From 717c168c4632205655ac583a1bb0e67ce4fa3014 Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Mon, 25 Aug 2025 13:56:09 +0200 Subject: [PATCH] Working --- .vscode/ltex.dictionary.de-DE.txt | 1 + .vscode/ltex.hiddenFalsePositives.de-DE.txt | 3 + 01-Wiederholung.tex | 257 +++++++++++++++++++- Funktionentheorie.tex | 2 + Notizen/220427-Vorlesung.pdf | Bin 199476 -> 0 bytes 5 files changed, 261 insertions(+), 2 deletions(-) create mode 100644 .vscode/ltex.dictionary.de-DE.txt create mode 100644 .vscode/ltex.hiddenFalsePositives.de-DE.txt delete mode 100644 Notizen/220427-Vorlesung.pdf diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt new file mode 100644 index 0000000..44e16a1 --- /dev/null +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -0,0 +1 @@ +Multiplikationsabbildung diff --git a/.vscode/ltex.hiddenFalsePositives.de-DE.txt b/.vscode/ltex.hiddenFalsePositives.de-DE.txt new file mode 100644 index 0000000..45be18f --- /dev/null +++ b/.vscode/ltex.hiddenFalsePositives.de-DE.txt @@ -0,0 +1,3 @@ +{"rule":"GERMAN_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QFür komplexe Zahlen \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q bezeichnet man den Winkel zwischen der \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q-Achse und der Achse \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q als das “Argument” von \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} +{"rule":"GERMAN_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QMan invertiert eine komplexe Zahl, indem man den Betrag invertiert und das Argument negiert.\\E$"} +{"rule":"GERMAN_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QEs ist \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} diff --git a/01-Wiederholung.tex b/01-Wiederholung.tex index a590012..927910d 100644 --- a/01-Wiederholung.tex +++ b/01-Wiederholung.tex @@ -1,11 +1,264 @@ % spell checker language \selectlanguage{german} -\chapter{Platzhalter} +\chapter{Komplexe Zahlen} -\section{Platzhalter} +\section{Konstruktion, Notation und elementare Eigenschaften} \sideremark{Vorlesung 1} +In der Vorlesung „Analysis I“ haben Sie Funktionen $f : ℝ → ℝ$ diskutiert. In +der Vorlesung „Funktionentheorie“ diskutieren wir Funktionen $f : ℂ → ℂ$. Damit +alle auf demselben Stand sind, wiederhole ich in aller Kürze die wesentlichen +Begriffe zu den komplexen Zahlen. + +\begin{konstruktion}[Konstruktion der komplexen Zahlen] + Wir definieren auf der reellen Ebene $ℝ²$ zwei Abbildungen (oder: + „Operationen“): + \begin{align*} + + : ℝ² ⨯ ℝ² &→ ℝ², & \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}, \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} &↦ \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix} \\ + · : ℝ² ⨯ ℝ² &→ ℝ², & \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}, \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} &↦ \begin{pmatrix} x_1 x_2 - y_1 y_2 \\ x_1 y_2 + x_2 y_1 \end{pmatrix} + \end{align*} +\end{konstruktion} + +\begin{erinnerung}[Komplexe Zahlen] + Das Tripel $ℂ := (ℝ², +, ·)$ ist ein Körper, den wir als „Körper der komplexen + Zahlen\index{komplexe Zahlen}“ bezeichnen. Für das neutrale Element der + Addition/Multiplikation gilt: + \[ + 0_ℂ = \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \quad + 1_ℂ = \begin{pmatrix} 1 \\ 0 \end{pmatrix}. + \] + Die Menge der invertierbaren Elemente wird mit $ℂ^*$ bezeichnet, $ℂ^* = ℂ ∖ + \{0_ℂ \}$. +\end{erinnerung} + + +\subsection{Notation} + +\begin{erinnerung}[Reelle Zahlen als Teilmenge der komplexen Zahlen] + Die Abbildung + \[ + ι : ℝ → ℂ, \quad x ↦ \begin{pmatrix} x \\ 0 \end{pmatrix} + \] + ist ein Körpermorphismus. Das bedeutet: Für alle $z_1, z_2 ∈ ℂ$ gilt: + \[ + ι(z_1 + z_2) = ι(z_1) + ι(z_2) + \quad\text{und}\quad + ι(z_1 · z_2) = ι(z_1) · ι(z_2). + \] + Weiter gilt: $ι(1) = 1$ und $ι(0) = 0$. Weil die Abbildung $ι$ injektiv ist, + identifiziert man die reellen Zahlen häufig mit der Bildmenge von $ι$, nämlich + der $x$-Achse in $ℝ²$, und fasst $ℝ$ als Teilmenge von $ℂ$ auf. +\end{erinnerung} + +\begin{notation} + Man bezeichnet die $x$-Achse als „reelle Achse“\index{reelle Achse}, die + $y$-Achse als „imaginäre Achse“\index{imaginäre Achse}. Das Element + $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ wird mit $i$ bezeichnet. Statt $z = + \begin{pmatrix} x \\ y \end{pmatrix}$ schreibe kurz $z = x + iy$. Dabei nenne + $x$ den „Realteil von $z$“\index{Realteil} und $y$ den „Imaginärteil von + $z$“\index{Imaginärteil}. Die Schreibweisen $x = \operatorname{Re}(z)$ und $y + = \operatorname{Im}(z)$ sind üblich. +\end{notation} + + +\subsection{Betrag und Argument} + +\begin{notation}[Betrag einer komplexen Zahl] + \index{Betrag}Die euklidische Norm auf $ℂ$ wird mit $|·|$ bezeichnet. Für + eine komplexe Zahl $z = x + y·i$ nennt man $|z|$ den „Betrag von $z$“. Es + gilt $|z| = \sqrt{x² + y²}$. +\end{notation} + +\begin{bemerkung}[Betrag eines Produkts] + Für alle $z_1, z_2 ∈ ℂ$ gilt: $|z_1 · z_2| = |z_1| · |z_2|$. +\end{bemerkung} + +\begin{bemerkung}[Dreiecksungleichung] + Für alle $z_1, z_2 ∈ ℂ$ gilt: $|z_1 + z_2| ≤ |z_1| + |z_2|$. +\end{bemerkung} + +\begin{notation}[Argument einer komplexen Zahl] + \index{Argument}Für komplexe Zahlen $z ∈ ℂ^*$ bezeichnet man den Winkel + zwischen der $x$-Achse und der Achse $z·ℝ$ als das „Argument“ von $z$. Die + Schreibweise $\arg z$ ist üblich. +\end{notation} + +\begin{beobachtung}[Darstellung von komplexen Zahlen durch Betrag und Argument] + Sei $z ∈ ℂ^*$. Dann ist + \[ + z = |z| · \begin{pmatrix} \cos (\arg z) \\ \sin (\arg z) \end{pmatrix}. + \] +\end{beobachtung} + +\begin{beobachtung}[Darstellung Argument des negierten] + Sei $z ∈ ℂ^*$. Dann ist + \[ + \arg (-z) = π + \arg z. + \] +\end{beobachtung} + + +\subsection{Geometrische Bedeutung der Körperoperationen} + +Es ist per Definition klar, dass die Addition im Körper $ℂ$ einfach die +Vektoraddition der $ℝ²$ ist. Um die Bedeutung der Multiplikation zu verstehen, +sei $z ∈ ℂ^*$ irgendeine Zahl, mit Betrag $λ := |z|$ und Argument $α := \arg z$. +Man rechne man nach, dass die Multiplikationsabbildung +\[ + m_z : ℂ → ℂ, \quad \begin{pmatrix} x \\ y \end{pmatrix} ↦ z · \begin{pmatrix} x \\ y \end{pmatrix} +\] +gleich der Abbildung +\[ + \begin{pmatrix} x \\ y \end{pmatrix} ↦ λ·\begin{pmatrix} \cos α & - \sin α \\ \sin α & \cos α \end{pmatrix} · \begin{pmatrix} x \\ y \end{pmatrix} +\] +ist. Dies ist eine Drehstreckung. + +\begin{merke}[Multiplikation ist Drehstreckung] + Multiplikation einer komplexen Zahl $w$ mit $z ∈ ℂ^*$ bedeutet: Drehe $w$ um + den Winkel $α = \arg z$ und strecke das Ergebnis um den Faktor $λ = |z|$. +\end{merke} + +\begin{kons}[Multiplikation zweier komplexer Zahlen] + Gegeben komplexe Zahlen $z_1, z_2 ∈ ℂ$, dann ist $|z_1 · z_2| = |z_1| · + |z_2|$. Falls $z_1$ und $z_2 ∈ ℂ^*$ sind, gilt $\arg (z_1 · z_2) = \arg (z_1) + + \arg (z_2)$. +\end{kons} + +\begin{merke}[Multiplikation zweier komplexer Zahlen] + Man multipliziert zwei komplexe Zahlen, indem man die Beträge multipliziert + und die Argumente addiert. +\end{merke} + +\begin{merke}[Division zweier komplexer Zahlen] + Man dividiert zwei komplexe Zahlen, indem man die Beträge dividiert und die + Argumente subtrahiert. +\end{merke} + +\begin{merke}[Multiplikatives Inverses] + Man invertiert eine komplexe Zahl, indem man den Betrag invertiert und das + Argument negiert. +\end{merke} + + +\subsection{Konjugation} + +\begin{notation}[Konjugation] + Die Spiegelung an der reellen Achse, + \[ + τ : ℂ → ℂ, \quad \begin{pmatrix} x \\ y \end{pmatrix} ↦ \begin{pmatrix} x \\ -y \end{pmatrix} + \] + wird als „Konjugation“ bezeichnet\index{Konjugation}. Anstelle von $τ(z)$ ist + die Schreibweise $\overline{z}$ üblich. +\end{notation} + +\begin{erinnerung}[Konjugation als Körpermorphismus] + Komplexe Konjugation ist ein Körpermorphismus. Das bedeutet: Für alle $z_1, + z_2 ∈ ℂ$ gilt: + \[ + \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} + \quad\text{und}\quad + \overline{z_1 · z_2} = \overline{z_1} · \overline{z_2}. + \] + Weiter gilt: $\overline{0} = 0$ und $\overline{1} = 1$. +\end{erinnerung} + +\begin{erinnerung}[Allererste Eigenschaften der Konjugation] + Sei $z ∈ ℂ$. Dann gilt Folgendes. + \begin{itemize} + \item Die Gleichung $\overline{z} = z$ gilt genau dann, wenn $z ∈ ℝ$ ist. + \item Es ist $|z| = |\overline{z}|$ und $\arg \overline{z} = - \arg z$. + \item Es ist $\operatorname{Re}(z) = \frac{z + \overline{z}}{2}$ und + $\operatorname{Im}(z) = \frac{z - \overline{z}}{2i}$. + \item Es ist $z·\overline{z} = |z|²$. + \item Falls $z ∈ ℂ^*$ ist, so ist $\overline{z^{-1}} = + \overline{z}^{-1}$. + \end{itemize} +\end{erinnerung} + + +\subsection{Geometrische Bedeutung des multiplikativen Inversen} + +Gegeben eine komplexe Zahl $z ∈ ℂ^*$, so kann man das multiplikative Inverse von +$z$ mithilfe der Konjugation leicht ausrechnen. Es ist nämlich +\[ + \frac{1}{z} = \frac{\overline{z}}{\overline{z}·z} = \frac{\overline{z}}{|z|²} = \overline{z/|z|²}. +\] +Aber was bedeutet das anschaulich? + +\begin{erinnerung}[Spiegelung am Einheitskreis] + Die Abbildung + \[ + ℝ² ∖ \{0\} → ℝ² ∖ \{0\}, \quad z ↦ \frac{z}{|z|²} + \] + heißt „Spiegelung am Einheitskreis“\index{Spiegelung am Einheitskreis}. Die + Spiegelung am Einheitskreis lässt das Argument unverändert und invertiert den + Betrag. +\end{erinnerung} + +\begin{merke}[Multiplikatives Inverses] + Man invertiert eine komplexe Zahl, indem man am Einheitskreis spiegelt und + konjugiert (=an der $x$-Achse spiegelt). +\end{merke} + + +\subsection{Geometrische Bedeutung des Quadrierens} + +Nach der vorhergehenden Diskussion ist klar, was die Abbildung $q : ℂ → ℂ$, $z ↦ +z²$ geometrisch macht: +\begin{itemize} + \item Der Betrag wird quadriert. + \item Argument wird verdoppelt. +\end{itemize} +Aus dieser Beschreibung ergibt sich sehr schnell, dass jede komplexe Zahl eine +Quadratwurzel besitzt. + +\begin{kons}[Existenz von Quadratwurzeln] + Jede komplexe Zahl hat eine Quadratwurzel. +\end{kons} +\begin{proof} + Gegeben eine Zahl $z ∈ ℂ$. Falls $z = 0$ ist, dann ist $w := 0$ eine + Quadratwurzel von $z$. Ansonsten sei $λ := |z|$ und $α := \arg z$. Weiter sei + $w ∈ ℂ^*$ die Zahl mit Betrag $\sqrt{λ}$ und Argument $α/2$. Mit dieser Wahl + ist $w² = z$, die Zahl $w$ ist also eine Quadratwurzel von $z$. +\end{proof} + +Quadratwurzeln sind natürlich nicht eindeutig, denn wenn $w$ eine Quadratwurzel +von $z$ ist, dann ist $-w$ ebenfalls eine Quadratwurzel. + +\begin{kons}[Surjektivität der Quadrat-Abbildung] + Die Abbildung $q : ℂ → ℂ$, $z ↦ z²$ ist surjektiv, aber nicht injektiv. Es + ist $q^{-1}(0) = \{0\}$. Jede andere Faser von $q$ enthält genau zwei Punkte, + die sich exakt um das Vorzeichen unterscheiden. \qed +\end{kons} + +Aber Achtung! Obwohl jede komplexe Zahl eine Quadratwurzel besitzt, gibt es +keine stetige Wurzelfunktion! Das folgende Lemma zeigt, dass es nicht einmal +auf der Teilmenge $ℂ^* ⊂ ℂ$ eine stetige Wurzelfunktion geben kann. + +\begin{lem}[Nichtexistenz einer stetigen Wurzelfunktion] + Es sei $w: ℂ^* → ℂ^*$ eine Funktion, sodass für alle $z ∈ ℂ^*$ gilt: $w(z)² = + z$. Dann ist die Abbildung $w$ nicht stetig. +\end{lem} +\begin{proof} + Sei eine Funktion $w$ gegeben. Wir argumentieren mit Widerspruch und nehmen + an, dass die Funktion $w$ stetig sei. Dann ist aber die Funktion + \[ + f : ℂ^* → ℂ^*, \quad z ↦ w(z²)/z + \] + aber ebenfalls stetig. Per Annahme gilt nun aber für jedes $z ∈ ℂ^*$ die + Gleichung $w(z²)² = z²$. Das bedeutet: die komplexen Zahlen $w(z²)$ und $z$ + unterscheiden sich nur um ein Vorzeichen. Also nimmt die stetige Funktion $f$ + nur die Werte $± 1$ an. Weil $f$ stetig und $ℂ^*$ zusammenhängend ist, muss + $f$ also konstant sein. Auf der anderen Seite gilt für jedes $z ∈ ℂ^*$ die + Gleichung + \[ + f(-z) = \frac{w((-z)²)}{-z} = \frac{w(z²)}{-z} = - f(z), + \] + also ist $f$ doch nicht konstant! +\end{proof} + + % !TEX root = LineareAlgebra2 diff --git a/Funktionentheorie.tex b/Funktionentheorie.tex index 01f7b79..ba043e4 100644 --- a/Funktionentheorie.tex +++ b/Funktionentheorie.tex @@ -42,10 +42,12 @@ \newcommand\video[1]{\href{https://cplx.vm.uni-freiburg.de/storage/Lehre/Vorlesungen/LA2/#1-Video.mp4}{Erklärvideo #1} \href{https://cplx.vm.uni-freiburg.de/storage/Lehre/Vorlesungen/LA2/#1-Skript.pdf}{(Skript)}} \theoremstyle{plain} +\newtheorem{merke}[thm]{Merke} \newtheorem{aufgabe}[thm]{Aufgabe} \newtheorem{satz}[thm]{Satz} \newtheorem{situation}[thm]{Situation} \newtheorem{lemma}[thm]{Lemma} +\newtheorem{kons}[thm]{Konsequenz} \newtheorem{kor}[thm]{Korollar} \newtheorem{definition}[thm]{Definition} \newtheorem{fakt}[thm]{Fakt} diff --git a/Notizen/220427-Vorlesung.pdf b/Notizen/220427-Vorlesung.pdf deleted file mode 100644 index 6765d0dcbb2c3f593a551ce8ab92ac836974cba8..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 199476 zcmbrk1ymecvoMN#aDv0&?hxGF-QC>>4el1)-QC??0t5)|?vmhckDTv(=ezg*|9bDP z^=3^^cU9M}uDwfkRZWvAh=|cK(X+#m^%wQm4OGCvF##9>_C{84000jUfI-E>!4$wC zYzQ>8u{Zy~eS8??U4RZQKuJ5GsU6T6pbhx)F+vHT`}bW1LvvGS0P7#z-xyg_prOeJ z2LSWmNIpI|Q#+GC5BY=npAY*t`QM|JU5tQ#385h?^EXPy)Xp4e0bpWc<@msSaB~Ja znHt){@$kDl7#drf0s%&*=9YH6#OED7!~jbZUSf4NSw>k0VN(lB2~Q_eWluR3V^3>i zP7`8&J~$qCE_Yi8+YccD?zT2|&Rp)i#D*WznR0zx|3Nble-JsDm~knIi2Y6S0r3)B z0D%r%3=D2=ZuD*}^!85X3{0GyoD7W249v`Q9~5-X9(F)OcRD*~k`E3a>-mR=h^e!& zlcfXD(%ug62R1aacLDNJQ2fo<;e#>7-;C{?>Hm_0-q_xj;o}Aa6Fmz9#UD-|ygpzq z1t)tG7h_W=K4)V?JD|P2jWf?5y8qz+Ln>|R;b!k-;>`CiihsaAVln&=x&F7ZeyrO6 zT~>EPhYwj9|B>~BkW1Fy#L~<|#1Lr8$ISS#K8&n%%wJTPn7P1I&_&4!i-t%F?f1C_J zDZ=%~Lm63_zc4Ygv3xk^AF}_%>_4IXPt4}w`D=Gz_$%NuNL!jXYcu`XJd^-Te?@-) z6Z>BfGvmKbf7U|@!2GY>f|>I#NS8s_-o*|GVE+$$wvQYCVh>>WJJ2(;vaTx!&7YWHYHV*}`d>BxchJgd@VPZD^pI`09G$sl+tp@UitCJlmA|f5o2v@% zlzNH|v2gp(p%nxSVrTWLfOVuUCkxV7@0{pypt)tgVYUyytMl=;#j~(JVO+={0~h2C z^dZGTgdECIQp9N{HqK^`oL`hul9O6Fu$@(CBvjzc-9i?+eJ1h^!H{$}^#{U1eXMDv z8MyZHkWaKj#I?@$=X;Ma8k9$UVkuxaO*mOmVTw(>IGm(p8DPG!nia(!7B!_YS#Cxe zYD^*xz+&QG8T+YT81Xx78-RT;uTv#GBG;%ih&pMk@~Skyyt6zHhuR*aT&sMtyauP5 zynNZJY$eyVjO3)Vy(rtlV!KI0ONp)^S)8U?b>YNPFa4EQ@B#ru3>5^VR^znxDaLly z741W18LBoPzpKp?*o??gkjlMawNPgu;(ekHNS$DtL?KY*p_&Fn8Uzb|69$+Z$kCxJ zTgF(;Mes0k+aPXo?}s~ zFb|&|3Smr+W&Z35b9);>HReg+QywUJ5|wFsnxR=7rwxI&&FnH%FC?kPjXMXw2Gt4 zzgDQ1*;`3jAduvf`tEF#X@&GIc?x(MpZuT!tvdTa7lo#*P8_SjIG2_RG< zI3pZ>3y#XIYjaG#%q)wAWo}}eVA5s8WGgsmJ_tUrPGG`*Fy! 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