Math

The Not So Short Introduction to LATEX2Eより

Add $a$ squared and $b$ squared to get $c$ squared. or, using a mathematical approach: $a^2 + b^2 + c^2$

$\TeX{}$ is pronounced as $\tau\epsilon\chi$
$\times$ $100~$ m $^3$ of water
This comes from my $\heartsuit$

$a^2 + b^2 = c^2$

$\lim_{n \to \infty}\sum_{k=1}^n \frac{1}{k^2}= \frac{\pi^2}{6}$

A $d_{e_{e_p}}$ mathematical expression followed by a $h^{i^{g^h}}$ expression. As opposed to a smashed $\smash{d_{e_{e_p}}}$ expression followed by a $\smash{h^{i^{g^h}}}$ expression.

$\forall x \in \mathbf{R}:\qquad x^{2} \geq 0$
$x^{2} \geq 0\qquad\text{for all }x\in\mathbf{R}$
$x^{2} \geq 0\qquad \text{for all } x \in \mathbb{R}$

$\lambda,\xi,\pi,\theta,\mu,\Phi,\Omega,\Delta$

$p^3_{ij} \qquad m_\text{Knuth}$
$a^x+y \neq a^{x+y}\qquad e^{x^2} \neq {e^x}^2$
$\sqrt{x} \Leftrightarrow x^{1/2} \quad \sqrt[3]{2} \quad \sqrt{x^{2} + \sqrt{y}} \quad \surd[x^2 + y^2]$
$\Psi = v_1 \cdot v_2 \cdot \ldots \qquad n! = 1 \cdot 2\cdots (n-1) \cdot n$
$0.\overline{3} = \underline{\underline{1/3}}$

$\underbrace{\overbrace{a+b+c}^6 \cdot \overbrace{d+e+f}^9} _\text{meaning of life} = 42$

$f(x) = x^2 \qquad f'(x) = 2x \qquad f''(x) = 2$
$\hat{XY} \quad \widehat{XY} \quad \bar{x_0} \quad \bar{x}_0$
$\vec{a} \qquad \vec{AB} \qquad \overrightarrow{AB}$

$\lim_{x \rightarrow 0} \frac{\sin x}{x}=1$
×　 $%\DeclareMathOperator{\argh}{argh} %\DeclareMathOperator*{\nut}{Nut} \[3\argh = 2\nut_{x=1}\]$
$a\bmod b$
$x\equiv a \pmod{b}$
$3/8 \qquad \frac{3}{8} \qquad \tfrac{3}{8}$
In text style:
×　 $1\frac{1}{2}$ ~hours $\qquad 1\dfrac{1}{2}$ ~hours

$\sqrt{\frac{x^2}{k+1}}\qquad x^\frac{2}{k+1}\qquad \frac{\partial^2f} {\partial x^2}$

Pascal’s rule is
$\binom{n}{k} =\binom{n-1}{k} + \binom{n-1}{k-1}$
$f_n(x) \stackrel{*}{\approx} 1$

数式の中央寄せはだめみたい

$\sum_{i=1}^n \qquad \int_0^{\frac{\pi}{2}} \qquad \prod_\epsilon$
×　 $\sum^n_{\substack{0<i<n j\subseteq i}} P(i,j) = Q(i,j)$
$\sum^n_{\substack{0 \lt; i\text{<}n j\subseteq i}} P(i,j) = Q(i,j)$
${a,b,c} \neq \{a,b,c\}$
$1 + \left(\frac{1}{1-x^{2}} \right)^3 \qquad \left. \ddagger \frac{~}{~}\right)$
$\Big((x+1)(x-1)\Big)^{2}$
$\big( \Big( \bigg( \Bigg( \quad \big\} \Big\} \bigg\} \Bigg\} \quad \big\| \Big\| \bigg\| \Bigg\| \quad \big\Downarrow \Big\Downarrow \bigg\Downarrow \Bigg\Downarrow$

align と　label はだめみたい。
$$\begin{align} f(x) &= (a+b)(a-b) \label{1} $= a^2-ab+ba-b^2 \tag{3.4} $= a^2+b^2 \tag{wrong} \end{align}$$

This is a reference to $\eqref{1}$ .

$\mathbf{X} = \left( \begin{array}{ccc} x_1 & x_2 & \ldots \\ x_3 & x_4 & \ldots \\ \vdots & \vdots & \ddots \end{array} \right)$

$$\|x| = \left\{ \begin{array}{rl} -x & \text{if } x < 0\\ 0 & \text{if } x = 0\\ x & \text{if } x > 0
\end{array} \right. $$

$\begin{equation*} |x| = \left\{ \begin{array}{rl} -x & \text{if } x < 0\\ 0 & \text{if } x = 0\\ x & \text{if } x > 0 \end{array} \right. \end{equation*}$