Is There a Way to Do Quadrupe Integral, Aribitary-Number Integrals, and Integrals with Ellipsis as Closed-Manifold Integral Symbols?

Question

Is there a way to do quadruple integrals, arbitrary-length integrals, and integrals with an ellipsis as closed-manifold integrals in LaTeX? If someone could define some macros, that would be great. This question How to insert a triple line integral? goes over triple integrals and this link https://www.overleaf.com/learn/latex/Integrals%2C_sums_and_limits goes over the topic for "regular" (non-closed-manifold) integrals.

EDIT: This is a partial answer but 1) the manifold of integration is set in the preamble and 2) the ellipses are not centered on the integral symbols

\documentclass{article}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{txfonts}
\usepackage{tikz}
\usetikzlibrary{math}
\usetikzlibrary{shapes}
\makeatletter
\ams@newcommand{\iiiiint}{\DOTSI\protect\MultiIntegral{5}}
\renewcommand{\MultiIntegral}[1]{%
  \edef\ints@c{\noexpand\intop
    \ifnum#1=\z@\noexpand\intdots@\else\noexpand\intkern@\fi
    \ifnum#1>\tw@\noexpand\intop\noexpand\intkern@\fi
    \ifnum#1>\thr@@\noexpand\intop\noexpand\intkern@\fi
    \ifnum#1>4 \noexpand\intop\noexpand\intkern@\fi % <---- added
    \noexpand\intop
    \noexpand\ilimits@
  }%
  \futurelet@let@token\ints@a
}
\makeatother
\newcommand*\oiiiint{%
   \tikz \node[draw, ellipse, inner xsep=-4pt, inner ysep=-9pt] {$\displaystyle\iiiint$};%
}
\newcommand*\oiiiintM{%
   \tikz \node[draw, ellipse, inner xsep=-4pt, inner ysep=-13pt] {$\displaystyle\iiiint\limits_M$};%
}
\newcommand*\oiiiiint{%
   \tikz \node[draw, ellipse, inner xsep=-4pt, inner ysep=-9pt] {$\displaystyle\iiiiint$};%
}
\newcommand*\oiiiiintM{%
   \tikz \node[draw, ellipse, inner xsep=-4pt, inner ysep=-13pt] {$\displaystyle\iiiiint\limits_M$};%
}
\newcommand*\oidotsint{%
   \tikz \node[draw, ellipse, inner xsep=-4pt, inner ysep=-9pt] {$\displaystyle\idotsint$};%
}
\newcommand*\oidotsintM{%
   \tikz \node[draw, ellipse, inner xsep=-4pt, inner ysep=-12pt] {$\displaystyle\idotsint\limits_M$};%
}
\begin{document}
\noindent $\displaystyle \oint\limits_C \omega$ \ $\displaystyle \oiint\limits_S \eta$ \ $\displaystyle \oiiint\limits_M \theta$ \ \indexspace $\oiiiint _M \alpha$ \ \indexspace $\oiiiintM \alpha$ \ \indexspace  $\oiiiiint _M \beta$ \ \indexspace $\oiiiiintM \beta$ \ \indexspace $\oidotsint _M \gamma$ \ \indexspace $\oidotsintM \gamma$ \ \indexspace Can this be done with $\ignorespaces $\backslash$oiiiint_M $\backslash$xi$, $\ignorespaces $\backslash$oint[n]_M $\backslash$phi$, and $\ignorespaces $\backslash$oidotsint_M $\backslash$lambda$?
\end{document}

Answer 1

pdfMsym provides a macro for rendering arbitrary length closed integrals with \oiNint, for example:

\[ \oiNint{5} \]

In order to create such a construction, pdfMsym utilizes two macros: \putexsym and \@skewedlim@op which place "extended symbols" (like the loop in \oiNint) over material and create large math operators which have skewed limits like \int respectively. The latter is of less interest, and details on it are provided in the pdfMsym documentation section 3.1, the same is true for \putexsym. To showcase the use of \putexsym I will demonstrate how to create \oidotsint:

\def\oidotsint{\putexsym{\int\mkern-8mu\cdots\mkern-8mu\int}\@BigLeftcirclecap\@BigRightcirclecap{2.5}{4}}
[ \oidotsint ]

This places an extended symbol capped by the linecaps/lineheads \@BigLeftcirclecap and \@BigRightcirclecap which are big half circle linecaps over \int\mkern-8mu\cdots\mkern-8mu\int. The value 2.5 is the height of these linecaps (and should not be changed), and 4 tells pdfMsym how much to reduce the width of the symbol (if we replace 4 with 0 the caps would be placed at the edges of \int...\int which doesn't look great). The exact dimensions of 4 and 2.5 are relative to the fontsize (given to \pdfmsymsetscalefactor) and the current mathmode. This definition will not properly set limits for \oidotsint, either surround this with a \mathop{...} or use \@skewedlim@op.

More information can be found in pdfMsym's documentation. pdfMsym is a package I wrote, and I don't have much experience writing packages so I don't know how reliable it is. Furthermore, I suspect that some features will change in the future so the details in this post may become outdated in the future.

Answer 2

With package esint:

\documentclass[border=12pt]{standalone}
\usepackage{esint}
\begin{document}
\def\arraystretch{2}
\begin{tabular}{c c c}
macro  & textstyle & displaystyle \\
\verb+\int+ & $\int$ & $\displaystyle \int$\\
\verb+\iint+ & $\iint$& $\displaystyle \iint$\\
\verb+\iiint+ & $\iiint$ & $\displaystyle \iiint$\\
\verb+\iiiint+ & $\iiiint$ & $\displaystyle \iiiint$\\
\verb+\idotsint+ & $\idotsint$ & $\displaystyle \idotsint$\\
%
\verb+\oint+ & $\oint$ & $\displaystyle \oint$\\
\verb+\oiint+ & $\oiint$ & $\displaystyle \oiint$\\
\verb+\varoiint+ & $\varoiint$ & $\displaystyle \varoiint$\\
\verb+\sqint+ & $\sqint$ &$\displaystyle \sqint$\\
\verb+\sqiint+ & $\sqiint$ & $\displaystyle \sqiint$\\
\verb+\ointctrclockwise+ & $\ointctrclockwise$ & $\displaystyle \ointctrclockwise$\\
\verb+\ointclockwise+ & $\ointclockwise$ & $\displaystyle \ointclockwise$\\
\verb+\varointclockwise+ & $\varointclockwise$ & $\displaystyle \varointclockwise$\\
\verb+\varointctrclockwise+ & $\varointctrclockwise$ & $\displaystyle \varointctrclockwise$\\
\verb+\fint+ & $\fint$  &$\displaystyle \fint$\\
\verb+\landupint+ & $\landupint$  &$\displaystyle \landupint$\\
\verb+\landdownint+ & $\landdownint$ & $\displaystyle \landdownint$
\end{tabular}
\end{document}

Answer 3

These commands are defined in unicode-math, so you can use them with your font of choice in LuaTeX or XeTeX.

\documentclass{article}
\tracinglostchars=3 % Error if a symbol is missing
\usepackage{unicode-math}
\usepackage{newcomputermodern}
\begin{document}
\begin{align}
  & \iiiint_a^b \ % quadruple integral
  & \oiiint_C \ % Triple contour integral
  & \idotsint_a^b % ellipsis integral
\end{align}
\end{document}

There aren’t aribrarily-large integral symbols. You could, I suppose, fake one with negative kerning between single integral signs, but if you end up needing to distinguish between quintuple and sextuple integration, I recommend you find a notation that’s easier to read. Possibly \int_B where $B$ is a higher-dimensional region, if that makes sense in your research area?

Not all fonts will have all these symbols, but New Computer Modern Math, STIX Two Math and XITS Math are three that are especially comprehensive. These fonts also let you choose between upright and slanted integrals as stylistic sets.

If your math font does not contain one of the symbols, you can swap it in with a command like

\setmathfont{STIX Two Math}[
  range={\iiiint,\oiiint},
  Scale=MatchUppercase]