STAT METHODOLOGY I
STAT METHODOLOGY I STAT 613
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Short Math Guide for MEX Michael Downes American Mathematical Society Version 109 20020322 currently available at httpwwwams orgtexshortmathguidehtml 1 Introduction This is a concise summary of recommended features in ETEX and a couple of extension packages for Writing math formulasi Readers needing greater depth of detail are referred to the sources listed in the bibliography especially Lamport LUG AMUG LFG LGG and LC A certain amount of familiarity with standard ETEX terminology is assumed if your memory needs refreshing on the ETEX meaning of command optional argument environment package and so forth see Lamportli The features described here are available to you if you use ETEX with two extension packages published by the American Mathematical Society amssymb and amsmathi Thus the source le for this document begins with documentclassfart icle us epac kagefams symb amsmat h The amssymb package might be omissible for documents whose math symbol usage is rela tively modest the easiest way to test this is to leave out the amssymb reference and see if any math symbols in the document produce Unde ned control sequence7 messages any noteworthy features found in other packages are not covered here see Section 10 Regarding math symbols please note especially that the list given here is not intended to be comprehensive but to illustrate such symbols as users will normally nd already present in their ETEX system and usable without installing any additional fonts or doing other setup work If you have a need for a symbol not shown here you will probably want to consult The Comprehensive E ITEX Symbols List Pakin http www ctan orgt exarchiveinfosymbolscomprehens ivei 2 Inline math formulas and displayed equations 21 THE FUNDAMENTALS Entering and leaving math mode in ETEX is normally done with the following commands and environments inline formulas displayed equations unnumbered beg inequat ion unnumbered endequat ion beginfequat ion automatically i i i numbered endfequat ion Note Alternative environments begin nath V V V end nath begindisp1aymath V V V endfdisplaymath are seldom needed in practice Using the plain TEX notation v V V for displayed equations is not recome mended Although it is not expressly forbidden in ETEX it is not documented anywhere in the ETEX book as being part of the ETEX command set and it interferes with the proper operation of various features such as the f leqn option Environments for handling equation groups and multiline equations are shown in Table l Short Math Guide for EMEX version 109 200203 22 2 Table 1 Multiline equations and equation groups vertical lines indicating nominal mar i beginequation1abe1xx beginsp1it aamp bcd ab67d amp quad ef e 7 f amp gh g h amp i endsp1it endequation beginmu1t1ine abcdef abcdef ijk1mn ijklmn 2 2 endmu1t1ine begingather a1b1c1 a1 121 Cl 23 a2b2c2d2e2 a2 b2 52 7 d2 e2 24 endgather begina1ign a1amp b1c1 a1 bl 01 2A5 a2amp b2c2d2e2 a2 b2 52 7 d2 e2 26 enda1ign begina1ign a11amp b11amp a12amp b12 an 1211 an 212 2 7 a21amp b21amp an b21 L122 b22 622 2 8 a22amp b22C22 enda1ign beginf1a1ign a11amp b11amp a12amp b12 all bu 12 1112 2 1 a21amp b21amp a22amp b22c22 endf1a1ign a21 1121 a22 1122 C22 Note 1 The split environment is something of a special case It is a subordinate environment that can be used as the contents of an equation environment or the contents of one line in a multipleequation structure such as align or gather Note 2 The eqnarray and eqnarrayxoK environments described in Lamport are not recommended because they produce inconsistent spacing of the equal signs and make no attempt to prevent overprinting of the equation body and equation number Short Math Guide for ETEX version 109 200203 22 3 22 AUTOMATIC NUMBERING AND CROSS REFERENCING To get an autonumbered equa tion7 use the equation environment to assign a label for crossreferencing7 use the 1abe1 comman beginequation1abe1reio endequation To get a crossreference to an autonumbered equation7 use the eqref command using equations eqrefax1 and eqrefb22 we can derive The above example would produce something like using equations 32 and 357 we can derive In other words7 eqrefax1 is equivalent to reffax1l To give your equation numbers the form mm sectionnumberequationnumber use the numberwithin command in the preamble of your document numberwithinequationsect ion For more details on custom numbering schemes see Lamport7 6l37 Cl8l4ll The subequat ions environment provides a convenient way to number equations in a group with a subordinate numbering scheme For example7 supposing that the current equation number is 217 write beginequation1abe1first abc endequation some intervening text beginsubequations1abe1grp beginalign aampbc1abe1second dampefg1abe1third hampij1abe1fourth endfalign endfsubequat ions to get a b 5 29 some intervening text a b 5 210a defg leOb h i j 210C By putting a 1abe1 command immediately after beginfsubequat ions you can get a reference to the parent number eqrefgrp from the above example would produce 210 while eqrefsecond would produce 210a 3 Math symbols and math fonts Sill CLASSES OF MATH SYMBOLS The symbols in a math formula fall into different classes that correspond more or less to the part of speech each symbol would have if the formula were expressed in words Certain spacing and positioning cues are traditionally used for the different symbol classes to increase the readability of formulas Short Math Guide for ETEX version 109 200203 22 4 Class Description number 39 part of speech Examples 0 Ord Simpleordinary noun A 0 39i39 00 1 Op pre x operator 2 H f 2 Bin binary operator conjunction U 3 Rel relationcomparison verb lt C 4 Open leftopening delimiter 5 Close rightclosing delimiter l 6 Pun post Xpunctuation i l Note 1 The distinction in TEX between class 0 and an additional class 7 has to do only with font selection issues and is immaterial here Note 2 Symbols of class Bin notably the minus Sign 7 are automatically coerced to class 0 no space if they do not have a suitable left operan V The spacing for a few symbols follows tradition instead of the general rule although is semantically speaking of class 2 we write 162 with no space around the slash rather than k 2 And compare plq plq no space with pmid q p l q class3 Spacing The proper way to de ne a new math symbol is discussed in E TEXZE font selection LFGli It is not really possible to give a useful synopsis here because one needs rst to the 39C 39 of font 39 32 SOME SYMBOLS INTENTIONALLY OMITTED HERE The following math symbols that are mentioned in the ETEX book Lamport are intentionally omitted from this discussion because they are superseded by equivalent symbols when the amssymb package is loaded If you are using the amssymb package anyway the only thing that you are likely to gain by using the alternate name is an unnecessary increase in the number of fonts used by your document Box see square D Diamond see 1ozenge ltgt 1eadsto see rightsquigarrow w Join see bowt ie gtlt1 1hd see vartriangleleft lt1 unlhd see trianglelefteq 1 rhd see vartriangleright D unrhd see trianglerighteq E Furthermore there are many7 many additional symbols available for ETEX use above and beyond the ones included here This list is not intended to be comprehensive For a much more comprehensive list of symbols including nonmathematically oriented ones such as phonetic alphabetic or dingbats see The Comprehensive LATEX Symbols List Pakin http www ctan orgt exarchiveinfosymbolscomprehens ivei 33 LATIN LETTERS AND ARABIC NUMERALS The Latin letters are simple symbols class 0 The default font for them in math formulas is italici ABCDEFGHIJKLMNOPQRSTUVWXYZ abodefghij klmnopqrstuvwzyz When adding an accent to an i or j in math dotleSS variants can be obtained with imath and jmath i imath jmath j hatjmath Arabic numerals 079 are also of class or Their default font is uprightromani 0123456789 Short Math Guide for EZEX version 109 200203 22 5 34 GREEK LETTERS Like the Latin letters the Greek letters are simple symbols class 0 For obscure historical reasons the default font for lowercase Greek letters in math formu las is italic while the default font for capital Greek letters is uprightroman In other elds such as physics and chemistry however the typographical traditions are somewhat different The capital Greek letters not present in this list are the letters that have the same appearance as some Latin letter A for Alpha B for Beta and so on In the list of lowercase letters there is no omicron because it would be identical in appearance to Latin 0 In practice the Greek letters that have Latin lookalikes are seldom used in math formulas to avoid confusion F Gamma 1 alpha 5 xi F digamma A Delta 6 beta 7r pi 5 varepsilon A Lambda 7 gamma p rho varkappa 39i39 Phi 6 delta 0 sigma go varphi H Pi e epsilon 739 tau 1 varpi 11 Psi Q zeta 7 upsilon g varrho 2 Sigma 77 eta lt15 Phi g varsigma 9 Theta t9 theta X chi 19 vartheta T Upsilon L iota 1 psi E Xi H kappa w omega 9 Omega A 1ambda u mu 1 nu 35 OTHER ALPHABETIC SYMBOLS These are also class 0 N aleph C complement 71 hslash circledS 3 Im 1 beth Z ell U mho lk Bbbk 9 Re 1 daleth 5 eth E partial j Finv J gimel h hbar p wp D Game 36 MISCELLANEOUS SIMPLE SYMBOLS These symbols are also of class 0 ordinary which means they do not have any builtin spacingi clubsuit ltgt 1ozenge D square amp amp diagdown 1 measuredangle surd 1 angle diagup V nabla T top backprime O diamondsuit h natural A triangle bigstar 0 emptyset neg V triangledown Q blacklozenge 3 exists E nexists varnothing I blacksquare l7 flat prime A blacktriangle V forall ll sharp V blacktriangledown Q7 heartsuit Q spadesuit l bot 00 infty lt1 sphericalangle Note 1 A common mistake in the use of the symbols D and is to try to make them serve as binary operators or relation symbols without using a properly de ned math symbol command If you merely use the existing commands square or the interesymbol spacing will be incorrect because those commands produce a class70 symbol Note 2 Synonyms lnot Shani Math Guide for EMEX version 109 200203 22 37 BINARY OPERATOR SYMBOLS 9 cdot centerdot 7 o circ H amalg QB circledast 9 ast circledcirc K barwedge G circleddash O bigcirc U cup V bigtriangledown M Cup A bigtriangleup D boxdot El boxminus EH boxplus X boxtimes o bullet cap rm Cap Y curlyvee A curlywedge i dagger i ddagger ltgt diamond div gt26 divideontimes 4 dotplus i doublebarwedge gt gtrdot T intercal X 1eftthreetimes lt 1essdot gtlt 1times 3F mp D odot 9 ominus EB oplus oslash otimes i pm A rightthreetimes gt4 rtimes setminus Synonyms A 1and V 1or LUJ doublecup rm doublecap 3 8 RELATION SYMBOLS lt gt gt N AND VARIANTS lt lt gt gt m approx cong curlyeqprec curlyeqsucc i doteq doteqdot W A HZ H ltgt E eqslantgtr lt eqslantless E equiv fallingdotseq gt geqslant gtgt gg gtgtgt ggg i gnapprox gtrapprox gt g gtreqless E gtreqqless 2 gtrless 2 gtrsim g gvertneqq S 1e 2 1eqq lt 1eqslant g 1essapprox lt E 1essqutr i 1esseqqgtr g 1essgtr S 1esssim ltlt 11 ltltlt 111 g 1napprox 5 ngeqslant Synonyms 7 ne S 1e 2 ge Doteq ltltlt llless gtgtgt gggtr y ngtr f nleq jg nleqq nleqslant nless g precapprox lt preccurlyeq j preceq g precnapprox precneqq j precnsim j precsim risingdotseq smallsetminus sqcap Ll sqcup star gtlt times lt1 triangleleft D triangleright L uplus V vee X veebar wedge Z wr N sim 2 simeq gt succ E succapprox gt succcurlyeq succeq succnapprox i succneqq succnsim succsim z thickapprox N thicksim triangleq Shani Math Guide for EMEX version 109 200203 22 39 RELATION SYMBOLS ARROWS See also Section 4 O circlearrowleft C circlearrowright n curvearrowleft m curvearrowright Li downdownarrows J downharpoonleft L downharpoonright lt gt hookleftarrow lt gt hookrightarrow lt7 1eftarrow 5 Leftarrow lt 1eftarrowtai1 w 1eftharpoondown 1 1eftharpoonup 1eft1eftarrows lt gt 1eftrightarrow W 1eftright squigarrow E Lleftarrow lt7 1ongleftarrow lt Longleftarrow lt gt 1ongleftrightarrow ltgt Longleftrightarrow gt gt 1ongmapsto A 1ongrightarrow gt Longrightarrow lt P 1ooparrow1eft 9 1ooparrowright 1 Lsh gt gt mapsto w multimap Qt nLeftarrow lt1 nLeftrightarrow 9b nRightarrow nearrow 6L nleftarrow ltgt nleftrightarrow a9 nrightarrow Synonyms lt gets a to I restriction 310 RELATION SYMBOLS MISCELLANEOUS 9 bac keps i1 on 4 blacktr iangl e1 eft gt blacktr iangl er ight gtlt1 bowt i e g nsubseteqq ntrianglelefteq 5 ntriangleright nwarrow A rightarrow Rightarrow gt gt rightarrowtail rightharpoondown A rightharpoonup rightleftarrows rightleftharpoons rightrightarrows w rightsquigarrow 3 Rrightarrow F Rsh searrow swarrow 6H twoheadleftarrow a twoheadrightarrow upharpoonleft I upharpoonright TT upuparrows therefore 31 trianglelefteq E trianglerighteq OC varpropto g varsubsetneq g varsubsetneqq v smallsmile v smile E sqsubset E sqsubseteq j sqsupset Q sqsupseteq dashv ntrianglerighteq C subset 2 varsupsetneq A frown V nvdash Subset 2 varsupsetneqq E in M4 anash Q subseteq A vartriangle mid nvDash g subseteqq lt1 vartriangleleft models Hf nVDash g subsetneq D vartriangleright 3 ni parallel subsetneqq F vdash f nmid L perp D supset F Vdash notin m pitchfork 3 Supset D vDash H nparallel Olt propto D supseteq HF Vvdash I nshortmid I shortmid 2 supseteqq H nshortparallel H shortparallel 2 supsetneq g nsubseteq A smallfrown 2 supsetneqq Synonyms 3 owns 311 CUMULATIVE VARIABLE SIZE OPERATORS f int Q bigodot L biguplus H prod oTnt QB ngopI39Lus ngvee f Smallint blgcap blgotlmes blgwedge U bigcup U bigsqcup H coprod 2 sum Short Math Guide for EMEX version 109 200203 22 8 312 PUNCTUATION i r 7 i i v dotsm vdots i colon dotsb dotso i I z i i dotsc l dotsi H ddots Note 1 The by itself produces a colon with class relation spacing The command colon produces special spacing for use in constructions such as fcolon Ato B f A a B N ate 2 Although the commands cdots and ldots are frequently used We recommend the more semani tically oriented commands dotsb dotsc dotsi dotsm dotso for most purposes see 46 313 PAIRING DELIMITERS EXTENSIBLE See Section 6 for more information ltgt 1Vert rVert 1group rgroup H lt gt 1ang1e rangle I 1 1moustache rmoustache gt 1brace rbrace 1cei1 rceil 1vert rvert 1floor rfloor 314 NONPAIRING EXTENSIBLE SYMBOLS vert Vert backslash arrowvert Arrowvert bracevert Note 1 Using vert I Vert or I for paired delimiters is not recommended see 62 Synonyms l 315 EXTENSIBLE VERTICAL ARROWS T uparrow W Uparrow l I l quot1 316 ACCENTS z acutex i barx E vecx ii widetildexxx i gravex i brevex z39 dotx m widehatxxx z ddotx i checkx i ddotx i t i1dex i hatx dddotx 317i NAMED OPERATORS These operators are represented by a multiletter abbreviation arccos arccos csc csc inj lirn injlim max max tan tan arcsin arcsin deg deg ker ker rnin min tanh tanh arctan arctan det det lg 1g Pr Pr varinjlim arg arg dirn dim lirn 1im proj lirn projlim hm varprojlim cos cos exp exp lirninf 1iminf sec sec varliminf cosh cosh gcd gcd hrn sup 1imsup sin sin m varlimsup cot cot horn hom ln 1n s1nh sinh coth coth inf inf log 1og sup sup To de ne additional named operators outside the above list7 use the DeclareMathOperat or command for exarnple7 after Short Math Guide for EMEX version 109 200203 22 9 DeclareMat hUperat orrankrank DeclareMathUperat oresssupess sup one could write rankx rankz esssupyz esssupyz The star form DeclareMathUperat or creates an operator that takes limits in a displayed formula like sup or maxi en prede ning such a named operator is problematic eg when using one in the title or abstract of an article there is an alternative form that can be used directly operat ornamefrankx A rankz 318i MATH FONT SWITCHES Not all of the fonts necessary to support comprehensive math font switching are commonly available in a typical ETEX setupi Here are the results of applying various font switches to a wide range of math symbols when the standard set of Computer Modern fonts is in user It can be seen that the only symbols that respond correctly to all of the font switches are the uppercase Latin letters In fact nearly all math symbols apart from Latin letters remain unaffected by font switches and although the lowercase Latin letters capital Greek letters and numerals do respond properly to some font switches they produce bizarre results for other font switchesi Use of alternative math font sets such as Lucida New Math may ameliorate the situation somewhatl default mathbf mathsf mathit mathcal mathbb mathfrak X X X X X X as z x X 9 n 1 0 0 0 0 l4 o H H l l l l H E E E E 7i g 5 5 5 5 5 5 5 N N N N N N N E E E E E E E H H H H H H H 9 9 9 9 9 9 9 A common desire is to get a bold version of a particular math symboll For those symbols where mathbf is not applicable the boldsymbol or pmb commands can be used A00 WAD N A00 739er N A00 1er 3 1 Ainfty pi A0 sim mathbffAboldsymbolinfty boldsymbol boldsymbolpi mathbfAboldsymbol0 simpmbApmbinfty pmbpmbpi pmbApmbO The boldsymbol command is obtained preferably by using the bm package which provides a newer more powerful version than the one provided by the amsmath packager Generally speaking it is illadvised to apply boldsymbol to more than one symbol at a timer Short Math Guide for ETEX version 109 200203 22 10 3181 Calligraphic letters cmsy no lowercase Usage mathcal CM ABC39DEfQHIJK MAOQRSTMVWXJZZ 3182 Blackboard Bold letters msbm no lowercase Usage mathbb CR ABCDEFGHHJKLMNOJPQRSTUVWXYZ 3183 Fraktur letters eufm Usage mathfrak CS Qt 36533 WMDQBQDQGTUEU23333 abcbefghij lmnopqrstunmmg 4 Notations 41 TOP AND BOTTOM EMBELLISHMENTS These are visually Similar to accents but gen erally span multiple symbols rather than being applied to a single base symbol For ease of reference7 widet ilde and widehat are redundantly included here and in the table of math accents ii widet i1dexxx riff overbracexxx zzz overrightarrowxxx m widehatxxx III underbracexxx LMgt underrightarrowxxx 7 V In overleftrightarrowxxx III Overllne CXXX III overleftarrowxxx zzz underleftr1ghtarrowxxx III underlinexxx III underleftarrowxxx 42 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 superscript7 possibly empty L B L51 0 41 x1 eftarr ownmu 1 quad xrightarrow T npm i 1 43 AFFIXING SYMBOLS TO OTHER SYMBOLS In addition to the standard accents Sec tion 3167 other symbols can be placed above or below a base symbol with the overset and underset commands For example7 writing oversetX will place a superscriptSize above the X7 thus X See also the description of sideset in Section 84 44 MATRICES The environments pmatrix7 bmatrix7 Bmatrix7 matrix and Vmatrix have respectively 7 H l l and delimiters built in There is also amatrix environ ment sans delimiters7 and an array environment that can be used to obtain left alignment or other variations in the column Specs beginpmatrix alphaamp beta 1 5 gamma amp delta lt7 6 endpmatrix To produce a small matrix suitable for use in text7 there is a smallmatrix environment eg7 that comes closer to tting within a Single text line than a normal matrix This example was produce y Short Math Guide for ETEX version 109 200203 22 11 bigl beginfsmallmatrix aampb campd endfsmallmatrix bigr To produce a row of dots in a matrix spanning a given number ofcolumns7 use hdot sf or For example7 hdotsforfS in the second column of a fourcolumn matrix will print a row of dots across the nal three columns For pieceWise function de nitions there is a cases environment Prjbegincases 0amp textfif rj is odd I 1 quotrj 2amp textif rj is even endfcases Notice the use of t ext and the embedded math Note The plain TEX form matrixf cr cr and the related commands pmatrix cases should be avoided in IA TEX and when the amsmath package is loaded they are disabled 45 MATH SPACING COMMANDS When the amsmath package is used7 all of these math spacing commands can be used both in and out of math mo e Abbrev Spelled out Example Abbrev Spelled out Example no space 34 no space 3 thinspace 34 negthinspace 3i medspace 3 4 negmedspace 3i thickspace 3 4 negthickspace 31 quad 4 qquad 3 4 For ner control over math spacing7 use mspace and math unitsl One math unit7 or mu7 is equal to 118 em Thus to get a negative half quad Write mspace90mu There are also three commands that leave a space equal to the height andor Width of a given fragment of ETEX material Example Result phantomXXX space as Wide and high as three X s hphantomeXX space as Wide as three X7s height 0 vphantomX space of Width 07 height height of X 46 DOTS For preferred placement of ellipsis dots raised or online in various contexts there is no general consensus It may therefore be considered a matter of taste By using the semantically oriented commands o dotsc for dots With commas77 o dotsb for dots With binary operatorsrelations77 o dotsm for multiplication dots77 o dotsi for dots With integrals77 o dotso for other dots77 none of the above instead of 1dots and cdots7 you make it possible for your document to be adapted to different conventions on the fly7 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 We have the series A1 A2dotsc We have the series Al7 A2 7 the re the regional sum A1A2d0tsb gional sum A1 142 7 the orthogonal the orthogonal product A1A2dotsm product 1411427 and the in nite hue and the infinite integral gral intA1intA2dotsi A1 A2 Short Math Guide for ETEX version 109 200203 22 12 437 NONBREAKING DASHES The command nobreakdash suppresses the possibility of a linebreak after the following hyphen or dashi For example if you write pages 1797 as pages 1nobreakdash9 then a linebreak will never occur between the dash and the 9 You can also use nobreakdash to prevent 39 l 39 in quot like padici For frequent use its advisable to make abbreviations eigi newc ommandppnobreakdash for quotpadicquot newc ommandNdashnobreakdasht ext endashgt o for quot pages 1Ndash 9quot quot0 r quotn dimensional quot quotndimensionalquot newc ommandn 1 n nobreakdashhspac e0pt The last example shows how to prohibit a linebreak after the hyphen but allow normal hyphenation in the following word It suf ces to add a zerowidth space after the hypheni 48 ROOTS The command sqrt produces a square root To specify an alternate radix give an optional argumenti sqrtfracnn1 S S sqrt 32 gi n 7 l 49 BOXED FORMULAS The command boxed puts a box around its argument like fbox except that the contents are in math mo e 42gt boxedeta 1eq Cde1taeta LambdaM0de1ta If you need to box an equation including the equation number see the FAQ that comes with the amsmath package 5 Fractions and related constructions 51 THE frac dfrac AND tfrac COMMANDS The frac command takes two ar gumentsinumerator and denominatoriand typesets them in normal fraction formi Use dfrac or tfrac to overrule ETEXls guess about the proper size to use for the fractions contents t textstyle d displaystyle l ElogQ log2 5 1 beginequation frac1k1og2 c f tfrac1k1og2 C f endequation 9 1 mrT 2 i 5 Z 9 it 2 11 B 2 l T t E 0g 2 beginequation Rez fracnpi dfractheta psi2 1eft dfractheta psi2right 2 1eft dfrac12 1og 1eft1vertdfracBArightrvertrightquot2 endequation 52 THE binom dbinom AND tbinom COMMANDS For binomial expressions such as there are binom dbinom and tbinom commands 2kltfgt2klltzgt2k2 2quotkbinomk12 k1binomk22quotCk2 Short Math Guide for EMEX version 109 200203 22 13 53 THE genfrac COMMAND The capabilities of frac binom and their variants are subsumed by a generalized fraction command genfrac with six arguments The last two correspond to fracls numerator and denominator the rst 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 ptiie invisible and the fourth argument is a mathstyle override integer values 073 select respectively displaystyle textstyle scriptstyle and scriptscriptstyle If the third argument is left empty the line thickness defaults to normall genf rac le delim ght delim thickness mathstyle numeratorH denominator To illustrate here is how frac tfrac and binom might be de ned newc ommandfrac 2 genfrac12 newc ommandtfrac 2 genfrac112 newc ommandbinom 2 genfrac Opt12 Note For technical reasons using the primitive fraction commands over atop above in a ETEX doc ument is not recommended see egv amsmathfaq 54 CONTINUED FRACTIONS The continued fraction 1 54 1 7 can be obtained by typing cfrac1sqrt2 cfrac1sqrt2 cfrac1sqrt2dotsb This produces betterlooking results than straightforward use of frac Left or right placement of any of the numerators is accomplished by using cfrac 1 or cfrac 1 instead of cfrac 6 Delimiters 61 DELIMITER SIZES Unless you indicate otherwise delimiters in math formulas will remain at the standard size regardless of the height of the enclosed material To get larger sizes you can either select a particular size using a big prefix see below or you can use 1eft and right pre xes for autosizing The automatic delimiter sizing done by 1eft 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 has fairly large quantum jumps This means that an expression that is in nitesimally too large for a given delimiter size will get the next larger size a jump of 6pt or so Spt top and bottom in normalsized text There are two or three situations where the delimiter size is commonly adjusted These adjustments are done using the following commands Delimiter size text 1eft bigl Bigl biggl Biggl size right bigr Bigr biggr Biggr Result 023 M3 W3 03 Short Math Guide for EZEX version 109 200203 22 14 The rst kind of adjustment is done for cumulative operators with limits such as summation signs With 1eft and right the delimiters usually turn out larger than necessary and using the Big or bigg sizes instead gives better results 17 117 E ai E zij versus a i J39 i p 117 E zij j bigg1sumi aiBigl1vertsumj xijBigrrvert pbiggr quot1p The second kind of situation is clustered pairs of delimiters where 1eft and r ight make them all the same size because that is adequate to cover the encompassed material but what you really want is to make some of the delimiters slightly larger to make the nesting easier to see albl 4a2b2a2bl t alb2 Versus albl a2b2a2bl t alb2f 1efta1 b1 a2 b2right 1efta2 b1 a1 b2right quadt extversusquad big1a1 b1 a2 b2bigr bigla2 b1 a1 b2bigr The third kind of situation is a slightly oversize object in running text such as I where the delimiters produced by 1eft and right cause too much line spreading In that case bigl and bigr can be used to produce delimiters that are larger than the base size but still able to t within the normal line spacing IWI 62 VERTICAL BAR NOTATIONS The use of the I character to produce paired delimiters is not recommended There is an ambiguity about the directionality of the symbol that will in rare cases produce incorrect spacing egq Ik I I kI produces IkI I E kI Using 1vert for a left vert bar77 and rvert for a right vert bar77 whenever they are used in pairs will prevent this problem compare IEkI produced by 1vert krvert For double bars there are analogous 1Vert rVert commands Recommended practice is to de ne suitable commands in the document preamble for any paireddelimiter use of vert bar symbols providecommandabs 1 1vert1rvert providecommandnorm 1 1Vert1rVert whereupon absz would produce and normv would produce 7 The text command The main use of the command text is for words or phrases in a display It is similar to mbox in its effects but unlike mbox automatically produces subscriptsize text if used in a subscript fhlibxl is monotonic i 1 c 1 71 f xi1xi text is monotonic quad i 1dotsc1 71 mod AND ITS RELATIVES Commands mod bmod pmod pod deal with the special spacing conventions of mod notation mod and pod are variants of med preferred by some authors mod omits the parentheses whereas pod omits the mod and retains the parentheses gcdnm mod n z E y mod 12 z E y mod 0 z E y d 72 gcdnmbmod nquad xequiv ypmod b quad xequiv ymod cquad xequiv ypod d Short Math Guide for EMEX version 109 200203 22 15 8 Integrals and sums 81 ALTERING THE PLACEMENT OF LIMITS The limits on integrals sums and similar symbols are placed either to the side of or above and below the base symbol depending on convention and context ETEX has rules for automatically choosing one or the other and most of the time the results are satisfactory In the event they are not there are three ETEX commands that can be used to in uence the placement of the limits 1imits nolimits displaylimits Compare 25 I 25t I iT TztiltX0 WM and irirztlltXo int absx X Z RX 0 int1imitsabsxxztltXO The 1imits command should follow immediately after the base symbol to which it applies and its meaning is shift the following subscript andor superscript to the limits position regardless of the usual convention for this symbol nolimits means to shift them to the side instead and displaylimits which might be used in de ning a new kind of base symbol means to use standard positioning as for the sum comman See also the description of the intlimits and nosumlimits options in AMUG 82 MULTIPLE INTEGRAL SIGNS iint iiint and iiiint give multiple integral signs with the spacing between them nicely adjusted in both text and display style idotsint is an extension of the same idea that gives two integral signs with dots between them fzydzdy fzyzdzdydz 81 A A fwzyzdwdzdydz Afzlzk 82 83 MULTILINE SUBSCRIPTS AND SUPERSCRIPTS The substack command can be used to produce a multiline subscript or superscript for example sumsubstack Z Pz j 01e i1e m ogigm oltjltn MK Pij 84 THE sideset COMMAND Therels also a command called sideset for a rather special purpose putting symbols at the subscript and superscript corners of a symbol like 2 or Note The sideset command is only designed for use with large operator symbols with ordinary symbols the results are unreliable With sideset you can write sideset Z nEn sumnltktextn odd nEn nltk n odd The extra pair of empty braces is explained by the fact that sideset has the capability of putting an extra symbol or symbols at each corner of a large operator to put an asterisk at each corner of a product symbol you would type sideset prod Hik Short Math Guide for ETEX version 109 200203 22 16 9 Changing the size of elements in a formula The TATEX mechanisms for changing font size inside a math formula are completely different from the ones used outside math formulas If you try to make something larger in a formula with one of the text commands such as 1arge or huge 1arge you will get a warning message Command 1arge invalid in math mode Such an attempt however often indicates a misunderstanding of how ETEX math symbols work If you want a symbol analogous to a summation sign in its typographical properties then in principle the best way to achieve that is to de ne it as a symbol of type mathop with the standard ETEX DeclareMathSymbol command see LFGM In this particular example it is currently unlikely that you will be able to lay your hands on a math font with a suitable textsizedisplaysize pair but that is probably best understood as a problem of inadequate fonts not as a ETEX problem Consider the expression Engt0 2 fracsumn gt O z n ngkgn 74k prod11eq k1eq n 1q k Using dfrac instead of frac wouldnlt change anything in this case if you want the sum and product symbols to appear full size you need the displaystyle command 22 ngto fracdisplaysty1esumn gt O z n H 174k displaysty1eprod11eq k1eq n 1q k 13kg And if you want fullsize symbols but with limits on the side use the nolimits command a so Zngt0 Zn fracdisplaysty1esumnolimitsngt 0 z n H 17 qk disp1aysty1eprodnolimits11eq k1eq n 1q k 1 k n There are similar commands textstyle scriptstyle and scriptscriptstyle to force ETEX to use the symbol size and spacing that would be applied in respectively inline math firstorder subscript or secondorder order subscript even when the current context would normally yield some other size Note These commands belong to a special class of commands referred to in the ETEX book as declarations In particular notice where the braces fall that delimit the effect of the comman Rightdisplaysty1e Wrongdisplaysty1e 10 Other packages of interest Many other TATEX packages that address some aspect of mathematical formulas are available from CTAN the Comprehensive TEX Archive Net work To recommend a few examples accents Under accents and accents using arbitrary symbols amsthm General theorem and proof setup bm Bold math package provides a more general and more robust implementation of boldsymbol cases Apply a large brace to two or more equations without losing the individual equation numbers Short Math Guide for ETEX version 109 200203 22 17 delarray Delimiters spanning multiple rows of an array kuVio Commutative diagrams and other diagrams xypic Commutative diagrams and other diagrams rsfs Ralph Smithls Formal Script font setup The TEX Catalogue http wwwtexac uktexarchivehe1pCataloguecataloguehtml is a good place to look if you know a packagels name 11 Other documentation of interest References AMUG American Mathematical Society User s Guide for the amsmath package amsldoc tex 1999 CLSL Pakin Scott The Comprehensive B TEX Symbols List httpwww ctanorg texarchiveinfosymbolscomprehensive July 2001 Lamport Lamport Leslie E ITEX a document preparation system 2nd edition Addison Wesley 1994 LC Goossens Michel Mittelbach Frank Samarin Alexander The E ITEX Com panion AddisonWesley 1994 LFG ETEXS Project Team E ITEXZg font selection fntguidetex 1994 LGC Goossens Michel Rahtz Sebastian Mittelbach Frank The B TEX Graphics Companion Addison Wesley Longman 1997 LGG Carlisle D P Packages in the graphics bundle grfguidetex 1995 LUG BTEXS Project Team B TEXZE for authors usrguidetex 1994
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