Math 233 In the following pages I have excerpted some pages from The Not So Short Introduction to LATEX 2, by Tobias Oetiker. The full text of this document is also posted on our class Blackboard page, under “Reference.”

1

54

Typesetting Mathematical Formulae placement of \bar for a variable with subscript. The apostrophe mark ’ gives a prime:

$f(x) = x^2 \qquad f’(x) = 2x \qquad f’’(x) = 2\\[5pt] \hat{XY} \quad \widehat{XY} \quad \bar{x_0} \quad \bar{x}_0$

f (x) = x2 ˆ XY

d XY

f 0 (x) = 2x x¯0

f 00 (x) = 2

x ¯0

Vectors are often specified by adding small arrow symbols on top of a variable. This is done with the \vec command. The two commands \overrightarrow and \overleftarrow are useful to denote the vector from A to B: $\vec{a} \qquad \vec{AB} \qquad \overrightarrow{AB}$

~a

~ AB

−−→ AB

Names of log-like functions are often typeset in an upright font, and not in italics as variables are, so LATEX supplies the following commands to typeset the most important function names: \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 \[\lim_{x \rightarrow 0} \frac{\sin x}{x}=1\]

sin x =1 x→0 x lim

For functions missing from the list, use the \DeclareMathOperator command. There is even a starred version for functions with limits. This command works only in the preamble so the commented lines in the example below must be put into the preamble. %\DeclareMathOperator{\argh}{argh} %\DeclareMathOperator*{\nut}{Nut} \[3\argh = 2\nut_{x=1}\]

3 argh = 2 Nut x=1

For the modulo function, there are two commands: \bmod for the binary operator “a mod b” and \pmod for expressions such as “x ≡ a (mod b):” $a\bmod b \\ x\equiv a \pmod{b}$

a mod b x ≡ a (mod b)

3.8 List of Mathematical Symbols

3.8

63

List of Mathematical Symbols

The following tables demonstrate all the symbols normally accessible from math mode. To use the symbols listed in Tables 3.12–3.8,8 the package amssymb must be loaded in the preamble of the document and the AMS math fonts must be installed on the system. If the AMS package and fonts are not installed on your system, have a look at CTAN:macros/latex/required/amslatex. An even more comprehensive list of symbols can be found at CTAN:info/ symbols/comprehensive. Table 3.1: Math Mode Accents. a ˆ a ` a ¯ a ´ ˚ a

a ˇ a˙ ~a a ˘

\hat{a} \grave{a} \bar{a} \acute{a} \mathring{a}

\check{a} \dot{a} \vec{a} \breve{a}

a ˜ a ¨ [ AAA ] AAA

\tilde{a} \ddot{a} \widehat{AAA} \widetilde{AAA}

Table 3.2: Greek Letters. There is no uppercase of some of the letters like \Alpha, \Beta and so on, because they look the same as normal roman letters: A, B. . . α β γ δ  ε ζ η Γ ∆ Θ

\alpha \beta \gamma \delta \epsilon \varepsilon \zeta \eta \Gamma \Delta \Theta

θ ϑ ι κ λ µ ν ξ Λ Ξ Π

\theta \vartheta \iota \kappa \lambda \mu \nu \xi \Lambda \Xi \Pi

o π $ ρ % σ ς τ Σ Υ Φ

o \pi \varpi \rho \varrho \sigma \varsigma \tau \Sigma \Upsilon \Phi

υ φ ϕ χ ψ ω

\upsilon \phi \varphi \chi \psi \omega

Ψ Ω

\Psi \Omega

8 These tables were derived from symbols.tex by David Carlisle and subsequently changed extensively as suggested by Josef Tkadlec.

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Typesetting Mathematical Formulae

Table 3.3: Binary Relations. You can negate the following symbols by prefixing them with a \not command. < ≤  ≺  ⊂ ⊆ @ v ∈ ` | ^ :

< \leq or \le \ll \prec \preceq \subset \subseteq \sqsubset a \sqsubseteq \in \vdash \mid \smile : a

> ≥    ⊃ ⊇ A w 3 a k _ ∈ /

> \geq or \ge \gg \succ \succeq \supset \supseteq \sqsupset a \sqsupseteq \ni , \owns \dashv \parallel \frown \notin

= ≡ . = ∼ ' ≈ ∼ = 1 ./ ∝ |= ⊥  6=

= \equiv \doteq \sim \simeq \approx \cong \Join a \bowtie \propto \models \perp \asymp \neq or \ne

Use the latexsym package to access this symbol

Table 3.4: Binary Operators. + ± · × ∪ t ∨ ⊕ ⊗ 4


+ \pm \cdot \times \cup \sqcup \vee , \lor \oplus \odot \otimes \bigtriangleup \lhd a \unlhd a

− ∓ ÷ \ ∩ u ∧ 

5  

\mp \div \setminus \cap \sqcap \wedge , \land \ominus \oslash \bigcirc \bigtriangledown \rhd a \unrhd a

/ . ? ∗ ◦ •  ] q † ‡ o

\triangleleft \triangleright \star \ast \circ \bullet \diamond \uplus \amalg \dagger \ddagger \wr

3.8 List of Mathematical Symbols

65

Table 3.5: BIG Operators. P Q ` R

L

\sum \prod \coprod \int \bigoplus

S T F H

N

\bigcup \bigcap \bigsqcup \oint \bigotimes

W V U J

\bigvee \bigwedge \biguplus \bigodot

Table 3.6: Arrows. ← → ↔ ⇐ ⇒ ⇔ 7→ ←( )

↑ l ⇓ % . ;

\leftarrow or \gets \rightarrow or \to \leftrightarrow \Leftarrow \Rightarrow \Leftrightarrow \mapsto \hookleftarrow \leftharpoonup \leftharpoondown \rightleftharpoons \uparrow \updownarrow \Downarrow \nearrow \swarrow \leadsto a a

←− −→ ←→ ⇐= =⇒ ⇐⇒ 7−→ ,→ * + ⇐⇒ ↓ ⇑ m & -

\longleftarrow \longrightarrow \longleftrightarrow \Longleftarrow \Longrightarrow \Longleftrightarrow \longmapsto \hookrightarrow \rightharpoonup \rightharpoondown \iff (bigger spaces) \downarrow \Uparrow \Updownarrow \searrow \nwarrow

Use the latexsym package to access this symbol

Table 3.7: Arrows as Accents. −−→ AB ←−− AB ←→ AB

\overrightarrow{AB} \overleftarrow{AB} \overleftrightarrow{AB}

AB −−→ AB ←−−

AB ←→

\underrightarrow{AB} \underleftarrow{AB} \underleftrightarrow{AB}

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Typesetting Mathematical Formulae

Table 3.8: Delimiters. ( [ { h | / b e

) ] } i k \ c d

( [ or \lbrack \{ or \lbrace \langle | or \vert / \lfloor \rceil

) ] or \rbrack \} or \rbrace \rangle \| or \Vert \backslash \rfloor \lceil

↑ ↓ l ⇑ ⇓ m

\uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow

Table 3.9: Large Delimiters.      

  w w

\lgroup \arrowvert \rmoustache

\rgroup \Arrowvert

     

\lmoustache \bracevert

Table 3.10: Miscellaneous Symbols. ... ~ < ∀ 0

∇ ⊥ ♦ ¬

\dots \hbar \Re \forall ’ \nabla \bot \diamondsuit \neg or \lnot a

··· ı = ∃ 0 4 > ♥ [

\cdots \imath \Im \exists \prime \triangle \top \heartsuit \flat

.. .  ℵ f ∅ 2 ∠ ♣ \

\vdots \jmath \aleph \mho a \emptyset \Box a \angle \clubsuit \natural

..

.

` ℘ ∂ ∞ 3 √ ♠ ]

\ddots \ell \wp \partial \infty \Diamond a \surd \spadesuit \sharp

Use the latexsym package to access this symbol

Table 3.11: Non-Mathematical Symbols. These symbols can also be used in text mode. † ‡

\dag \ddag

§ ¶

\S \P

© £

\copyright \pounds

® %

\textregistered \%

3.8 List of Mathematical Symbols

67

Table 3.12: AMS Delimiters. p |

z

\ulcorner \lvert

\digamma

q |

\urcorner \rvert

x k

\llcorner \lVert

y k

\lrcorner \rVert

Table 3.13: AMS Greek and Hebrew. κ

\varkappa

i

\beth

\gimel

‫ג‬

k

\daleth

Table 3.14: Math Alphabets. See Table 6.4 on 111 for other math fonts. Example ABCDEabcde1234 ABCDEabcde1234 ABCDEabcde1234 ABCDE A BC DE ABCDEabcde1234 ABCDEa1234

u n d Y
 | g

Command \mathrm{ABCDE abcde 1234} \mathit{ABCDE abcde 1234} \mathnormal{ABCDE abcde 1234} \mathcal{ABCDE abcde 1234} \mathscr{ABCDE abcde 1234} \mathfrak{ABCDE abcde 1234} \mathbb{ABCDE abcde 1234}

Required package

mathrsfs amsfonts or amssymb amsfonts or amssymb

Table 3.15: AMS Binary Operators.

\dotplus \ltimes \doublecup \veebar \boxplus \boxtimes \intercal \curlyvee

 o e Z

~ f

\centerdot \rtimes \doublecap \barwedge \boxminus \boxdot \circledast \curlywedge

> r [  } i h

\divideontimes \smallsetminus \doublebarwedge \circleddash \circledcirc \rightthreetimes \leftthreetimes

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Typesetting Mathematical Formulae

Table 3.16: AMS Binary Relations.

l 6 0 5 ≪ . / ≶ Q S 4 2 w j q J B I D C E

\lessdot \leqslant \eqslantless \leqq \lll or \llless \lesssim \lessapprox \lessgtr \lesseqgtr \lesseqqgtr \preccurlyeq \curlyeqprec \precsim \precapprox \subseteqq \shortparallel \blacktriangleleft \vartriangleright \blacktriangleright \trianglerighteq \vartriangleleft \trianglelefteq

m > 1 = ≫ & ' ≷ R T < 3 % v k c A ∵ b a p ∴

\gtrdot \geqslant \eqslantgtr \geqq \ggg \gtrsim \gtrapprox \gtrless \gtreqless \gtreqqless \succcurlyeq \curlyeqsucc \succsim \succapprox \supseteqq \Supset \sqsupset \because \Subset \smallfrown \shortmid \therefore

+ : ; P $ , l m ∼ ≈ u v w 

  ∝ G t ` @

\doteqdot \risingdotseq \fallingdotseq \eqcirc \circeq \triangleq \bumpeq \Bumpeq \thicksim \thickapprox \approxeq \backsim \backsimeq \vDash \Vdash \Vvdash \backepsilon \varpropto \between \pitchfork \smallsmile \sqsubset

3.8 List of Mathematical Symbols

L99 ⇔  W     " x (  

69

Table 3.17: AMS Arrows.

\dashleftarrow \leftleftarrows \leftrightarrows \Lleftarrow \twoheadleftarrow \leftarrowtail \leftrightharpoons \Lsh \looparrowleft \curvearrowleft \circlearrowleft \multimap \downdownarrows \upharpoonright \rightsquigarrow

99K ⇒  V  

 # y     !

\dashrightarrow \rightrightarrows \rightleftarrows \Rrightarrow \twoheadrightarrow \rightarrowtail \rightleftharpoons \Rsh \looparrowright \curvearrowright \circlearrowright \upuparrows \upharpoonleft \downharpoonright \leftrightsquigarrow

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Typesetting Mathematical Formulae

Table 3.18: AMS Negated Binary Relations and Arrows.

* $

\nless \lneq \nleq \nleqslant \lneqq \lvertneqq \nleqq \lnsim \lnapprox \nprec \npreceq \precneqq \precnsim \precnapprox \subsetneq \varsubsetneq \nsubseteq \subsetneqq

≯

 
        ) ! + %

\ngtr \gneq \ngeq \ngeqslant \gneqq \gvertneqq \ngeqq \gnsim \gnapprox \nsucc \nsucceq \succneqq \succnsim \succnapprox \supsetneq \varsupsetneq \nsupseteq \supsetneqq

& ' " # ∦ . /   0 2 1 3 6 7 5 4

\varsubsetneqq \varsupsetneqq \nsubseteqq \nsupseteqq \nmid \nparallel \nshortmid \nshortparallel \nsim \ncong \nvdash \nvDash \nVdash \nVDash \ntriangleleft \ntriangleright \ntrianglelefteq \ntrianglerighteq

8 :

\nleftarrow \nLeftarrow

9 ;

\nrightarrow \nRightarrow

=