Linking formulas and explanations

Question

I am preparing a collection of the most useful formulas for an exam and I would like to have a brief explanation for each one of them.

Since this is supposed to be a quick reference I'd prefer to pack all the formulas together with no text or whatsoever and then link them to their explanation that I could put in another page. What could I do to achieve this?

Manually cross-linking with \autoref kind of beats the purpose, it would take me too long and would be hard to maintain since there's an awful lot of equations to write.

---

Edit: What I currently have

As suggested in the comments, I'll add a section that I currently have.

\section{Pulsed Radar}
\subsection{Characteristics}
\begin{align}
    f_r &= \frac{1}{T}
        &\text{Pulse Repetition Frequency (PRF)} \label{basic:PRF_vs_PRI} \\
        &&\text{and Pulse Repetition Interval (PRI)} \nonumber
        \\
    \tau &= \frac{1}{B}
        &\text{Bandwidth}
        \label{basic:BW}
        \\
    d_t &= \frac{\tau}{T}
        &\text{Duty cycle}
\end{align}

\subsection{Range}
\begin{align}
    R &= \frac{c}{2t}
        &\text{Range}
        \\
    R_\text{max} &= \frac{c}{2T}
        &\text{Max. unambiguous range} \\
        &&\text{(see \autoref{basic:PRF_vs_PRI})} \nonumber
        \\
    R_\text{min} &= \delta R = \frac{c}{2\tau}
        &\text{Minimum range or range resolution} \\
        &&\text{(see \autoref{basic:BW})} \nonumber
        \\
    M &= \frac{R_\text{max} - R_\text{min}}{\delta R}
        &\text{Number of space bins (} R_\text{min} = \delta R \text{)}
\end{align}

Which gives me this:

As you can see, I currently have the comments right next to the equations, but I want to move them to their own section, so that I can make them just a bit more expressive.

What I have in mind is something like a \footnote, where I write the note and it gets automatically numbered, linked and moved at the bottom of the page - I don't want to manually label every single comment and put a link to it next to the relative equation. The only actual difference from a footnote is that I don't want the text at the bottom of the page - but at the end of the document - and I want it formatted normally.

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Edit 2: why I want it

I want to explain why I have the idea of separating equations from comments: this would allow me to write the formulas in more columns, so that everything is more compact!

Answer 1

(updated the answer to switch to multicolumn mode automatically if no explanations are to be shown)

My proposed solution is admittedly not a direct answer to your query. Instead, I suggest taking an indirect approach -- specifically, creating two separate but parallel documents: One with just the formulas, the other with the formulas and their (brief or wordy) explanations. Because all explanations reside inside \tbox ("textual parbox") macros, switching back and forth between the two manifestations is as easy as changing \ifshowtbox from false to true (or vice versa) and recompiling the document.

First, here's how the document would look if \ifshowtbox is set to true, i.e., if the explanations contained in the \tbox directives are supposed to be shown:

And here's how the document would look if \ifshowtbox is set to false. Observe that the columns are much narrower now since we've also switched to 3-column mode.

\documentclass{article}
\usepackage[letterpaper,margin=1in]{geometry} % set page parameters suitably
\usepackage{multicol} % for 'multicols' env.
\usepackage[mathindent]{amsmath} % for 'align' env.
\allowdisplaybreaks
\usepackage[colorlinks,allcolors=blue]{hyperref}
\usepackage[nameinlink]{cleveref} % for '\cref' macro

%% create a new conditional: '\showtbox'
\newif\ifshowtbox\showtboxfalse % or "\showtboxtrue"
\ifshowtbox
  \newcommand\tbox[1]{\parbox[t]{0.5\textwidth}{\raggedright #1}}
\else
  \newcommand\tbox[1]{}
  \setlength\mathindent{10pt} % default: 25pt
\fi

\begin{document}
\stepcounter{section} % just for this example
\addtocounter{equation}{2}

\ifshowtbox\else\begin{multicols*}{3}\fi 

\section{Pulsed Radar}
\subsection{Characteristics}
\begin{align}
    f_r &= \frac{1}{T}
        &&\tbox{Pulse Repetition Frequency (PRF) and Pulse Repetition Interval (PRI)} 
        \label{basic:PRF_vs_PRI} 
        \\
    \tau &= \frac{1}{B}
        &&\tbox{Bandwidth}
        \label{basic:BW} 
        \\[1ex]
    d_t &= \frac{\tau}{T}
        &&\tbox{Duty cycle}
\end{align}

\subsection{Range}
\begin{align}
    R &= \frac{c}{2t}
        &&\tbox{Range} 
        \\[1ex]
    R_{\max} &= \frac{c}{2T}
        &&\tbox{Max.\ unambiguous range; cf. \cref{basic:PRF_vs_PRI}} 
        \\
    R_{\min} &= \delta R = \frac{c}{2\tau}
        &&\tbox{Min.\ range or range resolution; cf.  \cref{basic:BW}} 
        \\
    M &= \frac{R_{\max} - R_{\min}}{\delta R}
        &&\tbox{Number of space bins ($R_{\min} = \delta R$)}
\end{align}

\ifshowtbox\else\end{multicols*}\fi

\end{document}

Answer 2

After Barbara's comment:

An "endnotes" approach might satisfy your requirements. Text is entered near its referent, and output separately at the end. There are several good packages; search for "endnotes" and "enotez" on CTAN. Unfortunately, I don't know how any of these fare when embedded in a display math structure. – barbara beeton

I checked the package enotez, which is fairly simple to use and does what I needed. So I edited my document like this:

\section{Pulsed Radar}
\subsection{Characteristics}
\begin{multicols}{2} \noindent
    $$ f_r = \frac{1}{T} \
        \endnote{Pulse Repetition Interval (PRF) versus Pulse Repetition
            Interval (PRI)}
        $$
    $$ \tau = \frac{1}{B} \
        \endnote{Bandwidth}
        $$
\end{multicols}

\subsection{Range}
\begin{multicols}{4} \noindent
    $$ R = \frac{c}{2t} \
        \endnote{Range}
        $$
    $$ R_\text{max} = \frac{c}{2T} \
        \endnote{Max. unambiguous range}
        $$
    $$ R_\text{min} = \delta R = \frac{c}{2 \tau} \
        \endnote{Min. range or range resolution}
        $$
    $$ \theta_B = R \frac{\lambda}{d} \
        \endnote{Azimuth resolution; d is the length of the antenna}
        $$
    $$ M = \frac{R_\text{max} - R_\text{min}}{\delta R} \
        \endnote{Number of space bins}
        $$
\end{multicols}

\subsection{Power}
\begin{multicols}{2} \noindent
    $$d_t = \frac{\tau}{T} \
        \endnote{Duty Cycle}
        $$
    $$ P_\text{avg} = P_t \cdot d_t \
        \endnote{Average power}
        $$
\end{multicols}
\printendnotes[itemize]

Which gives me this:

It's probably not the best solution because I had to give up the \align environment since, for some reason, all the notes were printed twice. I also gave up references (at least, for now) because defining an equation environment for each equation is too much.

But still, this is fair enough for a quick reference of formulas and comments. Thank you all for your suggestions!