\documentclass[DIV=9, pagesize=auto]{scrartcl}

\usepackage{lmodern}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\usepackage{amsmath}
\usepackage{cases}
\usepackage{microtype}
\usepackage{shortvrb}
\usepackage[colorlinks]{hyperref}
\MakeShortVerb"
\newcommand*\kp[1]{\kern#1pt}
\newcommand*{\fancybreak}{%
  \par \medbreak \vspace{0pt plus 1cm}\penalty 500 \vspace{0pt plus -1cm}%
}
\newcommand*{\mail}[1]{\href{mailto:#1}{\texttt{#1}}}
\newcommand*{\pkg}[1]{\textsf{#1}}
\newcommand*{\cmd}[1]{\texttt{\string#1}}
\newcommand*{\env}[1]{\texttt{#1}}
\newcommand*{\meta}[1]{\textlangle\textsl{#1}\textrangle}
\newcommand*{\opt}[1]{\texttt{[#1]}}

%\addtokomafont{title}{\rmfamily}

\title{The \pkg{cases} package%
  \textnormal{\thanks{~~This manual corresponds to \pkg{cases}~v3.2, dated 2020/03/29.}}}
\author{Donald Arseneau\\\mail{asnd@triumf.ca}}
\date{Mar 2020}


\begin{document}

\maketitle

\begin{small}
\noindent
Copyright \textcopyright~1993, 1994, 1995, 2000, 2002, 2020 by Donald Arseneau, \mail{asnd@triumf.ca}.
These macros may be freely transmitted, reproduced, or modified 
provided that this notice is left intact.  Sub-equation numbering
is based on \pkg{subeqn.sty} by Stephen Gildea; parts are based
on \LaTeX's \env{eqnarray} by Leslie Lamport and the \LaTeX3 team;
and some on amsmath.sty by the American Mathematical Society.
\par
\end{small}

\section*{Description}

The \pkg{cases} package provides a \LaTeX\ environment \env{numcases} to produce 
multi-case equations with a separate equation number for each case.  There is
also \env{subnumcases} which numbers each case with the overall equation
number plus a letter [8a, 8b, etc.].   The syntax is
%
\begin{flushleft}
"\begin{numcases}{"~\meta{left side}~"}"\\
\quad  \meta{case 1\kp1} "&" \meta{explanation 1\kp1} "\\" \\
\quad  \meta{case 2\kp2} "&" \meta{explanation 2\kp2} "\\" \\
\quad  .~.~.\\
\quad  \meta{case $n$} "&" \meta{explanation $n$} \\
"\end{numcases}"
\end{flushleft}
%
Each \meta{case} is a math formula, to be typeset in display-math mode, like
in a regular numbered equation. Each \meta{explanation} is a piece of lr-mode
text (which may contain math mode in "\("\,\dots"\)" or "$"\,\dots"$").  The explanations
are optional.  Equation numbers are inserted automatically, just as for
the \env{eqnarray} environment.  In particular, the \cmd{\nonumber} command suppresses
an equation number and the \cmd{\label} command allows reference to a particular 
case. In a \env{subnumcases} environment, a \cmd{\label} in the \meta{left side} of the 
equation gives the overall equation number, without any letter. 

To use this package, 
include "\usepackage{cases}" after \cmd{\documentclass}, and also after 
"\usepackage{amsmath}" if you are using that.

\fancybreak

\noindent \emph{Question:} Is there a \env{numcases*} environment for
unnumbered cases?\\
\emph{Answer:} That would have the natural name \env{cases}, and it is 
provided by \AmS-\LaTeX\ (\pkg{amsmath} package), or by this package given 
the \opt{cases} option. It can also be achieved by using an ordinary \LaTeX\ array 
between `"\left\lbrace\ "' and `"\right."'.

Speaking of \pkg{amsmath} and package options, there are differences between the 
style used for this package and the cases done by \pkg{amsmath} (see below), but cases.sty
has options to increase compatibility.  Here is the full list of options for 
this package.
\begin{description}
\item[\opt{subnum}] Force all \env{numcases} environments to be treated as 
    \env{subnumcases}.
\item[\opt{amsstyle}] For compatibility with \pkg{amsmath}'s \env{cases}, make \env{numcases}
    use cramped math style (\cmd{\textstyle}), and put explanations in the same math style.
\item[\opt{casesstyle}] Change \pkg{amsmath}'s \env{cases} environment to work in the text/math style
    of \env{numcases}.
\item[\opt{cases}] Define a \env{cases} environment for use without \pkg{amsmath}. (This 
    is actually the same as the \opt{casesstyle} option.)
\item[\opt{fleqn}] Flush-left equation alignment, indented by \cmd{\mathindent}
    or \cmd{\mathmargin}. (Usually inherited from the \cmd{\documentclass} options.)
\item[\opt{leqno}] Left-side equation numbering (usually inherited from the \cmd{\documentclass}
    options). This looks silly with numbered cases!
\end{description}

\section*{Examples}

A simple example is:
%
\begin{verbatim}
     \begin{numcases} {|x|=}
       x, & for $x \geq 0$\\
       -x, & for $x < 0$
     \end{numcases}
\end{verbatim}
%
Giving:
%
\begin{numcases} {|x|=}
  x, & for $x \geq 0$\label{x}\\
  -x, & for $x < 0$\label{-x}
\end{numcases}

\fancybreak
\noindent
Another example, employing sub-numbering, is calculating the square root 
of a complex number $c+id$. First compute
\begin{subnumcases} {\label{weqn} w\equiv}
  0       & for $c = d = 0$\label{wzero}\\
  \sqrt{|c|}\,\sqrt{\frac{1 + \sqrt{1+(d/c)^2}}{2}} & for $|c| \geq |d|$ \\
  \sqrt{|d|}\,\sqrt{\frac{|c/d| + \sqrt{1+(c/d)^2}}{2}} & for $|c| < |d|$
\end{subnumcases}
Then, using $w$ from eq.~(\ref{weqn}), the square root is
\begin{subnumcases}{\label{sqrteqn} \sqrt{c+id}=}
  0\,,              \label{aaa}   & $w=0$ (case \ref{wzero})\\
  w+i\frac{d}{2w}\,,   & $w \neq 0$, $c \geq 0$ \\
  \frac{|d|}{2w} + iw\,, & $w \neq 0$, $c < 0$, $d \geq 0$ \\
  \frac{|d|}{2w} - iw\,,\label{ddd} & $w \neq 0$, $c < 0$, $d < 0$ 
\end{subnumcases}
These equations, eq.~(\ref{weqn}) and (\ref{sqrteqn}), were produced by:
\begin{small}
\begin{verbatim}
Another example, employing sub-numbering, is calculating the square root 
of a complex number $c+id$. First compute
\begin{subnumcases} {\label{weqn} w\equiv}
  0       & for $c = d = 0$\label{wzero}\\
  \sqrt{|c|}\,\sqrt{\frac{1 + \sqrt{1+(d/c)^2}}{2}} & for $|c| \geq |d|$ \\
  \sqrt{|d|}\,\sqrt{\frac{|c/d| + \sqrt{1+(c/d)^2}}{2}} & for $|c| < |d|$
\end{subnumcases}
Then, using $w$ from eq.~(\ref{weqn}), the square root is
\begin{subnumcases}{\label{sqrteqn} \sqrt{c+id}=}
  0\,,                 & $w=0$ (case \ref{wzero})\\
  w+i\frac{d}{2w}\,,   & $w \neq 0$, $c \geq 0$ \\
  \frac{|d|}{2w} + iw\,, & $w \neq 0$, $c < 0$, $d \geq 0$ \\
  \frac{|d|}{2w} - iw\,, & $w \neq 0$, $c < 0$, $d < 0$ 
\end{subnumcases}
\end{verbatim}
\end{small}

\section*{Compatibilibility with amsmath}

When used in conjunction with amsmath.sty, the cases package will obey the the variant 
commands \cmd{\tag}, \cmd{\notag}, and \cmd{\mathmargin}, however the formatting details
differ between \pkg{amsmath}'s \env{cases} environment and \env{numcases}.
For comparison, equation~(\ref{weqn}) formatted by \pkg{amsmath} and its \env{cases} environment
may be  entered as
\begin{flushleft}\small
\begin{verbatim}
\begin{equation}
\label{wams} w \equiv
\begin{cases}
 0    & \text{for}\ c = d = 0\\
 \sqrt{|c|}\,\sqrt{\frac{1+\sqrt{1+(d/c)^2}}{2}} &\text{for}\ |c| \geq |d|\\
 \sqrt{|d|}\,\sqrt{\frac{|c/d| + \sqrt{1+(c/d)^2}}{2}} &\text{for}\ |c|<|d|
\end{cases}
\end{equation}
\end{verbatim}
\end{flushleft}
which produces
\begin{equation}
\label{wams} w\equiv
\begin{cases}
  0    & \text{for }c = d = 0\\
  \sqrt{|c|}\,\sqrt{\frac{1+\sqrt{1+(d/c)^2}}{2}} & \text{for}\ |c| \geq |d| \\
  \sqrt{|d|}\,\sqrt{\frac{|c/d| + \sqrt{1+(c/d)^2}}{2}} & \text{for}\ |c| < |d|
\end{cases}
\end{equation}
To get this more compact layout with \env{numcases} you can insert "\textstyle" 
at the beginning of each case, as needed, or use the \pkg{cases} 
package option \opt{amsstyle}. 
To have  the (unnumbered) \env{cases} environment give the more open layout of
eq.~(\ref{weqn}) you can put "\displaystyle" at the beginning of each case, or use
the option \opt{casesstyle} for the \pkg{cases} package. (Yes these go with the
\pkg{cases} package, they are not options for \pkg{amsmath}.)

Another slight difference is that the cases within \env{numcases} can be right-justified by
inserting \cmd{\hfill} at the beginning of each, which might be desired in rare situations,
like maybe the absolute value example numbered (\ref{x}) and (\ref{-x}) above.

For full disclosure, even without any relevant package options, cases.sty will 
slightly adjust the \env{cases} environment from \pkg{amsmath}, by adding a little
space after the left brace.

\section*{Sub-numbering}
For control of the sub-equation-numbering style, see the
\AmS-\LaTeX\ documentation for \env{subequations}, currently 
in section 3.11.3. If you are not using amsmath, that documentation
still mostly applies, except the name for regular equation numbers
is then `mainequation' instead of `parentequation'. Also, the
sub-numbering style can be controlled more easily by defining
\cmd{\thesubequation}. An example for capitalized letters is
\begin{verbatim}
\renewcommand\thesubequation{\themainequation.\Alph{equation}} % 13.C
\end{verbatim}
(noting that the counter to reference is `equation' not `subequation').

\end{document}