Comment equation operators

Describe an operator

$$f(x)\stackrel{\text{def}}{=}{x}^2+c$$

\[
f(x) 
\overset{\text{def}}{=} x^2 + c
\overset{\text{nutze \eqref{eq:innerfunction}}}{=} f(a_0 x^4)
\]
\[
g(x) = a_0 x^4 
\]

Reference a secondary calculation

$$\begin{array}{c}f(g(x))\stackrel{\text{nutze (1)}}{=}f(a_0{x}^4)\\ g(x)=a_0{x}^4\end{array}\text{\hspace{1em}(1)}$$

f(g(x)) \overset{\text{nutze \eqref{eq:inner}}}{=} f(a_0 x^4)
g(x) = a_0 x^4 \vphantom{eq:inner}\tag{*} 

f(g(x)) \overset{\text{nutze \eqref{eq:inner}}}{=} f(a_0 x^4)

g(x) = a_0 x^4 \label{eq:innerfunction}\label{eq:inner}\tag{\theequation}\refstepcounter{equation}

Comment stretchable arrows

$$\begin{array}{rl}\text{MoveEqLeft}0,99<1-\mathbb{P}(A{)}^k=1-p^k\stackrel{+p^k-0,99}{\Rightarrow }{p}^k<0,01\stackrel{{\log }_p()}{\Rightarrow }k>{\log }_p(0,01)& \\ & \stackrel{p=0,11}{\Rightarrow }k>{\log }_{0,11}(0,01)\approx 2,086\stackrel{k\in {\mathbb{N}}^{+}}{\Rightarrow }k\ge 3\end{array}$$

Don’t comment after line

• works on a blackboard, looks bad on paper

$$\begin{array}{rlrl}0,99& <1-\mathbb{P}(A{)}^k& & \\ 0,99& <1-p^k& & |+p^k-0,99\\ p^k& <0,01& & |{\log }_p()\\ k& >{\log }_p(0,01)& & |p=0,11\\ k& >{\log }_{0,11}(0,01)\approx 2,086& & |k\in {\mathbb{N}}^{+}\\ \Rightarrow k& \ge 3& & \end{array}$$

Use cases

$$\begin{array}{rl}\text{MoveEqLeft}f(x)\stackrel{\text{def}}{=}{x}^2+c\stackrel{\text{(1)}}{=}h(x)+c& \\ & \text{overunderset}\text{symmetrisch}\text{auf [0,1]}={\int }_{\infty }^{-\infty }\{\begin{array}{ll}{e}^{x-z}& \text{fu¨r }0\le x\le 1\wedge z\ge x\\ 0& \text{sonst}\end{array}dx\\ & ={\int }_0^z{e}^{x-z}dx\ll t\end{array}$$

Don’t write long overbraces

$$f(x)\stackrel{\text{def}}{=}{x}^2+c\stackrel{\text{(1)}}{=}h(x)+c\text{overunderset}\text{symmetrisch}\text{auf [0,1]}=\underset{\text{fu¨r x<0}}{\underbrace{{\int }_{-\infty }^{0}0dx}}+\stackrel{\text{fu¨r 0≤x≤1 und z≥x}}{\overbrace{{\int }_0^z{e}^{x-z}dx}}+\underset{\text{fu¨r 0≤x≤1 und z<x}}{\underbrace{{\int }_z^{1}0dx}}+\stackrel{\text{x>1}}{\overbrace{{\int }_1^{\infty }0dx}}$$

Comment matrix columns

\documentclass{article}
\usepackage{amsmath,amssymb,mathtools,aligned-overset,array}
\begin{document}
\def\rb#1{\rotatebox{90}{$\xleftarrow{#1}$}}
\begin{tabular}{c}
$\begin{matrix}
\rb{text1}&\rb{text1}&\rb{text1}&\rb{text1}\\
\end{matrix}$\\
$\begin{bmatrix}
X_x & Y_x & Z_x & T_x \\
X_y & Y_y & Z_y & T_y \\
X_z & Y_z & Z_z & T_z \\
0 & 0 & 0 & 1
\end{bmatrix}$
\end{tabular}
\end{document}

• https://mirror.physik.tu-berlin.de/pub/CTAN/obsolete/info/math/voss/mathmode/Mathmode.pdf#page=110

Overlapping braces

\documentclass{article}
\usepackage{amsmath,amssymb,mathtools,aligned-overset,array,xcolor}
\begin{document}
\begin{align}\label{eq:pqFormel}
y &= 2x^2 -3x +5\nonumber\\
& \hphantom{= \ 2\left(x^2-\frac{3}{2}\,x\right. }%
\textcolor{blue}{%
\overbrace{\hphantom{+\left(\frac{3}{4}\right)^2- %
\left(\frac{3}{4}\right)^2}}^{=0}}\nonumber\\[-11pt]
&= 2\left(\textcolor{red}{%
\underbrace{%
x^2-\frac{3}{2}\,x + \left(\frac{3}{4}\right)^2}%
}%
\underbrace{%
- \left(\frac{3}{4}\right)^2 + \frac{5}{2}}%
\right)\\
&= 2\left(\qquad\textcolor{red}{\left(x-\frac{3}{4}\right)^2}
\qquad + \ \frac{31}{16}\qquad\right)\nonumber\\
y\textcolor{blue}{-\frac{31}{8}}
&= 2\left(x\textcolor{cyan}{-\frac{3}{4}}\right)^2\nonumber
\end{align}
\end{document}

• https://mirror.physik.tu-berlin.de/pub/CTAN/obsolete/info/math/voss/mathmode/Mathmode.pdf#page=117

Vertical and horizontal aligned braces

\documentclass{article}
\pagestyle{empty}
\usepackage{amsmath,amssymb,amsfonts,mathtools,aligned-overset,array,xcolor}
\begin{document}
\def\num#1{\hphantom{#1}}
\def\vsp{\vphantom{\rangle_1}}
\begin{equation*}
\frac{300}{5069}%
\underbrace{\longmapsto\vphantom{\frac{1}{1}}}_{%
\mathclap{\substack{%
\Delta a=271\num9\vsp \\[2pt]
\Delta b=4579\vsp\\[2pt]
\text{$1$ iteration}%
}}} \frac{29}{490}%
\underbrace{\longmapsto \frac{19}{321}\longmapsto}_{%
\mathclap{\substack{%
\Delta a=10\num{9}=\langle271\rangle_{29}\num{20}\\[2pt]
\Delta b=169=\langle4579\rangle_{490}\\[2pt]
\text{$2$ iterations}
}}} \frac{9}{152}
\underbrace{\longmapsto \frac{8}{135}\longmapsto\dots\longmapsto}_{%
\substack{%
\Delta a=1\num{7}=\langle10\rangle_{9}\num{119}\\[2pt]
\Delta b=17=\langle169\rangle_{152}\\[2pt]
\text{$8$ iterations}
}} \frac{1}{16}
\underbrace{\longmapsto\dots\longmapsto\vphantom{\frac{8}{135}}}_{%
\substack{%
\Delta a=0=\langle1\rangle_{1}\num{76} \\[2pt]
\Delta b=1=\langle17\rangle_{16} \\[2pt]
\text{$8$ iterations}
}} \frac{1}{1}
\end{equation*}
\end{document}

• https://mirror.physik.tu-berlin.de/pub/CTAN/obsolete/info/math/voss/mathmode/Mathmode.pdf#page=118

More

$$\begin{array}{c}\text{Annahmen:}\\ (A1):n\equiv 3(\mathrm{mod}19)\\ (A2):n\equiv 1(\mathrm{mod}20)\\ (A3):n\equiv 2(\mathrm{mod}21)\\ (A4):n\equiv 0(\mathrm{mod}23)\\ (A5):n\equiv 1(\mathrm{mod}22)\end{array}\begin{array}{c}\text{Zu Zeigen:}\\ (Z1.1):n\equiv 3(\mathrm{mod}19)\\ (Z2.1):n\equiv 1(\mathrm{mod}20)\\ (Z3.1):n\equiv 2(\mathrm{mod}21)\\ (Z4.1):n\equiv 0(\mathrm{mod}23)\\ (Z5.1):n\equiv 1(\mathrm{mod}11)\end{array}$$ $$\begin{array}{rl}\text{MoveEqLeft}L((a+b{)}^{\ast })\stackrel{\text{FS 1.2.8 *}}{=}L(a+b{)}^{\ast }\stackrel{\text{FS 1.2.8 +}}{=}(L(a)\cup L(b){)}^{\ast }\stackrel{\text{FS 1.2.8 }a,b\in \Sigma }{=}(\{a\}\cup \{b\}{)}^{\ast }\stackrel{\text{Def. }\cup }{=}\{a,b{\}}^{\ast }& \\ & \stackrel{\text{Def. }\in }{=}\{w\in \{a,b{\}}^{\ast }\left\}\stackrel{\text{Def. }{|\cdot |}_{\cdot }}{=}\right\{w\in \{a,b,c{\}}^{\ast }\mid {\left|w\right|}_c=0\}\stackrel{\text{Def. }\Sigma }{=}\{w\in {\Sigma }^{\ast }\mid {\left|w\right|}_c=0\}\end{array}$$ $$X=\sum _{1\le i\le j\le n}{X}_{ij}X=\sum _{1\le i\le j\le n}{X}_{ij}$$