Cases with long math equation

Question

What would be the best formatting of the following align and cases where the middle case has a very long math equation?

\begin{align*}
      V_{ijk}& = \begin{cases}
          \dfrac{y_u+y_l}2 - \dfrac{x_u+x_l}2,& \text{if }x_u < y_l\\[15pt]
          \dfrac1C \bigg[\dfrac{x_u^3-y_l^3}{6}+\dfrac{y_l^2+x_u^2}{2}x_l + (x_u-y_l)\dfrac{x_l^2}{2}+ \dfrac{y_u^2-x_u^2}{2}(x_u-x_l)& - (y_u-x_u)\dfrac{x_u^2-x_l^2}{2} \bigg],
          \\[12pt]& \text{if }x_u\in\left[y_l,y_u\right]\\[15pt]
          \dfrac1{x_u-x_l}\bigg[\dfrac{(y_u+y_l)^2}6 - \dfrac{y_u+y_l}2 x_l + \dfrac{x_l^2}2\bigg],& \text{otherwise}
      \end{cases}
\end{align*}

The main default of this current workaround which is nearly ok, is that there is an extra space before -(y_u-x_u) that is annoying. Note that I put the second if text under second equation because it would be too width for my memoir otherwise. Please favour simpliest solutions if possible :)

Answer 1

with \mfrac from nccmath and multlined from the mathtools package:

\documentclass{article}
\usepackage{mathtools, nccmath}

%---------------- show page layout. don't use in a real document!
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
%---------------------------------------------------------------%

\begin{document}

\[
V_{ijk} = \begin{cases} 
  \mfrac{y_u+y_l}{2} - \mfrac{x_u+x_l}{2},  &\text{if }x_u < y_l    \\[3ex]
    \begin{multlined} 
    \mfrac{1}{C} \bigg[\mfrac{x_u^3-y_l^3}{6}+\mfrac{y_l^2+x_u^2}{2}x_l
      + (x_u-y_l)\mfrac{x_l^2}{2}\\[-2ex]
      + \mfrac{y_u^2-x_u^2}{2}(x_u-x_l) - (y_u-x_u)\mfrac{x_u^2-x_l^2}{2} \biggr]
    \end{multlined},
    & \text{if }x_u\in [y_l,y_u ]
    \\[5ex]
    \mfrac{1}{x_u-x_l}\biggl[\mfrac{(y_u+y_l)^2}{6} - \mfrac{y_u+y_l}{2} x_l
    + \mfrac{x_l^2}{2}\biggr],& \text{otherwise}
  \end{cases}
 \]
\end{document}

Answer 2

I'm pretty sure it is a duplicate. But it is also easy to solve using aligned

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{align*}
  V_{ijk}& = \begin{cases} \dfrac{y_u+y_l}2 - \dfrac{x_u+x_l}2,&
    \text{if }x_u < y_l
    \\[15pt]
    \begin{aligned}[b]
      &\dfrac1C \bigg[\dfrac{x_u^3-y_l^3}{6}+\dfrac{y_l^2+x_u^2}{2}x_l
      + (x_u-y_l)\dfrac{x_l^2}{2}
      \\
      &+ \dfrac{y_u^2-x_u^2}{2}(x_u-x_l) -
      (y_u-x_u)\dfrac{x_u^2-x_l^2}{2} \bigg]
    \end{aligned},
    & \text{if }x_u\in\left[y_l,y_u\right]
    \\[15pt]
    \dfrac1{x_u-x_l}\bigg[\dfrac{(y_u+y_l)^2}6 - \dfrac{y_u+y_l}2 x_l
    + \dfrac{x_l^2}2\bigg],& \text{otherwise}
  \end{cases}
 \end{align*}
\end{document}

Answer 3

My suggestion is not to use cases, because it increases the interline spaces also inside nested alignments.

Using multlined for the long middle equation does the job.

\documentclass{article}
\usepackage{amsmath,mathtools}

\begin{document}

\begin{equation*}
V_{ijk} = \left\{
\begin{aligned}
& \dfrac{y_u+y_l}2 - \dfrac{x_u+x_l}2, && \text{if }x_u < y_l
\\[2ex]
& \begin{multlined}[b]
    \dfrac{1}{C} \biggl[
      \dfrac{x_u^3-y_l^3}{6}+\dfrac{y_l^2+x_u^2}{2}x_l + (x_u-y_l)\dfrac{x_l^2}{2} \\
      + \dfrac{y_u^2-x_u^2}{2}(x_u-x_l) - (y_u-x_u)\dfrac{x_u^2-x_l^2}{2}
    \biggr],
  \end{multlined} && \text{if }x_u\in [y_l,y_u]
\\[2ex]
& \dfrac{1}{x_u-x_l}\biggl[
    \dfrac{(y_u+y_l)^2}6 - \dfrac{y_u+y_l}2 x_l + \dfrac{x_l^2}2
  \biggr], && \text{otherwise}
\end{aligned}
\right.
\end{equation*}

\end{document}

Avoid code such as \frac1C that's hard to read and to maintain. Also note \biggl and \biggr instead of the unqualified \bigg.

Also avoid align for single equations (which this is an example of).

Answer 4

Here's a solution that employs a dcases* environment instead of the cases environment. The d in dcases denotes "display style"; this means, among other things, that one can write \frac instead of \dfrac in order to get "large" fraction terms.

Note that I've removed unnecessary \left and \right directives.

\documentclass{article}
\usepackage{mathtools} % for 'dcases*' environment

\begin{document}
\[
V_{ijk} = 
\begin{dcases*} 
\frac{y_u+y_l}{2} - \frac{x_u+x_l}{2}\,,
   &if $x_u < y_l$\\[1ex]
\begin{aligned}[b]
&\frac{1}{C} \biggl[
   \frac{x_u^3-y_l^3}{6}+\frac{y_l^2+x_u^2}{2}x_l
   + (x_u-y_l)\frac{x_l^2}{2}\\
&\quad + \frac{y_u^2-x_u^2}{2}(x_u-x_l) 
   - (y_u-x_u)\frac{x_u^2-x_l^2}{2} \,\biggr]\,,
\end{aligned}
   &if $x_u\in[y_l,y_u]$ \\[1ex]
\frac{1}{x_u-x_l} \biggl[ 
   \frac{(y_u+y_l)^2}{6} 
   - \frac{y_u+y_l}{2} x_l 
   + \frac{x_l^2}{2} \,\biggr]\,,
   & otherwise
\end{dcases*}
\]
\end{document}