What is the command for a sum symbol superimposed on an integral sign?

Question

I want to express a sum over all discrete states and an integral over all continuum states. Usually, the sign for this is a $\sum$ superimposed on an $\int$. Is there a command to make this symbol? If not, can it be made by combining the sum and integral symbols in some way?

Answer 1

You can build the symbol:

\documentclass{article}
\usepackage{amsmath}
\usepackage{graphicx}

\DeclareMathOperator*{\SumInt}{%
\mathchoice%
  {\ooalign{$\displaystyle\sum$\cr\hidewidth$\displaystyle\int$\hidewidth\cr}}
  {\ooalign{\raisebox{.14\height}{\scalebox{.7}{$\textstyle\sum$}}\cr\hidewidth$\textstyle\int$\hidewidth\cr}}
  {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}
  {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}
}
\begin{document}

$\SumInt_{\SumInt} \displaystyle\SumInt$

\end{document}

The MnSymbol package offers \sumint (using it some symbols will change):

\documentclass{article}
\usepackage{MnSymbol}

\begin{document}

$\sumint_{\sumint} \displaystyle\sumint$

\end{document}

Answer 2

This is Unicode Character 'SUMMATION WITH INTEGRAL' (U+2A0B) (⨋). It is available with the STIX fonts. (LaTeX support in beta testing at the moment.) The relevant command is \sumint.

Answer 3

Other option is the mathtools package:

\documentclass{article}
  \usepackage{mathtools}
\begin{document}
  $\int\mathllap{\sum}$

  $\mathclap{\displaystyle\int}\mathclap{\textstyle\sum}$ 

  $F(y)=\mathclap{\displaystyle\int}\mathclap{\textstyle\sum}\;\;\; f(y)\,\mathrm dy$

\end{document}

Due to the problems of spacing around the symbol pointed out by tohecz comments, there are more useful examples in this tex file. The first example in huge characters obtain exactly the the same ouput that using \ooalign{$\textstyle\sum$\cr\hidewidth$\displaystyle\int$\hidewidth\cr}

\documentclass[10]{article}
  \usepackage{mathtools}
  \usepackage{xcolor}

% Reference shadow equation without \int
\newcommand{\refereq}{
\textcolor{cyan}{
$F(y)= \sum f(y)\,\mathrm dy$
}\par\vspace{-1.41\baselineskip}\par}

\begin{document}

\bigskip  
Test inside formula \emph{vs} 
normal position of $\scriptstyle\sum$ 
(cyan shadow without 
$\scriptstyle\int$):
\bigskip 

{\Huge \refereq
$F(y)=
\mathrlap{\displaystyle\int}
\mathrlap{\textstyle\sum}
\phantom{\mathrlap{\displaystyle\int}
\textstyle\sum}
f(y)\,\mathrm dy$}

\bigskip 
Note that different symbol combinations
with {\tt mathtools} have different
spacing effects: \\
\bigskip 

\begin{tabular}{l|l|ll}
 & Symbol & Size & Alignment \\
1 & $\sum$ x 
&  \textbackslash{textstyle} 
& normal  \\
& $\displaystyle\sum$ x 
&  \textbackslash{display} 
& normal\\
2 & $\int$ x 
& \textbackslash{textstyle} 
& normal \\
& $\displaystyle\int$ x 
&  \textbackslash{display} 
& normal\\
3 & $\int\mathllap{\sum}$ x 
& \textbackslash{textstyle} 
& \textbackslash{}math{\bf l}lap
\{$\scriptstyle\sum$\} \\   
4 & $\displaystyle\int\mathllap{\sum}$ x 
& \textbackslash{displaystyle} 
& \textbackslash{}math{\bf l}lap
\{$\scriptstyle\sum$\} \\ 
5 & ${\displaystyle\int}
\mathllap{\sum}$ x 
& mixed 
& \textbackslash{}math{\bf l}lap
\{$\scriptstyle\sum$\} \\
6 & $\mathclap{\displaystyle\int}\mathclap{\textstyle\sum}$ x 
& mixed 
& \textbackslash{}math{\bf c}lap (both) \\
7 & $\mathrlap{\displaystyle\int}\textstyle\sum$ x 
& mixed 
&  \textbackslash{}math{\bf r}lap\{$\int$\}\\
8 & $\mathrlap{\displaystyle\int}\mathrlap{\textstyle\sum} $ x 
& mixed 
& \textbackslash{}math{\bf r}lap (both) \\
9 & $\mathllap{\displaystyle\int}\mathllap{\textstyle\sum}$ x 
& mixed 
& \textbackslash{}math{\bf l}lap (both) \\
\end{tabular}

\bigskip
Comments:

Note that there are little differences
inside the combined symbol. With respect
(8) in (5) and (9) and even in (6) the
$\int$  is lightly displaced to the
rigth, but in (7) is left displaced.  
\bigskip

\refereq
$F(y)=
{\displaystyle\int}\mathllap
{\textstyle\sum}
f(y)\,\mathrm dy$ 
(5) f is spaced from 
$\scriptstyle\int$ but is too
near to $\scriptstyle\sum$ 

\refereq
$F(y)=
\hspace{.5em}\mathclap{\displaystyle
\int}\mathclap{\textstyle\sum} 
\hspace{.5em}
f(y)\,\mathrm dy$ 
(6) Need extra .5em in both sides 
to look as (5).

\refereq
$F(y)=
\mathrlap{\displaystyle\int}
\textstyle\sum
f(y)\,\mathrm dy$ 
(7) f well spaced from 
$\scriptstyle\sum$ (but 
$\scriptstyle\int$ is too left?) 

\refereq
$F(y)=
\mathrlap{\displaystyle\int}
\mathrlap{\textstyle\sum}
\phantom{\mathrlap{\displaystyle
\int}\textstyle\sum}
f(y)\,\mathrm dy$ 
(8) Need recover right space 
but then fit perfectly.

\refereq
$F(y)=
\phantom{\textstyle\sum}
\mathllap{\displaystyle\int}
\mathllap{\textstyle\sum}
\phantom{\mathrlap{
\displaystyle\int}}
f(y)\,\mathrm dy$ 
(9) Need recover left space 
to see just as (5).

So, better approach (with 
{\tt mathtools}, of course), 
seem to be (8) 

\end{document}

Answer 4

This is not a very elaborate answer, but playing with the kerning might be an option:

\int\kern-1em\sum f(x)dx

Also, make sure to always query Detexify² first, it usually will give you the symbol you're looking for, but unfortunately not in this case.