endmatrixp}
\matrix\format \c\qquad&\c\\
\1&\2\\
\vspace {20pt}
\3&\4
\endmatrix
$$
by typing
**
$$
\def\1{\matrixp 1&0\\0&1\endmatrixp}
\def\2{\matrixp 0&1\\1&0\endmatrixp}
\def\3{\matrixp 1&1\\1&0\endmatrixp}
\def\4{\matrixp 1&0\\1&1\endmatrixp}
\matrix\format \c\qquad&\c\\
\1&\2\\
\vspace {20pt}
\3&\4
\endmatrix
$$
**
Here we've used *\vspace* to specify extra vertical space between
lines.

\subheading{Dots} For things like
$$x_1+\dots+\|(a,\dots,b)\|+ (y_1y_2\dots y_n) + \int\dots\int f(x,y)\,dx\,dy$$
just use *\dots* to get the dots:
**
$$x_1+\dots+\|(a,\dots, b)\|+ (y_1y_2\dots y_n) + \int\dots\int f(x,y)\,dx\,dy$$
**
*\dots* figures out what kind of dots to give on the basis of the next
symbol, so it probably won't give you the right kind of dots if you type
something like
**
$$x_1 + x+2 + x_3 +\dots$$
**
So as a last resort, use *\dotsb*, the dots $\dotsb$\,, together
with the proper spacing, that go between
binary operators and relations, like $+$ and $=$; or *\dotsc*, the dots
and spacing
that go before a comma; or *\dotsj*, the dots
between juxtaposed things; or *\dotsi*, the dots between integral signs.
If you do something weird (even though perfectly legitimate) you may
confuse *\dots* so thoroughly that you'll just end up with an incomprehensible
error message.  Resort to specific *\dots...* in that case also.

\subheading{Accents} The accents in math require different names, and there
are some more of them:
$$\example
*$\dot x$*&$\dot x$\\
*$\dotii x$*&$\dotii x$\\
*$\dotiii x$*&$\dotiii x$\\
*$\dotiv x$*&$\dotiv x$\\
*$\hat x$*&$\hat x$\\
*$\check x$*&$\check x$\\
*$\tilde x$*&$\tilde x$\\
*$\breve x$*&$\breve x$\\
*$\vec x$*&$\vec x$\\
*$\bar z$*&$\bar z$\endexample
$$
The choice between $\bar z$ (*$\bar z$*) and $\overline z$
(*$\overline z$*) is one of taste.

There are also variable size versions of *\hat* and *\tilde*, called
*\widehat* and *\widetilde*:
$$\openup1.5\jot\example
*$\widehat x,\widetilde x$*&$\widehat x, \widetilde x$\\
*$\widehat {xy},\widetilde {xy}$*&$\widehat {xy}, \widetilde {xy}$\\
*$\widehat {xyz},\widetilde {xyz}$*&$\widehat {xyz}, \widetilde {xyz}$
\endexample
$$

Another way of accenting a long formula is to put the accent as a superscript
to the whole formula in parentheses:
$$(x_1+x_2+\dots+x_n)^\tildesymbol$$
You can't type *^\tilde* because *\tilde* isn't a symbol, it's an instruction
to put an accent on something.  But there's *\tildesymbol* which is a symbol:
**
$$(x_1+x_2+\dots+x_n)^\tildesymbol$$
**
Similarly, there's *\hatsymbol*, *\vecsymbol*, *\barsymbol* and
corresponding symbols for all the dot accents.  There also should be
*\checksymbol* and *\brevesymbol*, but I forgot to make them, so they'll
come later.

\subheading{Oversetting and Undersetting}
To get something like
$$\underset \alpha\beta \to {\bf M}$$
type
**
$$\underset \alpha\beta \to {\bf M}$$
**
(The underset formula is between *\underset* and *\to*, so doesn't
have to be in braces even if it's more than one symbol.) Similarly,
you can *\overset* something.

You can also *\underbrace* or *\overbrace* something:
$$\hbox{*\underbrace{x_1+x_2+\dots+x_n}*}\qquad\underbrace{x_1+x_2+\dots+x_n}$$

And to get something like
$$
\undersetbrace n \rm\ times\to {x+\dots+x}
$$
you don't have to *\underset* and *\underbrace*; you can simply use
*\undersetbrace*:
**
$$
\undersetbrace n \rm\ times\to {x+\dots+x}
$$
**

\subheading{Boxed formulas} *\boxed* makes a formula boxed. For example,
**
$$
\boxed{x\le y}\qquad\text{for all $x\in A$, $y\in B$}
$$
**
produces
$$
\boxed{x\le y}\qquad\text{for all $x\in A$, $y\in B$}
$$

\subheading{Roots}  To get
$$
\root \alpha + \beta \of {1+\frac ab}
$$
you type
**
$$
\root \alpha + \beta \of {1+\frac ab}
$$
**
If you don't like the position of the root, you can *\uproot* it
or *\rightroot* it.  For example,
**
$$
\uproot 3\rightroot {-3}\root \alpha + \beta \of {1+\frac ab}
$$
**
gives
$$
\uproot 5\rightroot {-3}\root \alpha + \beta \of {1+\frac ab}
$$
The amount of *\uproot*'ing or *\rightroot*'ing is given in terms
of an arbitrary unit, that you don't specify.

\subheading{Phantoms, Smashes, Etc} As the old ``Joy of \tex'' points
out, there are numerous nefarious tricks that one can play with
*\phantom{...}*, which produces an ``invisible symbol'' that takes
up just as much room as the formula *...*.  Even more useful is
*\vphantom{...}*, which takes up no horizontal space but sticks just
as far above and below the line as *...*, and there is also
*\hphantom{...}*, which takes up no vertical space but as much horizontal
space as *...*.

You can do even more ``eye-balling'' tricks now that \amstex\ has
*\smash{...}*.  This leaves the formula *...* in place but makes
\tex\ think that it doesn't stick above or below the line at all!
And *\topsmash{...}* just smashes the part above the line, while
*\botsmash{...}* smashes the part below.

There's still *\shave*, a specialty discussed in the old ``Joy of
\tex'', but it's much easier to use now:  Just say
**
\shave{\sum_{i=1}^Na_i}
**
to shave off any excess space above the formula
$$
\shave{\sum_{i=1}^Na_i}$$
There's also *\topshave* and *\botshave*, as before.


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