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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
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
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,
looksbetterforsmallamountsof‘fractionmaterial:’
In display style:
\begin{equation*}
\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*}
\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*}
\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}}
\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,
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:
a = b + c + d + e + f
+ g + h + i + j
+ k + l + m + n + o + p
\label{eq:equation_too_long}
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)
signlosesitsnaturalgreaterimportancewithrespecttotheplusoperatorin
frontofk. ThebettersolutionisprovidedbytheIEEEeqnarrayenvironment
thatwillbediscussedindetailinSection3.5.
3.5 MultipleEquations
Inthemostgeneralsituationwehaveasequenceofseveralequalitiesthat
donotﬁtontooneline. Hereweneedtoworkwithverticalalignmentin
Beforeweoﬀeroursuggestionsonhowtodothis,westartwithafew
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
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
towardstheglorious...
3.5.2 IEEEeqnarrayEnvironment
The IEEEeqnarray environmentisa verypowerful commandwithmany
options. Here,wewillonlyintroduceitsbasicfunctionalities. Formore
8
Firstofall,inordertobeabletousetheIEEEeqnarrayenvironment
9
IEEEtrantools.Includethefollowinglinein
\usepackage[retainorgcmds]{IEEEtrantools}
Thestrengthof IEEEeqnarrayistheabilitytospecifythenumberof
columns inthe equation array. . Usually, , thisspeciﬁcation willbe {rCl},
i.e.,threecolumns,theﬁrstcolumnright-justiﬁed,themiddleonecentered
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
to use for r comments. . Moreover, , beside lcrLCfor math mode
entriestherearealsos,t,uforleft,centered,andrighttextmodeentries.
Notethespacesaroundtheequalitysignsincontrasttothespaceproduced
bytheeqnarrayenvironment.
3.5.3 CommonUsage
InthefollowingwewilldescribehowweuseIEEEeqnarraytosolvethemost
commonproblems.
Ifalineoverlapswiththeequationnumberasin(3.17),thecommand
8The oﬃcial manual is called
CTAN://macros/latex/contrib/IEEEtran/IEEEtran_
9TheIEEEtrantools package maynotbeincludedinyoursetup, itcanbefoundon
CTAN.
10
FormorespacingtypesrefertoSection3.9.1.
3.5MultipleEquations
65
\IEEEeqnarraynumspace
thatthewholeequationarrayisshiftedbythesizeoftheequationnumbers
\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
& = & 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}.