Declare a flexible math operator (such as expectation operator)

Question

I would like to define a math operator, such as the expectation operator, which will be able to optionally receive subscript and/or superscript.

Currently what I have is

\newcommand{\E}[1]{\mathbb{E}\left(#1\right)}

and I call it with \E{X}. This is nice, but when I wish to add a subscript I fall to \mathbb{E}_x\left(X\right) and it's quite annoying.

I therefore defined another command:

\newcommand{\Ex}[2]{\mathbb{E}_{#1}\left(#2\right)}

but that's not really helpful for two reasons: (a) I need to remember two commands, and (b) the usage is not intuitive (I use it with \E{x}{X} instead of something like \E_x{X}.

Can I define a math operator whose usage will be either of the following:

• \E{X}
• \E_x{X}
• \E^2{X}

and whose output in each case would be

• \mathbb{E}\left(X\right)
• \mathbb{E}_x\left(X\right)
• \mathbb{E}^2\left(X\right)

It this possible?

Note that DeclareMathOperator* is not good for the task, as in display mode it puts the sub/superscripts below/above the operator.

Answer 1

Implementation for \E, which has optional sub- or superscripts before the mandatory argument. It uses \mleft and \mright to avoid additional horizontal spacing.

\documentclass{article}
\usepackage{mleftright}
\usepackage{amssymb}

\makeatletter
\newcommand*{\E}{%
  \def\E@sub{}%
  \def\E@sup{}%
  \E@scripts
}
\newcommand*{\E@scripts}{%
  \@ifnextchar_\E@subscript{%
    \@ifnextchar^\E@supscript\E@finish
  }%
}
\def\E@subscript_#1{%
  \ifx\E@sub\@empty
    \def\E@sub{#1}%
  \else
    \errmessage{E already has a subscript}%
  \fi
  \E@scripts
}
\def\E@supscript^#1{%
  \ifx\E@sup\@empty
    \def\E@sup{#1}%
  \else
    \errmessage{E already has a superscript}%
  \fi
  \E@scripts
}
\newcommand*{\E@finish}[1]{%
  \mathbb{E}%
  \ifx\E@sub\@empty\else _{\E@sub}\fi
  \ifx\E@sup\@empty\else ^{\E@sup}\fi
  \mleft(#1\mright)%
}
\makeatother

\begin{document}
  \[ \E{X}, \E_x{X}, \E^2{X}, \E_x^2{X}, \E^2_x{X} \]
\end{document}

Answer 2

I wouldn't make the macro \E take an explicit argument. Instead, I would define \E via a \DeclareMathOperator statement and then write \E(X), \E X, etc. That way, you can decide "on the fly" whether or not to use parentheses.

Note that this approach requires no extra typing during input: You simply replace \E{X} with \E(X) -- exact same number of characters. :-)

\documentclass{article}
\usepackage{amsmath,amssymb}
\DeclareMathOperator{\E}{\mathbb{E}}
\begin{document}
$\E(K)$, $\E_x(K)$, $\E^2(K)$, $\E_x^2(K)$

\medskip
$\E K$, $\E_x K$, $\E^2 K$, $\E_x^2 K$
\end{document}