÷ƒ’À;è TeX output 1995.04.25:1439‹ÿÿÿÿ ª ýs o ýÞ’¢†ªóDÓítG®G®cmr17¹The–7tó߆µTG® cmtt12ºeuler“¹pacŒqk‘ÿäageŽŸ’Ć}óX«Q cmr12»F‘ÿVrank‘ê¨JensenŸû¥2ó¾KÈ cmsy8ÂŽŽŽŽŽŽŽŸ37’É »1995/03/05ŽŸ9n‘5óÂÖN ff cmbx12Å1Ž‘MLËInŒÌtros3ductionŽŸç‘5óKñ`y cmr10²The––YóßêŸ'EŽ‘ãxX“doGcumenš¸ãts.‘O‰The“AMS‘žëEuler“family“w˜as“designed“b˜y“HermannŽ¡‘5Zapf,‘Gcommissioned–ãŠbš¸ãy“the“American“Mathematical“SoGciet˜y‘ÿ*ª.‘ e\The“underlyingŽ¡‘5philosophš¸ãy–¢ÿof“Zapf‘Ç's“Euler“design“w˜as“to“capture“the“ a˜v˜or“of“mathematics“as“itŽ¡‘5mighš¸ãt–UUbGe“written“b˜y“a“mathematician“with“excellen˜t“handwriting."“[2Ž‘]Ž¡‘DThe–pÆeuler“²pacš¸ãk‘ÿqÇage“is“based“on“Kn˜uth's“macros“for“the“b•Go“ok–p\Concrete“Math-Ž¡‘5ematics"–3[1Ž‘].‘Š`Knš¸ãuth's“macros“can“bGe“found“through“anon˜ymous“ftp“to“Ælabrea.Ž¡‘5stanford.edu²:‘*ÁloGok–±Òfor“the“ le“Ægkpmac.tex“²in“directory‘Õ_Æpub/tex/local/lib².Ž¡‘5The–ÓŽEuler“fonš¸ãts“can“bGe“found“through“anon˜ymous“ftp“to“Æe-math.ams.org²:‘n9loGokŽ¡‘5in‘UUdirectory‘xâÆpub/tex/amsfonts².Ž¡‘DThe–…purpGose“of“the“Æeuler“²pacš¸ãk‘ÿqÇage“is“to“pro˜vide“the“math“part“of“the“loGokŽ¡‘5of–~Ï[1Ž‘].‘î5The“other“part“(text“fonš¸ãts)“is“pro˜vided“b˜y“the“Æbeton“²pac˜k‘ÿqÇage.‘î5The“reasonŽ¡‘5for–_Gcreating“t•¸ãw“o›_Gpac“k‘ÿqÇages˜is˜to˜mak“e˜it˜easy˜to˜use˜the˜Euler˜math˜fon“ts˜togetherŽ¡‘5with–²´other“text“fonš¸ãts“(in“particular,‘Ê it“appGears“that“the“Euler“fon˜ts“matc˜h“man˜yŽ¡‘5of–UUthe“pGopular“Pš¸ãostScript“fon˜ts“prett˜y“w˜ell).Ž¡‘DBasically‘ÿ*ª,‘pæthe–80Æeuler“²pacš¸ãk‘ÿqÇage“pro˜vides“the“same“setup“(the“same“de nitions,Ž¡‘5math–Z³cošGdes,‘Ü etc.)“as“Ægkpmac.tex“²with“resp˜ect“to“the“Euler“fon•¸ãts.‘ âHo“w“ev“er,Ž¡‘5Knš¸ãuth–N[2Ž‘]“admits“that“the“macros“w˜ere“written“for“one“spGeci c“pro‘Ž8ject,‘D namelyŽ¡‘5to–emt¸ãypšGeset“the“\Concrete“Mathematics"“b˜o˜ok“[1Ž‘].‘¢So,‘isthe“Æeuler“²pac¸ãk‘ÿqÇage“actuallyŽ¡‘5došGes–Œƒa“little“more“than“Ægkpmac.tex²:‘à$for“example,‘šOsome“`exotic'“sym•¸ãb˜ols,‘šOpresen“tŽ¡‘5in–õ:CM‘õ"math“italic,‘sare“missing“from“the“Euler“fonš¸ãts;‘Cthe“Æeuler“²pac˜k‘ÿqÇage“tak˜es“careŽ¡‘5of–UUthis“(whereas“Ægkpmac.tex“²doGes“not).Ž¡‘DThe–ÿsuse“of“the“Æeuler“²pacš¸ãk‘ÿqÇage“requires“a“lot“of“care“when“en˜tering“theŽ¡‘5manš¸ãuscript,‘ýisince–uethe“pac˜k‘ÿqÇage“will“cause,–ýie.g.,“math–uen˜umerals“to“come“fromŽ¡‘5the–ºEuler“Roman“fonš¸ãts;‘ì€these“n˜umerals“are“easily“distinguished“from“the“normalŽ¡‘5text–ðÀnš¸ãumerals.‘DThis“implies“that“slopp˜y“t˜yping“is“clearly“expGosed:‘¨œimagine“ho˜wŽ¡‘5\Æ$x$–?ýis“either“1“or“$-1$²"–€åwill“appšGear“when“t¸ãyp˜eset!‘*÷In“general,‘«bthe“t¸ãypist“willŽ¡‘5in–°Òeacš¸ãh“case“ha˜v˜e“to“decide“whether“an“input“fragmen˜t“is“a“math“or“a“non-mathŽ¡‘5en•¸ãtit“y‘ÿ*ª.‘_ This–holds“true“evš¸ãen“for“the“tiniest“details,‘([suc˜h“as“punctuation“c˜haractersŽ¡‘5(paren¸ãtheses,–UUcolons,“semicolons,“commas,“and“p•Gerio“ds).Ž¡‘DFinally‘ÿ*ª,‘ua–<ÿfew“wš¸ãords“abGout“the“tec˜hnical“qualit˜y“of“the“fon˜ts“(there“are“no“doubtsŽ¡‘5abGout–öµtheir“artistic“qualitš¸ãy):‘BwThe“fon˜ts“are“not“tuned“for“lo˜w“resolution“rendering:Ž¡‘5I‘Z(recommend–Zkat“least“600dpi.‘ The“fonš¸ãts“emplo˜y“none“of“the“tuning“mac˜hineryŽ¡‘5of– hCM:“e.g.,‘;lthey“do“not“use“the“`blac•¸ãk“er'– hparameter“spGeci ed“in“eacš¸ãh“Metafon˜tŽ‘5Ÿ&=‰ffÿr‰ffÿrŸ @‘ ÷ˆŸý-:óq¡% cmsy6ÃŽŽ‘LÜó|{Ycmr8¼UpŸ'EŽ‘ãxX).Ž¡‘DSome–¹æusers“do“not“likš¸ãe“the“Æ\hat“²accen˜t“glyph“presen˜t“in“the“Euler“fon˜ts“(theŽ¡‘5Æeuler–´Ÿ²pacš¸ãk‘ÿqÇage“uses“this“accen˜t“glyph“bGe“default).‘¤T‘ÿ*ªo“use“the“Æ\hat“²accen˜t“glyphŽ¡‘5from–PIthe“text“fonš¸ãt“instead,‘load“the“Æeuler“²pac˜k‘ÿqÇage“with“the“Ætext-hat-accentŽ¡‘5²option.Ž¡‘DIn–'v[1Ž‘],‘[þthe“space“around“relations“in“displaš¸ãys“w˜as“increased“(from“`5m˜u“plusŽ¡‘55mš¸ãu'–ú|to“`10m˜u“min˜us“3m˜u').‘SThe“Æeuler“²pac˜k‘ÿqÇage“doGes“not“c˜hange“this“space,‘ §sinceŽ¡‘5this–­uis“(for“most“cases)“conš¸ãtrolled“b˜y“the“LŸýó5‘üff±AŽŽ‘͉²T‘þU>Ÿ'EŽ‘ãxX“Æeqnarray“²en˜vironmen˜t.‘z(Also,Ž¡‘5in–K[1Ž‘],‘!Mdisplaš¸ãys“are“left“justi ed“with“a“suitable“inden˜tation;‘)ùw˜e“lea˜v˜e“suc˜h“mattersŽ¡‘5to–UUthe“general“la•¸ãy“out.Ž¦‘5Å3Ž‘MLËHello‘ffwŒÌorldŽŸç‘5²First,–UUwš¸ãe“announce“the“pac˜k‘ÿqÇage.ŽŸ‘8üs±1Ž‘?ª®ó !",š cmsy10¸hóÒoÇw cmss9Êpack¾9ageŽ‘#ǸiŽ¤ ‘8üs±2Ž‘?ª®óߤN cmtt9É\NeedsTeXFormat{LaTeX2e}[1994/12/01]Ž¡‘8üs±3Ž‘?ª®É\ProvidesPackage{euler}[\filedate\space\fileversion]Ž¡‘8üs±4Ž‘?ª®É\typeout{Package:–¹–`euler'“\fileversion\space“<\filedate>“(FJ“and“FMi)}Ž¦‘5Å4Ž‘MLËF‘þ¦fonŒÌt–ffand“other“de nitionsŽŸç‘5²The–?AMS‘.Euler“family“consists“of:‘P¼Euler“Roman“(medium“and“bGold),‘ vEuler“F‘ÿ*ªrak-Ž¤ ‘5tur–EÛ(medium“and“bšGold),‘üEuler“Script“(medium“and“b˜old),‘üand“Euler“ExtensionŽ¡‘5(medium–<‹only).‘i„The“Æ.fd“² les“for“these“fon¸ãts“are“de ned“in“the“Æamsfonts.fdd“² leŽ¡‘5(part–UUof“the“AMSF‘ÿ*ªon¸ãts“distribution).ŽŸþ6‘5ó"ÂÖN  cmbx12Í4.1Ž‘SÀEuler‘€RomanŽŸuT‘5²The–v0Euler“Roman“fonš¸ãts“replace“the“Computer“MoGdern“Math“Italic“fon˜ts,‘~gloGcatedŽ¡‘5in–UUthe“`letters'“math“symš¸ãbGol“fon˜t:ŽŸ‘8üs±5Ž‘?ª®É\DeclareSymbolFont{letters}{U}{eur}{m}{n}ŽŸ ‘8üs±6Ž‘?ª®É\SetSymbolFont{letters}{bold}{U}{eur}{b}{n}ŽŽŸ’ä²2ŽŽŒ‹F ª ýs o ý´‘D²In–£¼math“moGde,‘ÇAdigits“should“come“from“the“Euler“Roman“fon¸ãts.‘6”As“in“standardŽ¤ ‘5LŸýó5‘üff±AŽŽ‘͉²T‘þU>Ÿ'EŽ‘ãxX,‘çwš¸ãe–ê—assign“the“t˜ypGe“Æ\mathalpha“²to“the“digits“whic˜h“means“that“they“willŽ¡‘5v‘ÿqÇary–Èÿwith“math“alphabšGets.‘ÌÄOne“of“the“reasons“for“this“is“that“fo˜otnote“n•¸ãum“b˜ersŽ¡‘5are–Êîset“in“math“moGde,‘æand“wš¸ãe“w˜an˜t“these“n˜um˜bGers“to“come“from“the“text“fon˜t;‘ùthisŽ¡‘5is–šaccomplished“bš¸ãy“setting“the“T‘þU>Ÿ'EŽ‘ãxX“primitiv˜e“Æ\fam“²to“zero.‘[Þ(See“Section“7“bGelo˜w.)Ž©‘8üs±7Ž‘?ª®É\DeclareMathSymbol{0}\mathalpha{letters}{"30}Ž¤ ‘8üs±8Ž‘?ª®É\DeclareMathSymbol{1}\mathalpha{letters}{"31}Ž¡‘8üs±9Ž‘?ª®É\DeclareMathSymbol{2}\mathalpha{letters}{"32}Ž¡‘5±10Ž‘?ª®É\DeclareMathSymbol{3}\mathalpha{letters}{"33}Ž¡‘5±11Ž‘?ª®É\DeclareMathSymbol{4}\mathalpha{letters}{"34}Ž¡‘5±12Ž‘?ª®É\DeclareMathSymbol{5}\mathalpha{letters}{"35}Ž¡‘5±13Ž‘?ª®É\DeclareMathSymbol{6}\mathalpha{letters}{"36}Ž¡‘5±14Ž‘?ª®É\DeclareMathSymbol{7}\mathalpha{letters}{"37}Ž¡‘5±15Ž‘?ª®É\DeclareMathSymbol{8}\mathalpha{letters}{"38}Ž¡‘5±16Ž‘?ª®É\DeclareMathSymbol{9}\mathalpha{letters}{"39}ŽŸ‘5²The–î¸uppGercase“greek“letters“are“also“takš¸ãen“from“the“Euler“Roman“fon˜ts.‘ =ñW‘ÿ*ªeŽ¤ ‘5makš¸ãe–Ø/them“ordinary“sym˜bGols“(i.e.,‘øåof“t˜ypšGe“Æ\mathord²)“as“opp˜osed“to“Æplain.tex²,Ž¡‘5Ægkpmac.tex²,–UUand“standard“LŸýó5‘üff±AŽŽ‘͉²T‘þU>Ÿ'EŽ‘ãxX,“whicš¸ãh“mak˜e“them“v‘ÿqÇary“according“to“Æ\fam².Ž¦‘5±17Ž‘?ª®É\DeclareMathSymbol\Gamma‘ s,\mathord{letters}{"00}Ž¤ ‘5±18Ž‘?ª®É\DeclareMathSymbol\Delta‘ s,\mathord{letters}{"01}Ž¡‘5±19Ž‘?ª®É\DeclareMathSymbol\Theta‘ s,\mathord{letters}{"02}Ž¡‘5±20Ž‘?ª®É\DeclareMathSymbol\Lambda‘¹–\mathord{letters}{"03}Ž¡‘5±21Ž‘?ª®É\DeclareMathSymbol\Xi‘Ÿî\mathord{letters}{"04}Ž¡‘5±22Ž‘?ª®É\DeclareMathSymbol\Pi‘Ÿî\mathord{letters}{"05}Ž¡‘5±23Ž‘?ª®É\DeclareMathSymbol\Sigma‘ s,\mathord{letters}{"06}Ž¡‘5±24Ž‘?ª®É\DeclareMathSymbol\Upsilon\mathord{letters}{"07}Ž¡‘5±25Ž‘?ª®É\DeclareMathSymbol\Phi‘æX\mathord{letters}{"08}Ž¡‘5±26Ž‘?ª®É\DeclareMathSymbol\Psi‘æX\mathord{letters}{"09}Ž¡‘5±27Ž‘?ª®É\DeclareMathSymbol\Omega‘ s,\mathord{letters}{"0A}ŽŸ‘5²Euler–UUdošGesn't“ha•¸ãv“e–UUthe“sp˜ecial“v‘ÿqÇarian¸ãts“of“Æ\sigma“²and“Æ\rho²:Ž¦‘5±28Ž‘?ª®É\let\varsigma=\sigmaŽ¡‘5±29Ž‘?ª®É\let\varrho=\rhoŽŸ‘ 3P\mathbfŽŽŽ‘5²If–¨Zthe“Æeuler“²pac¸ãk‘ÿqÇage“is“loaded“with“the“Æmathbf“²option,‘½Æ\mathbf“²should“proGduceŽŸ ‘5bGold–UUEuler“Roman“letters“and“digits.Ž¦‘5±30Ž‘?ª®É\DeclareOption{mathbf}Ž¡‘5±31Ž‘M×pÉ{\AtBeginDocument{\DeclareMathAlphabet\mathbf{U}{eur}{b}{n}}}ŽŸ‘5²Note–ƒthe“use“of“Æ\AtBeginDocument²;‘™ûit“ensures“that“the“rede nition“o•¸ãv“errides‘ƒtheŽ¤ ‘5default–UUsettings“(Section“6).ŽŸþ6‘5Í4.2Ž‘SÀEuler‘€F‘þàrakturŽŸuT‘À$É\mathfrakŽŽŽ‘5²The–ßÀEuler“F‘ÿ*ªraktur“fonš¸ãts“get“their“o˜wn“math“sym˜bGol“fon˜t.‘  W‘ÿ*ªe“de ne“a“mathŽ¡‘5alphabGet–¬>idenš¸ãti er|Æ\mathfrak²|to“access“this“sym˜bGol“fon˜t.‘vF‘ÿ*ªor“compatibilit˜yŽ¡‘5with–previous“vš¸ãersions“of“the“Æeuler“²pac˜k‘ÿqÇage,‘lw˜e“de ne“Æ\frak“²as“an“alias“forŽ¡‘5Æ\mathfrak–UU²(the“Æamsfonts“²pacš¸ãk‘ÿqÇage“also“pro˜vides“this“alias).Ž¦‘5±32Ž‘?ª®É\DeclareSymbolFont{EulerFraktur}{U}{euf}{m}{n}Ž¤ ‘5±33Ž‘?ª®É\SetSymbolFont{EulerFraktur}{bold}{U}{euf}{b}{n}Ž¡‘5±34Ž‘?ª®É\DeclareSymbolFontAlphabet\mathfrak{EulerFraktur}Ž¡‘5±35Ž‘?ª®É\@ifpackageloaded{amsfonts}{}{\newcommand\frak{\mathfrak}}ŽŽŸ’ä²3ŽŽŒ‹l ª ýs o ý´‘D²The– Ofolloš¸ãwing“c˜haracters“come“from“the“Euler“F‘ÿ*ªraktur“sym˜bGol“fon˜t“(as“de nedŽ© ‘5in‘UUÆgkpmac.tex²):ŽŸ‘5±36Ž‘?ª®É\DeclareMathSymbol{!}\mathord‘ s,{EulerFraktur}{"21}Ž¤ ‘5±37Ž‘?ª®É\DeclareMathSymbol{(}\mathopen‘¹–{EulerFraktur}{"28}Ž¡‘5±38Ž‘?ª®É\DeclareMathSymbol{)}\mathclose{EulerFraktur}{"29}Ž¡‘5±39Ž‘?ª®É\DeclareMathSymbol{+}\mathbin‘ s,{EulerFraktur}{"2B}Ž¡‘5±40Ž‘?ª®É\DeclareMathSymbol{-}\mathbin‘ s,{EulerFraktur}{"2D}Ž¡‘5±41Ž‘?ª®É\DeclareMathSymbol{=}\mathrel‘ s,{EulerFraktur}{"3D}Ž¡‘5±42Ž‘?ª®É\DeclareMathSymbol{[}\mathopen‘¹–{EulerFraktur}{"5B}Ž¡‘5±43Ž‘?ª®É\DeclareMathSymbol{]}\mathclose{EulerFraktur}{"5D}ŽŸ‘5²There–˜Eis“a“curious“detail“here:‘÷¦The“t¸ãypGe“of“`!'‘:–is“Æ\mathclose“²in“Æplain.tex“²(andŽ¦‘5standard–kLŸýó5‘üff±AŽŽ‘͉²T‘þU>Ÿ'EŽ‘ãxX)“but“Æ\mathord“²in“Ægkpmac.tex².‘ÄThe“reason“for“this“is“probablyŽ¦‘5that–'oone“should“not“use“Euler“punctuation“c¸ãharacters“as“part“of“the“text,Ÿü^ÿ±1ŽŽ‘Øh²andŽ¦‘5punctuation–àcš¸ãharacters“in“math“are“usually“opGerators.‘üE.g.,‘Àin“CSP‘ßî(Comm˜uni-Ž¦‘5cating–LƒSequen¸ãtial“ProšGcesses),‘ŠN`!'“denotes“an“output“op˜eration,‘ŠNa“tigh¸ãtly“bindingŽ¦‘5binary–ÔopšGerator;‘Y”there“is“usually“no“space“around“this“op˜erator“(to“indicate“theŽ¦‘5high‘UUprecedence).Ž¦‘DThe–ÎþEuler“F‘ÿ*ªraktur“symš¸ãbGol“fon˜t“con˜tains“more“punctuation“c˜haracters“thanŽ¦‘5de ned›*/abGo•¸ãv“e.‘ðTW‘ÿ*ªe˜w“an“t˜to˜use˜these˜c“haracters˜as˜they˜lo•Gok˜b“etter˜when˜theyŽ¦‘5appšGear–ʧin“math“form¸ãulas“(there“are“some“strange“lo˜oking“formš¸ãulas“in“[1Ž‘]).‘ѽEv˜enŽ¦‘5if–pdone“ma¸ãy“argue“that,–·(e.g.,“semicolon–pdis“most“often“used“in“the“text“sense“(inŽ¦‘5displaš¸ãys),‘Z¯it–ó7is“more“consisten˜t“and“easier“to“remem˜bGer“if“ev˜erything“in“mathŽ¦‘5moGde–UUis“Euler.ŽŸ‘5±44Ž‘?ª®É\DeclareMathSymbol{"}\mathord‘ s,{EulerFraktur}{"7D}Ž¡‘5±45Ž‘?ª®É\DeclareMathSymbol{&}\mathord‘ s,{EulerFraktur}{"26}Ž¡‘5±46Ž‘?ª®É\DeclareMathSymbol{:}\mathrel‘ s,{EulerFraktur}{"3A}Ž¡‘5±47Ž‘?ª®É\DeclareMathSymbol{;}\mathpunct{EulerFraktur}{"3B}Ž¡‘5±48Ž‘?ª®É\DeclareMathSymbol{?}\mathord‘ s,{EulerFraktur}{"3F}Ž¡‘5±49Ž‘?ª®É\DeclareMathSymbol{^}\mathord‘ s,{EulerFraktur}{"5E}Ž¡‘5±50Ž‘?ª®É\DeclareMathSymbol{`}\mathord‘ s,{EulerFraktur}{"12}ŽŸ‘5²Here,‘[wš¸ãe–YÞalso“c˜hanged“the“t˜ypšGe“of“`?'‘ato“Æ\mathord“²to“b˜e“consistenš¸ãt“with“the“t˜ypGeŽ¦‘5of–û2`!'.‘S»Regarding“the“math“coGdes“assigned“bš¸ãy‘¿ÆINITEX²:“The“Euler“fon˜ts“do“not“ha˜v˜eŽ¦‘5the–Öéglyphs“for“Æ\#²,–ð2Æ\$²,“Æ\%²,“and›ÖéÆ@²,“so˜these˜glyphs˜are˜still˜tak•¸ãen˜from˜the˜text˜fon“ts.Ž¦‘5Note–ðalso“that“wš¸ãe“do“not“set“the“math“coGde“for“the“other“quote“c˜haracter:‘§Nit“isŽ¦‘5activ¸ãe– in“math“mošGde“(co˜de“Æ"8000²);‘Ïit“pro˜duces“primes“(and“should“con•¸ãtin“ue– to“doŽ¦‘5so).ŽŸ‘5±51Ž‘?ª®É\DeclareMathDelimiter{(}{EulerFraktur}{"28}{largesymbols}{"00}Ž¡‘5±52Ž‘?ª®É\DeclareMathDelimiter{)}{EulerFraktur}{"29}{largesymbols}{"01}Ž¡‘5±53Ž‘?ª®É\DeclareMathDelimiter{[}{EulerFraktur}{"5B}{largesymbols}{"02}Ž¡‘5±54Ž‘?ª®É\DeclareMathDelimiter{]}{EulerFraktur}{"5D}{largesymbols}{"03}ŽŸ‘íÙÌ\oldstylenumsŽŽ¦‘íÙÌ\TextOldstyleŽŽŽŽ¦¦‘íÙÌ\MathOldstyleŽŽŽŽŽ‘5²W‘ÿ*ªe–²_proš¸ãvide“a“new“de nition“of“the“Æ\oldstylenums“²macro:‘+Ûthis“de nition“allo˜wsŽ¦‘5us–'to“moGdify“the“generation“of“non-aligning“(ak‘ÿqÇa“\oldstš¸ãyle")“n˜umerals“in“text“andŽ¦‘5in–UUmath“separately‘ÿ*ª,“using“t•¸ãw“o–UUhoGoks,“Æ\TextOldstyle“²and“Æ\MathOldstyle².ŽŸ‘5±55Ž‘?ª®É\DeclareRobustCommand\oldstylenums[1]%Ž¡‘5±56Ž‘IÚÉ{\begingroupŽ¡‘5±57Ž‘\2É\spaceskip\fontdimen\tw@\fontŽ‘5Ÿff‰ffÿr‰ffÿrŸ J=‘ "5Ÿý-:ó¹Aa¨cmr6½1ŽŽŽ‘LܼIn–4Bparticular,‘‹ýone“should“w•ÃŽatc“h–4Bout“for“punctuation“cšÃŽharacters“in“displa˜ys:‘„õIs,–‹ýe.g.,“anŽ¤ €exclamation–—zpŸ'EŽ‘ãxX,‘F/the“c¸ãharactersŽ¦‘5Æ{²,–£ÂÆ\²,“and–”Æ}“²are“assigned“math“cošGdes“(and“Æ\“²is“also“giv¸ãen“a“delimiter“co˜de);‘³qtheseŽ¦‘5are–zgthe“only“visible“ASCIGI‘z^(7-bit)“cš¸ãharacters“that“w˜e“don't“giv˜e“a“math“coGde“(w˜eŽ¦‘5don't–UUassign“math“coGdes“to“anš¸ãy“of“the“in˜visible“ASCIGI“c˜haracters).Ž¦‘DThe–UUEuler“Script“fonš¸ãts“also“con˜tain“a“section“sign“(¸x²):Ž¤‘5±94Ž‘?ª®É\DeclareMathSymbol\mathsection\mathord{EulerScript}{"78}ŽŸþ6‘5Í4.4Ž‘SÀEuler‘€ExtensionŽŸuT‘5²W‘ÿ*ªe–UUallošGcate“a“new“sym¸ãb˜ol“fonš¸ãt“group“for“this“fon˜t:Ž¡‘5±95Ž‘?ª®É\DeclareSymbolFont{EulerExtension}{U}{euex}{m}{n}ŽŸ‘5²This–j¦fonš¸ãt“con˜tains“some“alternativ˜e“v˜ersions“of“some“Computer“MoGdern“sym˜bGols.Ž¦‘5The–yPÆamsmath“²pac¸ãk‘ÿqÇage“(part“of“AMS-LŸýó5‘üff±AŽŽ‘͉²T‘þU>Ÿ'EŽ‘ãxX)“rede nes“the“Æ\coprod²,–ÂNÆ\prod²,“andŽ¦‘5Æ\sum–*²symš¸ãbGols;‘8|to“mak˜e“the“Æeuler“²pac˜k‘ÿqÇage“w˜ork“correctly“with“the“Æamsmath“²pac˜k-Ž¦‘5age,‘+cwš¸ãe– æneed“to“tak˜e“di eren˜t“actions“depGending“on“whether“the“Æamsmath“²pac˜k‘ÿqÇageŽ¦‘5has–q¶bšGeen“loaded“b˜efore“the“Æeuler“²pacš¸ãk‘ÿqÇage“or“not.‘ÆêNote:‘ª‰This“will“ó&ý': cmti10Ñnot‘dŽ²w˜ork“withŽ¦‘5the–UUÆamstex“²pac¸ãk‘ÿqÇage.Ž¡‘5±96Ž‘?ª®É\@ifpackageloaded{amsmath}Ž¤ ‘5±97Ž‘M×pÉ{\DeclareMathSymbol\coprod@\mathop{EulerExtension}{"60}%Ž¡‘5±98Ž‘R‘É\DeclareMathSymbol\prod@‘ s,\mathop{EulerExtension}{"51}%Ž¡‘5±99Ž‘R‘É\DeclareMathSymbol\sum@‘,Â\mathop{EulerExtension}{"50}}Ž¡‘1±100Ž‘M×pÉ{\DeclareMathSymbol\coprod‘¹–\mathop{EulerExtension}{"60}%Ž¡‘1±101Ž‘R‘É\DeclareMathSymbol\prod‘,Â\mathop{EulerExtension}{"51}%Ž¡‘1±102Ž‘R‘É\DeclareMathSymbol\sum‘æX\mathop{EulerExtension}{"50}}ŽŸ‘5²W‘ÿ*ªe–UUshall“respGectfully“ask“the“user“not“to“use“the“Æamstex“²pac¸ãk‘ÿqÇage.ŽŸ‘1±103Ž‘?ª®É\AtBeginDocumentŽ¡‘1±104Ž‘M×pÉ{\@ifpackageloaded{amstex}%Ž¡‘1±105Ž‘`½ÈÉ{\PackageWarningNoLine{euler}{Please–¹–do“not“use“`amstex'.Ž¡‘1±106Ž’ý¤ÉUse–¹–`amsmath'“instead}{}}}ŽŸ‘1±107Ž‘?ª®É\DeclareMathSymbol\intop‘ s,\mathop‘¹–{EulerExtension}{"52}Ž¡‘1±108Ž‘?ª®É\DeclareMathSymbol\ointop–¹–\mathop“{EulerExtension}{"48}Ž¡‘1±109Ž‘?ª®É\DeclareMathSymbol\braceld\mathord{EulerExtension}{"7A}Ž¡‘1±110Ž‘?ª®É\DeclareMathSymbol\bracerd\mathord{EulerExtension}{"7B}Ž¡‘1±111Ž‘?ª®É\DeclareMathSymbol\bracelu\mathord{EulerExtension}{"7C}Ž¡‘1±112Ž‘?ª®É\DeclareMathSymbol\braceru\mathord{EulerExtension}{"7D}Ž¡‘1±113Ž‘?ª®É\DeclareMathSymbol\infty‘ s,\mathord{EulerExtension}{"31}ŽŽŸ’ä²6ŽŽŒ‹Hl ª ýs o ý´‘5²It–õ is“not“enough“to“substitute“the“Euler“in¸ãtegral“sign“for“the“Computer“MoGdernŽ¤ ‘5inš¸ãtegral–Ò sign,‘ìNif“w˜e“are“using“the“Æamsmath“²pac˜k‘ÿqÇage.‘FThe“m˜ultiple“in˜tegral“signs“willŽ¡‘5loGok–UUbad.‘qÇThe“folloš¸ãwing“will“mak˜e“them“lošGok“b˜etter.Ž©‘1±114Ž‘?ª®É\AtBeginDocumentŽ¤ ‘1±115Ž‘M×pÉ{\@ifpackageloaded{amsmath}%Ž¡‘1±116Ž‘`½ÈÉ{\def\intkern@{\mkern–¹–-6mu“}%Ž¡‘1±117Ž‘ew^É\def\ints@#1{\mkern‘¹–-3muŽ¡‘1±118Ž‘s¤ É\mathop{\mkern‘¹–3muŽ¡‘1±119Ž’ÐâÉ\intopŽ¡‘1±120Ž’ÐâÉ\ifnum–¹–#1=\z@“\intdots@“\elseŽ¡‘1±121Ž’ý¤É\intkern@Ž¡‘1±122Ž’ý¤É\ifnum–¹–#1>\tw@“\intop\intkern@Ž¡‘1±123Ž’ž*fÉ\ifnum–¹–#1>\thr@@“\intop\intkern@“\fiŽ¡‘1±124Ž’ý¤É\fiŽ¡‘1±125Ž’ÐâÉ\fiŽ¡‘1±126Ž’ÐâÉ\intopŽ¡‘1±127Ž‘s¤ É}\ilimits@Ž¡‘1±128Ž‘ew^É}%Ž¡‘1±129Ž‘`½ÈÉ}{}}ŽŸ‘5²All–UUEuler“arroš¸ãws“come“from“the“Euler“Extension“fon˜t:Ž¦‘1±130Ž‘?ª®É\DeclareMathSymbol\nearrow\mathrel{EulerExtension}{"25}Ž¡‘1±131Ž‘?ª®É\DeclareMathSymbol\searrow\mathrel{EulerExtension}{"26}Ž¡‘1±132Ž‘?ª®É\DeclareMathSymbol\nwarrow\mathrel{EulerExtension}{"2D}Ž¡‘1±133Ž‘?ª®É\DeclareMathSymbol\swarrow\mathrel{EulerExtension}{"2E}Ž©‘1±134Ž‘?ª®É\DeclareMathSymbol\Leftrightarrow\mathrel{EulerExtension}{"2C}Ž¡‘1±135Ž‘?ª®É\DeclareMathSymbol\Leftarrow‘Ÿî\mathrel{EulerExtension}{"28}Ž¡‘1±136Ž‘?ª®É\DeclareMathSymbol\Rightarrow‘æX\mathrel{EulerExtension}{"29}Ž¡‘1±137Ž‘?ª®É\DeclareMathSymbol\leftrightarrow\mathrel{EulerExtension}{"24}Ž¡‘1±138Ž‘?ª®É\DeclareMathSymbol\leftarrow‘Ÿî\mathrel{EulerExtension}{"20}Ž¡‘1±139Ž‘?ª®É\let\gets=\leftarrowŽ¡‘1±140Ž‘?ª®É\DeclareMathSymbol\rightarrow‘æX\mathrel{EulerExtension}{"21}Ž¡‘1±141Ž‘?ª®É\let\to=\rightarrowŽ¦‘1±142Ž‘?ª®É\DeclareMathDelimiter\uparrowŽ¡‘1±143Ž’ÐâÉ\mathrel{EulerExtension}{"22}{largesymbols}{"78}Ž¡‘1±144Ž‘?ª®É\DeclareMathDelimiter\downarrowŽ¡‘1±145Ž’ÐâÉ\mathrel{EulerExtension}{"23}{largesymbols}{"79}Ž¡‘1±146Ž‘?ª®É\DeclareMathDelimiter\updownarrowŽ¡‘1±147Ž’ÐâÉ\mathrel{EulerExtension}{"6C}{largesymbols}{"3F}Ž¡‘1±148Ž‘?ª®É\DeclareMathDelimiter\UparrowŽ¡‘1±149Ž’ÐâÉ\mathrel{EulerExtension}{"2A}{largesymbols}{"7E}Ž¡‘1±150Ž‘?ª®É\DeclareMathDelimiter\DownarrowŽ¡‘1±151Ž’ÐâÉ\mathrel{EulerExtension}{"2B}{largesymbols}{"7F}Ž¡‘1±152Ž‘?ª®É\DeclareMathDelimiter\UpdownarrowŽ¡‘1±153Ž’ÐâÉ\mathrel{EulerExtension}{"6D}{largesymbols}{"77}Ž¦‘1±154Ž‘?ª®É\DeclareMathSymbol\leftharpoonup‘,Â\mathrel{EulerExtension}{"18}Ž¡‘1±155Ž‘?ª®É\DeclareMathSymbol\leftharpoondown‘¹–\mathrel{EulerExtension}{"19}Ž¡‘1±156Ž‘?ª®É\DeclareMathSymbol\rightharpoonup‘ s,\mathrel{EulerExtension}{"1A}Ž¡‘1±157Ž‘?ª®É\DeclareMathSymbol\rightharpoondown\mathrel{EulerExtension}{"1B}ŽŸ‘D²Note–UUthe“funnš¸ãy“mixture“of“the“Euler“Script“and“Extension“fon˜ts:ŽŸ‘1±158Ž‘?ª®É\DeclareMathDelimiter\lbraceŽ¡‘1±159Ž’ÐâÉ\mathopen{EulerScript}{"66}{EulerExtension}{"08}ŽŽŸ’ä²7ŽŽŒ‹VN ª ýs o ý´‘1±160Ž‘?ª®É\DeclareMathDelimiter\rbraceŽ© ‘1±161Ž‘}LÉ\mathclose{EulerScript}{"67}{EulerExtension}{"09}ŽŸ!Ä‘5Å5Ž‘MLËMore‘ffmathŽŸç‘5²Actually‘ÿ*ª,‘m½wš¸ãe–5¨still“need“some“Computer“MoGdern“fon˜ts:‘2nW‘ÿ*ªe“need“the“equals“signŽ¤ ‘5for–\çlong/extensible“(horizonš¸ãtal)“double“arro˜ws,‘žÌand“w˜e“need“the“min˜us“sign“forŽ¡‘5the–(Âsingle“arroš¸ãws!‘ìThe“Euler“Extension“fon˜t“is“based“on“the“Metafon˜t“coGde“forŽ¡‘5Computer–ÿ¾MošGdern,‘Ýand“w¸ãe“need“the“Computer“Mo˜dern“equals“and“min¸ãus“signs“toŽ¡‘5makš¸ãe–UUlong/extensible“arro˜ws.Ž¡‘DW‘ÿ*ªe– §allošGcate“math“alphab˜ets“to“pro˜duce“the“equals“and“minš¸ãus“signs“for“arro˜ws.Ž¡‘5Note–¥Rthat“since“Euler“arroš¸ãws“are“only“a˜v‘ÿqÇailable“in“medium“w˜eigh˜t“(they“comeŽ¡‘5from–§øthe“Euler“Extension“fonš¸ãt“whic˜h“is“only“a˜v‘ÿqÇailable“in“medium“w˜eigh˜t),‘¼¡w˜e“letŽ¡‘5all–UUmath“vš¸ãersions“of“these“math“alphabGets“refer“to“medium“w˜eigh˜t“fon˜ts.ŽŸ‘1±162Ž‘?ª®É\DeclareMathAlphabet\cm@equals@alphabet{OT1}{cmr}{m}{n}Ž¦‘1±163Ž‘?ª®É\DeclareMathAlphabet\cm@minus@alphabet{OMS}{cmsy}{m}{n}ŽŸ‘5²Note–¾Ethat“wš¸ãe“use“OT1“encoGding“for“the“Æ\cm@equals@alphabet²,‘Øev˜en“though“T1Ž¡‘5encošGding–UUma¸ãy“b˜e“the“default“for“the“users“system.Ž¡‘DAlso,‘ߘnote–Ãñthat“wš¸ãe“already“ha˜v˜e“the“Computer“MoGdern“Æcmsy“²family“assignedŽ¡‘5to–^Ía“symš¸ãbGol“fon˜t“(namely“the“`sym˜bGols'“sym˜bGol“fon˜t).‘Ž/Ho˜w˜ev˜er,‘a+taking“the“min˜usŽ¡‘5sign–™×directly“from“this“symš¸ãbGol“fon˜t“w˜ould“mak˜e“long/extensible“arro˜ws“come“outŽ¡‘5wrong–min“the“`bšGold'“math“v¸ãersion.‘¸Û(It“is“p˜erhaps“\o•¸ãv“erkill"–mto“allo˜cate“a“mathŽ¡‘5alphabšGet–UUjust“for“this“purp˜ose?)ŽŸ‘ 3PÉ\RelbarŽŽ¡‘ 3P\relbarŽŽŽŽŽ‘5²The–´%long“arroš¸ãws“are“made“using“the“macros“Æ\Relbar“²(for“double“arro˜ws)“andŽ¡‘5Æ\relbar–™8²(for“single“arro¸ãws).‘=oThe“standard“LŸýó5‘üff±AŽŽ‘͉²T‘þU>Ÿ'EŽ‘ãxX“de nitions“of“these“macros“useŽ¡‘5the–US`Æ=²'“and“`Æ-²'“cš¸ãharacters“to“proGduce“the“extension“part“of“the“long“arro˜ws.‘qÆSinceŽ¡‘5w•¸ãe›ùåha“v“e˜c“hanged˜the˜math˜coGdes˜of˜these˜c“haracters˜(they˜are˜no“w˜tak“en˜from˜theŽ¡‘5Euler–ð9F‘ÿ*ªraktur“fonš¸ãts“in“math“moGde),‘rw˜e“ha˜v˜e“to“rede ne“the“Æ\Relbar“²and“Æ\relbarŽ¡‘5²macros.ŽŸ‘1±164Ž‘?ª®É\def\Relbar{\mathrel{\cm@equals@alphabet{\mathchar"703D}}}Ž¦‘1±165Ž‘?ª®É\def\cm@minus{{\cm@minus@alphabet{\mathchar"7000}}}Ž¦‘1±166Ž‘?ª®É\def\relbar{\mathrel{\smash\cm@minus}}Ž¦‘1±167Ž‘nêŠÉ%–¹–\smash,“because“-“has“the“same“height“as“+ŽŸ‘é 6\leftarrowfillŽŽ¡‘äf \rightarrowfillŽŽŽŽŽ‘5²Unfortunately‘ÿ*ª,‘Œ since–Y¶the“de nitions“of“the“\arro¸ãw- lling"“macros“(Æ\leftarrowfillŽ¡‘5²and–ö Æ\rightarrowfill²)“in“standard“LŸýó5‘üff±AŽŽ‘͉²T‘þU>Ÿ'EŽ‘ãxX“still“use“explicit“min¸ãus“signs“(insteadŽ¡‘5of–UUÆ\relbar²),“wš¸ãe“also“ha˜v˜e“to“rede ne“those“macros:ŽŸ‘1±168Ž‘?ª®É\def\leftarrowfillŽ¦‘1±169Ž‘M×pÉ{$\m@th\mathord\leftarrow\mkern-6mu%Ž¦‘1±170Ž‘WJœÉ\cleaders\hbox{$\mkern-2mu\cm@minus\mkern-2mu$}\hfillŽ¦‘1±171Ž‘WJœÉ\mkern-6mu\cm@minus$}Ž¦‘1±172Ž‘?ª®É\def\rightarrowfillŽ¦‘1±173Ž‘M×pÉ{$\m@th\cm@minus\mkern-6mu%Ž¦‘1±174Ž‘WJœÉ\cleaders\hbox{$\mkern-2mu\cm@minus\mkern-2mu$}\hfillŽ¦‘1±175Ž‘WJœÉ\mkern-6mu\mathord\rightarrow$}ŽŸ‘5²W‘ÿ*ªe›Øãha•¸ãv“e˜de ned˜Æ\cm@minus˜²to˜pro•Gduce˜a˜Computer˜Mo“dern˜min•¸ãus˜sign˜of˜t“ypGeŽ¡‘5Æ\mathord².‘d«W‘ÿ*ªe–¦Ltherebš¸ãy“sa˜v˜e“a“few“Æ\mathord“²tok˜ens“(compared“to“the“de nitionsŽ¡‘5of–UUÆ\leftarrowfill“²and“Æ\rightarrowfill“²in“standard“LŸýó5‘üff±AŽŽ‘͉²T‘þU>Ÿ'EŽ‘ãxX).ŽŽŸ’ä8ŽŽŒ‹ b\ ª ýs o ý´‘D²There–Ýare“a“few“sym¸ãbšGols“in“the“original“Computer“Mo˜dern“Math“Italic“fon¸ãtsŽ© ‘5that–B÷are“not“presenš¸ãt“in“the“Euler“fon˜ts.‘k¨W‘ÿ*ªe“simply“tak˜e“them“from“the“CM‘Bófon˜ts.ŽŸ Ûº‘1±176Ž‘?ª®É\DeclareSymbolFont{cmmigroup}{OML}{cmm}{m}{it}Ž¤ ‘1±177Ž‘?ª®É\SetSymbolFont{cmmigroup}{bold}{OML}{cmm}{b}{it}Ž¡‘1±178Ž‘?ª®É\DeclareMathAccent\vec‘/?Ü\mathord{cmmigroup}{"7E}Ž¡‘1±179Ž‘?ª®É\DeclareMathSymbol\triangleleft‘¹–\mathbin{cmmigroup}{"2F}Ž¡‘1±180Ž‘?ª®É\DeclareMathSymbol\triangleright\mathbin{cmmigroup}{"2E}Ž¡‘1±181Ž‘?ª®É\DeclareMathSymbol\star‘*†F\mathbin{cmmigroup}{"3F}Ž¡‘1±182Ž‘?ª®É\DeclareMathSymbol\lhook‘%Ì°\mathrel{cmmigroup}{"2C}Ž¡‘1±183Ž‘?ª®É\DeclareMathSymbol\rhook‘%Ì°\mathrel{cmmigroup}{"2D}Ž¡‘1±184Ž‘?ª®É\DeclareMathSymbol\flat‘*†F\mathord{cmmigroup}{"5B}Ž¡‘1±185Ž‘?ª®É\DeclareMathSymbol\natural‘Y„\mathord{cmmigroup}{"5C}Ž¡‘1±186Ž‘?ª®É\DeclareMathSymbol\sharp‘%Ì°\mathord{cmmigroup}{"5D}Ž¡‘1±187Ž‘?ª®É\DeclareMathSymbol\smile‘%Ì°\mathrel{cmmigroup}{"5E}Ž¡‘1±188Ž‘?ª®É\DeclareMathSymbol\frown‘%Ì°\mathrel{cmmigroup}{"5F}ŽŸÛº‘5²Note–€zthat“the“arroš¸ãw“head“of“Æ\vec“²is“slan˜ted;‘Çnthis“loGoks“a“bit“strange“in“com˜binationŽ¦‘5with–UUan“uprighš¸ãt“fon˜t“suc˜h“as“Euler“Roman.Ž¦‘DAlso,›'\note–ýZthat“in“the“`bGold'“math“v•¸ãersion,˜arro“ws–ýZwith“hoGoks“will“come“outŽ¦‘5wrong–ö‚(the“hošGoks“will“b˜e“b˜old,‘Ìbut“the“arro¸ãw“heads“will“b˜e“of“medium“w•¸ãeigh“t).Ž¦‘5This–®can“bGe“repaired“in“a“similar“w•¸ãa“y–®as“the“long/extensible“arroš¸ãws,‘Ïbut“is“it“w˜orthŽ¦‘5it–UUto“allošGcate“a“math“alphab˜et“just“for“this“purp˜ose?Ž¦‘DFinally‘ÿ*ª,–UUKnš¸ãuth“w˜an˜ted“a“little“extra“space“bšGefore“the“prime“sup˜erscript:ŽŸ Ûº‘1±189Ž‘?ª®É\begingroupŽ¡‘1±190Ž‘M×pÉ\catcode`\'=\activeŽ¡‘1±191Ž‘M×pÉ\gdef'{^\bgroup\mskip2mu\prim@s}–¹–%“more“space“before“'Ž¡‘1±192Ž‘?ª®É\endgroupŽŸÛº‘D²OpšGen–)question:‘çoThe“Æ\dots“²and“Æ\ldots“²macros“pro˜duce“dots“(p˜erio˜ds)“fromŽ¦‘5the–x Euler“fonš¸ãts;‘ŠEthe“Æ\ddots“²and“Æ\vdots“²macros“proGduce“dots“from“the“text“fon˜t;Ž¦‘5and–#Œthe“Æ\cdots“²macro“prošGduces“dots“from“the“Computer“Mo˜dern“`sym¸ãb˜ols'“mathŽ¦‘5sym•¸ãbGol›UUfon“t.‘qÇShould˜w“e˜do˜something˜abGout˜this?ŽŸ òˆ‘5Å6Ž‘MLËT‘þ¦fext–fffonŒÌts“in“mathŽŸç‘5²The–}text“fon¸ãts“used“in“math“through“math“alphabšGets“and“the“`op˜erators'“sym¸ãb˜olŽ¦‘5fonš¸ãt–ìçshould“bGe“the“same“as“the“fon˜ts“used“for“the“main“text.‘8~(It“is“no˜w“easy“toŽ¦‘5replace–¯the“fonš¸ãts“for“the“`opGerators'“sym˜bGol“fon˜t,‘*jsince“the“uppGercase“greek“lettersŽ¦‘5are–UUnoš¸ãw“tak˜en“from“the“Euler“Roman“fon˜ts.)Ž¦‘DIn–ø„order“to“reduce“the“total“n•¸ãum“bGer–ø„of“fonš¸ãts“needed,‘ w˜e“shall|if“pGossible|useŽ¦‘5text–wtfon¸ãts“in“the“same“encošGding“for“b˜oth“text“and“math.‘Ø%W‘ÿ*ªe“therefore“determineŽ¦‘5whether–ɤT1“encošGding“has“b˜een“selected“as“the“default“enco˜ding.‘δW‘ÿ*ªe“de ne“aŽ¦‘5switc¸ãh|Æ\ifCorkEncoding²|for–UUthat“purpGose.ŽŸ Ûº‘1±193Ž‘?ª®É\newif\ifCorkEncodingŽ¡‘1±194Ž‘?ª®É\edef\@tempa{\encodingdefault}\def\@tempb{T1}Ž¡‘1±195Ž‘?ª®É\ifx\@tempa\@tempbŽ¡‘1±196Ž‘M×pÉ\CorkEncodingtrueŽ¡‘1±197Ž‘?ª®É\elseŽ¡‘1±198Ž‘M×pÉ\CorkEncodingfalseŽ¡‘1±199Ž‘?ª®É\fiŽŸÛº‘5²Note–{îthat“wš¸ãe“determine“the“encoGding“to“use“at“the“time“the“Æeuler“²pac˜k‘ÿqÇage“isŽ¦‘5loaded–§{(Ñnot‘šS²at“the“end“of“the“preamš¸ãble).‘h9This“allo˜ws“the“user“to“retain“OT1ŽŽŸ’ä9ŽŽŒ‹ q  ª ýs o ý´‘5²encoGding–mcfor“text“fonš¸ãts“used“in“math“(other“pac˜k‘ÿqÇages“ma˜y“depGend“on“that)“b˜yŽ¤ ‘5selecting–UUT1“encošGding“Ñafter‘hâ²the“Æeuler“²pac¸ãk‘ÿqÇage“has“b˜een“loaded.Ž¡‘DW‘ÿ*ªe–™2rede ne“the“`opšGerators'“sym¸ãb˜ol“fon¸ãt“and“all“math“alphab˜ets“de ned“b¸ãyŽ¡‘5the–Í@LŸýó5‘üff±AŽŽ‘͉²T‘þU>Ÿ'EŽ‘ãxX“kš¸ãernel.‘ÙˆDue“to“the“curren˜t“implemen˜tation“of“math“accen˜ts“that“tak˜eŽ¡‘5their–Âglyphs“from“text“fonš¸ãts,‘ÝMall“text“fon˜ts“m˜ust“bšGe“used“in“a“common“enco˜ding.Ž¡‘5The–only“LŸýó5‘üff±AŽŽ‘͉²T‘þU>Ÿ'EŽ‘ãxX“`base'“pac¸ãk‘ÿqÇage“that“de nes“extra“math“alphabGets“is“the“ÆoldlfontŽ¡‘5²pac•¸ãk›ÿqÇage.‘àÛHo“w“ev“er,‘ƒthis‘z\pac“k˜age–z\cannot“bGe“used“together“with“the“Æeuler“²pac¸ãk˜ageŽ¡‘5(it–£+allošGcates“to˜o“manš¸ãy“sym˜bGol“fon˜ts).‘[IIn“other“w˜ords,‘¶ the“follo˜wing“setup“shouldŽ¡‘5suce–UUfor“most“users.Ž© [n‘1±200Ž‘?ª®É\ifCorkEncodingŽ¤ ‘1±201Ž‘IÚÉ\AtBeginDocumentŽ¡‘1±202Ž‘R‘É{\DeclareSymbolFont{operators}{T1}\rmdefault\mddefault\updefaultŽ¡‘1±203Ž‘WJœÉ\SetSymbolFont{operators}{bold}{T1}\rmdefault\bfdefault\updefaultŽ¡‘1±204Ž‘WJœÉ\DeclareMathAlphabet\mathbf{T1}\rmdefault\bfdefault\updefaultŽ¡‘1±205Ž‘WJœÉ\DeclareMathAlphabet\mathsf{T1}\sfdefault\mddefault\updefaultŽ¡‘1±206Ž‘WJœÉ\DeclareMathAlphabet\mathit{T1}\rmdefault\mddefault\itdefaultŽ¡‘1±207Ž‘WJœÉ\DeclareMathAlphabet\mathtt{T1}\ttdefault\mddefault\updefaultŽ¡‘1±208Ž‘WJœÉ\SetMathAlphabet\mathsf{bold}{T1}\sfdefault\bfdefault\updefaultŽ¡‘1±209Ž‘WJœÉ\SetMathAlphabet\mathit{bold}{T1}\rmdefault\bfdefault\itdefault}Ž¡‘1±210Ž‘?ª®É\elseŽ¡‘1±211Ž‘IÚÉ\AtBeginDocumentŽ¡‘1±212Ž‘R‘É{\DeclareSymbolFont{operators}{OT1}\rmdefault\mddefault\updefaultŽ¡‘1±213Ž‘WJœÉ\SetSymbolFont{operators}{bold}{OT1}\rmdefault\bfdefault\updefaultŽ¡‘1±214Ž‘WJœÉ\DeclareMathAlphabet\mathbf{OT1}\rmdefault\bfdefault\updefaultŽ¡‘1±215Ž‘WJœÉ\DeclareMathAlphabet\mathsf{OT1}\sfdefault\mddefault\updefaultŽ¡‘1±216Ž‘WJœÉ\DeclareMathAlphabet\mathit{OT1}\rmdefault\mddefault\itdefaultŽ¡‘1±217Ž‘WJœÉ\DeclareMathAlphabet\mathtt{OT1}\ttdefault\mddefault\updefaultŽ¡‘1±218Ž‘WJœÉ\SetMathAlphabet\mathsf{bold}{OT1}\sfdefault\bfdefault\updefaultŽ¡‘1±219Ž‘WJœÉ\SetMathAlphabet\mathit{bold}{OT1}\rmdefault\bfdefault\itdefault}Ž¡‘1±220Ž‘?ª®É\fiŽŸ[n‘5²Note–that“wš¸ãe“pGerform“the“setup“at“the“end“of“the“pream˜ble“(and“not“while“theŽ¤ 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s,\mathalpha{operators}{"0A}Ž¡‘1±230Ž‘?ª®É\fiŽŸI‘`\hatŽŽŽ‘5²There–is“one“more“accenš¸ãt“that“is“normally“tak˜en“from“the“`opGerators'“sym˜bGol“fon˜t,Ž¤ ‘5namely–‰zthe“Æ\hat“²accenš¸ãt.‘5This“is“the“only“accen˜t“pro˜vided“b˜y“the“Euler“fon˜ts;‘£Œw˜eŽ¡‘5shall–use“this“accenš¸ãt“b˜y“default.‘¹É(A‘Îcurious“fact“is“that“Kn˜uth“dev˜oted“a“fairlyŽ¡‘5large–]pGortion“of‘HzÆgkpmac.tex“²to“the“construction“of“a“ó  b> cmmi10µ“²with“a“hat“accen¸ãt|usingŽ¡‘5the–UUÆ\widehat“²macro).ŽŽŸ’áÿ10ŽŽŒ‹ % ª ýs o ý´‘1±231Ž‘?ª®É\DeclareMathAccent\hat\mathalpha{EulerFraktur}{"5E}ŽŸ‘5²Ho•¸ãw“ev“er,‘ÜŽsome–Áƒusers“do“not“likš¸ãe“this“Æ\hat“²accen˜t.‘¶PThey“prefer“the“accen˜t“glyphŽ© ‘5to–UUbGe“takš¸ãen“from“the“text“fon˜t“(similar“to“the“other“accen˜ts).ŽŸ‘1±232Ž‘?ª®É\DeclareOption{text-hat-accent}Ž¤ ‘1±233Ž‘M×pÉ{\ifCorkEncodingŽ¡‘1±234Ž‘`½ÈÉ\DeclareMathAccent\hat\mathalpha{operators}{"02}%Ž¡‘1±235Ž‘R‘É\elseŽ¡‘1±236Ž‘`½ÈÉ\DeclareMathAccent\hat\mathalpha{operators}{"5E}%Ž¡‘1±237Ž‘R‘É\fi}ŽŸ‘¦|\hbarŽŽŽ‘5²The–ÕbÆ\hbar“²macro“in“standard“LŸýó5‘üff±AŽŽ‘͉²T‘þU>Ÿ'EŽ‘ãxX“is“compGosed“of“the“letter“`h'“from“the“`letters'Ž¦‘5sym•¸ãbGol›Ðìfon“t˜and˜the˜macron˜(`bar')˜accen“t˜from˜the˜`opGerators'˜sym“bGol˜fon“t.‘E¤WithŽ¦‘5the–i·`letters'“symš¸ãbGol“fon˜t“no˜w“bšGeing“Euler“Roman,‘nÏthe“bar“is“no“longer“p˜ositionedŽ¦‘5correctly–ÿ*ª.‘@ÕW“e–Âcš¸ãhoGose“the“follo˜wing“de nition“(the“Æ9mu“²in“the“original“de nition“hasŽ¦‘5bGeen–UUc¸ãhanged“to“Æ8mu²):ŽŸ‘1±238Ž‘?ª®É\ifCorkEncodingŽ¡‘1±239Ž‘M×pÉ\DeclareMathSymbol\euler@hbar\mathord{operators}{"09}Ž¡‘1±240Ž‘?ª®É\elseŽ¡‘1±241Ž‘M×pÉ\DeclareMathSymbol\euler@hbar\mathord{operators}{"16}Ž¡‘1±242Ž‘?ª®É\fiŽ¡‘1±243Ž‘?ª®É\def\hbar{{\euler@hbar\mkern-8muh}}Ž¤!Ä‘5Å7Ž‘MLËF‘þ¦fos3otnotesŽŸç‘ò“bÉ\@makefnmarkŽŽŽ‘5²F‘ÿ*ªošGotnote‘}5n•¸ãum“b˜ers–}5are“not“math“en¸ãtities,‘¨oso“they“should“b˜e“pro˜duced“b¸ãy“digits“fromŽ¦‘5the–Kètext“fonš¸ãt.‘n£Since“foGotnote“mark˜ers“(whether“they“are“n˜um˜bGers“or“sym˜bGols)“areŽ¦‘5prošGduced–?#in“math“mo˜de,‘y–wš¸ãe“can“accomplish“this“b˜y“selecting“Æ\operator@font²;Ž¦‘5this–áwill“cause“digits“to“bšGe“tak¸ãen“from“the“`op˜erators'“sym¸ãb˜ol“fonš¸ãt“(whic˜h“is“theŽ¦‘5same–UUas“the“fon¸ãt“used“for“text;“see“previous“section).Ž©‘1±244Ž‘?ª®É\def\@makefnmark{\hbox{$\operator@font^{\@thefnmark}\m@th$}}Ž¡‘5Å8Ž‘MLËPros3cessing‘ffoptionsŽŸç‘5²Finally‘ÿ*ª,–UUwš¸ãe“m˜ust“remem˜bšGer“to“pro˜cess“the“options:Ž¦‘1±245Ž‘?ª®É\ProcessOptionsŽŸ ‘1±246Ž‘?ª®¸hµ=ó$m#½R cmss10Ïpack¸ãageŽ‘&G"¸iŽ¡‘5ÅReferencesŽŸç‘5²[1]ŽŽ‘DŽ;ó'Œ-ø cmcsc10ÒR.–¿L.“Graham²,›«ÐÒD.“E.“Knuth²,˜and‘š…ÒO.“P‘ÿ a–ÿUTt“ashnik².‘=ÑConcr‘ÿ}'ete‘ÓŽMathematics².Ž¤ ‘DŽ;Addison-W–ÿ*ªesley“,‘UU1989.ŽŸ‘5[2]ŽŽ‘DŽ;ÒD.–¬#E.“Knuth².›øùT¸ãypGesetting–/wConcrete“Mathematics.˜ó(p®0J cmsl10ÓTUGbGoat‘g,²10(1):31{36,Ž¡‘DŽ;1989.ŽŽŸ’áÿ11ŽŽŒø-ƒ’À;誘 ó(p®0J cmsl10ó'Œ-ø cmcsc10ó&ý': cmti10ó$m#½R cmss10ó"ÂÖN  cmbx12óÒoÇw cmss9óߤN cmtt9ó߆µTff cmtt12óßê cmmi10óKñ`y cmr10óÙ“ Rcmr7ù™Tßßßßßßß