User's Guide for the amsmath Package (Version 2.0)
Contents
1. a21 021 a22 b22 C22 3 4 SPLIT EQUATIONS WITH ALIGNMENT 5 3 8 Split equations without alignment The multline environment is a variation of the equation environment used for equations that don t fit on a single line The first line of a multline 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 multlinegap Any additional lines in between will be centered independently within the display width unless the fleqn option is in effect Like equation multline has only a single equation number it is centered vertically on the total height unless the tbtags option is in effect Thus none of the individual lines need to be marked with notag It s possible to force one of the middle lines to the left or right with com mands Nshoveleft Nshoveright These commands take the entire line as an argument up to but not including the final NN for example 3 10 begin multline framebox 65 columnwidth A framebox 5 columnwidth B shoveright framebox 55 columnwidth C framebox 65 columnwidth D end multline The value of multlinegap can be changed with the usual IXTEX commands setlength or addtolength 3 4 Split equations with alignment Like multline the split environment is for single equations that are too long to fit on one line and hence must be split into multiple lines Unlike multline however the split environment prov2. Explanation In an alignat structure the number of alignment points per line is dictated by the numeric argument given after begin alignat If you use more alignment points in a line it is assumed that you accidentally left out a newline command and the above error is issued L Improper argument for math accent Example Package amsmath Error Improper argument for math accent amsmath Extra braces must be added to amsmath prevent wrong output See the amsmath package documentation for explanation Type H return for immediate help 1 415 tilde k_ lambda_j P_ tilde mathcal M Explanation Non simple arguments for any ATEX command should be enclosed in braces In this example extra braces are needed as follows P_ tilde mathcal M L Font OMX cmex m n 7 cmex7 not loadable Example Font OMX cmex m n 7 cmex7 not loadable Metric TFM file not found lt to be read again gt relax 1 8 a b b 2 h I wasn t able to read the size data for this font 24 10 ERROR MESSAGES AND OUTPUT PROBLEMS So I will ignore the font specification Wizards can fix TFM files using TFtoPL PLtoTF You might try inserting a different font spec e g type I font lt same font id gt lt substitute font name gt 2 Explanation Certain extra sizes of some Computer Modern fonts that were formerly available mainly through the AMSFonts distribution are considered part of standard TFX as of June 199
3. 4 2 Math spacing commands The amsmath package slightly extends the set of math spacing commands as shown below Both the spelled out and abbreviated forms of these commands are robust and they can also be used outside of math Abbrev Spelled out Example Abbrev Spelled out Example no space no space Xs thinspace gt lt negthinspace lt medspace gt lt negmedspace NS thickspace gt lt negthickspace quad gt qquad gt lt 3More precisely The maximum number of columns in a matrix is determined by the counter MaxMatrixCols normal value 10 which you can change if necessary using JATRX s setcounter or addtocounter commands 12 4 MISCELLANEOUS MATHEMATICAL FEATURES For the greatest possible control over math spacing use mspace and math units One math unit or mu is equal to 1 18 em Thus to get a negative Nquad you could write Nnspacet 18 0muJ 4 3 Dots For preferred placement of ellipsis dots raised or on line in various contexts there is no general consensus It may therefore be considered a matter of taste By using the semantically oriented commands e dotsc for dots with commas e Ndotsb for dots with binary operators relations e Ndotsm for multiplication dots e dotsi for dots with integrals e dotso for other dots none of the above instead of ldots and cdots you make it possible for your document to be adapted
4. aoaaa a 6 The text command CONTENTS Koo o ouo o N OO S S CONTENTS 7 Integrals and sums 7 1 Multiple integral signs ee 7 2 Multiline subscripts and superscripts 7 3 The sideset command 204 8 Commutative diagrams 9 Using math fonts 9 1 Introductions X 404 eee SS BS AEA ee ed 9 2 Recommended use of math font commands 9 3 Italic Greek letters 2 2 10 Error messages and output problems 10 1 General remarks 02 2000 020004 10 2 Errroranessages o uou ae al ey ly a al of LAUR oe ee Pe 10 3 Warning messages 2 20 08 6 eo a a y Rd d e do eie 10 4 Wrong output ek wk Re rore OREL Ree hd 11 Additional information 11 1 Converting existing documents 0 11 2 Technical notes 2 4 2 6 66 6 olco a er EE a a a a 11 3 Getting helpossa il WEE Re Xx ies 11 4 Of possible interest 2 Bibliography Index iii 19 19 19 19 20 20 21 21 22 22 22 22 28 28 29 29 29 29 30 30 32 1 Introduction The amsmath package is a TEX package that provides miscellaneous enhance ments for improving the information structure and printed output of documents that contain mathematical formulas Readers unfamiliar with TEX should refer to 7 If you have an up to date version of ATEX the amsmath package is nor mally provided along with it Upgrading when a newer version of the amsmath package
5. 1 Introduction The amsmath package provides a number of additional displayed equation struc tures beyond the ones provided in basic IATEX The augmented set includes equation equation align align gather gather flalign flalign multline multline alignat alignat split Although the standard eqnarray environment remains available it is better to use align or equation split instead Except for split each environment has both starred and unstarred forms where the unstarred forms have automatic numbering using IATEX s equation counter You can suppress the number on any particular line by putting Nnotag before the NN you can also override it with a tag of your own using tag label where label means arbitrary text such as or ii used to number the equation There is also a tag command that causes the text you supply to be typeset literally without adding parentheses around it tag and tag can also be used within the unnumbered versions of all the amsmath alignment structures Some examples of the use of tag may be found in the sample files testmath tex and subeqn tex provided with the amsmath package The split environment is a special subordinate form that is used only inside one of the others It cannot be used inside multline however In the structures that do alignment split align and variants relation symbols have an amp before them but not after unlike eqnarray Putting the amp after the relation symbol
6. 15 hat 13 hdotsfor 11 hom 18 horizontal space around operator names 17 in math mode 11 idotsint 19 iiiint 19 iiint 19 iint 19 inf 18 injlim 18 integrals multiple 19 placement of limits 2 intertext 9 intlimits option 2 ker 18 kuvio package 20 Mabel 6 10 langle 16 ldots 12 left 15 16 25 26 leftroot 13 leqno option 3 Mg 18 Min 1 2 17 18 liminf 18 limits see subscripts and superscripts limsup 18 Man 18 Mog 17 18 lVert 16 lvert 16 34 math fonts 21 math symbols see math fonts mathbb 21 mathbf 21 mathcal 21 mathfrak 21 mathit 21 mathrn 21 mathscr 21 mathsf 21 mathtt 21 matrices 10 ellipsis dots 11 matrix 27 matrix environment 10 max 18 mbox 18 medspace 11 Metafont source files 24 nin 18 mod 18 mspace 12 multline environment 3 5 25 29 multlinegap 5 namelimits option 2 negmedspace 11 negthickspace 11 negthinspace 11 newcommand 21 nobreakdash 12 nointlimits option 2 nolimits 19 nonamelimits option 2 nosumlimits option 2 notag 3 5 numberwithin 10 28 29 operator names 17 operatorname 18 Noperatorname 18 over 15 over 28 overleftarrow 13 overleftrightarrow 13 overrightarrow 13 INDEX overset 14 Noverwithdelims 15 page breaks 8 pagebreak 9 pmatrix 27 pmatrix environment 10 pmod 18 pod 18 PostScript fonts 24 Pr 18
7. There are two possible solutions 1 instead of using left and right use big delimiters of fixed size Nbigl bigr Nbiggl biggr see 4 14 1 or 2 use null delimiters to break up the left right pair into parts for each line AAA left BBB right left CCC right The latter solution may result in mismatched delimiter sizes ensuring that they match requires using vphantom in the line that has the smaller delimiter or possibly smash in the line that has the larger delimiter In the argument of vphantom put a copy of the tallest element that occurs in the other line e g xxx left int_t yyy right left vphantom int_t zzz right LS Paragraph ended before xxx was complete Example Runaway argument Paragraph ended before multline was complete lt to be read again gt par 1 100 h I suspect you ve forgotten a causing me to apply this control sequence to too much text How can we recover My plan is to forget the whole thing and hope for the best Explanation This might be produced by a misspelling in the end multline command e g begin multline end multiline or by using abbreviations for certain environments such as bal and eal for beginfalign and endfalign bal eal For technical reasons that kind of abbreviation does not work with the more complex displayed equation environments of the amsmath package gather align split etc cf technote
8. have the arguments for numberwithin in reverse order numberwithin section equation That means print the section number as equation number section number and reset to 1 every time an equation occurs when what you probably wanted was the inverse numberwithin equation section 11 83 GETTING HELP 29 10 4 2 The numberwithin command had no effect on equation num bers Are you looking at the first section in your document Check the section num bers elsewhere to see if the problem is the one described in 810 4 1 Additional information 11 1 Converting existing documents 11 1 1 Converting from plain ATEX A IATEX document will typically continue to work the same in most respects if usepackage amsmath is added in the document preamble By default how ever the amsmath package suppresses page breaks inside multiple line displayed equation structures such as eqnarray align and gather To continue allowing page breaks inside eqnarray after switching to amsmath you will need to add the following line in your document preamble Nallowdisplaybreaks 1 To ensure normal spacing around relation symbols you might also want to change eqnarray to align multline or equation split as appropriate Most of the other differences in amsmath usage can be considered optional refinements e g using DeclareMathOperator Hom Hom instead of newcommand Hom mbox Hom 11 1 2 Converting from AMS AT X 1 1 See diffs
9. named abbreviation for that notation along the lines of frac and binon The primitive generalized fraction commands over overwithdelims atop atopwithdelims above abovewithdelims produce warning messages if used with the amsmath package for reasons discussed in technote tex 4 12 Continued fractions The continued fraction 4 6 v2 Vm can be obtained by typing cfrac i sqrt 2 cfrac 1 sqrt 2 cfrac 1i sqrt 2 dotsb 1H This produces better looking results than straightforward use of frac Left or right placement of any of the numerators is accomplished by using cfrac 1 or Ncfrac r instead of Ncfrac 4 13 Smash options The command smash 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 amsmath package Nsmash has optional arguments t and b because occasionally it is advantageous to be able to smash only the top or only the bottom of something while retaining the natural depth or height For example when adjacent radical symbols are unevenly sized or positioned because of differences in the height and depth of their contents Nsmash can be employed to make them more consistent Compare V t 4 y z and yz y Vz where the latter was produced by sqrt x sqrt smash b y sqrt z 4 14 Delimiters 4 14 1 Delimiter sizes The automatic de
10. of AxxxB In the second argument of DeclareMathOperator the name text a pseudo text mode prevails the hyphen character will print as a text hyphen rather than a minus sign and an asterisk will print as a raised text asterisk instead of a centered math star Compare a b c and a b x c But otherwise the name text is printed in math mode so that you can use e g subscripts and superscripts there If the new operator should have subscripts and superscripts placed in limits position above and below as with lim sup or max use the form of the DeclareMathOperator command DeclareMathOperator Lim Lim The following operator names are predefined 18 6 THE NTEXT COMMAND Narccos arccos deg deg lg lg Mprojlim projlim arcsin arcsin det det Mim lim sec sec arctan arctan dim dim liminf liminf sin sin arg arg exp exp limsup lim sup sinh sinh cos cos gcd gcd ln In sup sup cosh cosh hom hom log log tan tan cot cot inf inf max max tanh tanh coth coth injlim injlim min min csc csc ker ker Pr Pr varlimsup lim varinjlim lim varliminf lim varprojlim lim There is also a command operatorname such that using operatorname abc in a math formula is equivalent to a use of abc defined by DeclareMathOper ator This may be occasionally useful for constructing more complex notation or other purposes Use the variant Noperatorname to get limits 5 2 mod and its relatives Commands m
11. processing of mathematical documents It is recommended there fore that there should be a one to one correspondence in any given document between the vert bar character and a selected mathematical notation and similarly for the double bar command This immediately rules out the use of and for delimiters because left and right delimiters are distinct usages that do not relate in the same way to adjacent symbols recommended prac tice is therefore to define suitable commands in the document preamble for any paired delimiter use of vert bar symbols providecommand abs 1 lvert 1 rvert providecommand norm 1 lVert 1 rVert whereupon the document would contain abs z to produce z and norm v to produce v B5 Operator names 5 1 Defining new operator names Math functions such as log sin and lim 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 Mog sin Mim and so forth but new ones come up all the time in mathematical papers so the amsmath package provides a general mechanism for defining new operator names To define a math function xxx to work like sin you write DeclareMathOperator xxx xxx whereupon ensuing uses of xxx will produce xxx in the proper font and au tomatically add proper spacing on either side when necessary so that you get Axxx B instead
12. projlim 18 qquad 11 quad 11 12 R 21 raisetag 8 rangle 16 ref 10 reqno option 3 right 15 16 25 26 rVert 16 rvert 16 scriptscriptstyle 14 scriptstyle 14 sec 18 setcounter 10 11 setlength 5 shoveleft 5 shoveright 5 sideset 14 19 20 sin 1 2 17 18 sinh 18 smallmatrix environment 11 smash 15 26 sphat 13 split environment 3 5 6 9 22 25 26 28 29 sptilde 13 stackrel 14 subarray environment 19 subeqn tex 3 subequations environment 10 subscripts and superscripts 14 INDEX multi line 19 on sums 19 placement 2 substack 19 sumlimits option 2 sup 18 superscripts see subscripts and su perscripts tabular environment 25 tag 3 tag 3 tan 18 tanh 18 tbinom 14 tbtags option 2 5 technote tex 15 26 28 29 testmath tex 3 TEX Users Group 30 text 2 7 9 18 text fragments inside math 9 18 textstyle 14 tfrac 14 theequation 9 10 thickspace 11 thinspace 11 TUGboat 30 underleftarrow 13 underleftrightarrow 13 underrightarrow 13 underset 14 uproot 13 usepackage 2 24 value 10 varDelta 22 varGamma 22 varinjlim 18 varLambda 22 varliminf 18 varlimsup 18 varOmega 22 varPhi 22 varPi 22 varprojlim 18 35 varPsi 22 varSigma 22 varTheta 22 varUpsilon 22 varXi 22 Vmatrix environment 10 vmatrix environment 10 vphantom 26 xleftar
13. tex LS Runaway argument See the discussion for the error message Paragraph ended before xxx was complete 10 2 ERROR MESSAGES 27 LS Unknown option xxx for package yyy Example LaTeX Error Unknown option intlim for package amsmath h The option intlim was not declared in package amsmath perhaps you misspelled its name Try typing return to proceed 2 Explanation This means that you misspelled the option name or the package simply does not have an option that you expected it to have Consult the documentation for the given package LE Old form pmatrix should be begin pmatrix Example Package amsmath Error Old form pmatrix should be begin pmatrix See the amsmath package documentation for explanation Type H lt return gt for immediate help pmatrix gt left matrix check pmatrix env matrix 1 16 pmatrix a amp b cr c amp dNcr 7 h pmatrix is old Plain TeX syntax whose use is ill advised in LaTeX Explanation When the amsmath package is used the old forms of pmatrix matrix and cases cannot be used any longer because of naming conflicts Their syntax did not conform with standard IATEX syntax in any case L Erroneous nesting of equation structures Example Package amsmath Error Erroneous nesting of equation structures amsmath trying to recover with aligned See the amsmath package documentation for expla
14. the ATEX manual 7 first edition Chap ter 6 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 LaTeX Error at the beginning and nonalphabetical characters such as V Where examples are given we show also the help messages that appear on screen when you respond to an error message prompt by entering h There is also a section discussing some output errors i e instances where the printed document has something wrong but there was no ATEX error during typesetting 10 2 Error messages L begin split won t work here Example Package amsmath Error begin split won t work here 1 8 begin split h Did you forget a preceding begin equation If not perhaps the aligned environment is what you want Explanation The split environment does not construct a stand alone displayed equation it needs to be used within some other environment such as equation or gather 10 2 ERROR MESSAGES 23 L Extra amp on this line Example Package amsmath Error Extra amp on this line See the amsmath package documentation for explanation Type H return for immediate help 1 9 end alignat 7 h An extra amp here is so disastrous that you should probably exit and fix things up
15. the matrix environments all 4 2 MATH SPACING COMMANDS 11 use more economical horizontal spacing than the rather prodigal spacing of the array environment Also unlike the array 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 If you need left or right alignment in a column or other special formats you must resort to array To produce a small matrix suitable for use in text there is a smallmatrix environment e g 4 5 that comes closer to fitting within a single text line than a normal matrix Delimiters must be provided there are no p b B v V versions of smallmatrix The above example was produced by bigl begin smallmatrix a amp bNN c amp d end smallmatrix bigr hdotsfor number produces a row of dots in a matrix spanning the given number of columns For example a b c d begin matrix a amp b amp c amp dNN C2 dodenus e amp hdotsfor 3 end matrix The spacing of the dots can be varied through use of a square bracket option for example hdotsfor 1 5 3 The number in square brackets will be used as a multiplier i e the normal value is 1 0 Dit Q42t2 g Q1nin a21t Dot e t 4 1 a2101 2 Q2nUn Aniti An2te Dnt 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 hdotsfor 2 4 a_ n1 t_1 amp a_ n2 t_2 amp dots amp D_nt end pmatrix
16. will interfere with the normal spacing it has to go before 3 2 Single equations The equation environment is for a single equation with an automatically gen erated number The equation environment is the same except for omitting the number 1Basic IATFX doesn t provide an equation environment but rather a functionally equiv alent environment named displaymath 4 3 DISPLAYED EQUATIONS Table 3 1 Comparison of displayed equation environments vertical lines in dicating nominal margins begin equation a b a b end equation begin equation a a b end equation begin equation label xx begin split ax b c d a b c d amp Nquad e fNN 2 deeem amp gth gth amp i end split end equation begin multline a b c d e f 3 a b c d e f a Sea ee i j k l m n end multline begin gather a_1 b_itc_1 a_2 b_2 c_2 d_2te_2 end gather A S ay b1 61 N ct wa a2 b2 c2 do e2 begin falign a_1 amp b_1 c_1 6 ay b1 c a_2 amp b_2 c_2 d_2 e_2 7 az bo 09 da z end falign beginfalign a_ 11 amp b_ 11 amp a_ 12 amp b_ 12 8 au bu a2 bie a_ 21 amp b_ 21 amp a_ 22 amp b_ 22 c_ 22 end falign begin flalign a_ 11 amp b_ 11 amp a_ 12 amp b_ 12 a_ 21 amp b_ 21 amp a_ 22 amp b_ 22 c_ 22 end flalign 9 a21 021 a22 b22 C22 a11 b11 12 512
17. 2b2 a2b1 a1b2 left a_1 b_1 a_2 b_2 right left a_2 b_1 a_1 b_2 right quad text versus quad Nbigl a 1 b 1 a 2 b_2 bigr Nbigl a 2 b 1 a 1 b_2 bigr The third kind of situation is a slightly oversize object in running text such as 5 where the delimiters produced by left and right cause too much line spreading In that case Nbigl and bigr can be used to produce delimiters that are slightly larger than the base size but still able to fit within the normal line spacing l In ordinary ATEX big bigg Big and Bigg delimiters aren t scaled properly over the full range of IATEX font sizes With the amsmath package they are 4 14 2 Vertical bar notations The amsmath package provides commands lvert rvert lVert rVert compare langle rangle to address the problem of overloading for the vert bar character This character is currently used in ATEX documents to represent a wide variety of mathematical objects the divides relation in a number theory expression like p g or the absolute value operation z or the such that condition in set notation or the evaluated at notation fe t ay The multiplicity of uses in itself is not so bad what is bad however is that fact 5 1 DEFINING NEW OPERATOR NAMES 17 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
18. 4 cmex7 9 cmmib5 9 and cmbsy5 9 If these extra sizes are missing on your system you should try first to get them from the source where you obtained IATEX If that fails you could try getting the fonts from CTAN e g in the form of Metafont source files directory tex archive fonts latex mf or in PostScript Type 1 format directory tex archive fonts cm ps type1 bakoma If the font name begins with cmex there is a special option cmex10 for the amsmath package that provides a temporary workaround I e change the usepackage to usepackage cmex10 amsmath This will force the use of the 10 point size of the cmex font in all cases Depend ing on the contents of your document this may be adequate Le Math formula deleted Insufficient extension fonts Example Math formula deleted Insufficient extension fonts 1 8 abtb 2 Explanation This usually follows a previous error Font not loadable see the discussion of that error above for solutions L Missing number treated as zero Example Missing number treated as zero to be read again a 1 100 end alignat h A number should have been here I inserted 0 If you can t figure out why I needed to see a number look up weird error in the index to The TeXbook Explanation There are many possibilities that can lead to this error However one possibility that is relevant for the amsmath package is that you forgot to give th
19. User s Guide for the amsmath Package Version 2 0 American Mathematical Society 1999 11 30 Contents 1 Introduction 2 Options for the amsmath package 3 Displayed equations 3 L ntrod ction icem ERE Ok A 3 2 Single equations ele 3 3 Split equations without alignment 3 4 Split equations with alignment 3 5 Equation groups without alignment 3 6 Equation groups with mutual alignment 3 7 Alignment building blocks 3 8 Adjusting tag placement 3 9 Vertical spacing and page breaks in multiline displays 3 10 Interrupting a display 0 0 3 11 Equation numbering 0 0 4 Miscellaneous mathematical features AE Matrices uoo ud nee ere SS BM a a ee 4 0 Math spacing commands ALS ADOUS S e entes nas decre tu dv Be Oe Meat ok 4 4 Nonbreaking dashes sn 4 5 Accents in math 0 0004 A 607 IROOGS 246 4 renidet i feret ar n i a ea t 4 7 Boxed formulas oaa aa a 4 8 Over and under arrows ooo o a a 4 9 Extensible arrows a 000084 4 10 Affixing symbols to other symbols 4 11 Fractions and related constructions 4 12 Continued fractions 04 4 13 Smash options lt aa m meci a i a ara e e AVA Delimtters eai A a ee ee aa E G a aa 3 5 Operator names 5 1 Defining new operator names 5 2 mod and its relatives
20. ace equation numbers level with the last resp first line if numbers are on the right resp left sumlimits default Place the subscripts and superscripts of summation sym bols above and below in displayed equations This option also affects other symbols of the same type J 69 and so forth but exclud ing integrals see below nosumlimits Always place the subscripts and superscripts of summation type symbols to the side even in displayed equations intlimits Like sumlimits but for integral symbols nointlimits default Opposite of intlimits namelimits default Like sumlimits but for certain operator names such as det inf lim max min that traditionally have subscripts placed under neath when they occur in a displayed equation nonamelimits Opposite of namelimits To use one of these package options put the option name in the optional ar gument of the Nusepackage command e g Nusepackage intlimits amsmath The amsmath package also recognizes the following options which are nor mally selected implicitly or explicitly through the documentclass command and thus need not be repeated in the option list of the usepackage amsmath statement 3 2 SINGLE EQUATIONS 3 leqno Place equation numbers on the left reqno Place equation numbers on the right fleqn Position equations at a fixed indent from the left margin rather than centered in the text column 3 Displayed equations 3
21. at each corner of a large operator to put an asterisk at each corner of a product symbol you would type sideset _ _ prod TL g amp Commutative diagrams Some commutative diagram commands like the ones in AMS T X are available as a separate package amscd For complex commutative diagrams authors will need to turn to more comprehensive packages like kuvio or XY pic but for simple diagrams without diagonal arrows the amscd commands may be more convenient Here is one example SY gT T IT S T I Z T J begin CD S mathcal W _ Lambda otimes T gt j gt gt T VVV VV End P V S otimes T I Q Z otimes T J end CD In the CD environment the commands gt gt gt lt lt lt VVV and AAA give respec tively right left down and up arrows For the horizontal arrows material between the first and second gt or lt symbols will be typeset as a superscript and material between the second and third will be typeset as a subscript Sim ilarly material between the first and second or second and third As or Vs of vertical arrows will be typeset as left or right sidescripts The commands and give horizontal and vertical double lines A null arrow command can be used instead of a visible arrow to fill out an array where needed 9 2 RECOMMENDED USE OF MATH FONT COMMANDS 21 9 Using math fonts 9 1 Introduction For more comprehensive information on f
22. aybreak takes an optional argument between 0 and 4 denoting the desirability of the pagebreak displaybreak 0 means it is permissible to break here without encouraging a break display break with no optional argument is the same as displaybreak 4 and forces a break 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 allowdisplaybreaks 1 in the preamble of your document An optional argument 1 4 can be used for finer control 1 means allow page breaks but avoid them as much as possi ble values of 2 3 4 mean increasing permissiveness When display breaks are enabled with allowdisplaybreaks the command can be used to prohibit a pagebreak after a given line as usual Note Certain equation environments wrap their contents in an unbreakable box with the consequence that neither displaybreak nor allowdisplaybreaks will have any effect on them These include split aligned gathered and alignedat 3 10 Interrupting a display The command intertext is used for a short interjection of one or two lines of text in the middle of a multiple line display structure see also the text command in 86 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 intertext may only appear right after a or command Notice the position of the word and in th
23. dy dz A A 7 2 LL 1e tana f ften A A 7 2 Multiline subscripts and superscripts The substack command can be used to produce a multiline subscript or su perscript for example sum_ substack 5 P i j O le i le m 0 lt i lt m 0 lt j lt n O lt j lt n P i j A slightly more generalized form is the subarray environment which allows you to specify that each line should be left aligned instead of centered as here sum_ begin subarray 1 5 P i j i in Lambda 0 lt j lt n i A end subarray aca P i j 7 3 The sideset command There s also a command called sideset for a rather special purpose putting symbols at the subscript and superscript corners of a large operator symbol such as gt or Note this command is not designed to be applied to anything other than sum class symbols 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 nolimits here s sum nolimits E_n in display mode 7 3 25 7 20 9 USING MATH FONTS If however you want not only the prime but also something below or above the sum symbol it s not so easy indeed without Nsideset it would be downright difficult With sideset you can write sideset S7 nEn sum_ n lt k text n odd nE_n n k n odd The extra pair of empty braces is explained by the fact that sideset has the capability of putting an extra symbol or symbols
24. e number argument of an alignat environment as in 10 2 ERROR MESSAGES 25 begin alignat ak b amp c amp d aE b amp c amp d end alignat where the first line should read instead begin falignat 2 Another possibility is that you have a left bracket character following a linebreak command in a multiline construction such as array tabular or eqnarray This will be interpreted by IATEX as the beginning of an additional vertical space request 7 C 1 6 even if it occurs on the following line and is intended to be part of the contents For example beginfarray atb g NN m n endfarray To prevent the error message in such a case you can add braces as discussed in the ATEX manual 7 C 1 1 beginfarray amp bNN L g 3NN m n endfarray Li Missing right inserted Example Missing right inserted lt inserted text gt Might 1 10 end multline h I ve inserted something that you may have forgotten See the inserted text gt above With luck this will get me unwedged But if you really didn t forget anything try typing 2 now then my insertion and my current dilemma will both disappear Explanation This error typically arises when you try to insert a linebreak inside a left right pair of delimiters in multline or split environment begin multline AAA Left BBB CCC right end multline 26 10 ERROR MESSAGES AND OUTPUT PROBLEMS
25. es a journal TUGboat holds meetings and serves as a clearing house of general informa tion about TEX and TpEX related software TEX Users Group PO Box 2311 Portland OR 97208 2311 USA Phone 1 503 223 9994 Email office tug org Membership in the TEX Users Group is a good way to support continued de velopment 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 There is a Usenet newsgroup called comp text tex that is a fairly good source of information about IATEX 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 Bibliography 1 AMSFonts version 2 2 user s guide Amer Math Soc Providence RI 1994 distributed with the AMSFonts package BIBLIOGRAPHY 31 2 Instructions for preparation of papers and monographs AAAS PTEX Amer Math Soc Providence RI 1996 1999 3 Using the amsthm Package Amer Math Soc Providence RI 1999 4 Michel Goossens Frank Mittelbach and Alexander Samarin The ATRX companion Addison Wesley Reading MA 1994 Note The 1994 edition is not a reliable guide for the amsmath package unless you refer to the errata for Chapter 8 file compan err distributed with A TRX 5 G Gratzer Math into ATRX An Introduction to TEX and AMS ETRX ht
26. ftarrow overrightarrow underrightarrow overleftrightarrow underleftrightarrow 4 9 Extensible arrows xleftarrow and xrightarrow produce arrows that extend automatically to accommodate unusually wide subscripts or superscripts These commands take one optional argument the subscript and one mandatory argument the su perscript possibly empty FEi 1 4 3 el aye T C xleftarrow n mu 1 quad xrightarrow T n pm i 1 14 4 MISCELLANEOUS MATHEMATICAL FEATURES 4 10 Affixing symbols to other symbols IATEX provides stackrel for placing a superscript above a binary relation In the amsmath package there are somewhat more general commands Noverset and Nunderset that can be used to place one symbol above or below another symbol whether it s a relation or something else The input overset X will place a superscript size above the X X Nunderset is the analog for adding a symbol underneath See also the description of Nsideset in 87 3 4 11 Fractions and related constructions 4 11 1 The frac dfrac and tfrac commands The frac command which is in the basic command set of IATEX takes two arguments numerator and denominator and typesets them in normal fraction form The amsmath package provides also dfrac and tfrac as convenient abbreviations for displaystyle frac and textstyle frac 1 1 1 4 4 k logs c f i logs c f T logs c f k log c f begin equation frac i
27. ides for alignment among the split lines using amp to mark alignment points Unlike the other amsmath equation structures the split environment provides no numbering because it is intended to be used only inside some other displayed equation structure usually an equation align or gather environment which provides the numbering For example n n eM D I 3 11 le pmi s 6 3 DISPLAYED EQUATIONS begin equation label e barwq begin split H_c amp frac 1 2n sum n_ 1 0 1 41 n 1 p 2 sum_ l _1 dots 1 _p 1 prod p_ i 1 binom n_i 1 _i amp NquadNcdot n 1 n i 1 _i n_i 1 _i cdot MBigl n 1 2 sum p_ j 1 m_i 1 _i 2 Bigr Nend split Nendfequation The split structure should constitute the entire body of the enclosing struc ture apart from commands like label that produce no visible material 3 5 Equation groups without alignment The gather environment is used for a group of consecutive equations when there is no alignment desired among them each one is centered separately within the text width see Table 3 1 Equations inside gather are separated by a bslash command Any equation in a gather may consist of a begin split end split structure for example begin gather first equation begin split second amp equation amp on two lines end split third equation end gather 3 6 Equation groups with mutual alignment The align environment is used for
28. is example 3 22 A No X A 9 3 23 Az A Q A 9 and 3 24 As Nu begin align A_1 amp N_0 lambda Omega Nphi lambda Omega NN A_2 amp phi lambda Omega phi lambda Omega NN intertext and A_3 amp mathcal N lambda omega end align 3 11 Equation numbering 3 11 1 Numbering hierarchy In ATEX if you wanted to have equations numbered within sections that is have equation numbers 1 1 1 2 2 1 2 2 in sections 1 2 and so forth you could redefine theequation as suggested in the IATEX manual 7 86 3 C 8 4 renewcommand theequation thesection arabicfequation 10 4 MISCELLANEOUS MATHEMATICAL FEATURES 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 setcounter To make this a little more convenient the amsmath package pro vides a command Nnumberwithin To have equation numbering tied to section numbering with automatic reset of the equation counter write numberwithin equation section As its name implies the numberwithin command can be applied to any counter not just the equation counter 3 11 2 Cross references to equation numbers To make cross references to equations easier an eqref command is provided This automatically supplies the parentheses around the equation number L e if ref abc pr
29. is released can be done via http www ams org tex amsmath html or ftp ftp ams org pub tex This documentation describes the features of the amsmath package and dis cusses how they are intended to be used It also covers some ancillary packages amsbsy amstext amscd amsxtra amsopn These all have something to do with the contents of math formulas For infor mation on extra math symbols and math fonts see 1 and http www ams org tex amsfonts html For documentation of the amsthm package or AMS document classes amsart amsbook etc see 3 or 2 and http www ams org tex author info html If you are a long time IATEX user and have lots of mathematics in what you write then you may recognize solutions for some familiar problems in this list of amsmath features e A convenient way to define new operator name commands analogous to sin and Mim including proper side spacing and automatic selection of the correct font style and size even when used in sub or superscripts e Multiple substitutes for the eqnarray environment to make various kinds of equation arrangements easier to write e Equation numbers automatically adjust up or down to avoid overprinting on the equation contents unlike eqnarray e Spacing around equals signs matches the normal spacing in the equation environment unlike eqnarray e A way to produce multiline subscripts as are often used with summation or product symbols e An easy way to subs
30. k log_2 c f tfrac i k log_2 c N sqrt frac i k log_2 c f sqrt dfrac i k log_2 c f end equation 4 11 2 The binom dbinom and tbinom commands For binomial expressions such as e amsmath has binom Ndbinom and tbinom 4 5 of Ge a 2 k binom k 1 2 k 1 binom k 2 2 k 2 4 11 3 The genfrac command The capabilities of frac binom and their variants are subsumed by a general ized fraction command genfrac with six arguments The last two correspond to frac s numerator and denominator the first two are optional delimiters as seen in binom the third is a line thickness override binom uses this to set the fraction line thickness to 0 i e invisible and the fourth argument is a mathstyle override integer values 0 3 select respectively displaystyle textstyle scriptstyle and scriptscriptstyle If the third argument is left empty the line thickness defaults to normal Ngenfractleft delim amp right delim thickness mathstyleY numerator denominator To illustrate here is how frac tfrac and binom might be defined newcommand frac 2 genfrac 1 2 newcommand tfrac 2 genfrac 1 1 2 newcommand binom 2 genfrac 0pt 1 2 4 14 DELIMITERS 15 If you find yourself repeatedly using Ngenfrac throughout a document for a particular notation you will do yourself a favor and your publisher if you define a meaningfully
31. limiter sizing done by left and right has two limitations First it is applied mechanically to produce delimiters large enough to encompass the largest contained item and second the range of sizes is not even approxi mately 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 16 4 MISCELLANEOUS MATHEMATICAL FEATURES three situations where the delimiter size is commonly adjusted using a set of commands that have big in their names Delimiter text left bigl MBigl biggl Biggl size size right bigr Bigr biggr Biggr ma Jos e 5 OG OO OG e 3 The first kind of situation is a cumulative operator with limits above and below With left and Nright the delimiters usually turn out larger than necessary and using the Big or bigg sizes instead gives better results j 1 p biggl sum_i a_i Bigl lvert sum_j x_ ij Bigr rvert p biggr 1 p p 1 p Qi Tij versus Qi Tij n z i i j The second kind of situation is clustered pairs of delimiters where left and right make them all the same size because that is adequate to cover the en compassed material but what you really want is to make some of the delimiters slightly larger to make the nesting easier to see a1b1 a2b2 a2b1 a1b2 versus a1b1 a
32. ltiline displays The environments of the amsmath 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 raisetag 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 raisetag 6pt 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 3 9 Vertical spacing and page breaks in multiline displays You can use the dimension command to get extra vertical space between lines in all the amsmath displayed equation environments as is usual in IATEX When the amsmath package is in use page breaks 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 displaybreak command is provided displaybreak is best placed immediately before the where it is to take 3 11 EQUATION NUMBERING 9 effect Like IATEX s pagebreak displ
33. m txt 11 2 Technical notes The file technote tex contains some remarks on miscellaneous technical ques tions that are less likely to be of general interest 11 3 Getting help Questions or comments regarding amsmath and related packages should be sent to American Mathematical Society Technical Support Electronic Products and Services P O Box 6248 Providence RI 02940 Phone 800 321 4AMS 321 4267 or 401 455 4080 Internet tech support ams org 30 BIBLIOGRAPHY If you are reporting a problem you should include the following information to make proper investigation possible 1 The source file where the problem occurred preferably reduced to mini mum size by removing any material that can be removed without affecting the observed problem 2 A IATEX log file showing the error message if applicable and the version numbers of the document class and option files being used 11 4 Of possible interest Information about obtaining AMSFonts or other TEX related software from the AMS Internet archive e math ams org can be obtained by sending a request through electronic mail to webmaster ams org Information about obtaining the amsmath distribution on diskette from the AMS is available from American Mathematical Society Customer Services P O Box 6248 Providence RI 02940 Phone 800 321 4AMS 321 4267 or 401 455 4000 Internet cust serv ams org The TFX Users Group is a nonprofit organization that publish
34. ment of the second accent in doubled math accents is often poor With the amsmath package you will get improved placement of the second accent A hat hat A The commands dddot and ddddot are available to produce triple and quadruple dot accents in addition to the dot and ddot accents already avail able in IATEX To get a superscripted hat or tilde character load the amsxtra package and use Nsphat or sptilde Usage is A sphat note the absence of the character To place an arbitrary symbol in math accent position or to get under accents see the accents package by Javier Bezos 4 6 Roots In ordinary TFX the placement of root indices is sometimes not so good Vk sqrt beta k In the amsmath package leftroot and uproot allow you to adjust the position of the root sqrt leftroot 2 uproot 2 beta k will move the beta up and to the right Wk The negative argument used with leftroot moves the 08 to the right The units are a small amount that is a useful size for such adjustments 4 7 Boxed formulas The command boxed puts a box around its argument like fbox except that the contents are in math mode 4 2 n C n An 0 0 boxed eta leq C delta eta MLambda M O Ndelta 4 8 Over and under arrows Basic IATEX provides overrightarrow and overleftarrow commands Some additional over and under arrow commands are provided by the amsmath package to extend the set overleftarrow underle
35. nation Type H return for immediate help 1 260 end alignat end equation 28 10 ERROR MESSAGES AND OUTPUT PROBLEMS Explanation The structures align alignat etc are designed for top level use and for the most part cannot be nested inside some other displayed equation structure The chief exception is that align and most of its variants can be used inside the gather environment 10 8 Warning messages LS Foreign command over or Natop or above Example Package amsmath Warning Foreign command over frac or genfrac amsmath Should be used instead Explanation The primitive generalized fraction commands of TEX Nover Natop Nabove are deprecated when the amsmath package is used because their syn tax is foreign to IXTEX and amsmath provides native IATEX equivalents See technote tex for further information L Cannot use split here Example Package amsmath Warning Cannot use split here amsmath trying to recover with aligned Explanation The split environment is designed to serve as the entire body of an equation or an entire line of an align or gather environment There cannot be any printed material before or after it within the same enclosing structure begin equation left lt Not allowed begin split end split right lt Not allowed end equation 10 4 Wrong output 10 4 1 Section numbers 0 1 5 1 8 1 instead of 1 2 3 This most likely means that you
36. od bmod pmod pod are provided to deal with the special spacing conventions of mod notation Nbmod and pmod are available in ATEX but with the amsmath package the spacing of pmod will adjust to a smaller value if it s used in a non display mode formula mod and pod are variants of pmod preferred by some authors mod omits the parentheses whereas pod omits the mod and retains the parentheses 5 1 ged n m mod n x y modb x y modc r y d gcd n m bmod n Nquad x equiv y pmod b quad x equiv y mod c quad x equiv y pod d 6 The text command The main use of the command text is for words or phrases in a display It is very similar to the TEX command mbox in its effects but has a couple of advantages If you want a word or phrase of text in a subscript you can type _ text word or phrase which is slightly easier than the mbox equivalent _ mbox scriptsize word or phrase The other advantage is the more descriptive name 6 1 fix 1 2 i monotonic i 1 c 1 7 3 THE SIDESET COMMAND 19 f_ x_ i 1 x_i text is monotonic quad i 1 dots ct1 7 Integrals and sums 7 1 Multiple integral signs iint iiint and iiiint give multiple integral signs with the spacing be tween them nicely adjusted in both text and display style idotsint is an extension of the same idea that gives two integral signs with dots between them 7 1 J rew dx dy JII f a y z dx
37. oduces 3 2 then eqref abc produces 3 2 3 11 3 Subordinate numbering sequences The amsmath package provides also a subequations environment to make it easy to number equations in a particular group with a subordinate numbering scheme For example begin subequations end subequations causes all numbered equations within that part of the document to be numbered 4 9a 4 9b 4 9c if the preceding numbered equation was 4 8 A label command immediately after begin subequations will produce a ref of the parent number 4 9 not 4 9a The counters used by the subequations environ ment are parentequation and equation and addtocounter setcounter value etc can be applied as usual to those counter names To get anything other than lowercase letters for the subordinate numbers use standard BTEX methods for changing numbering style 7 86 3 8C 8 4 For example redefining theequation as follows will produce roman numerals begin subequations renewcommand theequation theparentequation roman equation 4 Miscellaneous mathematical features 4 1 Matrices The amsmath package provides some environments for matrices beyond the basic array environment of I3TEX The pmatrix bmatrix Bmatrix vmatrix and Vmatrix have respectively and delimiters built in For naming consistency there is a matrix environment sans delimiters This is not entirely redundant with the array environment
38. ont use in IXTEX see the IATEX font guide fntguide tex or The ATEX Companion 4 The basic set of math font commands in TFX includes mathbf mathrm Nnathcal mathsf mathtt mathit Additional math alphabet commands such as mathbb for black board bold mathfrak for Fraktur and mathscr for Euler script are available through the packages amsfonts and euscript distributed separately 9 2 Recommended use of math font commands If you find yourself employing math font commands frequently in your document you might wish that they had shorter names such as mb instead of mathbf Of course there is nothing to keep you from providing such abbreviations for yourself by suitable newcommand statements But for IXTEX 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 newcommand vect 1 mathbf 1 you can write vectf a vect b to produce a b If you decide several months down the road that you want to use the bold font for some other purpose and mark vectors by a small over arrow instead then you can put the change into effec
39. r 26 bigl 26 bigr 26 binon 14 15 binomials 14 bm package 2 Bmatrix environment 10 bmatrix environment 10 bmod 18 boxed 13 bslash 6 delimiters AC 21 cases 27 cases environment 8 CD environment 2 20 cdots 12 centertags option 2 cfrac 15 cmbsy5 24 cmex 24 cmex10 24 cmex7 24 cmmib5 24 comp text tex 30 compan err 31 continued fractions 15 cos 18 cosh 18 cot 18 coth 18 csc 18 dbinom 14 ddddot 13 INDEX dddot 13 ddot 13 DeclareMathOperator 2 17 18 Meg 18 det 18 dfrac 14 diffs m txt 29 dim 18 displaybreak 8 9 displayed equations 3 displayed equations centering 3 displaymath environment 3 displaystyle 14 documentclass 2 dot 13 dots see ellipsis dots dotsb 12 dotsc 12 dotsi 12 dotsm 12 dotso 12 e math ams org 30 ellipsis dots in matrices 11 eqnarray environment 1 3 25 29 eqref 10 equation environment 1 3 5 22 29 equation numbers cross references 10 hierarchy 10 left or right placement 3 overriding 3 subordinate numbering 10 vertical placement 2 equation environment 3 equations see displayed equations euscript package 21 exp 18 fbox 13 fleqn option 3 5 fntguide tex 21 frac 14 15 33 fractions 14 fracwithdelims 2 function names see operator names gather environment 5 6 22 26 28 29 gathered environment 7 9 ged 18 genfrac 14
40. row 13 xrightarrow 13
41. t merely by changing the definition of vect otherwise you would have to replace all occurrences of mathbf throughout your document perhaps even needing to inspect each one to see whether it is indeed an instance of a vector It can also be useful to assign distinct command names for different letters of a particular font DeclareSymbolFont AMSb U msb m n or use amsfonts package DeclareMathSymbol C mathalpha AMSb 43 DeclareMathSymbol R mathalpha AMSb 52 These statements would define the commands C and R to produce blackboard bold letters from the AMSb math symbols font If you refer often to the complex numbers or real numbers in your document you might find this method more convenient than let s say defining a field command and writing field C field R But for maximum flexibility and control define such a field command and then define C and R in terms of that command usepackage amsfonts to get the mathbb alphabet newcommand field 1 mathbb 1 newcommand C field C 22 10 ERROR MESSAGES AND OUTPUT PROBLEMS newcommand R field R 9 3 Italic Greek letters For italic versions of the capital Greek letters the following commands are provided varGamma I varSigma X varDelta A varUpsilon Y varTheta O varPhi varLambda A varPsi Y varxXi varOmega 2 varPi II Error messages and output problems 10 1 General remarks This is a supplement to Chapter 8 of
42. titute a variant equation number for a given equation instead of the automatically supplied number e An easy way to produce subordinate equation numbers of the form 1 3a 1 3b 1 3c for selected groups of equations The amsmath package is distributed together with some small auxiliary pack ages amsmath Primary package provides various features for displayed equations and other mathematical constructs 2 2 OPTIONS FOR THE AMSMATH PACKAGE amstext Provides a text command for typesetting a fragment of text inside a display amsopn Provides NDeclareMath perator for defining new operator names like sin and Mim amsbsy For backward compatibility this package continues to exist but use of the newer bm package that comes with IATEX is recommended instead amscd Provides a CD environment for simple commutative diagrams no support for diagonal arrows amsxtra Provides certain odds and ends such as fracwithdelims and Nac centedsymbol for compatibility with documents created using version 1 1 The amsmath package incorporates amstext amsopn and amsbsy The fea tures of amscd and amsxtra however are available only by invoking those packages separately Options for the amsmath package The amsmath package has the following options centertags default For a split equation place equation numbers vertically centered on the total height of the equation tbtags Top or bottom tags For a split equation pl
43. 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 Then we have the series A 1 A_2 Then we have the series A1 A a dotsc the regional sum A_1 the regional sum A A2 the A_2 dotsb the orthogonal orthogonal product A A2 and product A 1 A 2 dotsm and the infinite integral the infinite integral int_ A_1 int_ A_2 dotsi 3 A1 J Aa 4 4 Nonbreaking dashes A command nobreakdash is provided to suppress the possibility of a linebreak after the following hyphen or dash For example if you write pages 1 9 as pages 1 nobreakdash 9 then a linebreak will never occur between the dash and the 9 You can also use nobreakdash to prevent undesirable hyphen ations in combinations like p adic For frequent use it s advisable to make abbreviations e g newcommand p p nobreakdash for p adic newcommand Ndash nobreakdash for pages 1 Ndash 9 For Xn dimensional n dimensional newcommand n 1 n nobreakdash hspace Opt 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 4 99 EXTENSIBLE ARROWS 13 4 5 Accents in math In ordinary IATEX the place
44. tp www ams org cgi bin bookstore bookpromo fn 91 amp argi bookvideo amp itmc MLTEX Birkh user Boston 1995 6 Donald E Knuth The TgXbook Addison Wesley Reading MA 1984 7 Leslie Lamport ATRX A document preparation system 2nd revised ed Addison Wesley Reading MA 1994 8 Frank Mittelbach and Rainer Sch pf The new font family selection user interface to standard ATRX TUGboat 11 no 2 June 1990 pp 297 305 9 Michael Spivak The joy of TEX 2nd revised ed Amer Math Soc Provi dence RI 1990 Index 11 Nog d 11 11 3 5 8 9 23 25 9 mathbb 21 VM 17 above 15 above 28 abovewithdelims 15 accentedsymbol 2 accents package 13 addtocounter 10 11 addtolength 5 align environment 3 5 7 26 28 29 alignat environment 7 23 24 28 aligned environment 7 9 alignedat environment 9 allowdisplaybreaks 9 amsart class 1 amsbook class 1 amsbsy package 1 2 amscd package 1 2 20 amsfonts package 21 AMSFonts collection 24 amsmath package i 1 3 5 8 10 11 13 18 24 26 31 amsopn package 1 2 amstext package 1 2 amsthm package 1 31 amsxtra package 1 2 13 Narccos 18 Narcsin 18 Narctan 18 Narg 18 array environment 10 11 25 arrows extensible 13 in commutative diagrams 20 32 atop 15 Natop 28 atopwithdelims 15 BaKoMa fonts 24 big MBig bigg 16 biggl 26 bigg
45. two or more equations when vertical align ment is desired usually binary relations such as equal signs are aligned see Table 3 1 To have several equation columns side by side use extra ampersands to sep arate the columns 3 12 Lr y X Y a b c 3 13 a y PX e a b 3 14 ata y y X X Y Y a b cb begin align x amp y amp X amp Y amp a amp b cNN x amp y amp PY amp a amp bNN xt x amp yty amp X X amp Y Y amp a bk c b end falign Line by line annotations on an equation can be done by judicious application of 3 7 ALIGNMENT BUILDING BLOCKS 7 text inside an align environment 3 15 Yi Y2 Y3 Ys t ys by 3 21 3 16 y o y by 4 1 3 17 y 0 y by Axiom 1 begin align xk y_1 y_2 y_3 y_5 y_8 dots amp amp text by eqref eq C amp y circ y amp amp text by eqreffeq D amp y 0 y amp amp text by Axiom 1 end falign A variant environment alignat allows the horizontal space between equations to be explicitly specified This environment takes one argument the number of equation columns count the maximum number of amp s in any row add 1 and divide by 2 3 18 Y1 yoty3 Ys5tys by 3 21 3 19 y oy by 4 1 3 20 y 0 y by Axiom 1 begin alignat H 2 x amp y_1 y_2 y_3 y_5 y_8 dots amp Nquad amp text by eqref eq C amp y Ncirc y amp amp text b
46. y eqref eq D amp y O y amp amp Ntext by Axiom 1 end alignat 3 7 Alignment building blocks Some other equation alignment environments such as aligned and gathered construct self contained units that can be used inside of other expressions or set side by side These environments take an optional argument to specify their vertical positioning with respect to the material on either side The default is middle placement with the vertical midpoint of the total unit falling on the math axis For example a aa B 408888 versus 06 i cif n mmm P begin equation begin aligned alpha amp alpha alpha The height of the cross bar in the symbol 8 3 DISPLAYED EQUATIONS betak beta beta beta beta beta gamma amp gamma end aligned NqquadNtext versus Nqquad begin aligned t deltak delta delta etak eta eta eta eta eta eta Nvarphi amp Nvarphi end aligned endfequation Cases constructions like the following are common in mathematics 0 if r j is odd 3 21 uj Qo 2 n 1 0 2 2 ifr j is even and in the amsmath package there is a cases environment to make them easy to write P_ r j begin cases O amp text if r j is odd r N C 1 r j 2 amp text if r j is even end cases Notice the use of text cf 6 and the nested math formulas 3 8 Adjusting tag placement Placing equation numbers can be a rather complex problem in mu
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