How can I influence the spacing of mathematical functions by an own macro?

Question

I would like to get a macro \of to define a function with an optional macro \at to define its restriction. The reason is, that I don't like the spacing for functions (as there is no spacing) and I would like to automate that. The idea is, to set functions like f \of x or f\of x which would be quick and handy. But I am not sure, if that is doable for syntax reasons. I think, something like \func{f}{x} would be much easier to provide, but that would be too much typing and quite a lot of change for already existing formulae.

However, the reasons for this macro are the following properties which I would like to achieve. In future I want to expand this to all kind of derivatives or alike (total, partial, normal, increment, infinite element...).

Here is the list:

• Always set argument (value behind \of in ()
• Recognize x, (x), and {x} as argument
• always set half space in front of the function when it is set behind a letter, number or bracket
• set half space behind the function, if followed by a (
• allow powers to the function and to the argument and set it properly
• allow nested functions
• allow \at-notation with three optional formats (this will get interesting later on for the derivatives...)

And here are some examples:

\documentclass{article}
\usepackage{mathtools}
\usepackage[partial=upright]{unicode-math}

\begin{document}
\begin{verbatim}f \of x=x^2\end{verbatim}
$f(x)=x^2$
\begin{verbatim}f \of (x+y)=x+y\end{verbatim}
$f(x+y)=x+y$
\begin{verbatim}f \of (x+y)x+y\end{verbatim}
$f(x+y)x+y$
\begin{verbatim}f \of (x+y)(x+y)(x+y)\end{verbatim}
$f(x+y)\,(x+y)(x+y)$
\begin{verbatim}xf \of xx\end{verbatim}
$x\,f(x)x$
\begin{verbatim}zf \of xg \of yz\end{verbatim}
$z\,f(x)\,g(y)z$
\begin{verbatim}f^2 \of x\end{verbatim}
$f^2(x) \vee f(x)^2 \vee (f(x))^2 \quad \text{don't know which one is correct}$
\begin{verbatim}f \of x^2\end{verbatim}
$f(x^2)$
\begin{verbatim}f \of g \of x\end{verbatim}
$f(g(x))$
\begin{verbatim}\frac{a}{b}f\of x\end{verbatim}
$\dfrac{a}{b}f(x)$
\begin{verbatim}(a+b)f\of x\end{verbatim}
$(a+b)\,f(x)$
\begin{verbatim}\setFunctionAtBar \end{verbatim}
\begin{verbatim}f \of x \at a\end{verbatim}
$f(x)|_{x=a}$
\begin{verbatim}\setFunctionAtBracket\end{verbatim}
\begin{verbatim}f \of x \at {a+b}\end{verbatim}
$(f(x))_{x=(a+b)}$
\begin{verbatim}\setFunctionAtInline\end{verbatim}
\begin{verbatim}f \of x \at (a+b)\end{verbatim}
$f(a+b)$
\end{document}

Answer 1

Aside from syntax it seems main issue is spacing before a function term. TeX will add space there automatically if it knows it is a function. \log \sin etc are \mathop atoms and get space, but f is a mathord so does not. If you declare a one-letter math operator though, it gets the space that I think you want:

\documentclass{article}

\begin{document}
\showoutput
$zf(x)$

$z\mathop{{}f}(x)$

\end{document}