todo_formulas.tex

\documentclass[a5paper,openany,9pt]{extbook}

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\usepackage{enumitem} 
\usepackage{booktabs} 
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\usepackage{amsmath,amsfonts,amssymb,amsthm}
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\begin{document}
\chapter*{Recommendations how to write formulas better}

This information is based on \href{http://www.ctex.org/documents/shredder/src/texbook.pdf}{\textit{Knuth's TeXbook}},  AMS documentation, \href{https://tex.stackexchange.com/}{\textit{tex.stackexchange.com}} answers,  \colorbox{prpl}{\textcolor{white}{AND MY OWN PREFERENCES}}. I want to say that here are the rules to use correct command to get right spacing in math formulas. The rules which \LaTeX\ does automaticly aren't discussed here.

Also look at the source code to understand those rules

\begin{itemize}
\item
Use punctuations symbols after \verb|$*formula*$| in inline math and use them inside noninline-environments.

$b=3$, где
\begin{flalign}
&b=3,&
\end{flalign}
где

\item 
Enumeration. In simple enumeration use ties with formula division in inline mode and space command \verb|\,| in noninline mode:

$i=1$,~2, \ldots,~$n$.$\Bigr|$ 
\begin{flalign}
&i=1,\,2,\,\ldots,\,n.\Bigr|&
\end{flalign}
\item 
You can also use space command \verb|\,| in inline mode, if you don't use formula separation at all. But try to avoid it as much as possible. For example, use \verb|\{$x$,\{$x$, $y$\}\}| rather than \verb|$\{x,\,\{x,\,y\}\}$|.

$\Bigr|\{x,\,\{x,\,y\}\}\Bigr|$

$\Bigr|$\{$x$, \{$x$, $y$\}\}$\Bigr|$

\item 
Remember that \verb|\ldots|,\verb|\cdots| produce very bad spacing in the following example \verb|$\ldots$,| 

The same pattern goes assets declared by enumeration. 

$X = \{x_1$,~$x_2$, \dots,~$x_n$, \dots $\},\Bigr|$
\begin{flalign}
&X = \{x_1,\,x_2,\,\dots,\,x_n,\,\dots \},\Bigr|&
\end{flalign}

\item
However, for vectors and particularry for ordered pairs we use opposite pattern: no spaces at all both inline and noninle mode:

$\vv X = (x_1, x_2, \dots, x_n,\dots),\Bigr|$
\begin{flalign}
&\vv X = (x_1,x_2,\dots,x_n,\dots),\Bigr|&
\end{flalign}

\item 
Also we dont use spaces when enumeration is a direct part of the formula:

$\forall a,b,c \in \bbR$

\item
As well, notice that numerical gaps (intervals, segments) should take no space between left and right elements both in inline and noninline mode \verb|(a;b)|,\verb|[a;b]|

$\Bigr|(a;b)\Bigr|$
\begin{flalign}
&\Bigr|(a;b)\Bigr|&
\end{flalign}
Reference: Inline: \href{http://www.ctex.org/documents/shredder/src/texbook.pdf}{\textit{Knuth's TeXbook}}, page 173. Noninline: It's my own preference.

\item 
Use \verb|\quad| between mathematics statements both in inline and noninline mode, however try to avoid these constructions in inline mode.

$\forall y\in Y\quad  y\in X\Bigr|$
\begin{flalign}
&\forall y\in Y\quad  y\in X\Bigr|&
\end{flalign}

However if there is any punctuational symbol, then use simple space with ties, space-command \verb|\| or (the most recommended) formula separation.
\item
Use ``\verb=\,\big|\,='' and in sets as relation symbol

$\{x \,\big|\ 1 < x < 2\}\Bigr|$
\begin{flalign}
&\{x \,\big|\ 1 < x < 2\}\Bigr|&
\end{flalign}
Notice that we have no additional spaces after the first bracket and before the final bracket.
\item
Use ``\verb=\setminus='' for set minus operation. Not backslashes or smth like this.

\item
Phrases like i.e., e.g. requires ties \~ or spacecommand \verb|\ | after it because otherwise the space will be slighty bigger than recommended.  

$\Bigr|$i.e.~author$\Bigr|$

$\Bigr|$i.e.\ author$\Bigr|$

$\Bigr|$i.e. author$\Bigr|$

Remember that т.е. doesnt need comma after it, because it is union.


\item
Parentheses. When you want to use it with the word ``respectively'' then use this word in the last bracket. Do it like in the following russian way:

Будем говорить, что на множестве $G$ определена операция сложения <<$+$>> (умножения <<$\cdot$>>), если любой упорядоченной паре элементов ($a$,~$b$) элементов $G$ поставлен в соответствие элемент $y = a + b$ (соотв., $y = a\cdot b$)

Try to avoid 3 or more parentheses brackets. Try to edit sentences or just make new one additional phrase.

\item 

Because of bug use this: 

\verb|\end{itemize}\vspace{-1.0\baselineskip}\end{defn}|

when itemize and defn come to the end of the page. !Actually I need to solve this question on stackexchange.

\item 

Use \verb|\dfrac{1}{a}| in inline maths, because i dont like standard latex text-style fractions. They are too small, aren't they?

$a / b =  \dfrac{a}{b} \triangleq a \cdot \dfrac{1}{b}$; 

$a / b =  \frac{a}{b} \triangleq a \cdot \frac{1}{b}$; 

\item 

Try to avoid \verb|\Longleftrightarrow|, \verb|\Rightarrow| in order to give more mathematical text rather than lecture's summary.

\item 
About text in noninline equations. In noninline equations use \verb|\,\text{где }| or \verb|\,\text{где }\ | or the simplest \verb|\text{, где }| like in examples below:

$\Bigl| \dfrac{(z - a)^n}{(\zeta - a)^{n + 1}} f(\zeta) \Bigr| \le \dfrac{M}{r_1} \cdot q^n$, где $M = \sup\limits_{\zeta \in \gamma_{r_1}} |f(\zeta)| < +\infty,\Bigr|$
\begin{flalign}
&\Bigl| \frac{(z - a)^n}{(\zeta - a)^{n + 1}} f(\zeta) \Bigr| \le \frac{M}{r_1} \cdot q^n, \,\text{где } M = \sup_{\zeta \in \gamma_{r_1}} |f(\zeta)| < +\infty,\Bigr|&
\end{flalign}
\begin{flalign}
&\Bigl| \frac{(z - a)^n}{(\zeta - a)^{n + 1}} f(\zeta) \Bigr| \le \frac{M}{r_1} \cdot q^n, \,\text{где}\ M = \sup_{\zeta \in \gamma_{r_1}} |f(\zeta)| < +\infty,\Bigr|&
\end{flalign}
\begin{flalign}
&\Bigl| \frac{(z - a)^n}{(\zeta - a)^{n + 1}} f(\zeta) \Bigr| \le \frac{M}{r_1} \cdot q^n\text{, где } M = \sup_{\zeta \in \gamma_{r_1}} |f(\zeta)| < +\infty,\Bigr|&
\end{flalign}

\item

\verb|\text{ и}\ | OR \verb|\ \text{и}\ | OR the simplest \verb|\text{ и }|

$\Bigr|+\infty + a = +\infty$ и $+\infty- a = +\infty,\, \forall a \in \bbR\cup\{+\infty\}; \Bigr|$
\begin{flalign}
&\Bigr|+\infty + a = +\infty \text{ и}\ {+\infty}- a = +\infty,\, \forall a \in \bbR\cup\{+\infty\}; \Bigr|&
\end{flalign}
\begin{flalign}
&\Bigr|+\infty + a = +\infty\ \text{и}\ {+\infty}- a = +\infty,\, \forall a \in \bbR\cup\{+\infty\}; \Bigr|&
\end{flalign}
\begin{flalign}
&\Bigr|+\infty + a = +\infty\text{ и } {+\infty}- a = +\infty,\, \forall a \in \bbR\cup\{+\infty\}; \Bigr|&
\end{flalign}

\item 
Remember about  \verb|{+\infty}|
\item
When you want to show 2 or more different equations on the same line in noninline mode then consider using \verb|;\qquad| 
\begin{flalign}
&a(y) = \lim_{l \to +\infty}\frac{1}{\pi}\int_{-l}^{l} f(t)\cos(yt)\,dt;\qquad 
b(y) = \lim_{l \to +\infty}\frac{1}{\pi}\int_{-l}^{l} f(t)\sin(yt)\,dt&
\end{flalign}

\item 
Use \verb|,\,| in noninline math with commas so it looks the same in inline mode and noninline mode:

$\phi_{\nu}(x) = e^{i\nu\frac{\pi}{l}x},\, \nu \in \bbZ,\Bigr|$
\begin{flalign}
&\phi_{\nu}(x) = e^{i\nu\frac{\pi}{l}x},\, \nu \in \bbZ,\Bigr|&
\end{flalign}

\item
Use \verb=\,= before differentials in integrals \verb=\int f(x)\,dx=
\begin{flalign}
&\int f(x)\, dx&
\end{flalign}

\item
Use ``\verb=\,='' after factorials in math formulas \verb=\frac{n!}{k!\,(n-k)!}=
\begin{flalign}
&\frac{n!}{k!\,(n-k)!}&
\end{flalign}

\item Remember about index bug with \verb|"--~|... so you should just use 

\begin{center}
\verb|Условия Коши"--~Римана\rindex{условия!Коши---Римана}|
\end{center}

\item USE ; in lists to separate items-elements:
\begin{enumerate}
\item
$M\in G$;
\item
$\forall x \in G\quad x \le M \quad (\text{соотв., } x \ge M)$.
\end{enumerate}
\item
Remember about \verb|r(f;x_0,x)|

\item 

THINK of the last expression or smth like n=1,n here pls

\begin{alignat*}{1}
&{+\infty} + a = +\infty \text{ и }  {+\infty}- a = +\infty\quad \forall a \in \bbR\cup\{+\infty\}; \\
&{-\infty} - a = -\infty \text{ и } {-\infty} + a = -\infty\quad \forall a \in \bbR\cup\{-\infty\}; \\
&{+\infty}\cdot a = +\infty \text{ и } {-\infty}\cdot a = -\infty\quad \forall a > 0,\,  a\in\bbR\cup\{+\infty\};\\ 
&{+\infty}\cdot a = -\infty \text{ и } {-\infty}\cdot a = +\infty \quad \forall a < 0,\,  a\in\bbR\cup\{-\infty\};\\
&a/ (-\infty) = 0 \text{ и } a / (+\infty) = 0 \quad \forall a \in \bbR;
\end{alignat*}
\end{itemize}






















\newpage

VERY OLD STUFF NEEED CHECHING BRO
\section{COLON}

\begin{itemize}
\item
Use \verb|\mathbin{:}| in ratios:
$$x\mathbin{:}y\mathbin{:}z = 3\mathbin{:}4\mathbin{:}5$$

\item
Use ``\verb=\colon='' in mappings:
$$f\colon A \to B$$

\item[\colorbox{prpl}{\textcolor{white}{MY}}\ $\bullet$]
Use ``\verb={:}\quad='' or macros ``\verb=\cquad='', when it's a punctuation symbol in \textbf{non-inline math} and between math statements like \verb|A\cquad B| anywere.
$$\forall x \in A\cquad f(x) = 12$$
But use simple \textbf{formula separation} \verb|$\forall x \in A$: $f(x) = 12$| like here: in \textbf{inline math}
\begin{center}
$\forall x \in A$: $f(x) = 12$
\end{center}


\end{itemize}
	
\section{SPACING}

\begin{itemize}

\item[\colorbox{prpl}{\textcolor{white}{MY}}\ $\bullet$]
Use \verb|\quad| between logic elements in \textbf{non-inline math}
$$
\forall A \in B \quad \mu A < \mu B 
$$

But use simple formulas divison ``\verb=$\forall A \in B$ $\mu A < \mu B$='' with usual spaces in inline math in \textbf{inline math} (or consider quad idk actually): 
\begin{center}
$\forall A \in B$ $\mu A < \mu B$.
\end{center}
Use punctuation symbols if needed after \$-symbol.
\item[\colorbox{prpl}{\textcolor{white}{MY}}\ $\bullet$]
Use ``\verb|\quad |'' before parenthesis in \textbf{non-inline math}:
$$
\forall M' < M \quad (\forall M' > M)\quad \exists x\in G\cquad x>M' \quad (x<M')
$$
(?)But again use simple formulas divison with usual spaces in \textbf{inline math}.

\item 
Use comma without any spaces like this \verb|$a,b$| when comma actually is a part of formula, for example
\begin{center}
$1<a,b<2$. 
\end{center}
But when it is not part of the formula (for example, when it is range 1, 2, 3, \dots) use spaces in \textbf{non-inline math} ``\verb|~|'' or formulas divison in \textbf{inline math}. Dont put \verb|\,| after \verb|\dots,\ldots,\cdots|
$$
\phi_1(x), \ \phi_2(x),\ \ldots,\ \phi_n(x), \ldots
$$
$$
\phi_1(x), \ \phi_2(x),\ \dots,\ \phi_n(x), \ldots
$$

$\phi_1(x)$,~$\phi_2(x)$, \ldots,~$\phi_n(x), \ldots$

$\phi_1(x)$,~$\phi_2(x)$, \dots,~$\phi_n(x), \ldots$

Prove that $(1-x)^{-1}=1+x+x^2+\cdots$.

Prove that $(1-x)^{-1}=1+x+x^2+\cdots$.

Clearly $a_i<b_i$ for $i=1$,~2, $\ldots$,~$n$.

Clearly $a_i<b_i$ for $i=1$,~2, $\dots\,$,~$n$.

Clearly $a_i<b_i$ for $i=1$,~2, $\dots$,~$n$.

Clearly $a_i<b_i$ for $i=1$,~2, $\cdots$,~$n$.


Сonclusion: USE \verb|dots| without any space-commands in this situations.


\item 
\verb|$x_1$,~$x_2$,~\dots, $x_n$,~\dots|.

\begin{center}
$x_1$,~$x_2$,~\dots, $x_n$,~\dots
\end{center}

\textit{Reference:} \href{https://tex.stackexchange.com/questions/115030/using-commas-in-mathematical-formulas}{$tex.stackexchange.com/...$}

Refence: \href{http://www.ctex.org/documents/shredder/src/texbook.pdf}{\textit{texbook.pdf}}, page 172

\item 
\verb|\exists!| is actual command for exists the only one! $\exists! с \in \bbR$
\end{itemize}

\section{CORRECT SYMBOLS}

\begin{itemize}
\item
Use ``\verb=\,\big|\,='' and in sets as relation symbol
$$\{\,x : 1 < x < 2\,\}.$$
\item
Use ``\verb=\setminus='' for set minus operation. Not backslashes or smth like this.
$$
A = B \setminus C
$$
\item 
Use \verb|\ldots| between binary relation symbols like ``+'' and \verb|\dots| with commas in enumeration.
$$
\phi_1(x),\ \ldots,\ \phi_n(x), \ldots,\quad x_1 + \ldots + x_2 + \ldots = 12.
$$
Use \verb|\dots| when you don't care (it's fine btw because \LaTeX\ replaces dots either with cdots or with ldots). Do not use simple \verb|$...$|.

Refence: \href{http://www.ctex.org/documents/shredder/src/texbook.pdf}{\textit{texbook.pdf}}, page 172
\item 
Please use \verb=\accentset{*}{L}^{R}_1=:
$$
\accentset{*}{L}^{R}_1
$$
\end{itemize}
\section{SMALL PLACEMENT DETAILS}
\begin{itemize}
\item[\colorbox{prpl}{\textcolor{white}{MY}}\ $\bullet$]
Do not use ``\verb=\limits='' with \verb=\int=,\verb=\int=, etc.. I am fine with integration interval to be a side of integral-symbol.
$$
\int_\Delta{f(x) \phi_k(x)}\,dx=a_k\int_\Delta|\phi_k(x)|^2\,dx,
$$
But \verb=$\iint\limits$=
\end{itemize}

\section{About <<Ёфикации>>}
\begin{itemize}
\item
Sorry bro, but we don't want to use letter <<ё>>. Use <<е>> instead.
\end{itemize}

\section{Formulas in longtables}

We use \verb=\textstyle= in formulas in longtable in our formulas appendix, if it improves quality of reading. Also I like to "rescale" tables with changing font sizes into \verb=\footnotesize=.
\end{document}