\chapter{Mathematical Formulas}
\label{sec:maths}

Mathematical formulas are always set in \emph{math mode}, which, within a paragraph (referred to as \emph{inline}), can be turned on or off with a dollar sign.
There is also a \emph{block} environment (cf. \cref{lst:sample-math-environment}).
Important packages for mathematical features are the \texttt{amsmath}, 
\texttt{amsthm}, and \texttt{amssymb} packages of the American Mathematical 
Society, as well as \texttt{mathtools}.
As with many other environments, adding an asterisk turns off the numbering.

\example{lst:sample-math-environment}{maths/sample-math-environment}{Exempary math environments}

\section{A few examples}
%Die \cref{tbl:maths-common-commands,tbl:maths-logic-sets-braces} listen einige häufig in Formeln verwendeten Befehle auf.

\begin{table}[H]
  \centering
  \begin{tabular}{ll}
  \toprule
  Source code & Result \\ \midrule
  \mintinline{latex}{\sqrt{16}} & $\sqrt{16}$ \\
  \mintinline{latex}{\frac{3}{4}} & $\frac{3}{4}$ \\
  \mintinline{latex}{e^{\pi}} & $e^{\pi}$ \\
  \mintinline{latex}{\sum_{i=1}\^{n}x^2} & $\sum_{i=1}^{n}x^2$ \\
  \mintinline{latex}{12 \leq 4 x^2 + 13} & $12 \leq 4 x^2 + 13$ \\
  \mintinline{latex}{{n \choose k}} & ${n \choose k}$ \\
  \bottomrule
  \end{tabular}
	\caption{Frequently used commands (square root, fraction, power, sum, inequation, binomial coefficient). 
	By \mintinline{latex}{^{…}} and \mintinline{latex}{_{…}}, the content get super- or subscript.}
  \label{tbl:maths-common-commands}
\end{table}

\begin{table}[H]
  \widebox{
    \centering
    \begin{tabular}{ll}
    \toprule
    Source code & Result \\ \midrule
    \mintinline{latex}{(x), [x], \lbrace x \rbrace, \lvert x \rvert} & $(x), [x], \lbrace x\rbrace, \lvert x\rvert$\\
    \mintinline{latex}{\exists,\forall,\in,\notin,\infty} & $\exists,\forall,\in,\notin,\infty$ \\
    \mintinline{latex}{\alpha, \beta, \Gamma, \Delta, \varepsilon, \pi} & $\alpha, \beta, \Gamma, \Delta, \varepsilon, \pi$ \\
    \mintinline{latex}{\rightarrow, \leftarrow, \Rightarrow, \Leftarrow, \Leftrightarrow} & $\rightarrow, \leftarrow, \Rightarrow, \Leftarrow, \Leftrightarrow$ \\
    \mintinline{latex}{(A \cup B) \cap C} & $(A \cup B) \cap C$ \\
    \mintinline{latex}{(A \lor B) \land C} & $(A \lor B) \land C$ \\
    \mintinline{latex}{(A \cdot B) \times C} & $(A \cdot B) \times C$ \\ \bottomrule
    \end{tabular}
    \caption{Brackets, quantifiers, greek letters, arrows, operators}
    \label{tbl:maths-logic-sets-braces}
  }
\end{table}

\section{Growing brackets}

Especially in combination with fractions, brackts should grow according to their content.
This can be achieved by prepending each bracket (\mintinline{latex}{(}, \mintinline{latex}{)}, \mintinline{latex}{[}, \mintinline{latex}{]}, \mintinline{latex}{\lbrace} and \mintinline{latex}{\rbrace}) with a position marker (\mintinline{latex}{\left} oder \mintinline{latex}{\right}).

\example{lst:growing-brackets}{maths/growing-brackets}{Example for growing brackets}

\section{Lower and upper bounds}

The \mintinline{latex}{\limits} command renders lower and upper bounds of integrals above and below the integral sign.
Sums, products, and \todo{quite confusing in English}limits do this 
automatically (c.\,f. \cref{lst:limits}).
For inline formulas, \mintinline{latex}{\limits} are of limited suitability.

\example{lst:limits}{maths/limits}{Lower and upper bounds of sums, products, limits and integrals}

\section{Aligning equations}

The \texttt{align} environment allows to align multiple equations horizontally, e.\,g. at the \texttt{=} sign (\cref{lst:math-alignment-example}).
As in tables, the \texttt{\&} sign is used to specify anchorage points.
Line breaks are denoted by two backslashes.

\Example{lst:math-alignment-example}{maths/align-example}{maths/align-example_crop}{Equations aligned at equals signs}

\section{Intensional set notation}\todo{Is that really what this section is about? Looks rather like a “text within maths” section to me.}

Sometimes sets have to be defined in terms of textual descriptions or longer function names.
The \LaTeX{} math mode assumes that letters are variables rather than text, which creates problems when they are indeed supposed to be entire words.
For this case, there is the \mintinline{latex}{\mathrm{}} command (\enquote{math roman}, c.\,f. \cref{lst:set-builder-notation}).

\example{lst:set-builder-notation}{maths/set-builder-notation}{Problems arising from intensional set notation and their solution}