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3.6ArraysandMatrices
67
Therealsoexistsa*-versionwhereallequationnumbersaresuppressed.
Inthiscaseanequationnumbercanbemadetoappearusingthecommand
\IEEEyesnumber:
\begin{IEEEeqnarray*}{rCl}
a & = & b + c \\
& = & d + e \IEEEyesnumber\\
& = & f + g
\end{IEEEeqnarray*}
a=b+c
=d+e
(3.40)
=f+g
Sub-numbersarealsoeasilypossibleusing\IEEEyessubnumber:
\begin{IEEEeqnarray}{rCl}
a & = & b + c
\IEEEyessubnumber\\
& = & d + e
\nonumber\\
& = & f + g
\IEEEyessubnumber
\end{IEEEeqnarray}
a=b+c
(3.40a)
=d+e
=f+g
(3.40b)
3.6 ArraysandMatrices
Totypesetarrays,usethearrayenvironment. Itworksinasimilarwayto
thetabularenvironment. The\\commandisusedtobreakthelines:
\begin{equation*}
\mathbf{X} = \left(
\begin{array}{ccc}
x_1 & x_2 & \ldots \\
x_3 & x_4 & \ldots \\
\vdots & \vdots & \ddots
\end{array} \right)
\end{equation*}
X=
x
1
x
2
...
x
3
x
4
...
.
.
.
.
.
.
.
.
.
Thearrayenvironmentcanalsobeusedtotypesetpiecewisefunctions
byusinga“.”asaninvisible\rightdelimiter:
\begin{equation*}
|x| = \left\{
\begin{array}{rl}
-x & \text{if } x < 0,\\
0 & \text{if } x = 0,\\
x & \text{if } x > 0.
\end{array} \right.
\end{equation*}
|x|=
ifx<0,
0 ifx=0,
ifx>0.
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68
TypesettingMathematicalFormulae
Thecasesenvironmentfromamsmathsimplifiesthesyntax,soitisworth
alook:
\begin{equation*}
|x| =
\begin{cases}
-x & \text{if } x < 0,\\
0 & \text{if } x = 0,\\
x & \text{if } x > 0.
\end{cases}
\end{equation*}
|x|=
ifx<0,
0
ifx=0,
x
ifx>0.
Matricescanbetypesetbyarray,butamsmathprovidesabettersolution
usingthedifferentmatrixenvironments. Therearesixversionswithdifferent
delimiters: matrix(none),pmatrix(,bmatrix[,Bmatrix{,vmatrix|and
Vmatrix. Youdon’thavetospecifythenumberofcolumnsaswitharray.
Themaximumnumberis10,butitiscustomisable(thoughitisnotvery
oftenyouneed10columns!):
\begin{equation*}
\begin{matrix}
1 & 2 \\
3 & 4
\end{matrix} \qquad
\begin{bmatrix}
p_{11} & p_{12} & \ldots
& p_{1n} \\
p_{21} & p_{22} & \ldots
& p_{2n} \\
\vdots & \vdots & \ddots
& \vdots \\
p_{m1} & p_{m2} & \ldots
& p_{mn}
\end{bmatrix}
\end{equation*}
1 2
3 4
p
11
p
12
... p
1n
p
21
p
22
... p
2n
.
.
.
.
.
.
.
.
.
.
.
.
p
m1
p
m2
... p
mn
3.7 SpacinginMathMode
IfthespacingwithinformulaechosenbyL
A
T
E
Xisnotsatisfactory,itcanbe
adjustedbyinsertingspecialspacingcommands: \,for 3
18
quad(
),\: for
4
18
quad(
)and\;for
5
18
quad(
). Theescapedspacecharacter\␣generates
amediumsizedspacecomparabletotheinterwordspacingand\quad(
)
and\qquad(
)producelargespaces. Thesizeofa\quadcorresponds
tothewidthofthecharacter‘M’ofthecurrentfont. \!producesanegative
spaceof−
3
18
quad(−
).
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3.7SpacinginMathMode
69
\begin{equation*}
\int_1^2 \ln x \mathrm{d}x
\qquad
\int_1^2 \ln x \,\mathrm{d}x
\end{equation*}
2
1
lnxdx
2
1
lnxdx
Notethat‘d’inthedifferentialisconventionallysetinroman. Inthe
nextexample,wedefineanewcommand\ud(uprightd)whichproduces“d”
(noticethespacing
beforethed),sowedon’thavetowriteiteverytime.
The\newcommandisplacedinthepreamble.
\newcommand{\ud}{\,\mathrm{d}}
\begin{equation*}
\int_a^b f(x)\ud x
\end{equation*}
b
a
f(x)dx
Ifyouwanttotypesetmultipleintegrals,you’lldiscoverthatthespacing
betweentheintegralsistoowide.Youcancorrectitusing\!,butamsmath
providesaneasierwayforfine-tuningthespacing,namelythe\iint,\iiint,
\iiiint,and\idotsintcommands.
\newcommand{\ud}{\,\mathrm{d}}
\begin{IEEEeqnarray*}{c}
\int\int f(x)g(y)
\ud x \ud d y y \\
\int\!\!\!\int
f(x)g(y) \ud x \ud d y y \\
\iint f(x)g(y) ) \ud d x \ud d y
\end{IEEEeqnarray*}
 
f(x)g(y)dxdy

f(x)g(y)dxdy

f(x)g(y)dxdy
Seetheelectronicdocumenttestmath.tex(distributedwithA
M
S-L
A
T
E
X)
orChapter8ofTheL
A
T
E
XCompanion[3]forfurtherdetails.
3.7.1 Phantoms
Whenverticallyaligningtextusing^and_LAT
E
Xissometimesjustalittletoo
helpful. Usingthe\phantomcommandyoucanreservespaceforcharacters
thatdonotshowupinthefinaloutput. Theeasiestwaytounderstandthis
istolookatanexample:
\begin{equation*}
{}^{14}_{6}\text{C}
\qquad \text{versus} \qquad
{}^{14}_{\phantom{1}6}\text{C}
\end{equation*}
14
6
C
versus
14
6
C
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70
TypesettingMathematicalFormulae
Ifyouwanttotypesetalotofisotopesasintheexample,themhchem
packageisveryusefulfortypesettingisotopesandchemicalformulaetoo.
3.8 FiddlingwiththeMathFonts
DifferentmathfontsarelistedonTable3.14onpage79.
$\Re \qquad
\mathcal{R} \qquad
\mathfrak{R} \qquad
\mathbb{R} \qquad $
R
R
R
Thelasttworequireamssymboramsfonts.
SometimesyouneedtotellL
A
T
E
Xthecorrectfontsize. Inmathmode,
thisissetwiththefollowingfourcommands:
\displaystyle(123),\textstyle(123),\scriptstyle(
123
)and
\scriptscriptstyle(
123
).
If
isplacedinafraction,it’llbetypesetintextstyleunlessyoutell
LAT
E
Xotherwise:
\begin{equation*}
P = \frac{\displaystyle{
\sum_{i=1}^n (x_i- x)
(y_i- y)}}
{\displaystyle{\left[
\sum_{i=1}^n(x_i-x)^2
\sum_{i=1}^n(y_i- y)^2
\right]^{1/2}}}
\end{equation*}
P=
n
i=1
(x
i
x)(y
i
y)
n
i=1
(x
i
x)
2
n
i=1
(y
i
y)
2
1/2
Changingstylesgenerallyaffectsthewaybigoperatorsandlimitsare
displayed.
3.8.1 BoldSymbols
ItisquitedifficulttogetboldsymbolsinLAT
E
X;thisisprobablyintentional
asamateurtypesetterstendtooverusethem. Thefontchangecommand
\mathbfgivesboldletters,buttheseare roman(upright)whereasmath-
ematicalsymbolsarenormallyitalic,andfurthermoreitdoesn’tworkon
lowercaseGreekletters. Thereisa\boldmathcommand,butthiscanonly
beusedoutsidemathmode. Itworksforsymbolstoo,though:
$\mu, M \qquad
\mathbf{\mu}, \mathbf{M}$
\qquad \boldmath{$\mu, M$}
µ,M
µ,M
µ,M
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3.9Theorems,Lemmas,...
71
Thepackageamsbsy(includedbyamsmath)aswellasthepackagebm
fromthetoolsbundlemakethismucheasierastheyincludea\boldsymbol
command:
$\mu, M \qquad
\boldsymbol{\mu}, \boldsymbol{M}$
µ,M
µ,M
3.9 Theorems,Lemmas,...
Whenwritingmathematicaldocuments,youprobablyneedawaytotypeset
“Lemmas”,“Definitions”,“Axioms”andsimilarstructures.
\newtheorem{name}[counter]{text}[section]
Thenameargumentisashortkeywordusedtoidentifythe“theorem”.
Withthetext argumentyoudefinetheactualnameofthe“theorem”,which
willbeprintedinthefinaldocument.
Theargumentsinsquarebracketsareoptional. Theyarebothusedto
specifythenumberingusedonthe“theorem”.Usethecounterargumentto
specifythenameofapreviouslydeclared“theorem”. Thenew“theorem”
willthenbenumberedinthesamesequence.Thesectionargumentallows
youtospecifythesectionalunitwithinwhichthe“theorem”shouldgetits
numbers.
Afterexecuting the \newtheorem command d in the preamble of f your
document,youcanusethefollowingcommandwithinthedocument.
\begin{name}[text]
Thisismyinterestingtheorem
\end{name}
Theamsthmpackage(partofA
M
S-LAT
E
X)providesthe\theoremstyle{style}
commandwhichletsyoudefinewhatthetheoremisallaboutbypicking
fromthreepredefinedstyles: definition(fattitle,romanbody),plain(fat
title,italicbody)orremark(italictitle,romanbody).
Thisshouldbeenoughtheory. Thefollowingexamplesshouldremove
anyremainingdoubt,andmakeitclearthatthe\newtheoremenvironment
iswaytoocomplextounderstand.
Firstdefinethetheorems:
\theoremstyle{definition} \newtheorem{law}{Law}
\theoremstyle{plain}
\newtheorem{jury}[law]{Jury}
\theoremstyle{remark}
\newtheorem*{marg}{Margaret}
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72
TypesettingMathematicalFormulae
\begin{law} \label{law:box}
Don’t hide in the witness box
\end{law}
\begin{jury}[The Twelve]
It could be you! So beware e and
see law~\ref{law:box}.\end{jury}
\begin{jury}
You will disregard the last
statement.\end{jury}
\begin{marg}No, No, No\end{marg}
\begin{marg}Denis!\end{marg}
Law1. Don’thideinthewitnessbox
Jury2(TheTwelve)Itcouldbeyou! So
bewareandseelaw1.
Jury 3. Youwilldisregardthelaststate-
ment.
Margaret. No,No,No
Margaret. Denis!
The“Jury”theoremusesthesamecounterasthe“Law”theorem,soit
getsanumberthatisinsequencewiththeother“Laws”.Theargumentin
squarebracketsisusedtospecifyatitleorsomethingsimilarforthetheorem.
\newtheorem{mur}{Murphy}[section]
\begin{mur} If there are two or
more ways to do something, , and
one of those ways can result in
a catastrophe, then someone
will do it.\end{mur}
Murphy 3.9.1If there aretwo or r more
waystodo something,andone of those
ways can result in n a catastrophe, , then
someonewilldoit.
The“Murphy”theoremgetsanumberthatislinkedtothenumberof
thecurrentsection. Youcouldalsouseanotherunit,forexamplechapteror
subsection.
Ifyouwanttocustomizeyourtheoremsdowntothelastdot,thentheorem
packageoffersaplethoraofoptions.
3.9.1 ProofsandEnd-of-ProofSymbol
Theamsthmpackagealsoprovidestheproofenvironment.
\begin{proof}
Trivial, use
\begin{equation*}
E=mc^2.
\end{equation*}
\end{proof}
Proof. Trivial,use
E=mc
2
.
Withthecommand\qedhereyoucanmovethe‘endofproof’symbol
aroundforsituationswhereitwouldendupaloneonaline.
3.9Theorems,Lemmas,...
73
\begin{proof}
Trivial, use
\begin{equation*}
E=mc^2. \qedhere
\end{equation*}
\end{proof}
Proof. Trivial,use
E=mc
2
.
Unfortunately,thiscorrectiondoesnotworkforIEEEeqnarray:
\begin{proof}
This is a proof that ends
with an equation array:
\begin{IEEEeqnarray*}{rCl}
a & = & b + c \\
& = & d + e. \qedhere
\end{IEEEeqnarray*}
\end{proof}
Proof. Thisisaproofthatendswithan
equationarray:
a=b+c
=d+e.
Thereason forthisistheinternalstructure of IEEEeqnarray: italways
putstwoinvisiblecolumnsatbothsidesofthearraythatonlycontaina
stretchablespace. BythisIEEEeqnarrayensuresthattheequationarray
ishorizontallycentered. The\qedhere commandshouldactuallybeput
outsidethisstretchablespace,butthisdoesnothappenasthesecolumns
areinvisibletotheuser.
Thereisaverysimpleremedy. Definethestretchingexplicitly!
\begin{proof}
This is a proof that ends
with an equation array:
\begin{IEEEeqnarray*}{+rCl+x*}
a & = & b + c \\
& = & d + e. & \qedhere
\end{IEEEeqnarray*}
\end{proof}
Proof. Thisisaproofthatendswithan
equationarray:
a=b+c
=d+e.
Notethatthe+in{+rCl+x*}denotesstretchablespaces,oneonthe left
of the equations(which, if notspecified, , willbe e done e automatically by
IEEEeqnarray!) andoneontherightofthe e equations. . Butnowon n the
right,afterthestretchingcolumn,weaddanemptycolumnx.Thiscolumn
willonlybeneededonthelastlineifthe\qedherecommandisputthere.
Finally, wespecifya*. Thisisa a null-spacethatpreventsIEEEeqnarray
fromaddinganotherunwanted+-space!
Inthecaseofequationnumbering,thereisasimilarproblem. Comparing
74
TypesettingMathematicalFormulae
\begin{proof}
This is a proof that ends
with a numbered equation:
\begin{equation}
a = b + c.
\end{equation}
\end{proof}
Proof. This is aproof f thatendswitha
numberedequation:
a=b+c.
(3.41)
with
\begin{proof}
This is a proof that ends
with a numbered equation:
\begin{equation}
a = b + c. \qedhere
\end{equation}
\end{proof}
Proof. This is aproof f thatendswitha
numberedequation:
a=b+c.
(3.42)
younoticethatinthe(correct)secondversionthe2ismuchclosertothe
equationthaninthefirstversion.
Similarly,thecorrectwayofputtingtheQED-symbolattheendofan
equationarrayisasfollows:
\begin{proof}
This is a proof that ends
with an equation array:
\begin{IEEEeqnarray}{+rCl+x*}
a & = & b + c \\
& = & d + e. \\
&&& \qedhere\nonumber
\end{IEEEeqnarray}
\end{proof}
Proof. Thisisaproofthatendswithan
equationarray:
a=b+c
(3.43)
=d+e.
(3.44)
whichcontrastswith
\begin{proof}
This is a proof that ends
with an equation array:
\begin{IEEEeqnarray}{rCl}
a & = & b + c \\
& = & d + e.
\end{IEEEeqnarray}
\end{proof}
Proof. Thisisaproofthatendswithan
equationarray:
a=b+c
(3.45)
=d+e.
(3.46)
3.10ListofMathematicalSymbols
75
3.10 ListofMathematicalSymbols
Thefollowingtablesdemonstrateallthesymbolsnormallyaccessiblefrom
mathmode.
Notethatsometablesshowsymbolsonlyaccessibleafterloadingthe
amssymbpackageinthepreambleofyourdocument12. IftheA
M
Spack-
age and fonts are e not installed d on your system, , have a a look k at CTAN:
pkg/amslatex.Anevenmorecomprehensivelistofsymbolscanbefound
atCTAN:info/symbols/comprehensive.
Table3.1: MathModeAccents.
ˆ\hat{a}
ˇ\check{a}
˜a
\tilde{a}
`\grave{a}
˙\dot{a}
¨a
\ddot{a}
¯\bar{a}
a \vec{a}
[
AAA
\widehat{AAA}
´\acute{a}
˘\breve{a}
]
AAA
\widetilde{AAA}
˚\mathring{a}
Table3.2:GreekLetters.
Thereisnouppercaseofsomeoftheletterslike\Alpha,\Betaandsoon,
becausetheylookthesameasnormalromanletters:A,B...
α
\alpha
θ \theta
o
υ \upsilon
β
\beta
ϑ
\vartheta
π
\pi
φ \phi
γ
\gamma
ι \iota
\varpi
ϕ
\varphi
δ
\delta
κ
\kappa
ρ
\rho
χ
\chi
 \epsilon
λ
\lambda
\varrho
ψ
\psi
ε \varepsilon
µ
\mu
σ
\sigma
ω
\omega
ζ
\zeta
ν \nu
ς \varsigma
η
\eta
ξ \xi
τ
\tau
Γ
\Gamma
Λ
\Lambda
Σ
\Sigma
Ψ
\Psi
\Delta
Ξ
\Xi
Υ
\Upsilon
\Omega
Θ
\Theta
Π
\Pi
Φ
\Phi
12
Thetableswerederivedfromsymbols.texbyDavidCarlisleandsubsequentlychanged
extensivelyassuggestedbyJosefTkadlec.
76
TypesettingMathematicalFormulae
Table3.3: BinaryRelations.
Youcannegatethefollowingsymbolsbyprefixingthemwitha\notcom-
mand.
<
<
>
>
=
=
\leq or\le
\geq or\ge
\equiv
\ll
\gg
.
=
\doteq
\prec
\succ
\sim
\preceq
\succeq
\simeq
\subset
\supset
\approx
\subseteq
\supseteq
=
\cong
@
\sqsubset
a
A
\sqsupset
a
1
\Join
a
\sqsubseteq
\sqsupseteq
 \bowtie
\in
\ni ,\owns
\propto
\vdash
\dashv
|=
\models
| \mid
\parallel
\perp
\smile
\frown
\asymp
: :
/
\notin
=
\neq or\ne
a
Usethelatexsympackagetoaccessthissymbol
Table3.4: BinaryOperators.
+
+
-
±
\pm
\mp
 \triangleleft
· \cdot
÷
\div
 \triangleright
×
\times
\
\setminus
 \star
\cup
\cap
∗ \ast
\sqcup
\sqcap
◦ \circ
\vee ,\lor
\wedge ,\land
• \bullet
\oplus
\ominus
 \diamond
\odot
\oslash
\uplus
\otimes
\bigcirc
\amalg
\bigtriangleup
\bigtriangledown
† \dagger
\lhd
a
\rhd
a
‡ \ddagger
\unlhd
a
\unrhd
a
 \wr
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