$\huge\rm\LaTeX$
Trigonometric functions:
\begin{align*}
\cos^2 x +\sin^ 2 x=1
\end{align*}


Radicals:
\[
\sqrt{\matrix{a & b\cr c & d}}
\sqrt{
\rlap{\smash{\rm Hi!}}
\phantom{\matrix{a & b\cr c & d}}
}
\]


Arrows:
\begin{align*}
A\iff B\\
\implies\\
\impliedby\\
\iff\\
\mapsto\\
\to\\
\gets\\
\rightarrow\\
\leftarrow\\
\Rightarrow\\
\Leftarrow\\
\hookrightarrow\\
\hookleftarrow
\end{align*}


Providing reasons for each line in align:
\begin{align*}
f(ab)&=(ab)^2 && (\text{by definition of $f$})\\
&=(ab)(ab)\\
&=a^2 b^2 && (\text{since $G$ is abelian})\\
&=f(a)f(b) && (\text{by definition of $f$}).
\end{align*}
\begin{gather}
a = a \tag{$*$}\\
\text{if } a=b \text{ then } b=a \tag{$\dagger$}\\
\text{if } a=b \text{ and } b=c \text{ then } a=c \tag{3.1}
\end{gather}


Cases:
\begin{align*}
\det(A)&=1+(-1)^{n+1} \\
&= \begin{cases}
2 & \text{ if } n \text{ is odd}\\
0 & \text{ if } n \text{ is even}.
\end{cases}
\end{align*}
\[
|x| =
\begin{cases}
x & \text{ if } x\ge 0 \\
-x & \text{ if } x \lt 0
\end{cases}
\]


System of equations:
\begin{align*}
\left\{
\begin{array}{c}
a_1x+b_1y+c_1z=d_1 \\
a_2x+b_2y+c_2z=d_2 \\
a_3x+b_3y+c_3z=d_3
\end{array}
\right.
\end{align*}


Brackets:
\[
\left\langle
\matrix{a & b\cr c & d}
\right\rangle
\]


Braces:
\[
\left\lbrace
\matrix{a & b\cr c & d}
\right\rbrace
\]


Matrix:
\[
A = \pmatrix{
a_{11} & a_{12} & \ldots & a_{1n} \cr
a_{21} & a_{22} & \ldots & a_{2n} \cr
\vdots & \vdots & \ddots & \vdots \cr
a_{m1} & a_{m2} & \ldots & a_{mn} \cr
}
\]


Augmented matrix:
\begin{align*}
\left[\begin{array}{rrr|rrr}
1 & 0 & 0 & 1 &1 & 1 \\
0 & 1 & 0 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 \\
\end{array} \right]
\end{align*}


Block matrix:
\begin{align*}
M=
\left[\begin{array}{c|c}
A & B\\
\hline
C & D
\end{array}
\right]
\end{align*}


Matrix with fractions:
\begin{align*}
A=\begin{bmatrix}
\frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\
\frac{2}{3} &\frac{-1}{3} &\frac{-1}{3} \\
\frac{1}{3} & \frac{1}{3} & \frac{-2}{3}
\end{bmatrix}
\end{align*}


Elementary row operations (Gauss-Jordan elimination):
\begin{align*}
\left[\begin{array}{rrrr|r}
1 & 1 & 1 & 1 &1 \\
0 & 1 & 2 & 3 & 5 \\
0 & -2 & 0 & -2 & 2 \\
0 & 1 & -2 & 3 & 1 \\
\end{array}\right]
\xrightarrow{\substack{R_1-R_2 \\ R_3-R_2\\R_4-R_2}}
\left[\begin{array}{rrrr|r}
1 & 0& -1 & -2 &-4 \\
0 & 1 & 2 & 3 & 5 \\
0 & 0 & 4 & 4 & 12 \\
0 & 0 & -4 & 0 & -4 \\
\end{array}\right]
\xrightarrow[\frac{-1}{4}R_4]{\frac{1}{4}R_3}
\left[\begin{array}{rrrr|r}
1 & 0& -1 & -2 &-4 \\
0 & 1 & 2 & 3 & 5 \\
0 & 0 & 1 & 1 & 3 \\
0 & 0 & 1 & 0& 1 \\
\end{array}\right]
\end{align*}


Use array for tabular:
\begin{array}{ |c|c|c| }
\hline
a & a^2 \pmod{5} & 2a^2 \pmod{5} \\
\hline
0 & 0 & 0 \\
1& 1 & 2 \\
2& 4 & 3 \\
3 & 4 & 3\\
4 & 1 & 2\\
\hline
\end{array}


Sums:
$$\sum_{k=1}^n a_k$$
\[
\sum_{
\substack{
1\lt i\lt 3 \\
1\le j\lt 5
}}
a_{ij}
\]


Sub-arrays:
\[
\prod_{\begin{subarray}{rl}
i\lt 5\quad & j\gt 1 \\
k\ge2,\,k\ne 5 \quad & \ell\le 5,\,\ell\ne 2
\end{subarray}}
x_{ijk\ell}
\]


Integration:
$$\int\limits_a^b f(x)\,dx$$
\begin{align*}
\int_{a}^{b} \! f(x)\,\mathrm{d}x
\end{align*}
\[
\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}dx
\]


Second derivative:
\begin{align*}
f^{\prime\prime}
\end{align*}


The length (magnitude) of vectors:
\begin{align*}
\|\mathbf{v}\| \text{ or } \left\|\frac{a}{b}\right \|
\end{align*}


Explanations under equations:
\begin{align*}
n=\underbrace{1+1+\cdots+1}_{\text{$n$ times}}
\end{align*}


Explanations over equations:
\[
\overbrace{x+\cdots+x}^{n\text{ times}}
\]
\[
\overparen a \quad
\overparen ab \quad
\overparen{ab} \quad
\overparen{abc} \quad
\overparen{abcdef} \quad
\overparen{\underparen{abcd}}
\]


Inline:
$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$


Displayed:
$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$
\[
(x+1)^2 % original expression
= (x+1)(x+1) % definition of exponent
= x^2 + 2x + 1 % FOIL, combine like terms
\\
\{x\,|\,x\in\Bbb Z\}
\]


Parentheses:
$$\Biggl(\biggl(\Bigl(\bigl((x)\bigr)\Bigr)\biggr)\Biggr)$$


Fractions:
$$\frac{100}{20} = 5$$
$${a+1\over b+1}$$
$$\displaystyle\frac ab + {\textstyle \frac cd + \frac ef} + \frac gh$$
$$\left\{\frac ab,c\right\}$$
$$|\frac ab|$$
$$\left(\frac{a+b}{\dfrac{c}{d}}\right)$$
$$\left(\vcenter{\frac{a+b}{\dfrac{c}{d}}}\right)$$
\[
\displaystyle
\frac{2x+3y-\phantom{5}z}
{\phantom{2}x+\phantom{3}y+5z}
\]
$${a+1 \overwithdelims \{ \} b+2}+c$$
$$\left(\frac ab,c\right)$$
$${a+1 \above 1.5pt b+2}+c$$
$$\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}$$
\[
\def\specialFrac#1#2{\frac{x + #1}{y + #2}}
\specialFrac{7}{z+3}
\]
$$\sqrt[3]{\frac xy}$$


Partial differentials:
\[
i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>
\]


Graphics:

cayley-graph-blurred.png
Figure1. Cayley Graph

From the Figure 1 we can see…


Definitions, Remarks, lemmas, etc.:

Definition 1. Suppose that $(X,\mathcal M)$ and $(Y,\mathcal N)$ are measurable spaces, and $f:X\to Y$ is a map. We call $f$ is measurable if for every $B\in\mathcal N$ the set $f^{-1}(B)$ is in $\mathcal M$.

Remark 1. If $Y$ is a topological space, and $\mathcal N$ is the $\sigma$-algebra of Borel sets, then $f$ is measurable if and only if the following condition satisfied:
  • For every open set $V$ in $Y$, the inverse image $f^{-1}(V)$ is measurable.
Lemma 2 (fundamental lemma of integration). Let $\set{f_n}$ be a Cauchy sequence of step mappings. Then there exists a subsequence which converges pointwise almost everywhere, and satisfies the additional property: given $\eps$ there exists a set $Z$ of measure $< \eps$ such that this subsequence converges absolutely and uniformly outside $Z$.
Lemma 2 (fundamental lemma of integration). Let $\set{f_n}$ be a Cauchy sequence of step mappings. Then there exists a subsequence which converges pointwise almost everywhere, and satisfies the additional property: given $\eps$ there exists a set $Z$ of measure $< \eps$ such that this subsequence converges absolutely and uniformly outside $Z$.

Referencing equations:
\[
\]
At first, we sample $f(x)$ in the $N$ ($N$ is odd) equidistant points around $x^*$:
\[
f_k = f(x_k),\: x_k = x^*+kh,\: k=-\frac{N-1}{2},\dots,\frac{N-1}{2}
\]
where $h$ is some step.
Then we interpolate points $\{(x_k,f_k)\}$ by polynomial
\begin{equation} \label{eq:poly}
P_{N-1}(x)=\sum_{j=0}^{N-1}{a_jx^j}
\end{equation}
Its coefficients $\{a_j\}$ are found as a solution of system of linear equations:
\begin{equation} \label{eq:sys}
\left\{ P_{N-1}(x_k) = f_k\right\},\quad k=-\frac{N-1}{2},\dots,\frac{N-1}{2}
\end{equation}
Here are references to existing equations: (eq:poly), (eq:sys).
Here is reference to non-existing equation (eq:unknown).
\[
\]


Math scripting:
$$\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\Bbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\cal ABCDEFGHIJKLMNOPQRSTUVWXYZ$$
$$\frak ABCDEFGHIJKLMNOPQRSTUVWXYZ$$
$$\huge AaBb\alpha\beta123\frac ab\sqrt x$$