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NEW
VIII.5. OPERATORS


\begin{equation*}
p_k(x)=\prod_{\substack{i=1\\i\ne k}}ˆn
\left(\frac{x-t_i}{t_k-t_i}\right)
\end{equation*}

The amsmath package has also a \sideset command which can be used to put
symbols at any of the four corners of a large operator. Thus
ul ur
produces
$\sideset{_{ll}ˆ{ul}}{_{lr}ˆ{ur}}\bigcup$
ll lr

produces .
$\sideset{}{™}\sum$


NEW
VIII.5. OPERATORS

Mathematical text is usually typeset in italics, and TEX follows this tradition. But certain
functions in mathematics such as log, sin, lim and so on are traditionally typeset in
roman. This is implemented in TEX by the use of commands like $\log$, $\sin$, $\lim$
and so on. The symbols classi¬ed as “Log-like symbols” in the table at the end of this
chapter shows such functions which are prede¬ned in LTEX.
A

Having read thus far, it may be no surprise to learn that we can de¬ne our own
“operator names” which receive this special typographic treatment. This is done by
the \DeclareMathOperator command. Thus if the operator cl occurs frequently in the
document, you can make the declaration

\DeclareMathOperator{\cl}{cl}

in the preamble and then type $\cl(A)$ to produce cl(A), for example.
Note that an operator de¬ned like this accommodates subscripts and superscripts in
the usual way, that is, at its sides. Thus
We denote the closure of $A$ in the subspace $Y$ of $X$ by
$\cl_Y(A)$

produces

We denote the closure of A in the subspace Y of X by clY (A)

If we want to de¬ne a new operator with subscripts and superscripts placed in the “lim-
its” position below and above, then we should use the starred form of the \DeclareMathOperator
as shown below
\DeclareMathOperator*{\esup}{ess\,sup}


For $f\in Lˆ\infty(R)$, we define
\begin{equation*}
||f||_\infty=\esup_{x\in R}|f(x)|
\end{equation*}

(Note that the declaration must be done in the preamble.) This produces the output

For f ∈ L∞ (R), we de¬ne
|| f ||∞ = ess sup | f (x)|
x∈R


(Why the \, command in the de¬nition?)
102 TYPESETTING MATHEMATICS
VIII.


THE
VIII.6. MANY FACES OF MATHEMATICS

We have noted that most mathematics is typeset in italics typeface and some mathematical
operators are typeset in an upright fashion. There may be need for additional typefaces
as in typesetting vectors in boldface.
LTEX includes several styles to typeset mathematics as shown in the table below
A


EXAMPLE
COMMAND
TYPE STYLE
INPUT OUTPUT
italic
x+y=z
\mathit $x+y=z$
(default)
x+y=z
roman \mathrm $\mathrm{x+y=z}$
x+y=z
bold \mathbf $\mathbf{x+y=z}$
x+y=z
sans serif \mathsf $\mathsf{x+y=z}$
x+y=z
typewriter \mathtt $\mathtt{x+y=z}$
calligraphic
X+Y=Z
\mathcal $\mathcal{X+Y=Z}$
(upper case only)

In addition to these, several other math alphabets are available in various packages (some
of which are shown in the list of symbols at the end of this chapter).
Note that the command \mathbf produces only roman boldface and not math italic
boldface. Sometimes you may need boldface math italic, for example to typeset vectors.
For this, amsmath provides the \boldsymbol command. Thus we can get

In this case, we de¬ne
a+b=c

from the input
In this case, we define
\begin{equation*}
\boldsymbol{a}+\boldsymbol{b}=\boldsymbol{c}
\end{equation*}

If the document contains several occurrences of such symbols, it is better to make a
new de¬nition such as

\newcommand{\vect}[1]{\boldsymbol{#1}}


and then use $\vect{a}$ to produce a and $\vect{b}$ to produce b and so on. the
additional advantage of this approach is that if you change your mind later and want
vectors to be typeset with arrows above them as ’, then all you need is to change the

a
\boldsymol part of the de¬nition of \vect to \overrightarrow and the change will be
effected throughout the document.
Now if we change the input of the above example as
In this case, we define
\begin{equation*}
\boldsymbol{a+b=c}
\end{equation*}

then we get the output
103
AND
VIII.7. THAT IS NOT ALL!



In this case, we de¬ne
a+b=c

Note that now the symbols + and = are also in boldface. Thus \boldsymbol makes bold
every math symbol in its scope (provided the bold version of that symbol is available in
the current math font).
There is another reason for tweaking the math fonts. Recently, the International
Standards Organization (ISO) has established the recognized typesetting standards in
mathematics. Some of the points in it are,
Simple variables are represented by italic letters as a, x.
1.
Vectors are written in boldface italic as a, x.
2.
Matrices may appear in sans serif as in A, X.
3.
The special numbers e, i and the differential operator d are written in upright roman.
4.
Point 1 is the default in LTEX and we have seen how point 2 can be implemented.
A

to ful¬ll Point 4, it is enough if we de¬ne something like
\newcommand{\me}{\mathrm{e}}
\newcommand{\mi}{\mathrm{i}}
\newcommand{\diff}{\mathrm{d}}

and then use $\me$ for e and $\mi$ for i and $\diff x$ for dx.
Point 3 can be implemented using \mathsf but it is a bit dif¬cult (but not impossible)
if we need them to be in italic also. The solution is to create a new math alphabet, say,
\mathsfsl by the command

\DeclareMathAlphabet{\mathsfsl}{OT1}{cmss}{m}{sl}

(in the preamble) and use it to de¬ne a command \matr to typeset matrices in this font by

\newcommand{\matr}[1]{\ensuremath{\mathsfsl{#1}}}

so that $\maqtr A$ produces A.

AND
VIII.7. THAT IS NOT ALL!

We have only brie¬‚y discussed the basic techniques of typesetting mathematics using
LTEX and some of the features of the amsmath package which helps us in this task. For
A

more details on this package see the document amsldoc.dvi which should be available
with your TEX distribution. If you want to produce really beautiful mathematical doc-
uments, read the Master”“The TEX Book” by Donald Knuth, especially Chapter 18,
“Fine Points of Mathematics Typing”.

SYMBOLS
VIII.8.


Table VIII.1: Greek Letters

± θ „
o
\alpha \theta o \tau
β ‘ π …
\beta \vartheta \pi \upsilon
γ ι φ
\gamma \iota \varpi \phi
δ κ ρ •
\delta \kappa \rho \varphi
» χ
\epsilon \lambda \varrho \chi
µ µ σ ψ
\varepsilon \mu \sigma \psi
104 TYPESETTING MATHEMATICS
VIII.


ζ ν ‚ ω
\zeta \nu \varsigma \omega
· ξ
\eta \xi

“ Λ Σ Ψ
\Gamma \Lambda \Sigma \Psi
∆ Ξ Υ „¦
\Delta \Xi \Upsilon \Omega
˜ Π ¦
\Theta \Pi \Phi



Table VIII.2: Binary Operation Symbols

± © •
\pm \cap \diamond \oplus

\mp \cup \bigtriangleup \ominus
— —
\times \uplus \bigtriangledown \otimes
· \div \sqcap \triangleleft \oslash
— \ast \sqcup \triangleright \odot
\lhd—

\star \vee \bigcirc
\rhd—
—¦ § †
\circ \wedge \dagger
\unlhd—
• \ ‡
\bullet \setminus \ddagger
\unrhd—
· \cdot \wr \amalg
+ ’
+ -

Not prede¬ned in LTEX 2µ . Use one of the packages latexsym, amsfonts or amssymb.
A


Table VIII.3: Relation Symbols

|=
¤ ≥ ≡
\leq \geq \equiv \models
∼ ⊥
\prec \succ \sim \perp
|
\preceq \succeq \simeq \mid
\ll \gg \asymp \parallel
‚ ⊃ ≈
\subset \supset \approx \bowtie
\Join—
⊆ ⊇
\subseteq \supseteq \cong
\sqsubset— \sqsupset— \neq \smile
\sqsubseteq \sqsupseteq \doteq \frown
=
∈ ∝
\in \ni \propto =
< >
\vdash \dashv < >
: :

Not prede¬ned in LTEX 2µ . Use one of the packages latexsym, amsfonts or amssymb.
A


Table VIII.4: Punctuation Symbols

, . ·
; :
, ; \colon \ldotp \cdotp



Table VIII.5: Arrow Symbols

← ←’ ‘
\leftarrow \longleftarrow \uparrow
⇐ ⇐= ‘
\Leftarrow \Longleftarrow \Uparrow
’ ’’ “
\rightarrow \longrightarrow \downarrow
=’
’ “
\Rightarrow \Longrightarrow \Downarrow
” ←’
\leftrightarrow \longleftrightarrow \updownarrow
” ⇐’
\Leftrightarrow \Longleftrightarrow \Updownarrow
’ ’’
\mapsto \longmapsto \nearrow
105
SYMBOLS
VIII.8.


← ’
\hookleftarrow \hookrightarrow \searrow
\leftharpoonup \rightharpoonup \swarrow
\leftharpoondown \rightharpoondown \nwarrow
\leadsto—
\rightleftharpoons

Not prede¬ned in LTEX 2µ . Use one of the packages latexsym, amsfonts or amssymb.
A


Table VIII.6: Miscellaneous Symbols

. ..
. .
... .
···
\ldots \cdots \vdots \ddots
„µ ∀ ∞
\aleph \prime \forall \infty
\Box—
… ∃
\hbar \emptyset \exists
± \Diamond—
¬
\imath \nabla \neg

 \jmath \surd \flat \triangle

\ell \top \natural \clubsuit
„˜ ⊥ ™¦
\wp \bot \sharp \diamondsuit
\ \backslash ™
\Re \| \heartsuit
‚ \partial ™

\Im \angle \spadesuit
.

|
\mho . |

Not prede¬ned in LTEX 2µ . Use one of the packages latexsym, amsfonts or amssymb.
A

Table VIII.7: Variable-sized Symbols

\sum \bigcap \bigodot
\prod \bigcup \bigotimes
\coprod \bigsqcup \bigoplus
\int \bigvee \biguplus
\oint \bigwedge


Table VIII.8: Log-like Symbols

\arccos \cos \csc \exp \ker \limsup \min \sinh
\arcsin \cosh \deg \gcd \lg \ln \Pr \sup
\arctan \cot \det \hom \lim \log \sec \tan
\arg \coth \dim \inf \liminf \max \sin \tanh


Table VIII.9: Delimiters

‘ ‘
( )
( ) \uparrow \Uparrow
“ “
[ ]
[ ] \downarrow \Downarrow
{ }
\{ \} \updownarrow \Updownarrow
\lfloor \rfloor \lceil \rceil
/ \
\langle \rangle / \backslash
| | \|



Table VIII.10: Large Delimiters
 ±  ±
\rmoustache  \lmoustache  \rgroup \lgroup
 
¦ 
¦ 
\arrowvert \Arrowvert  \bracevert
¦
¦ 
106 TYPESETTING MATHEMATICS
VIII.


Table VIII.11: Math Mode Accents

ˆ ´ ¯ ™

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