How do I make the symbol | larger like other delimiters?

Question

Consider the following code

\documentclass{amsbook}
\usepackage{amsmath}
\usepackage{amssymb}
\begin{document}
\[
E(X) = \delta \left[ \frac{1}{-\delta} x e^{-\delta x} \left| \right|_0^\infty \ldots  \right]
\]
\end{document}

It produces the following, which looks odd:

Reversing the order of the middle \left| and \right| solves the problem, but only partially (note the differently-sized final delimiter):

\[
E(X) = \delta \left[ \frac{1}{-\delta} x e^{-\delta x}  \right|_0^\infty \left| \ldots  \right]
\]

Why is that? Is there an appropriate way to do this?

Answer 1

For a one-off formula you need nothing special, just use \Bigl and friends.

For several appearances,

\documentclass{article}
\usepackage{amsmath}
\usepackage{mleftright}

\ExplSyntaxOn

\NewDocumentCommand{\brackeval}{O{}mmmm}
 {% #1 = size instruction
  % #2 = before the evaluation bar
  % #3 = lower bound
  % #4 = upper bound
  % #5 = after the evaluation bar
  \tonyk_brackeval:nnnnn { #1 } { #2 } { #3 } { #4 } { #5 }
 }

\cs_new_protected:Nn \tonyk_brackeval:nnnnn
 {
  \tl_use:c { c__tonyk_brackeval_open_#1_tl } [
  #2 \vphantom{#5}
  \tl_use:c { c__tonyk_brackeval_middle_#1_tl } |\sb{#3}\sp{#4}
  \;
  #5 \vphantom{#2}
  \tl_use:c { c__tonyk_brackeval_close_#1_tl } ]
 }

% case of no optional argument
\tl_const:cn { c__tonyk_brackeval_open__tl } { }
\tl_const:cn { c__tonyk_brackeval_middle__tl } { }
\tl_const:cn { c__tonyk_brackeval_close__tl } { }
% case of *
\tl_const:cn { c__tonyk_brackeval_open_*_tl } { \mleft.\kern-\nulldelimiterspace\mleft }
\tl_const:cn { c__tonyk_brackeval_middle_*_tl } { \mright }
\tl_const:cn { c__tonyk_brackeval_close_*_tl } { \mright }
% other cases
\clist_map_inline:nn { big, Big, bigg, Bigg }
 {
  \tl_const:cx { c__tonyk_brackeval_open_#1_tl } { \exp_not:c { #1l } }
  \tl_const:cx { c__tonyk_brackeval_middle_#1_tl } { \exp_not:c { #1 } }
  \tl_const:cx { c__tonyk_brackeval_close_#1_tl } { \exp_not:c { #1r } }
 }

\ExplSyntaxOff

\begin{document}

\begin{gather}
E(X)=\delta\brackeval{x}{a}{b}{\dots} \\
E(X)=\delta\brackeval[big]{x}{a}{b}{\dots} \\
E(X)=\delta\brackeval[Big]{-\frac{1}{\delta}xe^{-\delta x}}{0}{\infty}{\dots} \\
E(X)=\delta\brackeval[*]{-\frac{1}{\delta}xe^{-\delta x}}{0}{\infty}{\dots}
\end{gather}

\end{document}

The effect is similar to Davislor's answer, but it allows also automatic sizing (the syntax is slightly different, though). The trick for automatic sizing is to add a \left. at the beginning, so for the bar we can use \right that accepts limits. The sizes will be the same thanks to the phantoms.

Answer 2

Although you could normally use \left[ ... \middle| ... \right] or \left\lbrack ... \middle\vert ... \right\rbrack, that breaks on the limits here.

So, a good alternative is \DeclarePairedDelimiterX from \mathtools. (See section 3.6 of the manual.) This allows you to divide the inside of the expression into multiple segments and use \delimsize for the middle delimiter. You call it with a size argument, such as [\big] or [\bigg].

\documentclass{amsbook}
\usepackage{amsmath, amssymb}
\usepackage{mathtools}

\DeclarePairedDelimiterX{\expect}[4]{\lbrace}{\rbrace}{#1 \delimsize\vert_{#2}^{#3} #4}

\begin{document}
\[
E(X) = \delta \expect[\bigg]{\frac{1}{-\delta} x e^{-\delta x}}{0}{\infty}{\ldots}
\]
\end{document}

You’d normally be able to use a starred version of the macro to scale automatically, such as \expect*, but if you do that here, the limits will break.

Eta: And I just noticed I wrote \lbrace and \rbrace instead of \lbrack and \rbrack. Whoops!

Answer 3

I have used \Bigl and \Bigr delimiters than \left and \right to have a correct size with the whole equation.

\documentclass{amsbook}
\usepackage{amsmath}
\usepackage{amssymb}
\begin{document}
\[
E(X) = \delta \Bigl[ \frac{1}{-\delta} x e^{-\delta x}  \Bigl|_0^\infty \cdots  \Bigr]
\]
\end{document}

With a large size \Biggl and \Biggr:

\documentclass{amsbook}
\usepackage{amsmath}
\usepackage{amssymb}
\begin{document}
\[
E(X) = \delta \Biggl[ \frac{1}{-\delta} x e^{-\delta x}  \Biggl|_0^\infty \cdots  \Biggr]
\]
\end{document}