84 lines
4.7 KiB
TeX
84 lines
4.7 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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\pkg{amsthm}, and \pkg{amssymb} packages of the American Mathematical
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Society, as well as \pkg{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}{Exemplary 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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\code{latex}{\textbackslash sqrt\{16\}} & $\sqrt{16}$ \\
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\code{latex}{\textbackslash frac\{3\}\{4\}} & $\frac{3}{4}$ \\
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\code{latex}{e\textasciicircum\{\textbackslash pi\}} & $e^{\pi}$ \\
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\code{latex}{\textbackslash sum\_\{i=1\}\textasciicircum\{n\}x\textasciicircum 2} & $\sum_{i=1}^{n}x^2$ \\
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\code{latex}{12 \textbackslash leq 4 x\textasciicircum 2 + 13} & $12 \leq 4 x^2 + 13$ \\
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\code{latex}{{n \textbackslash 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 \code{latex}{\textasciicircum\{…\}} and \code{latex}{\_\{…\}}, the content is set in 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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\code{latex}{(x), [x], \textbackslash lbrace x \textbackslash rbrace, \textbackslash lvert x \textbackslash rvert} & $(x), [x], \lbrace x\rbrace, \lvert x\rvert$\\
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\code{latex}{\textbackslash exists,\textbackslash forall,\textbackslash in,\textbackslash notin,\textbackslash infty} & $\exists,\forall,\in,\notin,\infty$ \\
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\code{latex}{\textbackslash alpha, \textbackslash beta, \textbackslash Gamma, \textbackslash Delta, \textbackslash varepsilon, \textbackslash pi} & $\alpha, \beta, \Gamma, \Delta, \varepsilon, \pi$ \\
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\code{latex}{\textbackslash rightarrow, \textbackslash leftarrow, \textbackslash Rightarrow, \textbackslash Leftarrow, \textbackslash Leftrightarrow} & $\rightarrow, \leftarrow, \Rightarrow, \Leftarrow, \Leftrightarrow$ \\
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\code{latex}{(A \textbackslash cup B) \textbackslash cap C} & $(A \cup B) \cap C$ \\
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\code{latex}{(A \textbackslash lor B) \textbackslash land C} & $(A \lor B) \land C$ \\
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\code{latex}{(A \textbackslash cdot B) \textbackslash 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, brackets should grow according to their content.
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This can be achieved by pre-pending each bracket (\code{latex}{(}, \code{latex}{)}, \code{latex}{[}, \code{latex}{]}, \code{latex}{\textbackslash lbrace} and \code{latex}{\textbackslash rbrace}) with a position marker (\code{latex}{\textbackslash left} or \code{latex}{\textbackslash 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 \code{latex}{\textbackslash limits} command renders lower and upper bounds of integrals above and below the integral sign.
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Sums, products, and limits do this
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automatically (cf. \cref{lst:limits}).
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For inline formulas, \code{latex}{\textbackslash limits} are of bounded 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 \mono{align} environment allows to align multiple equations horizontally, e.\,g., at the \mono{=} sign (\cref{lst:math-alignment-example}).
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As in tables, the \mono{\&} 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{Text in math mode}
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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 \code{latex}{\textbackslash text\{\}} command (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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