Long enough vertical bar | depending on expression before and after it - Elliptic integral

Question

I want to write an elliptic integral F(\phi|k^2) with different expressions for \phi and k^2 including fractions. So I want a new command (macro?) for this. It should look like e.g. \F{\phi}{k^2}. This is no problem at all, without scaling the vertical bar |. But I want to autoscale it with the arguments.

What can I do to create a new command like this? Do I have to go the long way by trying to determine the height of the arguments? How would this look alike?

Answer 1

A simple implementation with a perhaps more natural syntax:

\documentclass{article}
\usepackage{mathtools}

\DeclarePairedDelimiterX{\EllInner}[1]{(}{)}{%
  \renewcommand{\with}{\,\delimsize|\,}%
  #1%
}
\newcommand{\Ell}{F\EllInner}
\providecommand{\with}{}

\begin{document}

\[
\Ell{\varphi\with k^2} = \int_0^\varphi\frac{dt}{\sqrt{1-k^4\sin^2t}}
\qquad
\Ell*{\frac{\pi^2}{4}\with k^2}
\qquad
\Ell[\Big]{\frac{\pi^2}{4}\with k^2}
\]

\end{document}

If you prefer the two argument syntax, you can do with

\documentclass{article}
\usepackage{mathtools}

\DeclarePairedDelimiterX{\EllInner}[2]{(}{)}{%
  #1\,\delimsize|\,#2%
}
\newcommand{\Ell}{F\EllInner}

\begin{document}

\[
\Ell{\varphi}{k^2} = \int_0^\varphi\frac{dt}{\sqrt{1-k^4\sin^2t}}
\qquad
\Ell*{\frac{\pi^2}{4}}{k^2}
\qquad
\Ell[\Big]{\frac{\pi^2}{4}}{k^2}
\]

\end{document}

You can also use | as the inner separator, with a slick trick. The caveat is that the expressions must not contain |, but they can contain \vert, \lvert or \rvert.

\documentclass{article}
\usepackage{mathtools}

\DeclarePairedDelimiterX{\EllInner}[1]{(}{)}{\activatebar #1}
\newcommand{\Ell}{F\EllInner}
\newcommand{\activatebar}{%
  \begingroup\lccode`~=`| \lowercase{\endgroup\def~}{\,\delimsize\vert\,}%
  \mathcode`|="8000
}

\begin{document}

\[
\Ell{\varphi | k^2} = \int_0^\varphi\frac{dt}{\sqrt{1-k^4\sin^2t}}
\qquad
\Ell*{\frac{\pi^2}{4} | k^2}
\qquad
\Ell[\Big]{\frac{\pi^2}{4} | k^2}
\]

\end{document}

Answer 2

Also with mathtools, I define a \Fellcommand with DeclarePairedDelimiterXPP which takes two arguments. The \Fell* command adds a pair of implicit \left···\right, and the optional argument in \Fell (\big, \Big, &c.) allows to fine-tune the size of the delimiters.

A variant which relies on xparse, named \fell, uses only one argument in two parts separated by a semicolon, so the syntax is closer to what one writes by hand.

\documentclass[12pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{amssymb}
\usepackage{lmodern}
\usepackage{mathtools}
\DeclarePairedDelimiterXPP\Fell[2]{\mathrm{F}}(){}%
{#1\,\delimsize\vert\,\mathopen{}#2}

\usepackage{xparse}

\DeclarePairedDelimiterXPP{\Fel}[1]{\mathrm{F}}(){}%
{\Felargs{#1}}
\NewDocumentCommand{\Felargs}{>{\SplitArgument{1}{;}}m}
{\Felargsaux#1}
\NewDocumentCommand{\Felargsaux}{mm}
{#1\,\delimsize\vert\,\mathopen{}#2}%

\begin{document}

 \[ \Fell{\varphi}{ k^2} = \int_0^\varphi\frac{\mathrm d\mkern1mu t}{\sqrt{1-k^4\sin^2t}}\qquad\Fell*{\frac{\pi^2}{4}}{k^2} \qquad\Fell[\Big]{\frac{\pi^2}{4}}{k^2}
  \]%

\[ \Fel{\varphi ; \ell} = \int_0^\varphi\frac{\mathrm d\mkern1mu t}{\sqrt{1-\ell^2\sin^2t}}\qquad\Fel*{\frac{\pi^2}{4}; \ell}\qquad\Fel[\Big]{\frac{\pi^2}{4}; \ell}
  \]%

 \end{document} 

Answer 3

Following up on @GustavoMezzetti's comment, here's a solution that makes use of the \DeclarePairedDelimiterX macro of the mathtools package.

Aside: You may want to come up with a mnemonic macro name than "\F".

\documentclass{article}
\usepackage{mathtools}
% See p. 27 of the user guide of the 'mathtools' package
% for the source of the following code
\providecommand\given{}
\newcommand\SetSymbol[1][]{%
   \nonscript\:#1\vert
   \allowbreak
   \nonscript\:
   \mathopen{}}
\DeclarePairedDelimiterX\PSet[1]\lparen\rparen{% 
   \renewcommand\given{\SetSymbol[\delimsize]}#1}
\newcommand\F[2]{F\PSet*{#1\given #2}}

\begin{document}
\[  
\F{\phi}{k} \quad  \F{\phi}{k^2} \quad \F{\phi}{\frac{u}{v}} 
\]
\end{document}

---

Addendum: As @egreg has pointed out in comment, using \DeclarePairedDelimiterX in the definition of the \F macro may be sort of overkill, as no use is made of the optional argument to specify fixed-sized fences. A more streamlined version of the \F macro would therefore be:

\newcommand{\F}[2]{%
   F\mleft(#1\nonscript\:\middle\vert\nonscript\:‌​#2\mright)}

With this approach, it's necessary to load the mleftright package for its \mleft and \mright macros. On the other hand, it's no longer necessary to load the mathtools package.

A full MWE:

\documentclass{article}
\usepackage{mleftright} % for "\mleft" and "\mright" macros
\newcommand{\F}[2]{%
   F\mleft(#1\nonscript\:\middle\vert\nonscript\:‌​#2\mright)}
\begin{document}
\[
\F{\phi}{k} \quad  \F{\phi}{k^2} \quad \F{\phi}{\frac{u}{v}}
\]
\end{document}