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3.3BuildingBlocksofaMathematicalFormula
57
Table3.3onpage76listsalotofbinaryrelationslike⊆and⊥.
$p^3_{ij} \qquad
m_\text{Knuth}\qquad
\sum_{k=1}^3 k \\[5pt]
a^x+y \neq a^{x+y}\qquad
e^{x^2} \neq {e^x}^2$
p
3
ij
m
Knuth
3
k=1
k
a
x
+y=a
x+y
e
x
2
=e
x
2
Thesquare rootisenteredas\sqrt; thenth rootisgeneratedwith
\sqrt[n].ThesizeoftherootsignisdeterminedautomaticallybyL
A
T
E
X.
Ifjustthesignisneeded,use\surd.
Seevariouskindsofarrowslike→and onTable3.6onpage77.
$\sqrt{x} \Leftrightarrow x^{1/2}
\quad \sqrt[3]{2}
\quad \sqrt{x^{2} + \sqrt{y}}
\quad \surd[x^2 + y^2]$
xx
1/2
3
2
x2+
y
[x
2
+y
2
]
Whilethedotsigntoindicatethemultiplicationoperationisnormally
leftout,itissometimeswrittentohelptheeyeingroupingaformula. Use
\cdottotypesetasinglecentereddot. \cdotsisthreecentereddotswhile
\ldotssetsthedotslow(onthebaseline). Besidesthat,thereare\vdots
for vertical and \ddots fordiagonaldots. . There e are more examples s in
section3.6.
$\Psi = v_1 \cdot v_2
\cdot \ldots \qquad
n! = 1 \cdot 2
\cdots (n-1) \cdot n$
Ψ=v
1
·v
2
·...
n!=1·2···(n1)·n
The commands\overline and \underline create horizontal lines
directlyoverorunderanexpression:
$0.\overline{3} =
\underline{\underline{1/3}}$
0.
3=1/3
Thecommands\overbrace and\underbracecreatelonghorizontal
bracesoverorunderanexpression:
$\underbrace{\overbrace{a+b+c}^6
\cdot \overbrace{d+e+f}^7}
_\text{meaning of life} = 42$
6

a+b+c·
7

d+e+f

meaningoflife
=42
Toaddmathematicalaccentssuchassmall arrowsortildesignsto
variables,thecommandsgiveninTable3.1onpage75mightbeuseful. Wide
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58
TypesettingMathematicalFormulae
hatsandtildescoveringseveralcharactersaregeneratedwith\widetilde
and\widehat. Noticethedifferencebetween\hatand\widehatandthe
placementof\barforavariable withsubscript. . Theapostrophemark
givesaprime:
$f(x) = x^2 \qquad f’(x)
= 2x \qquad f’’(x) = 2\\[5pt]
\hat{XY} \quad \widehat{XY}
\quad \bar{x_0} \quad \bar{x}_0$
f(x)=x
2
f
(x)=2x
f

(x)=2
ˆ
XY
XY
x¯
0
¯x
0
Vectorsareoftenspecifiedbyaddingsmallarrowsymbolsonthetops
of variables. . Thisisdonewiththe\vec command. . The e two commands
\overrightarrowand\overleftarrowareusefultodenotethevectorfrom
AtoB:
$\vec{a} \qquad
\vec{AB} \qquad
\overrightarrow{AB}$
a
AB
−−→
AB
Namesoffunctionsareoftentypesetinanuprightfont,andnotinitalics
asvariablesare,soL
A
T
E
Xsuppliesthefollowingcommandstotypesetthe
mostcommonfunctionnames:
\arccos \cos
\csc \exp
\ker
\limsup
\arcsin \cosh \deg \gcd
\lg
\ln
\arctan \cot
\det \hom
\lim
\log
\arg
\coth \dim \inf
\liminf \max
\sinh
\sup
\tan \tanh \min
\Pr
\sec
\sin
\begin{equation*}
\lim_{x \rightarrow 0}
\frac{\sin x}{x}=1
\end{equation*}
lim
x→0
sinx
x
=1
For functions missing from m the e list, use the \DeclareMathOperator
command. Thereisevenastarredversionforfunctionswithlimits. . This
commandworksonlyinthepreamblesothecommentedlinesintheexample
belowmustbeputintothepreamble.
%\DeclareMathOperator{\argh}{argh}
%\DeclareMathOperator*{\nut}{Nut}
\begin{equation*}
3\argh = 2\nut_{x=1}
\end{equation*}
3argh=2Nut
x=1
Forthemodulofunction,therearetwocommands: \bmodforthebinary
operator“amodb”and\pmodforexpressionssuchas“x(modb):”
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3.3BuildingBlocksofaMathematicalFormula
59
$a\bmod b \\
x\equiv a \pmod{b}$
amodb
x(modb)
Abuilt-upfractionistypesetwiththe\frac{...}{...}command. In
in-lineequations,thefractionisshrunktofittheline. Thisstyleisobtainable
indisplaystylewith\tfrac.Thereverse,i.e.displaystylefractionintext,
ismadewith\dfrac. Oftentheslashedform1/2ispreferable,becauseit
looksbetterforsmallamountsof‘fractionmaterial:’
In display style:
\begin{equation*}
3/8 \qquad \frac{3}{8}
\qquad \tfrac{3}{8}
\end{equation*}
Indisplaystyle:
3/8
3
8
3
8
In text style:
$1\frac{1}{2}$~hours \qquad
$1\dfrac{1}{2}$~hours
Intextstyle: 1
1
2
hours
1
1
2
hours
Herethe\partialcommandforpartialderivativesisused:
\begin{equation*}
\sqrt{\frac{x^2}{k+1}}\qquad
x^\frac{2}{k+1}\qquad
\frac{\partial^2f}
{\partial x^2}
\end{equation*}
x2
k+1
x
2
k+1
2
f
∂x2
Totypesetbinomialcoefficientsorsimilarstructures,usethecommand
\binomfromamsmath:
Pascal’s rule is
\begin{equation*}
\binom{n}{k} =\binom{n-1}{k}
+ \binom{n-1}{k-1}
\end{equation*}
Pascal’sruleis
n
k
=
n−1
k
+
n−1
k−1
Forbinaryrelationsitmaybeusefultostacksymbolsovereachother.
\stackrel{#1}{#2}putsthesymbolgivenin#1insuperscript-likesizeover
#2whichissetinitsusualposition.
\begin{equation*}
f_n(x) \stackrel{*}{\approx} 1
\end{equation*}
f
n
(x)
≈1
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60
TypesettingMathematicalFormulae
Theintegraloperatorisgeneratedwith\int,thesumoperatorwith
\sum,andtheproductoperatorwith\prod.Theupperandlowerlimits
arespecifiedwith^and_likesubscriptsandsuperscripts:
\begin{equation*}
\sum_{i=1}^n \qquad
\int_0^{\frac{\pi}{2}} \qquad
\prod_\epsilon
\end{equation*}
n
i=1
π
2
0
Togetmorecontrolovertheplacementofindicesincomplexexpressions,
amsmathprovidesthe\substackcommand:
\begin{equation*}
\sum^n_{\substack{0<i<n \\
j\subseteq i}}
P(i,j) = Q(i,j)
\end{equation*}
n
0<i<n
ji
P(i,j)=Q(i,j)
L
A
T
E
Xprovidesall sortsofsymbolsforbracketing andotherdelim-
iters(e.g.[ ). Roundandsquarebracketscanbeenteredwiththe
correspondingkeysandcurlybraceswith\{,butallotherdelimitersare
generatedwithspecialcommands(e.g.\updownarrow).
\begin{equation*}
{a,b,c} \neq \{a,b,c\}
\end{equation*}
a,b,c={a,b,c}
Ifyouput\leftinfrontofanopeningdelimiterand\rightinfrontof
aclosingdelimiter,L
A
T
E
Xwillautomaticallydeterminethecorrectsizeof
thedelimiter.Notethatyoumustcloseevery\leftwithacorresponding
\right. Ifyoudon’twantanythingontheright,usetheinvisible“\right.”:
\begin{equation*}
1 + \left(\frac{1}{1-x^{2}}
\right)^3 \qquad
\left. \ddagger \frac{~}{~}\right)
\end{equation*}
1+
1
1−x2
3
Insomecasesitisnecessarytospecifythecorrectsizeofamathematical
delimiterbyhand,whichcanbedoneusingthecommands\big,\Big,\bigg
and\Biggasprefixestomostdelimitercommands:
$\Big((x+1)(x-1)\Big)^{2}$\\
$\big( \Big( \bigg( \Bigg( ( \quad
\big\} \Big\} \bigg\} \Bigg\} \quad
\big\| \Big\| \bigg\| \Bigg\| \quad
\big\Downarrow \Big\Downarrow
\bigg\Downarrow \Bigg\Downarrow$
(x+1)(x−1)
2
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3.4SingleEquationsthatare TooLong: : multline
61
Foralistofalldelimitersavailable,seeTable3.8onpage78.
3.4 SingleEquationsthatareTooLong: : multline
Ifanequation istoolong, we have to wrapitsomehow. . Unfortunately,
wrappedequationsareusuallylesseasytoreadthannotwrappedones. To
improvethereadability,therearecertainrulesonhowtodothewrapping:
1. Ingeneraloneshouldalwayswrapanequationbeforeanequalitysign
oranoperator.
2. Awrap before an equality sign ispreferable to a wrap before any
operator.
3. Awrapbeforeaplus-orminus-operatorispreferabletoawrapbefore
amultiplication-operator.
4. Anyothertypeofwrapshouldbeavoidedifatallpossible.
The easiestwayto achieve such a wrapping isthe use of the multline
environment:
6
\begin{multline}
a + b + c + d + e + f
+ g + h + i
\\
= j + k + l + m + n
\end{multline}
a+b+c+d+e+f+g+h+i
=j+k+l+m+(3.4)
Thedifferencefromtheequationenvironmentisthatanarbitraryline-break
(oralsomultipleline-breaks)canbeintroduced. Thisisdonebyputtinga
\\onthoseplaceswheretheequationneedstobewrapped. Similarlyto
equation*therealsoexistsamultline*versionforpreventinganequation
number.
OftentheIEEEeqnarrayenvironment(seesection3.5)willyieldbetter
results.Considerthefollowingsituation:
\begin{equation}
a = b + c + d + e + f
+ g + h + i + j
+ k + l + m + n + o + p
\label{eq:equation_too_long}
\end{equation}
a=b+c+d+e+f+g+h+i+j+k+l+m+n+o+p
(3.5)
HereitisactuallytheRHSthatistoolongtofitononeline. Themultline
environmentcreatesthefollowingoutput:
6
Themultline-environmentisfromamsmath.
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62
TypesettingMathematicalFormulae
\begin{multline}
a = b + c + d + e + f
+ g + h + i + j \\
+ k + l + m + n + o + p
\end{multline}
a=b+c+d+e+f+g+h+i+j
+k+l+m+n+o+(3.6)
Thisisbetterthan(3.5),butithasthedisadvantagethattheequality
signlosesitsnaturalgreaterimportancewithrespecttotheplusoperatorin
frontofk. ThebettersolutionisprovidedbytheIEEEeqnarrayenvironment
thatwillbediscussedindetailinSection3.5.
3.5 MultipleEquations
Inthemostgeneralsituationwehaveasequenceofseveralequalitiesthat
donotfitontooneline. Hereweneedtoworkwithverticalalignmentin
ordertokeepthearrayofequationsinaniceandreadablestructure.
Beforeweofferoursuggestionsonhowtodothis,westartwithafew
badexamplesthatshowthebiggestdrawbacksofsomecommonsolutions.
3.5.1 ProblemswithTraditionalCommands
Togroupmultipleequationsthealignenvironment
7
couldbeused:
\begin{align}
a & = b + c \\
& = d + e
\end{align}
a=b+c
(3.7)
=d+e
(3.8)
thisapproachfailsonceasinglelineistoolong:
\begin{align}
a & = b + c \\
& = d + e + f + g + h + i
+ j + k + l \nonumber \\
& + m + n + o \\
& = p + q + r + s
\end{align}
a=b+c
(3.9)
=d+e+f+g+h+i+j+k+l
+m+n+o
(3.10)
=p+q+r+s
(3.11)
Here+mshouldbebelowdandnotbelowtheequalitysign. Ofcourse,one
couldaddsomespace(\hspace{...}),butthiswillneveryieldaprecise
arrangement(andisbadstyle...).
Abettersolutionisofferedbytheeqnarrayenvironment:
7The
align-environmentcanalsobeusedtogroupseveralblocksofequationsbeside
eachother.AnotherexcellentusecasefortheIEEEeqnarrayenvironment. Tryanargument
like{rCl+rCl}.
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3.5MultipleEquations
63
\begin{eqnarray}
a & = & b + c \\
& = & d + e + f + g + h + + i
+ j + k + l \nonumber \\
&& +\: m + n + o \\
& = & p + q + r + s
\end{eqnarray}
=
b+c
(3.12)
=
d+e+f+g+h+i+j+k+l
+m+n+o
(3.13)
=
p+q+r+s
(3.14)
Thisisstillnotoptimal. Thespacesaroundtheequalitysignsaretoo
big.Particularly,theyarenotthesameasinthemultlineandequation
environments:
\begin{eqnarray}
a & = & a = a
\end{eqnarray}
=
a=a
(3.15)
...andtheexpressionsometimesoverlapswiththeequationnumbereven
thoughtherewouldbeenoughroomontheleft:
\begin{eqnarray}
a & = & b + c
\\
& = & d + e + f + g + h^2
+ i^2 + j
\label{eq:faultyeqnarray}
\end{eqnarray}
=
b+c
(3.16)
=
d+e+f+g+h
2
+i
2
+(j3.17)
Whiletheenvironmentoffersacommand\lefteqnthatcanbeusedwhen
theLHSistoolong:
\begin{eqnarray}
\lefteqn{a + b + c + d
+ e + f + g + h}\nonumber\\
& = & i + j + k + l + m
\\
& = & n + o + p + q + r + + s
\end{eqnarray}
a+b+c+d+e+f+g+h
i+j+k+l+m
(3.18)
n+o+p+q+r+(3.19)
Thisisnotoptimaleitherasthe RHS istooshortand the arrayisnot
properlycentered:
\begin{eqnarray}
\lefteqn{a + b + c + d
+ e + f + g + h}
\nonumber \\
& = & i + j
\end{eqnarray}
a+b+c+d+e+f+g+h
=
i+j
(3.20)
64
TypesettingMathematicalFormulae
Havingbadmouthedthecompetitionsufficiently,Icannowsteeryougently
towardstheglorious...
3.5.2 IEEEeqnarrayEnvironment
The IEEEeqnarray environmentisa verypowerful commandwithmany
options. Here,wewillonlyintroduceitsbasicfunctionalities. Formore
informationpleaserefertothemanual.
8
Firstofall,inordertobeabletousetheIEEEeqnarrayenvironment
oneneedstoloadthepackage
9
IEEEtrantools.Includethefollowinglinein
theheaderofyourdocument:
\usepackage[retainorgcmds]{IEEEtrantools}
Thestrengthof IEEEeqnarrayistheabilitytospecifythenumberof
columns inthe equation array. . Usually, , thisspecification willbe {rCl},
i.e.,threecolumns,thefirstcolumnright-justified,themiddleonecentered
withalittlemorespacearoundit(thereforewespecifycapitalCinsteadof
lower-casec)andthethirdcolumnleft-justified:
\begin{IEEEeqnarray}{rCl}
a & = & b + c
\\
& = & d + e + f + g + h
+ i + j + k \nonumber\\
&& \negmedspace {} + l + + m m + n + o
\\
& = & p + q + r + s
\end{IEEEeqnarray}
a=b+c
(3.21)
=d+e+f+g+h+i+j+k
+l+m+n+o
(3.22)
=p+q+r+s
(3.23)
Anynumberofcolumnscanbespecified:{c}willgiveonlyonecolumn
withallentriescentered,or{rCll}wouldaddafourth,left-justifiedcolumn
to use for r comments. . Moreover, , beside lcrLCfor math mode
entriestherearealsos,t,uforleft,centered,andrighttextmodeentries.
Additionalspacecanbeaddedwithand/andinincreasingorder.10
Notethespacesaroundtheequalitysignsincontrasttothespaceproduced
bytheeqnarrayenvironment.
3.5.3 CommonUsage
InthefollowingwewilldescribehowweuseIEEEeqnarraytosolvethemost
commonproblems.
Ifalineoverlapswiththeequationnumberasin(3.17),thecommand
8The official manual is called
CTAN://macros/latex/contrib/IEEEtran/IEEEtran_
HOWTO.pdf. ThepartaboutIEEEeqnarraycanbefoundinAppendixF.
9TheIEEEtrantools package maynotbeincludedinyoursetup, itcanbefoundon
CTAN.
10
FormorespacingtypesrefertoSection3.9.1.
3.5MultipleEquations
65
\IEEEeqnarraynumspace
canbeused: ithastobeaddedinthecorrespondinglineandmakessure
thatthewholeequationarrayisshiftedbythesizeoftheequationnumbers
(theshiftdependsonthesizeofthenumber!): insteadof
\begin{IEEEeqnarray}{rCl}
a & = & b + c
\\
& = & d + e + f + g + h
+ i + j + k
\\
& = & l + m + n
\end{IEEEeqnarray}
a=b+c
(3.24)
=d+e+f+g+h+i+j+k(3.25)
=l+m+n
(3.26)
weget
\begin{IEEEeqnarray}{rCl}
a & = & b + c
\\
& = & d + e + f + g + h
+ i + j + k
\IEEEeqnarraynumspace\\
& = & l + m + n.
\end{IEEEeqnarray}
a=b+c
(3.27)
=d+e+f+g+h+i+j+(3.28)
=l+m+n.
(3.29)
IftheLHSistoolong,asareplacementforthefaulty\lefteqncommand,
IEEEeqnarray offersthe \IEEEeqnarraymulticol commandwhichworks
inallsituations:
\begin{IEEEeqnarray}{rCl}
\IEEEeqnarraymulticol{3}{l}{
a + b + c + d + e + f
+ g + h
}\nonumber\\ \quad
& = & i + j
\\
& = & k + l + m
\end{IEEEeqnarray}
a+b+c+d+e+f+g+h
=i+j
(3.30)
=k+l+m
(3.31)
Theusageisidenticaltothe\multicolumnscommandinthetabular-
environment. Thefirstargument{3}specifiesthatthreecolumnsshallbe
combinedintoonewhichwillbeleft-justified{l}.
Notethatbyinserting\quadcommandsonecaneasilyadaptthedepth
oftheequationsigns,
11
e.g.,
11
Ithinkthatonequadisthedistancethatlooksgoodformostcases.
66
TypesettingMathematicalFormulae
\begin{IEEEeqnarray}{rCl}
\IEEEeqnarraymulticol{3}{l}{
a + b + c + d + e + f
+ g + h
}\nonumber\\ \qquad\qquad
& = & i + j
\\
& = & k + l + m
\end{IEEEeqnarray}
a+b+c+d+e+f+g+h
=i+j
(3.32)
=k+l+m
(3.33)
Ifanequationissplitintotwoormorelines,L
A
T
E
Xinterpretsthefirst
+orasasigninsteadofoperator. Therefore,itisnecessarytoaddan
additionalspace\: betweentheoperatorandtheterm: : insteadof
\begin{IEEEeqnarray}{rCl}
a & = & b + c
\\
& = & d + e + f + g + h
+ i + j + k \nonumber\\
&& + l + m + n + o
\\
& = & p + q + r + s
\end{IEEEeqnarray}
a=b+c
(3.34)
=d+e+f+g+h+i+j+k
+l+m+n+o
(3.35)
=p+q+r+s
(3.36)
weshouldwrite
\begin{IEEEeqnarray}{rCl}
a & = & b + c
\\
& = & d + e + f + g + h
+ i + j + k \nonumber\\
&& \negmedspace {} + l + + m m + n + o
\\
& = & p + q + r + s
\end{IEEEeqnarray}
a=b+c
(3.37)
=d+e+f+g+h+i+j+k
+l+m+n+o
(3.38)
=p+q+r+s
(3.39)
Notethespacedifferencebetween+andl! Theconstruction{} + + lforces
the+-signtobeabinaryoperatorratherthanjustasign,andtheunwanted
ensuingspacebetween{}and+iscompensatedbyanegativemediumspace
\negmedspace.
Ifa particularline shouldnothaveanequationnumber, thenumber
canbesuppressedusing\nonumber(or\IEEEnonumber). Ifonsuchaline
alabel\label{eq:...}isdefined,thenthislabelispassedontothenext
equationnumberthatisnotsuppressed. Placethelabelsrightbeforethe
line-break\\orthenexttotheequationitbelongsto. Apartfromimproving
thereadabilityofthesourcecodethispreventsacompilationerrorwhena
\IEEEmulticolcommandfollowsthelabel-definition.
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