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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?

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4 Answers

up vote 19 down vote accepted

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}

enter image description here

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

\documentclass{article}
\usepackage{MnSymbol}

\begin{document}

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

\end{document}

enter image description here

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

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

enter image description here

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.

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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}

MWE ouput

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}

enter image description here

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I believe you did not really test your code. Your second option does not work as expected. Try e.g. $F(y)=\mathclap{\displaystyle\int}\mathclap{\textstyle\sum} f(y)\,\mathrm dy$. –  tohecz Aug 23 '12 at 6:34
    
What happens after mixing characters without providing its own space is another problem :). But the fix is easy: add some space. I modified the MWE to solve your example. –  Fran Aug 23 '12 at 9:44
    
Sorry that I'm harsh, but this is not the way it should be done. I'm not sure that the spacing is correct now, and even it were correct, in some other math font it would probably come out incorrect again. I would recommend you to check the solution by Gonzalo, who uses \ooalign to make this correctly. You can find more details on this powerful command here. And please don't take this personally, I learn a lot from egreg's answers and this is an opportunity for you to learn something new! :) –  tohecz Aug 23 '12 at 10:15
    
Okay, add spaces to a guess is wrong for strigents typographers, but changing to '$F(y)=\mathrlap{\displaystyle\int}\mathrlap{\textstyle\sum}\phantom{\mathrlap{\‌​displaystyle\int}\textstyle\sum}f(y)\,\mathrm dy$', I think that also does the right job (I cannot see differences with respect '$F(y)= \sum f(y)\,\mathrm dy$'). Do not apologize for your feedback. I'm just learning from my own answers, not teaching. –  Fran Aug 23 '12 at 11:34
    
This: \mathrlap{\displaystyle\int}\mathrlap{\textstyle\sum}\phantom{\mathrlap{\‌​disp‌​laystyle\int}\textstyle\sum} is equivalent to \mathrlap{\‌​displaystyle\int}\textstyle\sum I think... –  tohecz Aug 23 '12 at 11:36
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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.

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