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diff --git a/komfour_zf.tex b/komfour_zf.tex index 13d46e6..c1a0813 100644 --- a/komfour_zf.tex +++ b/komfour_zf.tex @@ -3,7 +3,7 @@ % !TeX root = komfour_zf.tex %% TODO: publish to CTAN -\documentclass[]{tex/hsrzf} +\documentclass[twocolumn, small]{tex/hsrzf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Packages @@ -13,7 +13,7 @@ %% Language configuration \usepackage{polyglossia} -\setdefaultlanguage[variant=swiss]{german} +\setdefaultlanguage[variant=uk]{english} %% Math \usepackage{amsmath} @@ -28,7 +28,7 @@ type={CC}, modifier={by-nc-sa}, version={4.0}, - lang={german}, + lang={english}, ]{doclicense} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -79,11 +79,13 @@ { } \theoremstyle{komfourzf} -\newtheorem{theorem}{Satz} +\newtheorem{theorem}{Theorem} +\newtheorem{definition}{Definition} +\newtheorem{lemma}{Lemma} \setlist[description]{% align=right, labelwidth=2cm, leftmargin=!, % - format={\normalfont\slshape}} + format={\normalfont\itshape}} \setlist[itemize]{% align=right, labelwidth=5mm, leftmargin=!} @@ -93,104 +95,146 @@ \begin{document} -\maketitle -\tableofcontents +\section{Complex Numbers} -\section{Komplexe Zahlen} - -\begin{theorem}[Komplexe Einheit] -\( - j \defeq +\sqrt{-1} \iff j^2 = -1 -\) -\end{theorem} +\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{theorem}[Summe] Seien \(a, b \in \Cset\), - \(a = a_1 + ja_2, a_1,a_2 \in \Rset\) und \"ahnlich f\"ur \(b\) +\begin{definition}[Negation and Sum] Let \(z, w \in \Cset\) \[ - a \oplus b = (a_1 + b_1) + j (a_2 + b_2) + -z = (-z_1, -z_2) \quad + z \oplus w = (z_1 + w_1, z_2 + w_2) \] -\end{theorem} - -\begin{theorem}[Multiplikation] Seien \(a, b \in \Cset\) - \(\arg a = \phi, \arg b = \theta\) - \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] Seien \(a, b \in \Cset\) - mit \(\arg a = \phi, \arg b = \theta\), - dann \(a / b = |a|/|b|\exp{j(\phi - \theta)}\) -\end{theorem} - -\begin{theorem}[Potenzen]~ - \begin{itemize} - \item F\"ur \(n \in \Nset\) gilt - \(\cjs(x)^n = \cjs(nx) \iff \left(\ej{x}\right)^n = \ej{nx}\) - \item - \end{itemize} -\end{theorem} - -\begin{theorem}[Wurzeln] Sei \(\Cset \ni z = r\ej{\phi}\). - \(z\) hat genau \(n\) verschiedene \(n\)-te Wurzeln - (\(n \in \Nset\)) +\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 \(z, w \in \Cset\) \[ - w_{k+1} = \sqrt[n]{r}\exp \frac{j(\phi + 2k\pi)}{n} - \qquad k = 0,1,\ldots,n-1 + (a,b) \odot (c,d) = (ac - bd, ad + bc) \] - Beachtung! Allgemein \(a,b \in \Cset: \sqrt[n]{ab} \neq \sqrt[n]{a}\sqrt[n]{b}\) -\end{theorem} - -\begin{theorem}[Polynome in \(\Cset\)]~ % - \begin{itemize} - \item Jedes komplexe Polynom vom Grad \(\geq 1\) hat mindestens eine Nullstelle. - - \item Ein komplexes Polynom \(p(z) = a_n z^n + \cdots + a_1 z + a_0\) vom - Grad \(n\) zerf\"allt in \(\Cset\) in lauter lineare Faktoren, wobei \(z_k - \in \Cset\) als Nullstellen von \(p(z)\) nicht unbedingt verschieden sein - m\"ussen. +\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} - \item Ein komplexes Polynom \(p(z)\) vom Grad \(n\) hat in \(\Cset\) genau - \(n\) (verschiedene) Nullstellen, wenn diese mit ihrer Vielfachheit - gez\"ahlt werden. - \end{itemize} -\end{theorem} +\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}[Polynome mit reellen Koeffizienten]~ % - \begin{itemize} - \item F\"ur Polynome mit reellen Koeffizienten \(p(z)\) treten nicht-reelle - Nullstellen nur als \emph{konjugiert-komplexe} Paare \(w, \conj{w}\). - In der komplexen Linearfaktor-Zerlegung von \(p(z)\) k\"onnen dan wei - Faktoren \((z-z_0)\) und \((z-\conj{z_0})\) jeweils zu einem - quadratischen Faktor \[ - z^2 - 2 \Re(z_0) z + \len{z}^2 - \] mit \emph{reellen} Koeffizienten zusammengefasst werden. - - \item Ein Polynom mit reellen Koeffizienten von \emph{ungeraden} Grad hat - mindestens eine \emph{reelle} Nullstelle. - - \item Alle Nullstellen des Polynoms \(p(z) = a_n z^n + \cdot + a_1 z + a_0\) - liegen in der Gauss'schen Zahlenebene in einer Kreisscheibe um der - Ursprung mit Radius \[ - R = \sum_{k=0}^n \left\lvert\frac{a_k}{a_n}\right\rvert - \] - - %% TODO: kubische Gleichung - - \item F\"ur allgemeine Gleichungen vom Grad 5 und gr\"osser existieren - prinzipiell \emph{keine} nur aus den 4 Grundoperationen und Wurzeln - zusammengesetzten L\"osungsformeln. - \end{itemize} +\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{Komplexwertige Funktionen} -\section{Fourierreihen} -\section{Spektren} -\section{Diskrete Fouriertransformation} +\section{Complex valued functions} -\section{Lizenz} +\section{License} \doclicenseThis \end{document} -% vim: set et ts=2 sw=2 spelllang=de spell linebreak : +% vim: set et ts=2 sw=2 spelllang=us spell linebreak : |