\documentclass{article}
\usepackage{amssymb}
\usepackage{mathtools}
\mathtoolsset{
mathic % or mathic = true
}
\begin{document}
\section{Sums and Limits}
\paragraph{mathclap \& friends}
\[
X = \sum_{1\le i\le j\le n} X_{ij}
\]
\[
X = \sum_{\mathclap{1\le i\le j\le n}} X_{ij}
\]
\[
X = \sum_{\mathrlap{1\le i\le j\le n}} X_{ij}
\]
\[
X = \sum_{\mathllap{1\le i\le j\le n}} X_{ij}
\]
\paragraph{Cramped}
\[
\cramped{x^2} \leftrightarrow x^2 \quad
\cramped[\scriptstyle]{x^2} \leftrightarrow {\scriptstyle x^2}
\]
\paragraph{Smashoperator}
\[
V = \sum_{1\le i\le j\le n}^{\infty} V_{ij} \quad
X = \sum_{1\le i\le j\le n}^{3456} X_{ij} \quad
Y = \sum\limits_{1\le i\le j\le n} Y_{ij} \quad
Z = \mathop{T}_{1\le i\le j\le n} Z_{ij}
\]
\[
V = \sum_{1\le i\le j\le n}^{\infty} V_{ij} \quad
X = \smashoperator{\sum_{1\le i\le j\le n}^{3456}} X_{ij} \quad
Y = \smashoperator[r]{\sum\limits_{1\le i\le j\le n}} Y_{ij} \quad
Z = \smashoperator[l]{\mathop{T}_{1\le i\le j\le n}} Z_{ij}
\]
\paragraph{Adjustlimits}
\[
\text{a)} \lim_{n\to\infty} \max_{p\ge n} \quad
\text{b)} \lim_{n\to\infty} \max_{p^2\ge n} \quad
\text{c)} \lim_{n\to\infty} \sup_{p^2\ge nK} \quad
\text{d)} \limsup_{n\to\infty} \max_{p\ge n}
\]
\[
\text{a)} \adjustlimits\lim_{n\to\infty} \max_{p\ge n} \quad
\text{b)} \adjustlimits\lim_{n\to\infty} \max_{p^2\ge n} \quad
\text{c)} \adjustlimits\lim_{n\to\infty} \sup_{p^2\ge nK} \quad
\text{d)} \adjustlimits\limsup_{n\to\infty} \max_{p\ge n}
\]
\section{Tags}
\newtagform{bad}{((}{))}
\newtagform{good}[\textbf]{[}{]}
\begin{quote}\renewcommand*\rmdefault{ppl}\normalfont\itshape
\begin{equation*}
a=b \label{eq:example}\tag*{Q\&A}
\end{equation*}
See \ref{eq:example} or is it better with \refeq{eq:example}?
In the star form \verb|\ref*| becomes \ref*{eq:example} (\verb|\refeq*| is not defined).
\end{quote}
\usetagform{good}
\begin{equation*}
a=b \label{eq:example2}\tag*{Q\&A}
\end{equation*}
\begin{equation*}
a=b \label{eq:example3}\tag{Q\&A}
\end{equation*}
\usetagform{default}
% showonlyrefs not supported.
%\mathtoolsset{showonlyrefs,showmanualtags}
Normal tags.
\begin{equation}
a=a\label{eq:a0}
\end{equation}
That was equation \eqref{eq:a0}.
\usetagform{good}
OK tags.
\begin{equation}
a=a\label{eq:a1}
\end{equation}
That was equation \eqref{eq:a1}, but recall \eqref{eq:a0}
\renewtagform{good}{\{}{\}}
odd tag.
\begin{equation}
a=a\label{eq:a2}
\end{equation}
That was equation \eqref{eq:a2}, but recall \eqref{eq:a0} and \eqref{eq:a1}.
\usetagform{bad}
weird tag.
\begin{equation}
b=b\label{eq:b1}
\end{equation}
That was equation \eqref{eq:b1}, but recall \eqref{eq:a0}, \eqref{eq:a1} and \eqref{eq:a2}.
\usetagform{default}
Normal tags again.
\begin{equation}
c=c\label{eq:c0}
\end{equation}
Non-textual
\begin{equation}
d=d \label{eq:nt1}\tag{n$^{\mathit{th}}$}
\end{equation}
That was equation \eqref{eq:c0}, but recall \eqref{eq:a0}, \eqref{eq:a1}, \eqref{eq:a2},
\eqref{eq:b1} and \eqref{eq:nt1}.
\begin{gather}
a=a \label{eq:a} \\
b=b \label{eq:b} \tag{**}
\end{gather}
This should refer to the equation containing $a=a$: \eqref{eq:a}.
Then a switch of tag forms.
\begin{align}
c&=c \label{eq:c} \\
d&=d \label{eq:d}
\end{align}
This should refer to the equation containing $d=d$: \eqref{eq:d}
(but recall \eqref{eq:a}).
\begin{equation}
e=e
\end{equation}
%Back to normal.\mathtoolsset{showonlyrefs=false}
\begin{equation}
f=f
\end{equation}
%\mathtoolsset{showonlyrefs=true}
\begin{equation*}
1+1=2 \label{eq:much}
\end{equation*}
\begin{equation*}
2+2=4 \label{eq:wow}
\end{equation*}
\noeqref{eq:much}
Blabla \eqref{eq:wow}.
%\mathtoolsset{showonlyrefs=false}
\section{Arrows}
\[
A \xLeftarrow[under]{over} B \xRightarrow[under]{over} C
\]
\[
x \xleftrightarrow[under]{overlooooooong} y \xLeftrightarrow[underloooooooong]{over} z
\]
\[
x \xhookleftarrow[bar]{foo} y \xhookrightarrow[bluuuuuuuuub]{baz} t \xmapsto{heeereee} k
\]
\[
k \xleftharpoonup[.]{} l \xleftharpoondown{..} m \xrightharpoondown[...]{} n \xrightharpoonup{....} o
\]
\[
x \xrightleftharpoons{bluuuuub} y \xleftrightharpoons[blaaaaaab]{} z
\]
\[
z = \LaTeXoverbrace{
   \underbracket{x}_\text{real} + i
      \underbracket[1pt][7pt]{y}_\text{imaginary}
        }^\text{complex number}
\quad
\underbrace{1+1}_{=2}
\]
\section{Matrices}
\[
\begin{matrix*}
c & cocococococo \\
c & c
\end{matrix*}
\]
\[
\begin{matrix*}[l]
lalalalalala & l \\
l & l
\end{matrix*}
\]
\[
\begin{matrix*}[r]
rererererere & r \\
r & r
\end{matrix*}
\]
\[
\begin{pmatrix*}[l]
ppppppp & foo \\
l & ppppppppppppppp
\end{pmatrix*}
\]
\[
\begin{bmatrix*}
b & b \\
b & b
\end{bmatrix*}
\]
\[
\begin{Bmatrix*}[r]
B & B \\
B & BBBBBBBrBBBBBB
\end{Bmatrix*}
\]
\[
\begin{vmatrix*}
v & v \\
v & v
\end{vmatrix*}
\]
\[
\begin{Vmatrix*}[c]
V & V \\
VVVVVVcVVVVVV & bar
\end{Vmatrix*}
\]
\[
\begin{vsmallmatrix*}[l]
a & blblblbllbblblblblblblbl \\
c & d
\end{vsmallmatrix*}
\]
\[
\begin{bsmallmatrix} a & -b \\ -c & d \end{bsmallmatrix}
\begin{bsmallmatrix*}[r] a & -b \\ -c & d \end{bsmallmatrix*}
\]
\[
\begin{Vsmallmatrix} e & -f \\ -g & h \end{Vsmallmatrix}
\begin{Vsmallmatrix*}[r] e & -f \\ -g & h \end{Vsmallmatrix*}
\]
\[
\begin{bsmallmatrix*} a & -bbbbb \\ -c & d \end{bsmallmatrix*}
\begin{bsmallmatrix*}[r] a & -bbbbb \\ -c & d \end{bsmallmatrix*}
\]
\[
\begin{Vsmallmatrix*} e & -fffff \\ -g & h \end{Vsmallmatrix*}
\begin{Vsmallmatrix*}[r] e & -fffff \\ -g & h \end{Vsmallmatrix*}
\]
\mathtoolsset{smallmatrix-align=l}
\[
\begin{bsmallmatrix*} a & -bbbbb \\ -c & d \end{bsmallmatrix*}
\begin{bsmallmatrix*}[r] a & -bbbbb \\ -c & d \end{bsmallmatrix*}
\]
\[
\begin{Vsmallmatrix*} e & -fffff \\ -g & h \end{Vsmallmatrix*}
\begin{Vsmallmatrix*}[r] e & -fffff \\ -g & h \end{Vsmallmatrix*}
\]
\section{Cases}
\[
\begin{cases}
E = m c^2 & \text{Nothing to see here} \\
\int x-3\, dx & \text{Integral is text style}
\end{cases}
\]
\[
\begin{dcases}
E = m c^2 & c \approx 3.00\times 10^{8}\,\mathrm{m}/\mathrm{s} \\
\int x-3\, dx & \text{Integral is display style}
\end{dcases}
\]
\[
a= \begin{dcases*}
E = m c^2 & Nothing to see here (text in math) \\
\int x-3\, dx & Integral is display style (text in math)
\end{dcases*}
\]
\[
\begin{rcases}
E = m c^2 & 5^6 \quad \text{and so on} \\
\int x-3\, dx & \int x \, dx
\end{rcases} = b
\]
\[
\begin{rcases*}
x^2 & for $\int x\,dx>0$\\
x^3 & else
\end{rcases*} \Rightarrow \cdots
\]
\[
\begin{drcases}
E = m c^2 & 5^6 \quad \text{and so on} \\
\int x-3\, dx & \int x \, dx
\end{drcases} = b
\]
\[
\begin{drcases*}
x^2 & for $\int x\,dx>0$\\
\int x^3\,x & else
\end{drcases*} \Rightarrow \cdots
\]
\[
\text{foo} = \begin{cases*}
\pi & if something\\
\int \Omega^\Xi\,\Omega & otherwise
\end{cases*}
\]
\section{Gathered}
\[
A = \begin{gathered}[t]
\framebox[4cm]{first} \\
\framebox[4cm]{last}
\end{gathered} B
\]
\[
\begin{gathered}
a=b+c\\
b=c+d\\
...
\end{gathered}
\]
\[
\boxed{hello}
\]
\begin{align*}
\Aboxed{ f(x) & = \int h(x)\, dx} \\
& = g(x)
\end{align*}
\begin{align}
a&=b \intertext{Some text}
c&=d \shortintertext{Some short text}
e&=f
\end{align}
\section{Delimiters}
\[
\lvert \frac{a}{c} \rvert \quad \left\lvert \frac{a}{c} \right\rvert \quad \Bigg\lvert \frac{a}{b} \Bigg\rvert
\]
\DeclarePairedDelimiter\abs{\lvert}{\rvert}
\[
\abs{\frac{a}{b}} \quad \abs*{\frac{a}{b}} \quad \abs[\big]{\frac{a}{b}} \quad \abs[\Bigg]{\frac{a}{b}}
\]
\reDeclarePairedDelimiterInnerWrapper\abs{star}{#1\_#2\_#3}
\reDeclarePairedDelimiterInnerWrapper\abs{nostar}{#1@#2@#3}
\[
\abs{\pi} \quad \abs*{\phi}
\]
\DeclarePairedDelimiterX\innerp[2]{\langle}{\rangle}{#1,#2}
\DeclarePairedDelimiterX\braket[3]{\langle}{\rangle}{#1\,\delimsize\vert\,#2\,\delimsize\vert\,#3}
\[
\innerp*{A}{ \frac{1}{2} } \quad
\braket[\Big]{B}{\sum_{k} f_k}{C}
\]
\providecommand\given{}
\newcommand\SetSymbol[1][]{#1\vert\allowbreak}
\DeclarePairedDelimiterX\Set[1]\{\}{\renewcommand\given{\SetSymbol[\delimsize]}#1}
\[ \Set*{ x \in X \given \frac {\sqrt{x}}{x^2+1} > 1 } \]
\DeclarePairedDelimiterX\braketdelim[3]{\langle}{\rangle}%
{#1\,\delimsize\vert\,#2\,\delimsize\vert\,#3}
$\braketdelim{1}{\frac{8}{\frac{4}{1}}}{3}$
$\braketdelim*{1}{\frac{8}{\frac{4}{1}}}{3}$
$\braketdelim[\big]{1}{\frac{8}{\frac{4}{1}}}{3}$
% % XPP not available in v1.13
% \DeclarePairedDelimiterXPP\lnorm[1]{}\lVert\rVert{_2}{#1}
% \DeclarePairedDelimiterXPP\Prop[1]{\mathbb{P}}(){}{\renewcommand\given{\delimsize\vert}#1}
% \[
% \lnorm{\frac{\pi}{\int x dx}} \quad \Prop{foo}
% \]
\[
\left( \frac{\pi}{\omega} \right) \cdot \left[ \int x dx \right] \ldots \lbrack \sqrt{\frac{\sin x}{\cos z}} \rbrack \cdots \lparen \frac{\frac{foo}{bar}}{\frac{baz}{qux}} \rparen
\]
\paragraph{Operators}
\[
a := b \quad
a \vcentcolon= b \quad
a \ordinarycolon= b
\]
\[
a \coloneqq b \quad c \Colonapprox d \quad e \dblcolon f
\]
\[
\bigtimes \times \nuparrow \ndownarrow \otimes \bigotimes
\]
\section{Prescripts}
\[
{}^{4}_{12}\mathbf{C}^{5+}_{2} \quad
\prescript{14}{2}{\mathbf{C}}^{5+}_{2} \quad
\prescript{4}{12}{\mathbf{C}}^{5+}_{2} \quad
\prescript{14}{}{\mathbf{C}}^{5+}_{2} \quad
\prescript{}{2}{\mathbf{C}}^{5+}_{2}
\]
\newcommand*\myisotope[3]{%
\begingroup % to keep changes local. We cannot use a brace group
% as it affects spacing!
\mathtoolsset{
prescript-sup-format=\mathit,
prescript-sub-format=\mathbf,
prescript-arg-format=\mathrm,
}%
\prescript{#1}{#2}{#3}%
\endgroup
}
\[
\myisotope{A}{Z}{X}\to \myisotope{A-4}{Z-2}{Y}+
\myisotope{4}{2}{\alpha}
\]
\[
    a=\frac{
        \splitfrac{xy + xy + \int xy\, \text{dx} + xy + xy}
                  {+ xy + xy + xy + xy}
      }
      {z}
    =\frac{
        \splitdfrac{xy + xy + \int xy\, \text{dx} + xy + xy}
                   {+ xy + xy + xy + xy}
      }
      {z}
\]
\section{Multlines}
\begin{multline*}
p(x) = 3x^6 + 14x^5y + 590x^4y^2 + 19x^3y^3\\
- 12x^2y^4 - 12xy^5 + 2y^6 - a^3b^3
\end{multline*}
\[
A = \begin{multlined}[t]
\boxed{first} \\
\boxed{last}
\end{multlined} B
\]
\[
A = \begin{multlined}
\boxed{first} \\
\boxed{last}
\end{multlined} B
\]
\[
A = \begin{multlined}[b]
\boxed{first} \\
\boxed{last}
\end{multlined} B
\]
\mathtoolsset{multlined-pos=t,multlined-width=12em}
\[
A = \begin{multlined}[c][1cm]
\boxed{first} \\
\boxed{last}
\end{multlined} B
\]
\[
A = \begin{multlined}[c][0.5\textwidth]
\boxed{first} \\
\boxed{last}
\end{multlined} B
\]
\[
A = \begin{multlined}[c][2\textwidth]
\boxed{first} \\
\boxed{last}
\end{multlined} B
\]
\[
A = \begin{multlined}
\boxed{first} \\
\boxed{last}
\end{multlined} B
\]
\[
A = \begin{multlined}[b]
\boxed{first} \\
\boxed{last}
\end{multlined} B
\]
\[
A = \begin{multlined}[c][1cm]
\boxed{first} \\
\boxed{last}
\end{multlined} B
\]
\begin{equation}
foo\Coloneqq\begin{lgathered}[t]
x=1,\quad x+1=2 \\
y=2
\end{lgathered}
\end{equation}
\begin{equation}
bar\Coloneqq\begin{rgathered}[b]
x=1,\quad x+1=2 \\
y=2
\end{rgathered}
\end{equation}
\section{Spread-lines}
Spread it
\begin{spreadlines}{30pt}
  \[
  \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i
  \end{matrix}
  \]
  \[
  \begin{pmatrix}
   a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\
   a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\
   \vdots  & \vdots  & \ddots & \vdots  \\
   a_{m,1} & a_{m,2} & \cdots & a_{m,n}
  \end{pmatrix}
  \]
  $\begin{smallmatrix} a&b \\ c&d \end{smallmatrix}$
  $\begin{cases} n/2 & \quad \text{if } n \text{ is even}\\ -(n+1)/2 & \quad \text{if } n \text{ is odd}\\ \end{cases}$
  \begin{equation}\label{xx}
  \begin{split}
  a& =b+c-d\\
  & \quad +e-f\\
  & =g+h\\
  & =i
  \end{split}
  \end{equation}
  \begin{multline}
  a+b+c+d+e+f\\
  +i+j+k+l+m+n
  \end{multline}
  \begin{gather}
  a=b\\
  c=d
  \end{gather}
  \begin{align}
  a_1& =b_1+c_1\\
  a_2& =b_2+c_2-d_2+e_2
  \end{align}
  \begin{flalign*}
  a_{11}& =b_{11}&
  a_{12}& =b_{12}\\
  a_{21}& =b_{21}&
  a_{22}& =b_{22}+c_{22}
  \end{flalign*}
  \begin{alignat}{2}
  x& = y_1-y_2+y_3-y_5+y_8-\dots
  &\quad& \text{by foo}\\
  & = y'\circ y^*  && \text{by baz}\\
  & = y(0) y'      && \text {by Axiom 1.}
  \end{alignat}
  \begin{equation*}
  \left.\begin{aligned}
  B'&=-\partial\times E,\\
  E'&=\partial\times B - 4\pi j,
  \end{aligned}
  \right\}
  \qquad \text{Maxwell's equations}
  \end{equation*}
  \[
  \bigl( \begin{smallmatrix}
  a&b\\ c&d
  \end{smallmatrix} \bigr)
  \]
  \[
  \bigl( \begin{smallmatrix*}
  a&b\\ c&d
  \end{smallmatrix*} \bigr)
  \]
  \[
  \sum_{\begin{subarray}{l}
  i\in\Lambda\\ 0<j<n
  \end{subarray}}
  P(i,j)
  \]
  \begin{eqnarray}
       y & = & ax^2 + bx + c \\
    f(x) & = & x^2 + 2xy + y^2
  \end{eqnarray}
\end{spreadlines}
\[
  \begin{multlined}[c][\textwidth]
    \boxed{First line}     \\
    \boxed{Second line}    \\
    \shoveleft{L+E+F+T}    \\
    \shoveright{R+I+G+H+T} \\
    \shoveleft{L+E+F+T}    \\
    \shoveright{R+I+G+H+T} \\
    \boxed{Wup Wup}    \\
    \boxed{Last line}
  \end{multlined}
\]
\section{Stepped lines}
\newcounter{steplinecnt}
\newcommand\stepline{\stepcounter{steplinecnt}\thesteplinecnt}
\newgathered{stargathered}
  {\llap{\stepline}$*$\quad\hfil}%
  {\quad\textbf{over}}%
  {\setcounter{steplinecnt}{42}}
\[
  \begin{stargathered}
    x=1,\quad x+1=2 \\
    y=2
  \end{stargathered}
\]
\thesteplinecnt
\renewgathered{stargathered}
  {See:}%
  {the end}%
  {\setcounter{steplinecnt}{1337}}
\[
  \begin{stargathered}
    s=2.8,\quad s+0.2=3 \\
    t=4.5
  \end{stargathered}
\]
\thesteplinecnt
\section{Shifting equations}
\begin{align*}
\MoveEqLeft[3] \framebox[10cm][c]{Part 1}\\
= {} & \framebox[8cm][c]{2nd line}\\
19
& + \framebox[4cm][c]{ last part}
\end{align*}
\begin{alignat}{2}
&& \framebox[1.5cm]{1} &= \framebox[3cm]{2}\\
\ArrowBetweenLines
&& \framebox[1.5cm]{3} &= \framebox[2cm]{4}
\end{alignat}
\begin{align*}
a &= b              \\
\MTFlushSpaceAbove
& \vdotswithin{=} \\
\MTFlushSpaceBelow
& = c             \\
\shortvdotswithin{=}
& = d
\end{align*}
\end{document}