How can I create macros that always have the same kind of behavior?

Question

So this question is actually two, but I think they'll be solved in the same way, so I'm putting them together. The form \textstyle\lim_{n\to\infty} always bothers me, because it takes up lots of horizontal space, while \displaystyle\lim_{n\to\infty} doesn't take up too much extra vertical space. I tried to use \DeclareMathOperator*{\mylim}{{\displaystyle\lim}} to create a new limit command, so that I wouldn't have to type \displaystyle a million times, but it doesn't work. Can anyone help with this?

Also, I tried to do a similar thing with \DeclareMathOperator*{\mysum}{{\textstyle\sum}}, but that didn't work either. Is the fix the same as above?

MWE:

\documentclass[12pt]{letter}
\usepackage{amsmath,amssymb,amsthm}

\DeclareMathOperator*{\mylim}{{\displaystyle\lim}}
\DeclareMathOperator*{\mysum}{{\textstyle\sum}}

\begin{document}
$\mylim_{n\to\infty} x_n = \displaystyle\lim_{n\to\infty} x_n$

\begin{equation*}
     \mysum_{n=1}^\infty x_n = \displaystyle\sum_{n=1}^\infty x_n
\end{equation*}
\end{document}

Answer 1

By amsmath’s default (more specifically, amsopn):

• The starred version \DeclareMathOperator* creates a large operator which places subscript/superscript like \sum and \lim; that is, below/above in display math while, lower-/upper-right in inline math.
• The non-starred version \DeclareMathOperator creates a large operator which places subscript/superscript like \sin and \log; that is, always lower-/upper-right.

So, if you want a \sum that 1) always puts scripts on the right and 2) is always in \textstyle (the latter being the tricky part), you can simply do

\DeclareMathOperator{\mysum}{\textstyle\sum}% <- The effect of \textstyle is local!

---

Okay, the “scripts always below/above” part is the actual tricky part, and you want the operator to always be in \displaystyle (the latter is questionable in your current setting, so I’ll give two different approaches).

Update: I will use a macro \mustlimits@, which mirrors \nolimits@ from amsopn, in the following solutions. Since the macro name contains @, the following code snippets should be wrapped inside \makeatletter ... \makeatother.

1. You just want \limits and don’t actually want \displaystyle. Since \lim is essentially just Roman text, it comes with only one style. To force below/above scripts, simply do

   \newcommand{\mylim}{\lim\mustlimits@}
   

2. You do want \displaystyle (although \lim has only one style!). Well, as you wish:

   \newcommand{\mydisplaylim}{%
     \mathop{\hbox{$\displaystyle\m@th\operator@font lim$}}\mustlimits@
   }
   \newcommand{\mydisplaysum}{%
     \mathop{\displaystyle\sum}\mustlimits@
   }
   

3. So what is \mustlimits@ exactly? Here is its definition:

   \newcommand*{\mustlimits@}{\@ifnextchar\nolimits{\limits\@gobble}{\limits}}
   

---

MWE:

\documentclass[12pt]{letter}
\usepackage{amsmath,amssymb,amsthm}

\DeclareMathOperator{\mysum}{\textstyle\sum}% <- The effect of \textstyle is local!
\makeatletter
\newcommand{\mylim}{\lim\mustlimits@}
\newcommand{\mydisplaylim}{%
  \mathop{\hbox{$\displaystyle\m@th\operator@font lim$}}\mustlimits@
}
\newcommand{\mydisplaysum}{%
  \mathop{\displaystyle\sum}\mustlimits@
}
\newcommand*{\mustlimits@}{\@ifnextchar\nolimits{\limits\@gobble}{\limits}}
\makeatother

\begin{document}
$\mylim_{n\to\infty} x_n = \lim_{n\to\infty} x_n$

\begin{equation*}
     \mysum_{n=1}^\infty x_n = \sum_{n=1}^\infty x_n
\end{equation*}

$\mydisplaylim_{n\to\infty} x_n = \lim_{n\to\infty} x_n$

$\mydisplaysum_{n=1}^\infty x_n = \sum_{n=1}^\infty x_n$
\end{document}

---

Added: As @DavidCarlisle pointed out in the comments, \mylim and \mydisplaylim will very likely increase line spacing. I would not recommend using any of \mylim, \mysum, \mydisplaylim or \mydisplaysum.

Answer 2

You may use \newcommand instead, together with \limits and \nolimits.

\documentclass[12pt]{letter}
\usepackage{amsmath,amssymb,amsthm}

\newcommand{\mylim}{\lim\limits}
\newcommand{\mysum}{\sum\nolimits}

\begin{document}
$\mylim_{n\to\infty} x_n = \displaystyle\lim_{n\to\infty} x_n$

\begin{equation*}
     \mysum_{n=1}^\infty x_n = \displaystyle\sum_{n=1}^\infty x_n
\end{equation*}
\end{document}