Define command with optional characters

Question

I have the following command definition (in the MWE)

\documentclass{article}
\usepackage{amsmath}

\def\hE_#1^#2#3{\hat {\textrm{E}}_{#1}^{#2}\,[#3]}

\begin{document}
\[
  \hE_{1}^{2}{3},\qquad
  % \hE^{2}{3},\qquad
  % \hE_{1}{3},\qquad.
  % \hE^{2}_{1}{3},\qquad
  % \hE{3}
\]
\end{document}

Which outputs a particular expression with superscripts and subscripts. I want to define other usages of this command that allow for the commented variants. I thought I could do this

\def\hE^#1_#2#3{\hE_{#2}^{#1}{#3}}
\def\hE^#1#2{\hE_{}^{#1}{#2}}
\def\hE_#1#2{\hE_{#1}^{}{#2}}

But that gave me a TeX capacity exceeded error. Is there a way to do this? I prefer to use internal tex definitions rather than external packages if possible.

Answer 1

You can do it with xparse. The e argument type takes a list of tokens and in the body #1 refers to the item after the first token, #2 to the item after the second token and so on. You can test whether they are present with \IfValueT.

I'd not add the \, thin space. If you prefer, add it in the following code.

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

\NewDocumentCommand{\hE}{e{^_}m}{%
  \mathop{}\!\hat{\mathrm{E}}%
  \IfValueT{#1}{^{#1}}%
  \IfValueT{#2}{_{#2}}%
  [#3]%
}

\begin{document}

\[
  \hE_{1}^{2}{3},\quad
  \hE^{2}{3},\quad
  \hE_{1}{3},\quad
  \hE^{2}_{1}{3},\quad
  \hE{3}
\]

\end{document}

The output with \, for comparison:

I believe it's evident that the thin space is out of place, but you're the final judge.

On the other hand, I'd simply do

\newcommand{\hE}{\mathop{}\!\hat{\mathrm{E}}}

and type in the brackets.

\hE_{1}^{2}[3],\quad
\hE^{2}[3],\quad
\hE_{1}[3],\quad
\hE^{2}_{1}[3],\quad
\hE[3]

As you see, there is no gain with the proposed syntax (other than the possibility to insert the thin space).

More importantly, \textrm{E} is not the right tool; \mathrm{E} is.

Why \mathop{}\!? Because this way \hE is treated as a math operator like \sin or \log. I opted to not doing \operatorname{\hat{E}}, because in this way the superscript would be too high.