; TeX output 1996.06.28:2033ARrҠ.3{G ptmr7tEssentialQLyy#3{ ptmr7tATSEyX++^ZJonW BarbrickwithadditionsbyDa“vidCarlisle,MichelGoossens,SebastianRahtz,AdrianClark%d)January1994(W4e9Vff phvb7tContents:5ߌ ptmb7t1Introduction) b> cmmi10::::::::::::::::::::::::::::::::::::::p22HogwDoesLff>ߌ ptmb7tA;TU>NEXW@ork?呍::::::::::::::::::::::::::::::::,23ASampleLffA;TU>NEXFilex::::::::::::::::::::::::::::::::::*31SimpleText,a:::::::::::::::::::::::::::::::::::::::sn54DocumentClassesandOptionsT::::::::::::::::::::::::::::5b65EnvironmentsN::::::::::::::::::::::::::::::::::::::76TBypeStyles1:::::::::::::::::::::::::::::::::::::::x97SectioningCommandsandTablesofContentsFݍ::::::::::::::::::::108ProducingSpecialSymbolsn䍑::::::::::::::::::::::::::::::109Titles::::::::::::::::::::::::::::::::::::::::::*1110TabularMaterialFa::::::::::::::::::::::::::::::::::::n1111TablesandFigures:::::::::::::::::::::::::::::::::::1212Cross-ReferencesandCitations'썑::::::::::::::::::::::::::::n1313MathematicalTBypesetting졍:::::::::::::::::::::::::::::::31414Whatabout$'Gs?Ӎ::::::::::::::::::::::::::::::::::::A1615Letters\R:::::::::::::::::::::::::::::::::::::::::`1716ErrorsdH:::::::::::::::::::::::::::::::::::::::::: U1817AFinalReminderZ:::::::::::::::::::::::::::::::::::19AMathematicalsymbols4:::::::::::::::::::::::::::::::::A20BHorribleMathematicalExamplestoStudym::::::::::::::::::::::z25ҍ3{ ptmr7t1*ARrҍ21N8 ptmrc7tNEssentialL`ffO8 ptmrc7tOAMNTU>?EfX.{Bibliograph٠y::::::::::::::::::::::::::::::::::::::::N26 hListofTab$lesw1&Greekletters.....................................20 Db2&Binaryoperationsymbols+3..............................203&Relationsymbols㍍..................................204&Largedelimiters[󍍑...................................215&DelimitersG......................................216&ArrowsymbolsO...................................217&Miscellaneoussymbolsc...............................218&Log-likgesymbolsQ..................................229&Mathmodeaccents=).................................2210&Variable-sizedsymbols...............................2211&Lff?3{ ptmr7tA;TU>?E1fXmathconstructs\................................2212&AMSmiscellaneoussymbols4............................2213&AMSGreekandHebrew..............................2314&AMSdelimitersK...................................2315&AMSarrowsv.....................................2316&AMSne٠g7atedarrows................................2317&AMSbinaryoperators................................2318&AMSbinaryrelations................................2419&AMSne٠g7atedbinaryrelations<...........................241hIntrJoductionlCThisEdocumentisanattempttogiv٠eyouallthe@Kj ptmri7t@essentialGinformationthatyouwillneedinor- der9tousetheLffA;TU>?E1fXdocumentpreparationsystem[1].^1.GVOnlybasicfeaturesareco٠vered,hdand9avast0amountofdetailhasbeenomitted.,BInadocumentofthissizeitisnotpossibletoincludeev٠erythingthatyoumightneedtoknowY,Gandifyouintendtomakgee٠xtensiveuseoftheprogramyoul@shouldrefertoamorecompletereference.Attemptingtoproducecomple٠xdocumentsusingonly|theinformationfoundbelowwillrequiremuchmorewgorkthanitshould,}Yandwillprobablyproducealessthansatisfgactoryresult. 8ThemainreferenceforLffA;TU>?E1fXis@TheLffCKj ptmri7tCA-S@TU>4xEXUser'sguideandReferGenceManual냻byLeslieLamport[1]. .ThisTcontainsalltheinformationthatyouwillev٠erneedtoknowabouttheprogram,andyouwillneedaccesstoacopgyifyouaretouseLffA;TU>?E1fXseriouslyY.BothYtheManualandthisdocumentavoidYmentioningan٠ythingthatdependsontheparticularcomputersystemthatyouareusing.ThisisbecauseLffA;TU>?E1fXisavailableonanumberofsystemsandthe٠yalldi erinonewgayoranothers8.۸Instead,theybothrefertoa@localguidefortheirparticularsystem. h2hHoȽwDoesLفKe9V phvb7tKAFTyԒEgyXW"ork?lCIn?E1fX,kTyougeneratea lecontainingboththete٠xtthatyouwishtotypesetandinstructionsstotellLffA;TU>?E1fXhowyouwgantittoappears8.YYgouwillnormallycreatethis leusingyoursystem's8s9ste٠xteditor.Ygoucangiv٠ethe leanynameyoulikge,Gbutitshouldend\L0 pcrr7tL.tex"toidentifythet le's8scontents.YgouthengetLffA;TU>?E1fXtoprocessthe le,anditcreatesanew leoftypesetting Vff֟ 荍 h-:B3{ ptmr7tB1fhA3{ ptmr7tAFMor`thosereaderswhoha vesomefamiliaritywithLBA@ATEX,thisdocumentassumestheLBAATEX302&2cmmi8"{AreleaseofJanuary1994 orlaterjARrҍNEssentialL`ffOAMNTU>?EfX13.{commands;;this/hasthesamenameasyour lebutthe\L.tex"endingisreplacedby\L.dvi". ThisTstandsfor`LDeLviceLIndependent'and,asthenameimplies,this lecanbeusedtocreateoutputonarangeofprintingdevices.Ygour@localguidewillgointomoredetail. 卑Rather/thanencourageyoutodictatee٠xactlyhowyourdocumentshouldbelaidout,[LffA;TU>?E1fXinstructionsdsallowyoutodescribeits@loggicalstructurGe.iF٠orexample,iyoucanthinkofaquotationembeddedѾwithinyourte٠xtasanelementofthislogicalstructure:youwgouldnormallyexpectaquotationKtobedisplayedinarecognizablestyletosetito fromtherestofthete٠xt.XzA0humantypesetternwgouldrecognizethequotationandhandleitaccordinglyY,#IbutsinceLffA;TU>?E1fXisonlyacomputerprogramitrequiresyourhelp.GTherearethereforeLffA;TU>?E1fXcommandsthatallowyoutoidentifyquotationsandconsequentlyallowLffA;TU>?E1fXtotypesetthemcorrectlyY.Fundamental:toLffA;TU>?E1fXistheideaofa@documentclassthatdeterminese٠xactlyhowadocumentwilļbeformatted.LffA;TU>?E1fXpro٠videsstandarddocumentclassesthatdescribehowcommonlogicalstructures(suchasquotations)shouldbeformatted.6XYgoumayhav٠etosupplementthesestylesbyspecifying+theformattingoflogicalstructurespeculiartoyourdocument, suchasmathematicalformulae. kYgouUxcanalsomodifythestandarddocumentclassesorev٠encreateanentirelynewone,thoughyoushouldknowthebasicprinciplesoftypographicaldesignbeforecreatingaradicallynewstyle.Thereȁareanumberofgoodreasonsforconcentratingonthelogicalstructureratherthanonthevappearanceofadocument.CDoingsoprev٠entsyoufrommakingelementarytypographicaler-rorsl inthemistakgenbeliefthatthe٠yimprovethestheticsofadocument|youshouldrememberthattheprimaryfunctionofdocumentdesignistomakgedocumentseasiertoread,C notprettiers8.It!isalsomore e٠xible,sinceyouneedonlyalterthede nitionofthequotationstyletochangetheappearanceof@allXthequotationsinadocument.. Mostimportantofall,logicaldesignencour-agesx}betterwriting.Ax{visualsystemmakgesiteasiertocreatevisuale ectsratherthanacoherentstructure;logical=designencouragesyoutoconcentrateonyourwritingandmakgesithardertouseformattingasasubstituteforgoodwriting.$捍3hASampleLفKAFTyԒEgyXFileIHav٠ealookattheexampleLffA;TU>?E1fX leinFigure1.fItisaslightlymodi edcopgyofthestandardLffA;TU>?E1fXme٠xample leLsmall.tex9mȻ.Ygourlocalguidewilltellyouhowyoucanmakgeyourowncopgyofthis le.iThelinenumbersdowntheleft-handsidearenotpartofthe le,,buthav٠ebeenaddedtomakgeiteasiertorefertovariousportions.Alsohav٠ealookatFigure2whichshows,moreorless,theresultofprocessingthis le.k捍Rl phvr7tR3.1RunningUT`ext>JMostCdocumentsconsistalmostentirelyofrunningte٠xt|wgordsformedintosentences,Owhicharein2%turnformedintoparagraphs|andthee٠xample leisnoexception.0Describingrunningtextposes noproblems,§youjusttypeitinnaturallyY.Intheoutputthatitproduces,LffA;TU>?E1fXwill lllinesmandadjustthespacingbetweenwgordstogiv٠etidyleftandrightmargins..ThespacinganddistributionΒofthewgordsinyourinput lewillhav٠enoe ectatallontheeventualoutput.NAnynumberofspacesinyourinput learetreatedasasinglespacebyLffA;TU>?E1fX.yItalsore٠g7ardstheendofXeachlineasaspacebetweenwgords(seelines15{17). ~AXnewparagraphisindicatedbyablanklineinyourinput le,sodon'tleav٠eanyblanklinesunlessyoureallywishtostartaparagraph.LffA;TU>?E1fXMkreserv٠esanumberofthelesscommonkgeyboardcharactersforitsownuse.ThetencharactersʍL# $%&~_^\{}shouldnotappearaspartofyourte٠xt|iftheydo,LffA;TU>?E1fXwillgetconfused.DARrҍ41NEssentialL`ffOAMNTU>?EfX.q?ffQ0 pcrr7tQ1:%SMALL.TEX--Released5July1985 ff2:%USETHISFILEASAMODELFORMAKINGYOUROWNLaTeXff3:%INPUTFILE.EVERYTHINGTOTHERIGHTOFA %ISffAff4:%REMARKTO YOUANDISIGNOREDBYLaTeX.ff5:%ff6:%WARNING! DONOTTYPEANYOFTHEFOLLOWING10CHARACTERSff7:%EXCEPTASDIRECTED:+30&32$#%_{}^~\ff8:ff9:\documentclass[11pt]{article}%YOURINPUTFILEMUSTCONTAIN10:ff\begin{document}F3.%THESETWOLINESPLUS11:%ffTHE\endCOMMANDATTHEEND12:13:ff\section{SimpleText}%THISCOMMANDMAKESASECTIONTITLE.14:15:ffWordsareseparatedbyoneormore fdspaces. Paragraphsare16:separatedffbyoneormoreblanklines. Theoutputisnotaffected17:ffbyaddingextraspacesorextrablanklinestotheinputfile.18:19:20:ffDoublequotesaretypedlikethis:``quotedtext''.21:ffSinglequotesaretypedlikethis:`single-quotedtext'.22:23:ffLongdashesaretypedasthreedashcharacters---likethis.24:25:ffItalictextistypedlikethis:\emph{thisisitalictext}.26:ffBold32textistypedlikethis:\textbf{thisis boldtext}.27:28:ff\subsection{AWarningorTwo}+30%THISMAKESASUBSECTIONTITLE.29:30:ffIfyougettoomuchspaceafteramid-sentenceperiod---abbreviations31:fflikeetc.\arethecommonculprits)---thentypeabackslashfollowedby32:ffaspaceaftertheperiod,asinthissentence.33:34:ffRemember,don'ttypethe10specialcharacters(suchasdollarsignand35:ffbackslash)exceptasdirected! Thefollowingsevenareprintedby36:fftypingabackslashinfrontofthem: \$\&\#\%\_\{and\}.37:ffThemanualtellshowtomakeothersymbols.38:39:ff\end{document}k%THEINPUTFILEENDSLIKETHIS%Figure1:ASampleLffA;TU>?E1fXFileXARrҍNEssentialL`ffOAMNTU>?EfX15. wuэ1hSimpleT"eȽxtW37ordsOareseparatedbyoneormorespaces. P٠aragraphsareseparatedbyoneormoreblanklines. Theoutputisnota ectedbyaddinge٠xtraspacesorextrablanklinestotheinput le.Doublequotesaretypedlikgethis: \quotedte٠xt".Singlequotesaretypedlikethis: `single-quotedte٠xt'.Longdashesaretypedasthreedashcharacters|likgethis.Italicte٠xtistypedlikgethis:@thisisitalictext.Boldte٠xtistypedlikgethis:thisisboldtext._R1.1AUW&arLningorT`wIoIfyougettoomuchspaceafteramid-sentenceperiod|abbreviationslikgeetc.arethecommonculprits)|thentypeabackslashfollowedbyaspaceaftertheperiod,asinthissentence.Remember,rTdon'tntypethe10specialcharacters(suchasdollarsignandbackslash)e٠xceptasdirected! TheOfollowingsev٠enareprintedbytypingabackslashinfrontofthem:t$&#%Qff9 !", cmsy10fandg.Themanualtellshowtomakgeothersymbols.Z;Figure2:TheResultofProcessingtheSampleFileR3.2L+Sl phvr7tSA7LRT#SEXUCommandsԉThereUareanumberofwgordsinthe lethatstartwith`L\'(seelines9, 10and13).Theseare LffA;TU>?E1fXk@commandsandthe٠ydescribethestructureofyourdocument.Thereareseveralthingsthatyoushouldrealizeaboutthesecommands:ꍍAllLffA;TU>?E1fXcommandsconsistofa`L\'followedbyoneormorecharacters.s򍍍LffA;TU>?E1fX`commandsshouldbetypedusingthecorrectmixtureofupper-andlower-caseletters.L\BEGINandL\Beginare@notthesameasL\begin.Some2commandsareplacedwithinyourte٠xt.Theseareusedtoswitchthings,Bblikgedi erenttypestyles,G on8ando .TheL\smallcommandisusedlikgethistoemphasizete٠xt,normallybychangingtoan@italictypestyle(seeline25).Thecommandandthete٠xtarealgwaysenclosedbetween`L{'and`L}'|the`L{\em'turnsthee ectonandandthe`L}'turnsito .ҌAllthetypefgace-changingswitcheslikethishav٠eacorresponding@commandXwithanargument|allthepossibilitiesareshownonpage9.ThereareothercommandsthatlooklikgeꍑGL\command{text}In-thiscase,Ltext&iscalledthe\argument"ofthecommand. TheL\sectioncommandisklikgethis(seeline13).aSometimesyouhav٠etousecurlybrackets`L{}'toenclosetheargument,sometimessquarebrackgets`L[]',andsometimesboth.QThereismethodbehindthis?apparentmadness,obutforthetimebeingyoushouldbesuretocopgythecommandse٠xactlyasgiven.s򍍍WhenFacommand's8snameismadeupentirelyofletters,REyoumustmakgesurethattheendofthecommandismarkgedbysomethingthatisn'taletters8.Thisisusuallyeithertheopeningbrackgetaroundthecommand's8sargument,%ooraspace.Whenit'saspace,%othatspaceisalgwaysignoredbyLffA;TU>?E1fX.W37eshallseelaterthatthiscansometimesbeaproblem.`WARrҍ61NEssentialL`ffOAMNTU>?EfX.{R3.3Ov3erIallUStr. ucturexThere~aresomeLffA;TU>?E1fXcommandsthatmustappearinev٠erydocument.Theactualtextofthedocu- menthalgwaysstartswithaL\begin{document}commandandendswithanL\end{document}command((seelines10and39).An٠ythingthatcomesaftertheL\end{document}commandisignored.BEv٠erythingthatcomesbeforetheL\begin{document}commandiscalledthe@prGeamble.ThepreambleshouldcontainonlyLffA;TU>?E1fXcommandstodescribethedocument's8sstyle. . One;commandthatmustappearinthepreambleistheL\documentclasscommand(seeline9).hThiscommandspeci estheo٠verallstyleforthedocument.Oure٠xample leisasimpletechnicalHdocumentandusestheLarticlestyle,modi edtoprintinelev٠en-pointfonts.%oThereareotherstylesthatyoucanuse,asyouwilldisco٠verinSection4.]3R3.4OtherUThingstoLookAtLffA;TU>?E1fXcanprintbothopeningandclosingquotecharacters,andcanmanageeitheroftheseeithersingleordouble.T37odothis,itusesthetwgoquotecharactersfromyourke٠yboard:k2L`(whichsometimes'resemblesagrav٠eaccentorback-quote)andL'(apostrophe).Ygoutypethesecharactersonce܉forsinglequote(seeline21),andtwicefordoublequotes(seeline20)./3ThedoublequotecharacteritselfL"isalmostnev٠erused. . LffA;TU>?E1fXcanproducethreedi erentkindsofdashes.A^longdash,foruseasapunctuationsymbol,SasGistypedasthreedashcharactersinarowY,likgethis`L---'(seeline23).AGshorterdash,usedFbetweennumbersasin`10{20',R#istypedastwgodashcharactersinarowY,whileasingledashcharacterisusedasah7yphen.FromXtimetotime,-youwillneedtoincludeoneormoreoftheLffA;TU>?E1fXspecialsymbolsinyourite٠xt.Sevenofthemcanbeprintedbymakingthemintocommandsbyproceedingthemby}backslash(seeline36).TTheremainingthreesymbolscanbeproducedbymoreadvancedcommands,ascansymbolsthatdonotappearonyourkge٠yboardsuchasy,z,x,$,Gc ,]and|.ItissometimesusefultoincludecommentsinaLffA;TU>?E1fX le,^toremindyouofwhatyouhav٠edone`orwh7yyoudidit.Ev٠erythingtotherightofaL%-signisignoredbyLffA;TU>?E1fX,ַandsoitcanbeusedtointroduceacomment. }34hDocumentClassesandOptionsWwTherearefourstandarddocumentclassesavailableinLffA;TU>?E1fX:T0 pcrb7tTarticle/intendedforshortdocumentsandarticlesforpublication.Articlesdonothav٠echap-ters,yIandwwhenL\maketitleisusedtogenerateatitle(seeSection9),thetitleappearsatthetopofthe rstpageratherthanonapageofitsown.\Treport)intendedforlongertechnicaldocuments.vItissimilartoLarticle,e٠xceptthatitcon-tainschaptersandthetitleappearsonapageofitsown.TbookintendedBasabasisforbookpublication.A_P٠agelayoutisadjustedassumingthattheoutputwillev٠entuallybeusedtoprintonbothsidesofthepapers8.Tletter)intendediGforproducingpersonalletters.lThisstylewillallowyoutoproducealltheelementsofawelllaidoutletter:addresses,date,signature,etc..These!|standardstylescanbemodi edbyanumberof@options.The٠yappearinsquarebrackgetsaftervtheL\documentclasscommand.Onlyoneclasscanbeusedinadocumentbutyoucanhav٠ermorethanoneoption, inwhichcasetheirnamesshouldbeseparatedbycommas.Thestandardoptionsare:T11ptprintsthedocumentusingelev٠en-pointtypefortherunningtextratherthattheten-pointtypenormallyused.Elev٠en-pointtypeisabouttenpercentlargerthanten-point.ogARrҍNEssentialL`ffOAMNTU>?EfX17.{T12ptprintsUthedocumentusingtwelv٠e-pointtypefortherunningtextratherthantheten-point typenormallyused.T37welv٠e-pointtypeisabouttwentypercentlargerthanten-point.Ttwoside/causesdocumentsinthearticleorreportstylestobeformattedforprintingonbothsidesofthepapers8.Thisisthedefgaultforthebookstyle.Ttwocolumn;producestwgocolumnsofte٠xtoneachpage.Ttitlepage;causes|theL\maketitlecommandtogenerateatitleonaseparatepagefordoc-uments.intheLarticle0]style.&yA.separatepageisalgways.usedinboththeLreport*]andLbook3styles.Theres)isfurtheroptionwhichisv٠eryusefulinEuropeancountries.TheLa4paper0Roptioncauses3theoutputinallofthestandardstylestobeadjustedto tcorrectlyonA4papers8.@1(LffA;TU>?E1fXwgasjWdesignedinAmericawherethestandardpaperisshorterandslightlywiderthanA4;qwithoutthisoptionyouwill ndthatyouroutputlooksalittlestrange.)۷5hEnlvirJonmentsW37etmentionedearliertheideaofidentifyingaquotationtoLffA;TU>?E1fXsothatitcouldarrangetotypesetithcorrectlyY.fT37odothisyouenclosethequotationbetweenthecommandsL\begin{quotation}and>L\end{quotation}.YQThisisane٠xampleofaLffA;TU>?E1fXconstructioncalledan@environment.Anumberofspeciale ectsareobtainedbyputtingte٠xtintoparticularenvironments.R5.1QuotationsTherearetwgoenvironmentsforquotations:YLquote#zandLquotation8.yzLquote$pisusedeitherforashortquotationorforasequenceofshortquotationsseparatedbyblanklines:oWUSpresidentshav٠ebeenknownfortheirpith7yremarks:Thebuckstopshere.Iamnotacrook.ҍQUSffpresidentshavebeenknownfor ҍtheirffpithyremarks:ҍ\begin{quote}ҍTheffbuckstopshere.ҍIffamnotacrook.ҍ\end{quote}]UsebtheLquotation<,environmentforquotationsthatconsistofmorethanoneparagraph.P٠aragraphsintheinputareseparatedbyblanklinesasusual:Hereissomeadvicetoremember:(EnvironmentsJformakingquotationscanbeusedforotherthingsaswell.(Man٠y~Kproblemscanbesolvedbynovelap-plicationsofe٠xistingenvironments.ҍQHereffissomeadvicetoremember: ҍ\begin{quotation}ҍEnvironmentsffformakingquotationsҍcanffbeusedforotherthingsaswell.ҍManyffproblemscanbesolvedbyҍnovelffapplicationsofexistingҍenvironments.ҍ\end{quotation}zR5.2Center. ingUandFlushingTLe٠xt canbecentredonthepagebyputtingitwithintheLcenter)@environment,(anditwillappear ushLag7ainsttheleftorrightmarginsifitisplacedwithintheLflushleft:8orLflushright@8en-vironments. NoticeuXthespellingofLcenter&uX|unfortunatelyLffA;TU>?E1fXdoesn'tunderstandtheBritishEnglishspelling.XARrҍ81NEssentialL`ffOAMNTU>?EfX.{TLe٠xtZwithintheseenvironmentswillbeformattedinthenormalwgay;ginparticular,btheendsof the>linesthatyoutypearere٠g7ardedasspaces.ST37oindicateanewline,youtypetheL\\command.F٠orexample:u"䍑^onetwgothreeqfourr{ v٠eҍQ\begin{center} ҍoneҍtwoҍthreeff\\ҍfourff\\ҍfiveҍ\end{center}tR5.3ListsBThereV!arethreeenvironmentsforconstructinglists.Ineachone,eachnewitemisbe٠gunwithan\YL\itemcommand.IntheLitemize0environment,othestartofeachitemisgiv٠enamarkger,while,*intheLenumerate;ܻenvironment,eachitemismarkgedbyanumbers8.cTheseenvironmentscanfbenestedwithineachother,kinwhichcasetheamountofindentationusedandthemarkgerareadjustedaccordingly:䍍ItemizedlistsarehandyY.Howev٠er,don'tforget"1./The`item'command."2./The`end'command.ҍQ\begin{itemize} ҍ\itemffItemizedlistsarehandy.ҍ\itemffHowever,don'tforgetY\begin{enumerate}Y\itemffThe`item'command.Y\itemffThe`end'command.Y\end{enumerate}ҍ\end{itemize}sThethirdlist-makingenvironmentisLdescriptionD.InaLdescriptionG:youspecifytheitemlabelsinsidesquarebrackgetsaftertheL\itemcommand.F٠orexample:Threeanimalsthatyoushouldknowaboutare:gnatлAsmallanimalthatcausesnoendoftrouble.gnuAlargeanimalthatcausesnoendoftrouble.armadillo.eAmedium-sizedanimal.ҍQThreeffanimalsthatyoushould ҍknowffaboutare:ҍ\begin{description}Y\item[gnat]ffAsmallanimalthatcausesl%noffendoftrouble.Y\item[gnu]ffAlargeanimalthatcausesl%noffendoftrouble.Y\item[armadillo]ffAmedium-sizedanimal.ҍ\end{description}~R5.4V BerbatimUOutputBSometimesyouwillwganttoincludete٠xtexactlyasitappearsonaterminalscreen.6Forexample,youmightwganttoincludepartofacomputerprogram.NotonlydoyouwantLffA;TU>?E1fXtostopplayingaroundwiththelayoutofyourte٠xt,youalsowganttobeabletotypeallthecharacterson qARrҍNEssentialL`ffOAMNTU>?EfX19.{yourkge٠yboardwithoutconfusingLffA;TU>?E1fX.TheLverbatim5environmenthasthise ect: Thesectionofprograminquestionis:L{thisfinds%a&%b} fori:=1to27dobegintable[i]:=fn(i);process(i)end;ҍQTheffsectionofprogramin ҍquestionffis:ҍ\begin{verbatim}ҍ{ffthisfinds%a&%b}ҍforffi:=1to27dobegintable[i]ff:=fn(i);process(i)end;ҍ\end{verbatim}h6hT"ypeStyles^)W37eThav٠ealreadycomeacrosstheL\emphcommandforchangingthetypefgacetoemphasis./Here isafulllistoftheavailabletypefgacechangingcommands:Wd@CommanduorE ectL\textrm{...}u{\rmfamily...}TLe٠xtissetinromanfgamilyL\textsf{...}u{\sffamily...}TLe٠xtissetinUl phvr7tUsansǧser&_iffgamilyL\texttt{...}u{\ttfamily...}TLe٠xtissetinLtypewriterfgamilyL\textmd{...}u{\mdseries...}TLe٠xtissetinmediumseriesL\textbf{...}u{\bfseries...}TLe٠xtissetinboldseriesL\textup{...}u{\upshape...}TLe٠xtissetinuprightshapeL\textit{...}u{\itshape...}TLe٠xtissetin@italicshapeL\textsl{...}u{\slshape...}TLe٠xtissetinM3{ ptmro7tMslantedлshapeL\textsc{...}u{\scshape...}TLe٠xtissetinNsmallcapsshapeL\emph{...}u{\em...}TLe٠xtisset@emphasizedL\textnormal{..}u{\normalfont..}TLe٠xtissetinthedocumentfontWύRememberthatthedeclarationv٠ersions(secondcolumn)areused@insideapairofbracestolimitxstheamountofte٠xtthattheya ect.Itisrecommendedthatyouusethecommands( rstcolumn)J9withaparameterofthete٠xttotypesetdi erentlyY.Inadditiontothetypefgacecommands,thereareasetofcommandsthatalterthesizeofthetype.Thesecommandsare:, W3{ ptmr7tWtinyjscriptsizedAfootnotesizeP3{ ptmr7tPsmall^normalsize larȹge 23{ff ptmr7tLargeLARGEQX3{q ptmr7tXhuge/\V3{H ptmr7tVHugeҍQ\tinyfftiny ҍ\scriptsizeffscriptsizeҍ\footnotesizefffootnotesizeҍ\smallffsmallҍ\normalsizeffnormalsizeҍ\largefflargeҍ\LargeffLargeҍ\LARGEffLARGEҍ\hugeffhugeҍ\HugeffHuge >ARrҍ10,NEssentialL`ffOAMNTU>?EfX.`7hSectioningCommandsandTab$lesofContentsTLechnicaldocuments,;Slikgethisone,areoftendividedintosections.,Eachsectionhasaheading containingzatitleandanumberforeasyreference.lLffA;TU>?E1fXhasaseriesofcommandsthatwillallowyyoutoidentifydi erentsortsofsections.JOnceyouhav٠edonethis,LffA;TU>?E1fXtakgesontheresponsibilityoflayingoutthetitleandofpro٠vidingthenumbers. mThecommandsthatyoucanuseare:ڍL\chapter&\subsection-\paragraph\section&\subsubsection\subparagraphThenamingoftheselasttwgoisratherunfortunate,sincethe٠ydonotreallyhaveanythingtodowith`paragraphs'inthenormalsenseofthewgord:+the٠yarejustlowerlev٠elsofsection.Inmostdocumentastyles,headingsmadewithL\paragraphandL\subparagrapharenotnumbered.L\chapterIisnotavailableindocumentstyleLarticle,I.sThecommandsshouldbeusedintheordergiv٠en,sincesectionsarenumberedwithinchapters,subsectionswithinsections,etc.. mAsev٠enthsectioningcommand,:L\part,isalsoavailable.!VItsuseisalgwaysoptional,:anditisusedhtodividealargedocumentintoseriesofparts.Itdoesnotalterthenumberingusedforan٠yoftheothercommands.IncludingthecommandL\tableofcontentsinyourdocumentwillcauseacontentslistto8=beincluded,Fcontaininginformationcollectedfromthevarioussectioningcommands.Ygouwillnotice&thateachtimeyourdocumentisrunthroughLffA;TU>?E1fXthetableofcontentsisalgways&madeup]ooftheheadingsfromthe@prGe٠vioustimeyouranLffA;TU>?E1fXonit.ThisisbecauseLffATU>?E1fXcollectsinformationryforthetableasitprocessesthedocument,u.andthenincludesitthene٠xttimeitisrun.ThisjcansometimesmeanthatthedocumenthastobeprocessedthroughLffA;TU>?E1fX@twicetogetacorrecttableofcontents."T8hPrJoducingSpecialSymbolsYgouqcanincludeinyourLffA;TU>?E1fXdocumentawiderangeofsymbolsthatdonotappearonyourkge٠yboard.Forastart,youcanaddanaccenttoan٠yletter:)卍b߀fffd ffşYfdofhL\`{o} ffB3~AoWL\~{o}  ff337o38L\v{o} ff fffdofhL\c{o} ffB3AoWL\'{o}  ff337o38L\={o} ff ffşY}fdofhL\H{o} ffAϻoA?C .WL\d{o}  ff3^37o38L\^{o} ff ffşY_fdofhL\.{o} ffAճlAϻooWL\t{oo} ff37o%#38L\b{o} ff ffşYfdofhL\"{o} ffB3AoWL\u{o}  ff38 ffff($Sev٠eralothersymbolsareavailableandaregeneratedusingthefollowingcommands:ڍyzL\dag0xxL\S?G׻c< UAL\copyrightzzL\ddag*{L\P<$AL\poundsL\oe62qL\OEjL\AE C2qL\AE5p36aL\aarAL\AAAL\o<L\ONL\l2CL\E<AL\ss6>L?`<L!`<...CL\ldots$LffA;TU>?E1fXL\LaTeXThere3-isalsoaL\todaycommandthatprintsthecurrentdate.Whenyouusethesecommandsre-memberPthatLffA;TU>?E1fXwillignorean٠yspacesthatfollowthem,Zasothatyoucantype`L\pounds20'tohget`$20'.fHowev٠er, ifhyoutype`L\LaTeXiswonderful'hyouwillget`LffA;TU>?E1fXiswgonderful'|noticethelackofspaceafterLffA;TU>?E1fX.1T37oo٠vercomethisproblemyoucanfollowan٠yofthesecom- ARrҍNEssentialL`ffOAMNTU>?EfX,11.{mandsrbyapairofemptybracesandthenan٠yspacesthatyouwishtoinclude,ߎandyouwillsee thatL\LaTeX{}reallyiswonderful!(LffA;TU>?E1fXreallyiswgonderful!). +0FinallyY,kLffA;TU>?E1fXˉ`math'mode,normallyusedtolayoutmathematicalformulae,giv٠esaccesstoanev٠enlargerrangeofsymbols,2includingtheupperandlowercasegreekmathematicalalphabet,calligraphicyletters,zmathematicaloperatorsandrelations,arrowsandawholelotmore.sThiswillbediscussedinSection13. m9hTitlesR MostXdocumentshav٠eatitle.T37otitleaLffA;TU>?E1fXdocument,.youincludethefollowingcommandsinyourdocument,usuallyjustafterL\begin{document}. `Y0 pcrr7tY\title{requiredtitle} \author{requiredauthor}\date{requireddate}\maketitleV`IfƤtherearesev٠eralauthors,LthentheirnamesshouldbeseparatedbyL\andintheL\author command;the٠yqcanalsobeseparatedbyL\\ifyouwgantthemtobecentredondi erentlines.IftheL\datecommandisleftout,thenthecurrentdatewillbeprinted.#sj>L\title{Essential\LaTeX}\author{JonWarbrick\andANOther}\date{14thFebruary1988}\maketitleSٍ@EssentialLffA;TU>?E1fXCJW37arbrick ANOtheriB14thFebruary1988#{TheCe٠xactappearanceofthetitlevariesdependingonthedocumentstyle.IntheLreport(andLbookxstyles,zthextitleappearsonapageofitsown.IntheLarticle.&style,itnormallyappearsatthetopofthe rstpage;usetheclassoptionLtitlepage;toalterthis(seeSection4).10iTabJularMaterialR Because~LffA;TU>?E1fXwillalmostalgways~conv٠ertasequenceofspacesintoasinglespace,&itcanberatherdiculttolayouttables.Seewhathappensinthise٠xampleV`IncomeExpenditureResult20s0d19s11dhappiness20s0d20s1dmiseryҍQ\begin{flushleft} ҍIncome ExpenditureffResult\\ҍ20sff0d 19s11dhappiness\\ҍ20sff0d 20s1d fdmiseryҍ\end{flushleft}KHmThetLtabbing/environmentto٠vercomesthisproblem.+Withinit,Qyousettab-stopsandtabtothem0muchasyouwgoulddoonatypewriters8.,"T37ab-stopsaresetwiththeL\=command,]andtheL\>(commandmo٠ves(tothene٠xtstop.JTheL\\commandisusedtoseparateeachline.A$linethatendsinL\killproducesnooutput,andcanbeusedtosettab-stops:fdIncomeExpenditureResult20s0d19s11d^Happiness20s0d20s1d^MiserydQ\begin{tabbing} dIncomeff\=Expenditure\=%\killdIncomeff\>Expenditure\>Result\\d20sff0d\>19s11d\>Happiness \\d20sff0d\>20s1d \>Miseryd\end{tabbing} TARrҍ12,NEssentialL`ffOAMNTU>?EfX.{Unlikgefatypewriter's8stabke٠yY,&theL\>commandalwaysmo٠vesftothene٠xttab-stopinsequence, ev٠en6ifthismeansmovingtotheleft.;Thiscancausetexttobeoverwritteniftheg7apbetweentwgotab-stopsistoosmall.Ifjyouhav٠emoresophisticatedtabularwgork,youwillneedtheLtabular/ Իenvironment.JThisdi ersfromLtabbing/inthatyouneednotwgorryaboutwidthsofcolumns|LffA;TU>?E1fXwilllookatthe-wholetableandseehowwidecolumnsneedtobetocopewiththewidestentriesinthem.Ygoulmust,&howev٠er,telllLffA;TU>?E1fXhowman٠ycolumnsthereareandhowyouwishtolaythemout.Ygoudothisbysupplyinga@templateafterL\begin{tabular}withaletterforeachcolumn:卍dhLlzmeansthecolumnwillbeleftjusti edhLrzmeansthecolumnwillberightjusti edhLczmeansthecolumnwillbecentredItemsǪwithinarowareseparatedbythe&character(nowyouknowwh7yitistreatedspecially),andutherowisended,Rasbefore,withL\\.kAnsimpletablewiththreecolumnsandtwgorowslookslikge:1d@Name8TXAggeSǑHeightSebastian=LJ45SLj195cmMathewBLJ1XLj68cmҍQ\begin{tabular}{lrr} ҍ\emffName&\emAge&\emHeight\\ҍSebastianff&45&195cm0\\ҍMathew&ff1&68cmҍ\end{tabular}HVariousotherbellsandwhistlescanbeusedtomakgethelayoutnicer:/ 1.YgoucanputL\hlinebetweenrowstodra٠walineacrossthetable./ 2.T37ypeaL|inthetemplatewhereyouwgantav٠erticalbardownev٠eryrowofthetable. 3.Ygou)canspecifythatacolumnisofa x٠edwidthwithtext owingwithinitbyusingthe symbol\Lpinthetemplate,cfollowedbyawidthbetweenL{andL}. Thewidthcanbegiv٠eninan٠yof`cm',`mm'or`in'(butnot`inches'!)./Thus,amoresophisticatedtablemightlooklikgethis:QL̍fffd fffd@Group9G ff_eTBype ffhSherds ffff fffdGroups1{9$ ff_eGre٠ywgares ff*218 ff $fffdGroups40{44$ff_eBlackaz(mostly_e`black/bur-_enished')wgares$ff*116$ff$fffdGroups61{67ff_eBu -red-orange_ewgaresffš*46fffffdGroups81{85ff_eColour-coated_e newgaresffš*67fffffdGroups91{2,93{4ff_eMortariaandmis-_ecellaneousffš*35ff fffdGroup96,С ff_eSamian ffš*56 fffffd ffX ff ff*538 ffffҍQ\begin{tabular}{|l|p{1in}|r|} \hline ҍ\emffGroup&\emType&Sherds \\\hlineҍGroupsff1--932&Greywares&218\\ҍGroupsff40--44&Black(mostlyҍ`blackffburnished') wares&116\\ҍGroupsff61--67&Buff-red-orangeҍwaresff&46f\\ҍGroupsff81--85&Colour-coatedҍfineffwares&67K\\ҍGroupsff91--2,93--4&Mortariaҍandffmiscellaneous&35%\\ҍGroupff96&Samian&56+30\\\hline8&&538|3*\\\hlineҍ\end{tabular}bNotethatline-endingsinwhatyoutypearenotimportant;rowsendonlywhenaL\\isfound."?11iTab$lesandFiguresWhenyoulayoutsomete٠xtandnumbersinatable,itisalmostalgwaysintendedthatthewholeunit?staystogether,%Onotbreakingo٠ver?apage.VWhatdoyoudowhenyouareatthebottomofa şARrҍNEssentialL`ffOAMNTU>?EfX,13.{pageɭwithnotenoughspaceleft?SimilarlyY,whatdoyoudowhenyouwganttoleav٠espacefora photograph1ktobepastedin?gWhenyouaretypingthete٠xt,A#youdonotknowwhereLffA;TU>?E1fXwillstartaznewpage,Ysoitisoftenv٠erydiculttoleavespacejustwhereyouwgant. LffA;TU>?E1fXsolvesthisbyasystemof` oats',+objectsthatwillbeplacedinanicepositiononthepage,butnotnecessarilye٠xactly whereyouputitinyour le..Theywilloftenbeplacedatthebottomofthecurrentpage,orratthetopofthene٠xt,unoronapagebythemselves,uniftheyneedthatmuchspace.0LffA;TU>?E1fXletsyouspecifytwgosortsof oat,_namely@tablesand@ gurGes;?ineachofthemyoucanspecifyacaption.The@followingshowshowyouwgouldsetupspacefora gureinwhichyouwantedtoleav٠e3inchesofspace,followedbyacaption: L\begin{figure} \vspace{3in} \caption{APhotographofmySubject} \end{figure}pFiguresA]andtablesareautomaticallynumbered,Mlikgesections;V>justasL\tableofcontentslistsyourhsections, subsections,etc.withtheirpagenumbers,L\listoffiguresandL\listoftableswilllistthe guresandtableswiththeircaptions. 12iCrJoss-Ref$erencesandCitationskOneofthemostusefulthingsLffA;TU>?E1fXcandoforyouistotoautomaticallygenerateasection,page,A gure,table,orequationnumberinacross-reference.)F٠orexample,AtorefertoSection4ofthisreport(ondocumentstyles)andFigure2(theoutputofprocessingLsample.tex>)inthissentence,thecommandsusedwere:LtorefertoSection~\ref{sec:styles}ofthisreport(ondocumentstyles)andFigure~\ref{fig:result}(theoutputofprocessingpYgouwillseethattheL\refcommandinsertedthecross-referencesnumberintotheprinteddoc-ument.TheL~simplyinsertsaspacewhileprohibitingLffA;TU>?E1fXfrombreakingaline. 8_Ofcourse,torefertoacross-reference,theremustbeanindicationintheLffA;TU>?E1fXinput leofthe&pointbeingreferredto. wThisisdoneviatheL\labelcommand.Thelabelforasectionismostconv٠enientlydoneimmediatelyafterthesectionheading,asin:L\section{DocumentClassesandOptions}\label{sec:styles}pandthatfora gureortableinorimmediatelyfollowingtheL\captioncommand:L\caption{TheResultofProcessingtheSampleFile}\label{fig:result}pAn٠ytextmaybeusedinL\labelandL\refcommands,thoughtheconventionimpledaboveisaBgoodonetofollowY.aP٠agereferencesuseL\labelinthesamewgay,sSbutarereferredtousingL\pagerefinsteadofL\ref. 8_IndmuchthesamewgaythatyoucanrefertootherpartsofyourdocumentfromtheLffA;TU>?E1fX lesymbolicallyY,youOmayciteotherdocuments.T37odothis,youincludetheL\citecommandinyourte٠xt,asinLThebookbyLamport\cite{Lamport-LaTeX}istheprincipalreferenceworkon\LaTeX.pTheɃargumenttotheL\citecommand,knownasthe@citationkge7y,isusedtouniquelyidentifyapaper,book,orotherdocumentthathasbeendescribedina@biblioggr٠aphicdatabase. 8_HavingVtcitedotherwgorkfromyourdocument,^youcangeneratealistofreferencesbyinclud-ingthecommandsYARrҍ14,NEssentialL`ffOAMNTU>?EfX.{L\bibliographystyle{plain} \bibliography{mybib1,mybib2}ҍwherezfyouwgantthelisttoappears8.TheL\bibliographystylecommandtellsLffA;TU>?E1fXhowyouwgantthecitationstoappear,thestandardonesbeing:Tplain#CitationsYaresortedalphabeticallyinthelist,witheachentrybeingassignedanumber;entriesinthete٠xtareindicatedinsquarebrackgets.Tunsrt#CitationsCareorderedinthereferencelistintheorderoftheir rstappearanceandas-signedanumber;entriesinthete٠xtareindicatedinsquarebrackgets.Talpha#Citationsqaresortedalphabeticallyinthelistbuthav٠elabelslikge\Lam86"insteadofanumber;thesamelabelappearsinthete٠xtinsquarebrackgets.Tabbrev)CitationsaresimilartoLalpha#butaremorecompact(e.g.,byabbreviatingmonthandjournalnames).ThehsargumentoftheL\bibliographycommandisacomma-separatedlistof lenames,which;shouldendinL.bibwH(i.e.,Lmybib1.bib@wHandLmybib2.bibinthee٠xampleabove).These lescontainthee٠xpandedreferencesreferredtobytheL\citecommands.HAfulldescriptionofthe,:formatoftheseL.bibXt lesandhowthecitationsareconv٠ertedtoaLffA;TU>?E1fX-compatibleformare׭be٠yondthescopeofthis(introductory)guide;detailsaregiveninAppendixBחofLamport's8sbook./13iMathematicalT"ypesettingR13.1#ZMath,UDisplay-mathandEquation%MathematicsistreatedbyTU>?E1fXcompletelydi erentlyfromordinaryte٠xt.WTherearetwgospecial@modesformathematics,knownas@mathmodeand@displaymathmode.鍑MathmodecommandsaresurroundedbyL\(...L\)orbyL$...L$.эSome mathematicssetinlineK`y cmr102r3W=6.Note thatspacesintheinput leareignoredinmathmode.ҍQSomeffmathematicssetinline ҍ\(ff2\times3=326\).ҍNoteffthatspacesintheinputfileҍareffignoredinmathmode.IDisplaymathmodecommandsaresurroundedbyL\[...L\].Alargerequationtobedisplayedonalinebyitself.ɍQf(x)= O!cmsy71 u cmex10X t 0ercmmi7iٓRcmr7=0<$f^(i) ݲ(x)wfe n (֍ i/i!ҍQAfflargerequationtobedisplayedonalinebyitself. ҍ\[fff(x)=\sum_{i=0}^{\infty}\frac{f^{(i)}(x)}{i!}\]VThereisavariantofdisplaymathmode,theLequation5environment,whichautomaticallygen-ARrҍNEssentialL`ffOAMNTU>?EfX,15.{eratesanequationnumbers8.j/S^d;a1Jb2 ;a0Jb1Tc^]^dj2y0 j1y3^7=^d#4#6 #1#3(#^)(1)ҍQ\begin{equation} ҍ\left(\begin{array}{cc}ҍ1ff&2\\0&1ҍ\end{array}\right)ҍ\left(\begin{array}{cc}ҍ2ff&0\\1&3ҍ\end{array}\right)ҍ=ҍ\left(\begin{array}{cc}ҍ4ff&6\\1&3ҍ\end{array}\right)ҍ\end{equation}L6Theseoshorte٠xamplesshowthemaintypesofcommandsavailableinmathmode.A"few thingstonoteare: 1.SubscriptsandsuperscriptsareproducedwithL_andL^,asinLx_{1}=p^{2}forx1,=p^2|s.j 2.FractionsareproducedbytheL\fraccommand:L\(\frac{a+b}{c}\)giv٠es 3a+b3&fe D5c䪻. 3.Variouscommandsgiv٠enamestomathematicalsymbols: L\infty\Rightarrow\surd\bigotimesgenerate10)fpUGN 4.ArraysareproducedbytheLarray#environment.M.ThisisidenticaltotheLtabular/envi-ronmentXdescribedinSection10e٠xceptthattheentriesaretypesetinmathmodeinsteadofLR-mode.q5Note4thattheLarray#:henvironmentdoesnotputbrackgetsarroundthearrayY,soitcanqwalsobeusedforsettingdeterminants|orev٠ensetsofequationsinwhichyouwgantthecolumnstolineup. 5.ThecommandsL\leftandL\rightproducedelimitersthatgrowaslargeasneeded.FThe٠ycanbeusedwithavarietyofsymbols,e.g.,L\left(\left\{\left|.MThefullsetofthesedelimitersisshownintables5and4belowY.EIR13.2#ZSpacing؄AllYspacesintheinput leareignoredinmathmode.)Sometimesyoumaywganttoadjustthespacing;useoneofthefollowingcommandstoachiev٠ethis.dj L\,thinspaceК=L\:mediumspacej L\!ne٠g7ativethinspaceК=L\;thickspaceqAgoode٠xampleofwhereLffA;TU>?E1fXneedssomehelpwithspacingisj<īRR UVz?dxdy[::īR RUNzpdxdyҍQ\(\int\!\!\intffz\,dxdy..\int\intzdxdy\)'wR13.3#ZChangingUFontsinMathModeTheKdefgaultmathmodefontismathital2`ic.%Thisshouldnotbeconfusedwithordinary@textitalic.The5fontfor`ordinary'letterscanbechangedwiththeusualcommands,DL\emph,L\textbf,etc..NotethatlowercaseGreekletters(L\alpha,etc.)arere٠g7ardedasmathematicalsymbols(whichmeansthatthe٠ymustbetypedinmathmode)andarenota ectedbythesecommands.ARrҍ16,NEssentialL`ffOAMNTU>?EfX.{L\mathbfproducesboldfaceromanletters.7IIfyouwishtohav٠e]DF cmmib10]boldfacemathitalic letters,dRand6boldfgaceGreeklettersandmathematicalsymbols,usetheL\boldmathcommand@beforGegoingintomathmode.Thischangesthedefgaultmathfontstobold. pqǍxV=2>/)x'6:28Z"V cmbx10Zx=2)x'6:28]xQZ=2]`Ɍ cmbsy10`)]x`'Z6]:Z28ҍQ\(ffx=2\pi\Rightarrowx\simeq6.28\) ҍ\(\mathbf{x}ff=2\pi\Rightarrowx\simeq6.28\)ҍ{\boldmathff\(x=\mathbf{2}\pi\Rightarrowx98\simeq{\mathbf{6.28}}K\)}@=bThere3isalsoacalligraphicfontforuppercaseletters;KtheseareproducedbytheL\mathcalcommand:TFhҍQ\(ff\mathcal{F}\)&z14iWhatabout$'"s? ۍIf-&youhav٠econvertedtoLffA;TU>?E1fXfromplainorAU>?M S-TU>?EX,Xpyouwillprobablybewgonderingwh7ytherehasbeennomentionof$and$$. In'thesesystemsmathmodeissurroundedbyL$'s8sanddisplaymathmodeissurroundedbyL$$.Thisshascertaindra٠wbacksovertheLffA;TU>?E1fXsystemasitisdicultforyourtexteditortomatchL$'s8sasitishardtotellwhichonesarestartingmathmodeandwhichareendingit.xTU>?E1fXwillalsogetconfusedifyoumissa$out.The(incorrect)input(Llet(a,b,c)$beaPythagoreantriple,i.e.\threeintegerssuchthat$a^{2}+b^{2}=c^{2}$. pproducestheslightlymysteriouserrormessageL!Missing$inserted.`$r^l.56...triple,i.e.\threeintegerssuchthat$a^,{2}+b^{2}=c^{2}$? pNotethatitreportsthewrongerrorandinthewrongplace,theuseoftheL^commandoutofmath'Rmode.TU>?E1fXhastypeset`bea...Qsuch'Rthat'inmathmodeand@exitedljmathmodeattheL$after`suchthat'.uIfyouhadmadetheequivalentLffA;TU>?E1fXerror,OLffATU>?E1fXhasabetterideaofwhatyouindended:Llet(a,b,c)\)beaPythagoreantriple,i.e.\threeintegerssuchthat\(a^{2}+b^{2}=c^{2}\) pTheerrormessagemaystillbeunintellig7able,Dbutatleastitreportstheerrorintherightplace,you hav٠eusedL\)toendmathmodewhenyouwerenotinmathmode(asyouomittedtheL\(whichshouldhav٠ebeenbeforetheL(a,b,c)).LLaTeXerror. SeeLaTeXmanualforexplanation.TType Hforimmediatehelp.!Badmathenvironmentdelimiter.\@latexerr...forimmediatehelp.}\errmessage{#1}ARrҍNEssentialL`ffOAMNTU>?EfX,17.{L\)...ifinner$\else\@badmath\fi\else\@badmath ,\fil.56let(a,b,c)\)rbeaPythagoreantriple,i.e.\threeintegerssuchthat\...?T\Thesingledollarissometimesusefulforsmallsectionsofmath.T\TLetGbeap-groupҍQLetff$G$bea$p$-groupčThedoubledollaris@notalgwaysequivalenttoL\[...\],andsoshouldnotbeusedifyouwgantFEyourLffA;TU>?E1fX letobecompatiblewithdi erentstylesandoptions(trythe@ eqnclassoption).GR14.1#ZSymbols#ThefollowingtablesshowmostofthesymbolsavailablefromthestandardLffA;TU>?E1fXsymbolfonts.Ne٠g7ationsoftherelationalsymbolscanbemadewiththeL\notcommand:ǍG6HҍQ$Gff\not\equivH$čIntheappendixwehav٠ebroughttogetherasetoftablesregroupingallstandardLffA;TU>?E1fXsymbols(tables(|1to11. T37ables13toP12showsupplementaryAU>?M S-TU>?E1fXsymbols,RwhichareavailablewhenyouspecifytheLamssymbpackage. g15iLetterȽsP"ProducinglettersissimplewithLffA;TU>?E1fX.U*T37odothisyouusethedocumentstyleLletter.Ygoucanmakge:Zan٠ynumberofletterswithasingleinput le.aYgournameandaddress,HHwhicharelikelytobethepsameforallletters,saregiv٠enonceatthetopofthe le.EachletterisproducedbyaLletterenvironment,8having';thenameandaddressoftherecipientasitsargument.Theletteritselfbe٠ginswithanL\openingcommandtogeneratethesalutation. *.The2letterendswithaL\closingcommand./YgoucanusethecommandsL\enclandL\cctogeneratelistsofenclosuresandpeopletowhomyouaresendingcopies.?EfX.{Ybeaststhistime. \closing{BestRegards,}\cc{JimmyCarter\\RichardM.Nixon}\end{letter}\end{document} o16iErrJorȽsR㍻When͡youcreateanewinput leforLffA;TU>?E1fX, youwillprobablymakgemistakes.{Ev٠erybodydoes, andTit's8snothingtobewgorriedabout.WAswithmostcomputerprograms,%therearetwosortsofmistakge5thatyoucanmake:thosethatLffA;TU>?E1fXnoticesandthosethatitdoesn't.:T37otakearathersilly.e٠xample,YsinceLffA;TU>?E1fXdoesn'tunderstandwhatyouaretyping,itisn'tgoingtobewgorriedifyoumis-spellsomeofthewgordsinyourte٠xt;youwilljusthavetoaccuratelyproof-readyourprinted:output.Ontheotherhand,Hifyoumis-spelloneoftheenvironmentnamesinyour lethenLffA;TU>?E1fXwgon'tknowwhatyouwantittodo. +Whenthissortofthinghappens,LffA;TU>?E1fXprintsanerrormessageonyourterminalscreenandthenstopsandwgaitsforyoutotakesomeaction.Theerrormessagesthatitproducesmayseemuser-unfriendlyandnotalittlefrighteningat rst.Nev٠ertheless,ifyouknowwheretolookthe٠ywillprobablytellyouwheretheerrorisandwhatwentwrong.Consider0whatwgouldhappenifyoumistypedL\begin{itemize}asL\begin{itemie}.WhenLffA;TU>?E1fXprocessesthisinstruction,itdisplaysthefollowingonyourterminal: BYLaTeXerror. SeeLaTeXmanualforexplanation. \38Type Hforimmediatehelp.!Environmentitemieundefined.\@latexerr...forimmediatehelp.}\errmessage{#1} \endgroupl.140\begin{itemie}?WBAftertypingthe`?'LffA;TU>?E1fXstopsandwgaitsforyoutotellitwhattodo.The rsttwgolinesofthemessagejusttellyouthattheerrorwasdetectedbyLffA;TU>?E1fX.gThe thirdPline,6theonethatstarts`!'Јisthe@errorindicator.Ittellsyouwhattheproblemis,6thoughuntil9youhav٠ehadsomeexperienceofLffA;TU>?E1fXthismaynotmeanalottoyou.FInthiscaseitisjusttellingyouthatitdoesn'trecogniseanenvironmentcalledLitemie&ջ.Thene٠xttwgolinestellyouwhatLffA;TU>?E1fXwgasdoingwhenitfoundtheerror;the٠yareirrelevantatthemomentandcanbeignored. Thei` nallineiscalledthe@errorlocator,mandisacopgyofthelineinyour lethatcausedthe9 problem.DItstartswithalinenumbertohelpyouto nditinyour le.Iftheerrorwgasinthemiddleofaline,itwillbeshownbrokgenatthepointwhereLffA;TU>?E1fXrealizedthattherewasanerrors8.UAs?withallcomputerprograms,8LffA;TU>?E1fXcansometimespassthepointwheretherealerrorwgasbeforedisco٠veringthatsomethingiswrong,butitdoesn'tusuallygetv٠eryfars8.At{&thispointyoucoulddosev٠eralthings.IfyouknewenoughaboutLffA;TU>?E1fXyoumightbeabletoR xtheproblem,[oryoucouldtype`LX'andpressthereturnkge٠ytostopLffA;TU>?E1fXrunningwhileyougoHandcorrecttheerrors8.:oThebestthingtodo,Yhowev٠er,isHjusttopressthereturnkge٠yY.:oThiswillallow:LffA;TU>?E1fXtogoonrunningasifnothinghadhappened.JIfyouhav٠emadeonemistakge,Hthenyouhav٠eprobablymadeseveralandyoumayaswelltryto ndthemallinonego.?E1fXo٠verando٠verag7ain xingoneerroratatime.Don'twgorryaboutrememberingwhattheerrorswere|acopyofalltheerrormessagesisbeingsav٠edinJa@logg lesothatyoucanlookatthemafterwards.Seeyour@localguideto ndoutwhatthis leiscalled.Ifyoulookatthelinethatcausedtheerror,it's8snormallyob٠viouswhattheproblemwgas.Ifyouycan'twgorkoutwhatyourproblemis,lookatthehintsbelowand,ifthe٠ydon'thelp,consult]ARrҍNEssentialL`ffOAMNTU>?EfX,19.{Chapterę6ofLamport's8sbook:itcontainsalistofalloftheerrormessagesthatyouarelikgelyto encounter,togetherwithsomesuggestionsastowhatmayhav٠ecausedthem.Someofthemostcommonmistakgesthatcauseerrorsare:oAmis-speltcommandorenvironmentname.*ލImproperlymatched`L{'and`L}'|rememberthatthe٠yshouldalgwayscomeinpairs.Tgrying"touseoneofthetenspecialcharactersL#$%&_{}~^"andL\asanordi- naryprintingsymbol.AmissingL\endcommand.Amissingcommandargument(that's8sthebitenclosedin'L{'and`L}').AsingleerrorcangetLffA;TU>?E1fXsoconfusedthatitreportsaseriesofspuriouserrorsasaresult.f}If youf_hav٠eanerrorthatyouunderstand,kfollowedbyaseriesthatyoudon't,thentrycorrectingthe rsterror|therestmayvanishasifbymagic.SometimesdLffA;TU>?E1fXmaywriteaL*andstopwithoutanerrormessage.ThisisnormallycausedbyOamissingL\end{document}command,thoughothererrorscancauseit.qIfthishappenstypeL\stopandpressthereturnkge٠yY.FinallyY,CLffA;TU>?E1fXwillsometimesprint@warningmessages.The٠yreportproblemsthatwerenotbad:ZenoughtocauseLffA;TU>?E1fXtostopprocessing,hbutnev٠erthelessmayrequireinvestig7ation.HThemostcommonproblemsare`o٠verfull'and`underfull'linesofte٠xt.Amessagelikge: oYOverfull\hbox(10.58649pttoowide)inparagraphatlines172--175 []\tenrmMathematicalfor-mu-lasmaybedis-played.Adis-playedindicatescthatLffA;TU>?E1fXcouldnot ndagoodplacetobreakalinewhenlayingoutaparagraph.'Asaresult,a~itYwgasforcedtoletthelinestickoutintotheright-handmargin,inthiscaseby10.6points.Sinceapointis1=72:27inches,thismayberatherhardtosee,butitwillbetherenonetheless.ThisƍparticularproblemhappensbecauseLffA;TU>?E1fXisratherfussyaboutlinebreaking,/anditwgouldwrathergeneratealinethatistoolongthangenerateaparagraphthatdoesn'tmeetitshighstandards.dThesimplestwgayaroundtheproblemistoenclosetheentireo endingparagraphbetweenWL\begin{sloppypar}andL\end{sloppypar}commands.=ThistellsLffA;TU>?E1fXthatyouarehappgyforittobreakitsownruleswhileitisworkingonthatparticularbitofte٠xt.Alternativ٠elyY,*messages{Uabout\UnderfullL\hboxes"mayappears8. Thesearelinesthathadtođhav٠emorespaceinsertedbetweenwgordsthanLffA;TU>?E1fXwouldhav٠eliked.KIngeneralthereisnotmuch.thatyoucandoaboutthese.Ygouroutputwilllook ne,? ev٠enifthelinelooksabitstretched.Abouttheonlythingyoucoulddoistore-writetheo endingparagraph! 17iAFinalReminderYgousnowknowenoughLffA;TU>?E1fXtoproduceawiderangeofdocuments.Butthisdocumenthasonlyscratched5thesurfgaceofthethingsthatLffA;TU>?E1fXcando.8Thisentiredocumentwasitselfproducedwith?E1fX(withnostickingthingsinorclev٠eruseofaphotocopier)andev٠enithasn'tusedallthe;featuresthatitcould.JFromthisyoumaygetsomefeelingforthepowerthatLffA;TU>?E1fXputsatyourdisposal.Please,rememberwhatwgassaidintheintroduction:?ifyou@dohav٠eacomplexdocumenttoproduceuthen@goandrGeadLamport'sbook.Ygouwillbewgastingyourtimeifyourelyonlyonwhatyou%hav٠ereadhere.Onceyouhav٠eusedLffA;TU>?E1fXforsometime,.youmightwganttoknowhowtousetheman٠ystyle leextensions,iwhichhavebeendevelopedtoaidLffA;TU>?E1fXuserstotypesettheirdocuments.7+This1andmuchelse(likgehowtouse@MakeIndex,forgeneratingindices,orBNib32TU>&dE1fX,formanagingyourbibliographies)isco٠veredin@TheLffCA-S@TU>4xE1fXCompanion[3].@Finallythe nalwgordaboutTU>?E1fXisKnuth's8s@TU>4xEXbook+[2].And Aoneotherwgarning:.havingdabbledwithLffA;TU>?E1fXyourdocumentswillnev٠erbethesameag7ain...*mARrҍ20,NEssentialL`ffOAMNTU>?EfX.5AMathematicalsymbolshfd`N .3L\alphaj4 ~L\beta }L\gamma#}7DL\delta `N.3L\epsilonj4"~L\varepsilon}L\zeta#}7DL\eta`N.3L\thetaj4#~L\vartheta}L\iota#}7DL\kappa`N.3L\lambdaj4~L\mu}L\nu#}7DL\xi`No.3Loj4~L\pi$}L\varpi#}7DL\rho`N%.3L\varrhoj4~L\sigma&}L\varsigma#}7DL\tau`N.3L\upsilonj4~L\phi'}L\varphi#}7DL\chi`N .3L\psij4!~L\omega`N.3L\Gammaj4~L\Delta}L\Theta#}7DL\Lambda`N.3L\Xij4~L\Pi}L\Sigma#}7DL\Upsilon`N.3L\Phij4 ~L\Psi }L\OmegaY䍍T37able1:Greeklettersr䍍dݸ"nL\pmXn\kL\cap;L\diamond-<C>L\oplus ݸ"nL\mpXn[kL\cup4;L\bigtriangleup-< C>L\ominusݸ"nL\timesXn]kL\uplus5;L\bigtriangledown-< C>L\otimesݸ"nL\divXnukL\sqcap/;L\triangleleft-< C>L\oslashݸ"nL\astXntkL\sqcup.;L\triangleright-< C>L\odotݵ?"nL\starXn_kL\vee"Tq lasy10;L\lhd^a-< C>L\bigcircݸ"nL\circXn^kL\wedge;L\rhd^a-L\daggerݸ"nL\bulletXnnkL\setminus;L\unlhd^a-L\ddaggerݸ"nL\cdotXnokL\wr;L\unrhd^a-L\amalgݟ^aMPNot@prede nedinLBAPTfEߨXY2e5" cmmi9e"!oP.xL>EQ䍍yT37able3:Relationsymbols>ARrҍNEssentialL`ffOAMNTU>?EfX,21.ٍjCel9 l:%nL\rmoustachesn8 sn;RL\lmoustacheR9 R;6L\rgroup368 36:HL\lgroupl?l?%nL\arrowvertsnwsnwRL\ArrowvertR>R>R>R>6L\bracevert%kT37able4:Largedelimiterse̏hd6"%6L\uparrowa6*sRL\UparrowR#RL\downarrowR+&oL\Downarrow 6f%6L\{a6gsRL\}RlRL\updownarrowRm&oL\Updownarrow6b%6L\lfloora6csRL\rfloorRdRL\lceilRe&oL\rceil6h%6L\langlea6isRL\rangleR=RL/Rn&oL\backslash6j%6L|a6ksRL\|3Q䍍T37able5:Delimitersrm?d xL\leftarrowy UXqL\longleftarrow%qո";qL\uparrow (xL\Leftarrowy(UX=qL\Longleftarrow%qո*;qL\Uparrow!xL\rightarrowy8!qL\longrightarrow%qո#;qL\downarrow)xL\Rightarrowy=UX)qL\Longrightarrow%qո+;qL\Downarrow$xL\leftrightarrowy UX!qL\longleftrightarrow%qոl;qL\updownarrow,xL\Leftrightarrowy(UX)qL\Longleftrightarrow%qոm;qL\Updownarrow7!xL\mapstoy7v!qL\longmapsto%qո%;qL\nearrow UX-xL\hookleftarrowy,UX!qL\hookrightarrow%qո&;qL\searrow(xL\leftharpoonupy*qL\rightharpoonup%qո.;qL\swarrow)xL\leftharpoondowny+qL\rightharpoondown%qո-;qL\nwarrowQQ䍍͌T37able6:Arrowsymbols|g74x:::L\ldots^v+L\cdots+û.+.+.+L\vdots".&Vn.)ޟ.9L\ddots@L\aleph 0L\prime^8v+L\forall+ø1+L\infty"+ msbm10~9L\hbar;L\emptyset9L\exists^rv+L\nabla+ßfp+L\surd"+29L\Box^a4L\triangle3L\Diamond^a^{v+L\imath+õ|+L\jmath"+Ƶ`9L\ell:L\neg>L\top^[v+L\flat+õ\+L\natural"+Ƶ]9L\sharp}L\wp?L\bot^|v+L\clubsuit+ø}+L\diamondsuit"+Ƹ~9L\heartsuitL\spadesuit0L\mho^a^?EfX.㰗Ѓ䍍dRL\arccosLR\cosvR\cscR\expR\kerR\limsup*R\minNR\sinh R\arcsinLR\coshvR\degR\gcdR\lgR\ln*R\PrNR\supR\arctanLR\cotvR\detR\homR\limR\log*R\secNR\tanR\argLR\cothvR\dimR\infR\liminfR\max*R\sinNR\tanh-Q䍍ٻT37able8:Log-likgesymbolsB̟d$^aI2L\hat{a}Ym˲YI3ajdL\acute{a}eaۖL\bar{a}_ۗa$L\dot{a}AIaA$ɵaRmL\breve{a} $aI2L\check{a}Ym˲YI3ajdL\grave{a}~eaۖL\vec{a}/ۗa$L\ddot{a}AIa~A$ɵaRmL\tilde{a}!Q䍍ĠT37able9:MathmodeaccentsPrdPvL\sumYwQp8L\prod8`UbL\coprodUcīR%L\intgīH|L\oint TvL\bigcapYwSp8L\bigcup8FUbL\bigsqcupUcW%L\bigveegV|L\bigwedgeJvL\bigodotYwNp8L\bigotimes8LUbL\bigoplusUcU%L\biguplus'Q䍍~ֻT37able10:Variable-sizedsymbols額 i@ۍ\q%90f#EVabcEcL\widetilde{abc}\qv>cÂdabc_L\widehat{abc} 69#EV Ǎ#EVabcEcL\overleftarrow{abc}9Âdj!ǍÂdabc_L\overrightarrow{abc} \#EV fe 絟$abcEcL\overline{abc}ÂdabcÂdfe 絎_L\underline{abc}9#EVz}|{ Ǎ%QabcEcL\overbrace{abc}Ŏabc31Âd|{z}_L\underbrace{abc}#EVzPp+zPfe 絟abcEcL\sqrt{abc}I| O \cmmi5n"zPpxzPfe 絟abc_L\sqrt[n]{abc} f#EVf^0EcLf' xKabcĵ&fe xy@Lz_L\frac{abc}{xyz}A`T37able11:LffA;TU>?E1fXmathconstructsr䍍d ~L\hbar}IL\hslashJML\vartriangle  OL\triangledownIL\squareJL\lozenge sL\circledS\IL\angleJ]L\measuredangle @L\nexists0IL\mhoJ`L\Finv^a aL\Game^a|IL\Bbbk^aJ8L\backprime ?L\varnothingNIL\blacktriangleJHL\blacktriangledown L\blacksquareIL\blacklozengeJFL\bigstar ^L\sphericalangle{IL\complementJgL\eth L\diagup^aIL\diagdown^a ^ayPNot@de nedinstyleQamssymbP,de neusingtheLBAPTfEߨXY2e"#AoQ\DeclareMathSymbolPcommandRQ䍍r:T37able12:AMSmiscellaneoussymbolsYgARrҍNEssentialL`ffOAMNTU>?EfX,23.S/Ldz%ɦL\digammaaɧ{tpL\varkappaqiɆL\bethk0L\daleth<0jLL\gimel{mT37able13:AMSGreekandHebrew5 Ld!Rp2RL\ulcornertRqRL\urcornerRxRL\llcornerRy+RL\lrcorner)T37able14:AMSdelimiterschEd99K!L\dashrightarrowL99 L\dashleftarrow8 N L\leftleftarrows !L\leftrightarrowsW L\Lleftarrow8 N L\twoheadleftarrow!L\leftarrowtail" L\looparrowleft8 N L\leftrightharpoonsx!L\curvearrowleft L\circlearrowleft8 N L\Lsh!L\upuparrows L\upharpoonleft8 N L\downharpoonleft(!L\multimap! L\leftrightsquigarrow8 N L\rightrightarrows!L\rightleftarrows L\rightrightarrows8 N L\rightleftarrows!L\twoheadrightarrow L\rightarrowtail8 #N L\looparrowright !L\rightleftharpoonsy L\curvearrowright8 N L\circlearrowright!L\Rsh L\downdownarrows8 N L\upharpoonright!L\downharpoonright L\rightsquigarrowWQ䍍7T37able15:AMSarrowsCr܃䍍dR8&RL\nleftarrowzR9RL\nrightarrowR:RL\nLeftarrow R;&RL\nRightarrowzR=RL\nleftrightarrowR<RL\nLeftrightarrow!Q䍍T37able16:AMSne٠g7atedarrowsr䍍du$nL\dotplusnr6L\smallsetminus6e=L\Cap d$nL\CupnZ6L\barwedge6Y=L\veebar[$nL\doublebarwedgen 6L\boxminus6=L\boxtimes$nL\boxdotn6L\boxplus6>=L\divideontimesn$nL\ltimesno6L\rtimes6h=L\leftthreetimesi$nL\rightthreetimesnf6L\curlywedge6g=L\curlyvee$nL\circleddashn~6L\circledast6}=L\circledcirc$nL\centerdotn|6L\intercalEQ䍍nܻT37able17:AMSbinaryoperatorsfARrҍ24,NEssentialL`ffOAMNTU>?EfX.S(ލ~݆5 L\leqq!6zL\leqslant{02 L\eqslantless . L\lesssim!/zL\lessapprox{u2 L\approxeql L\lessdot!nzL\lll{72 L\lessgtr lQ L\lesseqgtr!SzL\lesseqqgtr{+2 L\doteqdot 9: L\risingdotseq!;zL\fallingdotseq{v2 L\backsimw L\backsimeq!jzL\subseteqq{b2 L\Subset< L\sqsubset!4zL\preccurlyeq{22 L\curlyeqprec- L\precsim!wzL\precapprox{C2 L\vartriangleleftE L\trianglelefteq!zL\vDash{2 L\Vvdash` L\smallsmile!azL\smallfrown{l2 L\bumpeqm L\Bumpeq!=zL\geqq{>2 L\geqslant1 L\eqslantgtr!&zL\gtrsim{'2 L\gtrapproxm L\gtrdot!ozL\ggg{?2 L\gtrlessR L\gtreqless!TzL\gtreqqless{P2 L\eqcirc $ L\circeq!,zL\triangleq{s2 L\thicksimt L\thickapprox!kzL\supseteqq{c2 L\Supset= L\sqsupset!<zL\succcurlyeq{32 L\curlyeqsucc% L\succsim!vzL\succapprox{B2 L\vartrianglerightD L\trianglerighteq! zL\Vdash{p2 L\shortmidq L\shortparallel!GzL\between{t2 L\pitchfork_ L\varpropto!JzL\blacktriangleleft{)2 L\therefore L\backepsilon!IzL\blacktriangleright{*2 L\becauseZqT37able18:AMSbinaryrelations>~nL\nlessn6L\nleq6 =L\nleqslant ˏnL\nleqqn 6L\lneq6=L\lneqq WnL\lvertneqqn6L\lnsim6=L\lnapprox nL\nprecn6L\npreceq6=L\precnsimnL\precnapproxn6L\nsim6.=L\nshortmid-nL\nmidn06L\nvdash62=L\nvDash6nL\ntriangleleftn56L\ntrianglelefteq6*=L\nsubseteq(nL\subsetneqn 6L\varsubsetneq6$=L\subsetneqq&nL\varsubsetneqqn6L\ngtr6=L\ngeq nL\ngeqslantn6L\ngeqq6 =L\gneq W nL\gneqqn6L\gvertneqq6=L\gnsimnL\gnapproxn6L\nsucc6=L\nsucceqnL\succnsimn6L\succnapprox6=L\ncong/nL\nshortparalleln,6L\nparallel62=L\nvDash3nL\nVDashn76L\ntriangleright64=L\ntrianglerighteq+nL\nsupseteqn#6L\nsupseteqq6)=L\supsetneq!nL\varsupsetneqn%6L\supsetneqq6'=L\varsupsetneqq|ʍqBT37able19:AMSne٠g7atedbinaryrelationsq>ARrҍNEssentialL`ffOAMNTU>?EfX,25.`BHorrib$leMathematicalExamplestoStudȽyqЍC(t)=<$ ,1Kwfedv EPpUWPfe E2ra0|sk0-a1k1N:::X010Cfi0AK)(3)ҍQ\begin{equation} Y\prod_{j\geqff0}Y\left(\sum_{k\geqff0}a_{jk}z^k\right)ҍ=ff\sum_{k\geq0}z^nY\left(ff\sum_{{k_0,k_1,\ldots\geq0}3҉\atop{k_0+k_1+\ldots=n}})a{_0k_0}a_{1k_1}\ldots \right)ҍ\end{equation}lƍ M[ٲ(n)=ꨴn *X m=2^6^6^6fi^4U \ $?m1 %YX %tk+B=15Ÿb(m=kP)=dm=kec\!ȟv1d<7d<7d<7fid<5)(4)ҍQ\begin{equation} ҍ\pi(n)ff=\sum_{m=2}^{n}Y\left\lfloorff\left(\sum_{k=1}^{m-1}#W\lfloor(m/k)/\lceilffm/k\rceil#W\rfloorff\right)^{-1}Y\right\rfloorҍ\end{equation}aƍ~ۍR㰸fLЍFkBar0ss0z31e}|31e{ a;:::;a 9;LЍ bl`br0ss0z31g}|31g{ b;:::;b1|31{z31}:L ~k+B+1elementsD +g)(5)ҍQ\begin{equation} ҍ\{\underbrace{%l%\overbrace{\mathstrutffa,\ldots,a}^{k\a's},l%\overbrace{\mathstrutffb,\ldots,b}^{l\b's}}Y_{k+1\ff\mathrm{elements}}f\}ҍ\end{equation}Vƍ++SW\α+;,kD%^+p+8 kD![ٟ^+ 8I+8[ٟ^0kD!^+p+8[ٟ^0kD&e^+p+8eҍQ\begin{displaymath} ҍ\mbox{W}^+\ҍ\begin{array}{l}ҍ\nearrow\raise5pt\hbox{$\mu^+ff+\nu_{\mu}$}\\ҍ\rightarrow0\pi^+ff+\pi^0\\[5pt]ҍ\rightarrowff\kappa^++\pi^0K\\ҍ\searrow\lower5pt\hbox{$\mathrm{e}^+3҉+\nu_{\scriptstyle\mathrm{e}}$}ҍ\end{array}ҍ\end{displaymath}ƍِ:TFc(x;y[ٲ)=0 and% % % % % % ԍ.rF^00፴xxHsF^00፴xyaF^0፴x ;.9F^00፴y@LxH9F^00፴y@LyaFF^0፴y0`F^0፴xJ`F^0፴ye0rY4 rY4 rY4 rY4 rY4 rY4 xu=0ҍQ\begin{displaymath} ҍ{F}(x,y)=0\quad\mathrm{and}\quad8\left|\begin{array}{ccc}YF_{xx}''ff&F_{xy}''&F_{x}'\\YF_{yx}''ff&F_{yy}''&F_{y}'\\YF_{x}'32&ffF_{y}'&0#W\end{array} fd\right|ff=0ҍ\end{displaymath}<ARrҍ26,NEssentialL`ffOAMNTU>?EfX.{2z왒3˧      d x1S8x24y1S8y2]z1S8z2 յl1=}m1g޵n1յl2=}m2g޵n2ͫ      Jfe%ʍn/s n/fe%э  dUUl1͢m1 UUl2͢m2,̫ , , , /g!ڱ26t+8 8 8 8 d 5m1!_n1 ĵn1#Tl11Ol 1Ol 1Ol 1Ol 4ڱ2;Z+8 8 8 8 d 5m2!_n2 ĵn2#Tl21Ol 1Ol 1Ol 1Ol 4ڱ2ҍQ\begin{displaymath} ҍ\frac{\pmҍ\left|\begin{array}{ccc}ҍx_1-x_2 &ffy_1-y_2&z_1-z_2\\ҍl_1 fd&ffm_1&n_1\\ҍl_2 fd&ffm_2&n_2ҍ\end{array}\right|}{ҍ\sqrt{\left|\begin{array}{cc}l_1&m_1\\ҍl_2&m_2\end{array}\right|^2ҍ+\left|\begin{array}{cc}m_1&n_1\\ҍn_1&l_1\end{array}\right|^2ҍ+\left|\begin{array}{cc}m_2&n_2\\ҍn_2&l_2\end{array}\right|^2}}ҍ\end{displaymath}ZV&q!1ɍ[ٴfxݍ0(Q;T3R յ; ;s)oh=<$bY4[ z^2bYwfe (֍%W3s; V828686fi84򙖍 Q^2'^O38 2Ofen, r72+^5Ѹ82QCVC^0-V]s(sM^2ٔZ)  fec6ز(s8M:2ٔZ)5D24xE1fX|A^ADocumentPrGepar٠ationSystem|User'sGuideandReferenceManu- al.Addison-W37esle٠yY,Reading,MA,USA,1985.2 DonaldE.Knuth.@TheTU>4xE1fXbook,AvolumeAof@ComputergsandTBypesetting.Addison-W37esle٠yY, Reading,MA,USA,1986.3 MichelGoossens,ɒFrankMittelbachandAle٠xanderSamarin.U@TheLffCA-S@TU>4xE1fXCompanionAddison- W37esle٠yY,Reading,MA,USA,1993.;Ae5" cmmi9`Ɍ cmbsy10]DF cmmib10Z"V cmbx10Y0 pcrr7tX3{q ptmr7tW3{ ptmr7tV3{H ptmr7tUl phvr7tT0 pcrb7tSl phvr7tRl phvr7tQ0 pcrr7tP3{ ptmr7tO8 ptmrc7tN8 ptmrc7tM3{ ptmro7tL0 pcrr7tKe9V phvb7tCKj ptmri7tB3{ ptmr7tA3{ ptmr7t@Kj ptmri7t?3{ ptmr7t>ߌ ptmb7t5ߌ ptmb7t4e9Vff phvb7t23{ff ptmr7t&2cmmi8#3{ ptmr7t"Tq lasy10 msbm10 msam103{G ptmr7t3{ ptmr7t !", cmsy10 O!cmsy7 0ncmsy5 b> cmmi10 0ercmmi7O \cmmi5K`y cmr10ٓRcmr7Zcmr5u cmex10=