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| -rw-r--r-- | komfour_zf.tex | 74 |
1 files changed, 68 insertions, 6 deletions
diff --git a/komfour_zf.tex b/komfour_zf.tex index fd4b33d..b9ed253 100644 --- a/komfour_zf.tex +++ b/komfour_zf.tex @@ -15,12 +15,20 @@ \usepackage{polyglossia} \setdefaultlanguage[variant=swiss]{german} +%% Math +\usepackage{amsmath} +\usepackage{amsthm} + +%% Layout +\usepackage{multicol} +\usepackage{enumitem} + %% License configuration \usepackage[ - type={CC}, - modifier={by-nc-sa}, - version={4.0}, - lang={german}, + type={CC}, + modifier={by-nc-sa}, + version={4.0}, + lang={german}, ]{doclicense} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -31,12 +39,38 @@ \semester{Fr\"uhlingssemester 2020} \authoremail{npross@hsr.ch} -\author{\textsl{Naoki Pross} -- \texttt{\theauthoremail}} +\author{Naoki Pross -- \texttt{\theauthoremail}} \title{\texttt{\themodule} Zusammenfassung} \date{\thesemester} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Macros and settings + +%% Equal by definition +\newcommand\defeq{\overset{\mathrm{def.}}{=}} + +%% number sets +\newcommand\Rset{\mathbb{R}} +\newcommand\Cset{\mathbb{C}} + +%% Complex operators +\DeclareMathOperator\cjs{cjs} +\newcommand\cjsl[1]{\cos #1 + j\sin #1} + +\newcommand\ej[1]{e^{j#1}} +\newcommand\conj[1]{\overline{j #1}} + +\renewcommand\Re{Re} +\renewcommand\Im{Im} + +%% Theorems +\newtheorem{theorem}{Satz} +\setlist[description]{% + align=right, labelwidth=2cm, leftmargin=!, % + format={\normalfont\slshape}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Document \begin{document} @@ -45,9 +79,37 @@ \tableofcontents \section{Komplexe Zahlen} +\begin{theorem}[Komplexe Einheit] +\( + j \defeq +\sqrt{-1} \iff j^2 = -1 +\) +\end{theorem} +\begin{theorem}[Multiplikation] \(a, b \in \Cset\) \(\arg a = \phi, \arg b = \theta\) +\begin{description} + \item[Kartesich] \(a \odot b = (a_1 b_1 - a_2 b_2) + j (a_1 b_2 + a_2 b_1)\) + \item[Polar] \(a\odot b &= |a|\cdot|b|\exp{j(\phi + \theta)}\) +\end{description} +\end{theorem} +\begin{theorem}[Division] \(a, b \in \Cset\) \(\arg a = \phi, \arg b = \theta\) +\begin{description} + \item[Kartesich] + \item[Polar] \(a / b &= |a|/|b|\exp{j(\phi - \theta)}\) +\end{description} +\end{theorem} + + +\subsection{Algebra} +Seien \(a, b \in \Cset\) und \(a = a_1 + ja_2, a_1,a_2 \in \Rset\) und \"ahnlich f\"ur \(b\) +\begin{align*} + a \oplus b &= (a_1 + b_1) + j (a_2 + b_2) \\ +\end{align*} + +\subsection{Neue Operationen} +\subsection{Graphische Darstellung} +\subsubsection{Ebene Geometrie} \section{Lizenz} \doclicenseThis - \end{document} +% vim: set et ts=2 sw=2 spelllang=de spell wrap linebreak : |