Formula Gallery

7.9 Formula Gallery

In this section I present a collection of formulas—some simple, some complex—that

illustrate the power of L A TEX.

Some of these examples require the amssymb package, so it is a good idea to include the line

\usepackage{amssymb,latexsym} following the \documentclass line of any article. Formula 1

A set-valued function

x 7→ { c ∈ C | c ≤ x }

\[ x \mapsto \{\, c \in C \mid c \leq x \,\} \]

To equalize the spacing around c ∈ C and c ≤ x, a thin space (\,) was added inside each brace (see Section 8.1). The same technique is used in several other formulas in this section.

Formula 2

[ (I j |j∈J)

\left| \bigcup (\, I_{j} \mid j \in J \,) \right|

< \mathfrak{m} \]

We use the delimiters \left| and \right| (see Section 7.5.1). The Fraktur m is introduced in Section 8.3.2.

7.9 Formula Gallery 181

Formula 3 Note that you have to add spacing both before and after the text fragment for some in the following example. The argument of \text is typeset in text mode, so spaces are recognized.

A= {x∈X|x∈X i , for some i ∈I} \[

A = \{\, x \in X \mid x \in X_{i}, \text{ for some $i \in I$} \,\} \]

Formula 4 Space to show logical structure:

\[ \langle a_{1}, a_{2} \rangle \leq \langle a’_{1}, a’_{2}\rangle \qquad \text{if{f}} \qquad a_{1} < a’_{1} \quad \text{or} \quad a_{1} = a’_{1} \text{ and } a_{2} \leq a’_{2}

\] Note that in if{f} (in the argument of the first \text) the second f is enclosed in

braces to avoid the use of the ligature—the merging of the two f’s. For the proper way of typesetting iff without a ligature, see Section 5.4.6.

Formula 5 Here are some examples of Greek letters:

⊆u ′ } \[

{ γ | γ < 2χ, B ′ *u ,B γ

\Gamma_{u’} = \{\,\gamma \mid \gamma < 2\chi,\ B_{\alpha} \nsubseteq u’, \ B_{\gamma} \subseteq u’ \,\} \]

See Section B.1 for a complete listing of Greek letters. We use the command \ to properly space the formula. This command can be used both in text and in math.

Formula 6 \mathbb allows you to use the blackboard bold math alphabet, which only provides capital letters:

A=B 2 ×Z

A = B^{2} \times \mathbb{Z} \]

182 Chapter 7 Typing math

Formula 7 \left[ and \right] provide stretched delimiters:

\[ y^C \equiv z \vee \bigvee_{ i \in C } \left[ s_{i}^{C} \right] \pmod{ \Phi } \]

Notice how the superscript is set directly above the subscript in s C i . Formula 8

A complicated congruence: _

y ∨ x ( [B γ ] |γ∈Γ)≡z∨ ( [B γ ] | γ ∈ Γ ) (mod Φ ) \[

y \vee \bigvee (\, [B_{\gamma}] \mid \gamma \in \Gamma \,) \equiv z \vee \bigvee (\, [B_{\gamma}] \mid \gamma \in \Gamma \,) \pmod{ \Phi^{x} }

Formula 9 Use \nolimits to force the “limit” of the large operator to display as a subscript (see Section 7.6.4):

m (x j |j∈I i ) |i<ℵ α \[

f (x) = m

f(\mathbf{x}) = \bigvee\nolimits_{\!\mathfrak{m}} \left(\,

\bigwedge\nolimits_{\mathfrak{m}} (\, x_{j} \mid j \in I_{i} \,) \mid i < \aleph_{\alpha} \,\right)

\] Notice that a negative space (\!) was inserted to bring the m a little closer to the big

join symbol W . Formula 10 The \left. command gives a blank left delimiter, which is needed to

balance the \right| command:

F (x) =b F (b)

a −b F (a)

7.9 Formula Gallery 183

\[ \left. \widehat{F}(x) \right|_{a}^{b} = \widehat{F}(b) - \widehat{F}(a) \]

Formula 11 The \underset and \overset commands build new symbols (see Sec- tion 8.2.1):

1 u+ 2 v

∼w ∼z

\[ u \underset{\alpha}{+} v \overset{1}{\thicksim} w \overset{2}{\thicksim} z \]

Note that the new symbols 1 ∼ and 2 ∼ are binary relations and + is a binary operation.

Formula 12 Small size bold def:

f (x) def =x 2 −1

\[ f(x) \overset{ \mathbf{def} }{ = } x^{2} - 1 \]

Formula 13 Math accents run amok:

\overbrace{a\spcheck + b\spcheck + \dots + z\spcheck}^ {\breve{\breve{n}}}

\] Recall that for the \sp commands you need the amsxtra package.

Formula 14

a+b+c

\[ \begin{vmatrix}

a + b + c & uv\\ a+b&c+d \end{vmatrix} =7

184 Chapter 7 Typing math

a+b+c

\[ \begin{Vmatrix}

a + b + c & uv\\ a+b&c+d \end{Vmatrix} =7

\boldsymbol{\alpha}^2\sum_{j \in \mathbf{N}} b_{ij}

\hat{y}_{j} = \sum_{j \in \mathbf{N}} b^{(\lambda)}_{ij}\hat{y}_{j} + (b_{ii} - \lambda_{i}) \hat{y}_{i} \hat{y}

\] \mathbf{N} makes a bold N and \boldsymbol{\alpha} produces a bold α (see

Section 8.3.2). Formula 16 To produce the formula

try typing \[

\left( \prod^n_{\, j = 1} \hat{ x }_{j} \right) H_{c}= \frac{1}{2} \hat{k}_{ij} \det \hat{ \mathbf{K} }(i|i)

\] which typesets as

This is not quite right. You can correct the overly large parentheses by using the \biggl and \biggr commands in place of \left( and \right), respectively (see

Section 7.5.2). Adjust the small hat over K by using \widehat:

7.9 Formula Gallery 185

\[ \biggl( \prod^n_{\, j = 1} \hat{ x }_{j} \biggr) H_{c} = \frac{1}{2}\hat{ k }_{ij} \det \widehat{ \mathbf{K} }(i|i)

\] which gives you the desired formula. Formula 17 In this formula, I have used \overline{I} to get

I. You could, instead, use \bar{I}, which is typeset as ¯ I.

det K(t = 1, t 1 ,...,t n )=

\[ \det \mathbf{K} (t = 1, t_{1}, \dots, t_{n}) = \sum_{I \in \mathbf{n} }(-1)^{|I|} \prod_{i \in I}t_{i} \prod_{j \in I} (D_{j} + \lambda_{j} t_{j}) \det \mathbf{A}^{(\lambda)} (\,\overline{I} | \overline{I}\,) = 0

\] Formula 18 The command \| provides the k math symbol in this formula:

− BH(z)(v − v

(v,v )→(0,0)

kv − v k

\[ \lim_{(v, v’) \to (0, 0)} \frac{H(z + v) - H(z + v’) - BH(z)(v - v’)} {\| v - v’ \|} = 0 \]

Formula 19 This formula uses the calligraphic math alphabet (see Section 8.3.2): Z

2 α|z|

2 α|z| 2 |∂u| 2 Φ 0 (z)e ≥c 4 α |u| Φ 0 e +c 5 δ −2 2 Φ 0 e |u| α|z|

\[ \int_{\mathcal{D}} | \overline{\partial u} |^{2} \Phi_{0}(z) e^{\alpha |z|^2} \geq c_{4} \alpha \int_{\mathcal{D}} |u|^{2}\Phi_{0} e^{\alpha |z|^{2}} + c_{5} \delta^{-2} \int_{A} |u|^{2} \Phi_{0} e^{\alpha |z|^{2}}

186 Chapter 7 Typing math

Formula 20 The \hdotsfor command sets dots that span multiple columns in a matrix. The \dfrac command is the displayed variant of the \frac command (see Section 7.4.1), used here because the matrix entries with \frac would look too small.

ϕ ·X n,n−1

\[ \mathbf{A} = \begin{pmatrix}

\dfrac{\varphi \cdot X_{n, 1}} {\varphi_{1} \times

\varepsilon_{1}} & (x + \varepsilon_{2})^{2} & \cdots & (x + \varepsilon_{n - 1})^{n - 1} & (x + \varepsilon_{n})^{n}\\[10pt]

\dfrac{\varphi \cdot X_{n, 1}} {\varphi_{2} \times \varepsilon_{1}} & \dfrac{\varphi \cdot X_{n, 2}} {\varphi_{2} \times \varepsilon_{2}} & \cdots & (x + \varepsilon_{n - 1})^{n - 1} & (x + \varepsilon_{n})^{n}\\

\hdotsfor{5}\\ \dfrac{\varphi \cdot X_{n, 1}} {\varphi_{n} \times

\varepsilon_{1}} & \dfrac{\varphi \cdot X_{n, 2}} {\varphi_{n} \times \varepsilon_{2}} & \cdots & \dfrac{\varphi \cdot X_{n, n - 1}} {\varphi_{n} \times \varepsilon_{n - 1}} & \dfrac{\varphi\cdot X_{n, n}}

{\varphi_{n} \times \varepsilon_{n}} \end{pmatrix} + \mathbf{I}_{n} \]

Recall the discussion of \dots vs. \cdots and \ldots in Section 7.4.3. In this for- mula, we have to use \cdots. Matrices are discussed in detail in Section 9.7.1.

Note the use of the command \\[10pt]. If you use \\ instead, the first and second lines of the matrix are set too close.

I show you in Section 15.1.2 how to rewrite the formula to make it shorter and more readable.

CHAPTER

More math

In the previous chapter, we discuss the basic building blocks of a formula and how to put them together to form more complex formulas. This chapter starts out by going one step lower, to the characters that make up a formula. We discuss math symbols and math alphabets.

L A TEX was designed for typesetting math, so it is not surprising that it contains a very large number of math symbols. Section 8.1 classifies and describes them. Sec- tion 8.2 discusses how to build new symbols from existing ones. Math alphabets and symbols are discussed in Section 8.3. Horizontal spacing commands in math are de- scribed in Section 8.4.

L A TEX provides a variety of ways to number and tag equations. These techniques are described in Section 8.5. We conclude the chapter with two minor topics: general- ized fractions (Section 8.6.1) and boxed formulas (Section 8.6.2).