% !TeX program = xelatex % !TeX encoding = utf8 % !TeX root = komfour_zf.tex %% TODO: publish to CTAN \documentclass[twocolumn, small]{tex/hsrzf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Packages %% TODO: publish to CTAN \usepackage{tex/hsrstud} %% Language configuration \usepackage{polyglossia} \setdefaultlanguage[variant=uk]{english} %% Math \usepackage{amsmath} \usepackage{amsthm} %% Layout \usepackage{enumitem} %% License configuration \usepackage[ type={CC}, modifier={by-nc-sa}, version={4.0}, lang={english}, ]{doclicense} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Metadata \course{Elektrotechnik} \module{KomFour} \semester{Fr\"uhlingssemester 2020} \authoremail{npross@hsr.ch} \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\Nset{\mathbb{N}} \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{#1}} \newcommand\len[1]{\lvert#1\rvert} \renewcommand\Re{\operatorname{Re}} \renewcommand\Im{\operatorname{Im}} %% Theorems \newtheoremstyle{komfourzf} % name of the style to be used {\topsep} {\topsep} {} {0pt} {\bfseries} {.} { } { } \theoremstyle{komfourzf} \newtheorem{theorem}{Theorem} \newtheorem{definition}{Definition} \newtheorem{lemma}{Lemma} \setlist[description]{% align=right, labelwidth=2cm, leftmargin=!, % format={\normalfont\itshape}} \setlist[itemize]{% align=right, labelwidth=5mm, leftmargin=!} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Document \begin{document} \section{Complex Numbers} \begin{definition}[Complex Unit and Zero] \[ j \defeq +\sqrt{-1} \iff j^2 = -1 \] \[ 1 = (1,0) \quad 0 = (0,0) \quad j = (0,1) \] \end{definition} \begin{definition}[Negation and Sum] Let \(z, w \in \Cset\) \[ -z = (-z_1, -z_2) \quad z \oplus w = (z_1 + w_1, z_2 + w_2) \] \end{definition} \begin{lemma} The complex numbers form an additive group. Let \(z, w, v \in \Cset\), we have \begin{description}[leftmargin=3cm] \item[Identity] \(z + 0 = z\) \item[Commutativity] \(z + w = w + z\) \item[Associativity] \(z + (w + v) = (z + w) + v\) \item[Inverse property] \(z + (-z) = (-z) + z = 0\) \end{description} \end{lemma} \begin{definition}[Multiplication] Let \(z, w \in \Cset\) \[ (a,b) \odot (c,d) = (ac - bd, ad + bc) \] \end{definition} \begin{lemma} The complex numbers form a commutative ring. Let \(z,w,v \in\Cset\) \begin{description}[leftmargin=3cm] \item[Identity] \(1\cdot z = z\) \item[Commutativity] \(z \cdot w = w \cdot z\) \item[Associativity] \(z (w v) = (z w) v\) \item[Distributivity] \(z (w + v) = zw + zv\) \end{description} \end{lemma} \begin{definition}[Real and imaginary part and conjugation] Let \(z = a + jb\). The \emph{real} part of \(z\) is \(\Re(z) = a\), similarly the \emph{imaginary} part is \(\Im(z) = b\). We can thus define the \emph{complex conjugate} \(\conj{z}\). \[ z = \Re(z) + j\Im(z) \quad \conj{z} = \Re(z) - j\Im(z) \] \end{definition} \begin{definition}[Absolute value] If \(z = a + jb\) we define the \emph{abolute value} \(\len{z} = \sqrt{a^2 + b^2}\) \end{definition} \begin{lemma}[Properties of absolute value] Let \(z,w\in\Cset\). We have \(\len{z} \in\Rset^+_0\), \(z\conj{z} = \len{z}^2\) and as a consequence \(\len{zw} = \len{z}\cdot\len{w}\) and \(\len{\conj{z}} = \len{z}\). In additon we have the inequalities \begin{align*} -\len{z} \leq &\Re(z) \leq \len{z} & \len{z} &\leq \len{\Re(z)} + \len{\Im(z)} \\ -\len{z} \leq &\Im(z) \leq \len{z} & \len{z + w} &\leq \len{z} + \len{w} \end{align*} The last one is the \emph{triangle inequality}. \end{lemma} \begin{definition}[Reciprocal and quotients] If \(z\) is a non-zero complex number we define the \emph{reciprocal} \(z^{-1}\) of \(z\) to be \(z^{-1} = \len{z}^{-2}\conj{z}\). If \(z = 0\) the reciprocal \(0^{-1}\) is left undefined. It is now possible to define \(z/w = zw^{-1}\) with \(z,w \in\Cset\) and \(w \neq 0\). \end{definition} \begin{lemma}[More properties of conjugation] Let \(z,w \in\Cset\). \(\conj{z} = z\) iff \(z \in \Rset\) and \(\conj{z} = \conj{w}\) iff \(z = w\). Furthermore: \begin{align*} \conj{\conj{z}} &= z & \conj{z \pm w} &= \conj{z} \pm \conj{w} & \Re(z) &= (z + \conj{z})/2 \\ \conj{z\cdot w} &= \conj{z}\cdot\conj{w} & \conj{z/w} &= \conj{z}/\conj{w} & \Im(z) &= (z - \conj{z})/2j \end{align*} \end{lemma} \begin{definition}[Polar notation] An alternative rapresentation of a complex number \(z = a + jb\) is its \emph{polar form} \(z = r \angle \phi\), where \(r = \len{z}\) and \(\phi = \arg{z} = r(\cos\phi + j\sin\phi)\). \begin{align*} a &= r\cos\phi & b &= r \sin\phi & r &= \sqrt{z\conj{z}} \end{align*} For \(a = 0\) we define \(\phi = \lim_{a\to 0} \arctan(b/a)\) and otherwise \begin{align*} \phi = \arg(z) &= \begin{cases} \arctan(b/a) & a > 0 \\ \arctan(b/a) + \pi & a < 0 \end{cases} \\ &= \begin{cases} \arccos(a/r) & b \geq 0 \\ -\arccos(b/r) & b < 0 \\ \end{cases} \end{align*} Another variant of this notation is \[ z = r\cjs\phi = r(\cos\phi + j\sin\phi) \] \end{definition} \begin{lemma}[Arithmetic in polar notation] Let \(z,w\in\Cset\) then the product \(zw\) has \[ \len{zw} = \len{z}\cdot\len{w} \quad \arg(zw) = \arg z + \arg w \] Similarly the quotient \(z/w\) follows \[ \len{z/w} = \len{z}/\len{w} \quad \arg(z/w) = \arg z - \arg w \] \end{lemma} \begin{theorem}[De Moivre's formula] Let \(n \in\Nset\) \[ \left(\cos\phi + j\sin\phi\right)^n = \cos(n\phi) + j\sin(n\phi) \] As a consequence with the binomial formula \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^n\) we have \[ \sin(n\phi) = \] \end{theorem} \section{Complex valued functions} \section{License} \doclicenseThis \end{document} % vim: set et ts=2 sw=2 spelllang=us spell linebreak :