84 lines
4.4 KiB
TeX
84 lines
4.4 KiB
TeX
\chapter{Mathematical Formulas}
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\label{sec:maths}
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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.
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There is also a \emph{block} environment (cf. \cref{lst:sample-math-environment}).
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Important packages for mathematical features are the \texttt{amsmath},
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\texttt{amsthm}, and \texttt{amssymb} packages of the American Mathematical
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Society, as well as \texttt{mathtools}.
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As with many other environments, adding an asterisk turns off the numbering.
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\example{lst:sample-math-environment}{maths/sample-math-environment}{Exempary math environments}
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\section{A few examples}
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%Die \cref{tbl:maths-common-commands,tbl:maths-logic-sets-braces} listen einige häufig in Formeln verwendeten Befehle auf.
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\begin{table}[H]
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\centering
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\begin{tabular}{ll}
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\toprule
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Source code & Result \\ \midrule
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\mintinline{latex}{\sqrt{16}} & $\sqrt{16}$ \\
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\mintinline{latex}{\frac{3}{4}} & $\frac{3}{4}$ \\
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\mintinline{latex}{e^{\pi}} & $e^{\pi}$ \\
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\mintinline{latex}{\sum_{i=1}\^{n}x^2} & $\sum_{i=1}^{n}x^2$ \\
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\mintinline{latex}{12 \leq 4 x^2 + 13} & $12 \leq 4 x^2 + 13$ \\
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\mintinline{latex}{{n \choose k}} & ${n \choose k}$ \\
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\bottomrule
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\end{tabular}
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\caption{Frequently used commands (square root, fraction, power, sum, inequation, binomial coefficient).
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By \mintinline{latex}{^{…}} and \mintinline{latex}{_{…}}, the content get super- or subscript.}
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\label{tbl:maths-common-commands}
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\end{table}
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\begin{table}[H]
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\widebox{
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\centering
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\begin{tabular}{ll}
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\toprule
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Source code & Result \\ \midrule
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\mintinline{latex}{(x), [x], \lbrace x \rbrace, \lvert x \rvert} & $(x), [x], \lbrace x\rbrace, \lvert x\rvert$\\
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\mintinline{latex}{\exists,\forall,\in,\notin,\infty} & $\exists,\forall,\in,\notin,\infty$ \\
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\mintinline{latex}{\alpha, \beta, \Gamma, \Delta, \varepsilon, \pi} & $\alpha, \beta, \Gamma, \Delta, \varepsilon, \pi$ \\
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\mintinline{latex}{\rightarrow, \leftarrow, \Rightarrow, \Leftarrow, \Leftrightarrow} & $\rightarrow, \leftarrow, \Rightarrow, \Leftarrow, \Leftrightarrow$ \\
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\mintinline{latex}{(A \cup B) \cap C} & $(A \cup B) \cap C$ \\
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\mintinline{latex}{(A \lor B) \land C} & $(A \lor B) \land C$ \\
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\mintinline{latex}{(A \cdot B) \times C} & $(A \cdot B) \times C$ \\ \bottomrule
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\end{tabular}
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\caption{Brackets, quantifiers, greek letters, arrows, operators}
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\label{tbl:maths-logic-sets-braces}
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}
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\end{table}
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\section{Growing brackets}
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Especially in combination with fractions, brackts should grow according to their content.
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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}).
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\example{lst:growing-brackets}{maths/growing-brackets}{Example for growing brackets}
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\section{Lower and upper bounds}
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The \mintinline{latex}{\limits} command renders lower and upper bounds of integrals above and below the integral sign.
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Sums, products, and \todo{quite confusing in English}limits do this
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automatically (c.\,f. \cref{lst:limits}).
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For inline formulas, \mintinline{latex}{\limits} are of limited suitability.
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\example{lst:limits}{maths/limits}{Lower and upper bounds of sums, products, limits and integrals}
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\section{Aligning equations}
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The \texttt{align} environment allows to align multiple equations horizontally, e.\,g. at the \texttt{=} sign (\cref{lst:math-alignment-example}).
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As in tables, the \texttt{\&} sign is used to specify anchorage points.
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Line breaks are denoted by two backslashes.
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\Example{lst:math-alignment-example}{maths/align-example}{maths/align-example_crop}{Equations aligned at equals signs}
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\section{Intensional set notation}\todo{Is that really what this section is about? Looks rather like a “text within maths” section to me.}
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Sometimes sets have to be defined in terms of textual descriptions or longer function names.
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The \LaTeX{} math mode assumes that letters are variables rather than text, which creates problems when they are indeed supposed to be entire words.
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For this case, there is the \mintinline{latex}{\mathrm{}} command (\enquote{math roman}, c.\,f. \cref{lst:set-builder-notation}).
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\example{lst:set-builder-notation}{maths/set-builder-notation}{Problems arising from intensional set notation and their solution}
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