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# [LaTeX] Sums, Integrals & More

category: Writing | course: LaTeX Math | difficulty:

Some mathematical “functions” exist, that have their own special symbol, and aren’t written by simply using their (shortened) name. I’m talking about: fractions, binomials, (square) roots, sums, products, integrals and logic/set operations.

## Fractions

A fraction is created with the command

\frac{numerator}{denominator}

Fractions can be nested within fractions as often as you like, but those nested fractions keep getting smaller and smaller, which is why I don’t recommend nesting them too deeply. If you want nested fractions to all stay at the same size, use the \cfrac{num}{denom} (continued fraction) command.

Alternatively, if you want your fractions displayed with a diagonal slash, you can achieve this effect by means of the xfrac package. After you’ve included it, use

\sfrac{numerator}{denominator}

$\frac{2}{3} \text{ or } \sfrac{2}{3} \text{ or } x^{\frac{2}{3}}$


## Binomial

The command for creating binomials – sometimes also used for column vectors – works similarly:

\binom{top}{bottom}

$\binom{6}{4} = \frac{6!}{4! \cdot 2!}$


## (Square) Roots

Any type of root can be created with:

\sqrt[n]{equation}

If you leave out the optional parameter, it’s a square root. Otherwise, it’s the n-th root. The symbol automatically scales with the equation.

$\sqrt{a^2 + b^2} \ \sqrt[4]{a^2 + b^2}$


## Sums & Products

The syntax for creating a sum symbol is:

\sum_{subscript}^{superscript}

The syntax for creating a product symbol is:

\prod_{subscript}{superscript}

$\sum_{i=1}^{n} 2i \not= \prod_{i=1}^{n} 2i$


## Integrals

A single integral can be created with

\int_{subscript}^{superscript}

If you want more integrals, you can just place these after each other. But, if you want multiple integrals with a single subscript – for example, a double integral over an area A – you can use the \iint, \iiint and \iiiint commands. These create two, three or four integrals after each other, respectively. For more integrals, you can use \idotsint, which displays two integral symbols with the familiar dots between them.

For cyclic integrals, you need to include the esint package. The syntax is

\oint_{subscript}^{superscript}

For a double cyclic integral, use \oiint.

\usepackage{esint}

\begin{document}
% Special command to make the differential in roman letters
% Not necessary, but highly recommended
\newcommand*\diff{\mathop{}\!\mathrm{d}}

% The actual integrals
$\int_{a}^{b} 4x \diff x \not= \idotsint 4x \diff x$
$\oint_{a}^{b} 4x \diff x \not= \oiint 4x \diff x$
\end{document}


## Logic & Set Operations

For operations on sets (unions and intersections), use the \bigcup and \bigcap commands.

For logical operations (AND and OR), use the \bigwedge and \bigvee commands.

$A \bigcup B = \left\{ x \in \mathbb{R} \middle| x \in A \bigvee x \in B \right\}$
$A \bigcap B = \left\{ x \in \mathbb{R} \middle| x \in A \bigwedge x \in B \right\}$


## Leftovers

Besides these, there are 6 other “big” symbols you can use:

 Command Visual \bigoplus \bigotimes \bigodot \bigsqcup \biguplus \coprod

## Stacking Subscripts

If you want multiple subscripts on top of each other under a big symbol, you could use the atop command, but a much better and easier solution is at hand:

\substack{something \\ something}

$\sum_{\substack{ i=1 \\ i \not= j}}^{n} i$


## Regular Superscripts

A problem arises if you try to get the subscript in display style, but want the superscript in regular text style. To solve this, you can set regular (and other) superscripts for big symbols with

\sideset{left superscripts}{superscript}

$\sideset{_a^b}{'}\sum_{\substack{ i=1 \\ i \not= j}}^{n} i$

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