From fe2b4220eb6c829bf2cdbc2969ec9e81345e6401 Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Mon, 17 Nov 2025 14:24:59 +0100 Subject: [PATCH] =?UTF-8?q?Working=E2=80=A6?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .vscode/ltex.dictionary.de-DE.txt | 4 + .vscode/ltex.hiddenFalsePositives.de-DE.txt | 4 + 07-nullstelle.tex | 3 +- 09-annulus.png | Bin 0 -> 52119 bytes 09-singularities.tex | 290 ++++++++++++++------ Notizen/220617-Vorlesung.pdf | Bin 281445 -> 0 bytes 6 files changed, 217 insertions(+), 84 deletions(-) create mode 100644 09-annulus.png delete mode 100644 Notizen/220617-Vorlesung.pdf diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt index 0083c71..d599b1e 100644 --- a/.vscode/ltex.dictionary.de-DE.txt +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -55,3 +55,7 @@ Casteggio Hebbarkeitsatz Laurentreihe .te +Laurentreihen +Laurentreihenentwicklung +Alphonse +funktionentheoretischen diff --git a/.vscode/ltex.hiddenFalsePositives.de-DE.txt b/.vscode/ltex.hiddenFalsePositives.de-DE.txt index 763371b..89b4e89 100644 --- a/.vscode/ltex.hiddenFalsePositives.de-DE.txt +++ b/.vscode/ltex.hiddenFalsePositives.de-DE.txt @@ -27,3 +27,7 @@ {"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QInsbesondere gibt es offene Umgebung \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, sodass auf \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q eine \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q-te Wurzelfunktion existiert, also eine Funktion \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q sodass für jedes \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q die Gleichung \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q gilt.\\E$"} {"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QWegen \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q gibt es eine offene Umgebung \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q auf der eine \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q-te Wurzelfunktion existiert, also eine Funktion \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q sodass für jedes \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q die Gleichung \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q gilt.\\E$"} {"rule":"UNPAIRED_BRACKETS","sentence":"^\\QDie Funktionen \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q und \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q stimmen demnach auf \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q überein, sind nach Korollar\\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q (“Identitätssatz für holomorphe Funktionen”) also gleich!\\E$"} +{"rule":"GERMAN_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QDer Punkt \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q ist genau dann eine wesentliche Singularität von \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, wenn der Hauptteil unendlich viele viele Summanden hat.\\E$"} +{"rule":"DE_CASE","sentence":"^\\QWege im Kreisring \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Als erstes werde ich kann den Funktionswert \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q als Integral über den Weg \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q ausdrücken.\\E$"} +{"rule":"DE_CASE","sentence":"^\\QWege im Kreisring \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Als Erstes werde ich kann den Funktionswert \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q als Integral über den Weg \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q ausdrücken.\\E$"} +{"rule":"GERMAN_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QDer Punkt \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q ist genau dann eine wesentliche Singularität von \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, wenn der Hauptteil unendlich viele Summanden hat.\\E$"} diff --git a/07-nullstelle.tex b/07-nullstelle.tex index 6bf9b5c..a3dbe9f 100644 --- a/07-nullstelle.tex +++ b/07-nullstelle.tex @@ -139,7 +139,8 @@ abgeschlossen, also eine ganze Zusammenhangskomponente von $U$! \end{proof} -\section*{Beispielanwendung: Selbstabbildungen der Kreisscheibe} +\section{Beispielanwendung: Automorphismen der Kreisscheibe} +\label{sec:7-3}% \sideremark{Vorlesung: 11} Im Folgenden bezeichnen wir mit $Δ := B_1(0)$ die offene Einheitskreisscheibe in diff --git a/09-annulus.png b/09-annulus.png new file mode 100644 index 0000000000000000000000000000000000000000..cdb0895d98d80c7a52aa62cc7f8c8667c6b9589a GIT binary patch literal 52119 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Gegeben eine holomorphe Funktion $f ∈ 𝒪(U ∖ ρ)$. Was kann ich @@ -214,90 +215,22 @@ Kreisringen. Kreisring $K_{r,R}(0)$ gegeben, $f ∈ 𝒪(K_{r,R}(0))$. \end{situation} -\begin{konstruktion}[Potenzreihenentwicklung auf Kreisring] +\begin{konstruktion}[Potenzreihenentwicklung auf Kreisring]\label{kons:9-2-4}% In Situation~\ref{situation:9-2-2} sei ein Punkt $w ∈ K_{r,R}(0)$ gegeben. Dann wähle reelle Zahlen $a$ und $A$ mit \[ r < a < |w| < A < R. \] - und betrachte den folgenden Weg $γ$ in $K_{r,R}(0)$: - \begin{center} - \begin{tikzpicture}[scale=2] + und betrachte den in Abbildung~\ref{fig:9-2-1} gezeigten Weg $γ$ in $K_{r,R}(0)$. + \begin{figure} + \begin{center} + \includegraphics[width=10cm]{09-annulus.png} + \end{center} -% Radii -\def\r{0.6} -\def\a{1.0} -\def\R{1.8} - -% Outer and inner circles of the annulus -\draw[thick] (0,0) circle (\R); -\draw[thick] (0,0) circle (\a); - -% Coordinate axes (optional, remove if not needed) -\draw[->] (-2,0)--(2,0); -\draw[->] (0,-2)--(0,2); - -% Labels for radii on x-axis -\draw (\a,0) node[below] {$a$}; -\draw (\R,0) node[below] {$R$}; - -% Point w -\coordinate (w) at (0.65,0.55); -\fill (w) circle (0.03) node[above right] {$w$}; - -% small epsilon circle around w -\draw[dashed] (w) circle (0.25) node[right] {$B_\varepsilon(w)$}; - -% Path γ (stylized as a wiggly circle between a and R) -\begin{scope} - \draw[thick,->,domain=0:360,smooth,variable=\t] - plot ({1.45*cos(\t) +0.12*cos(5*\t)}, - {1.45*sin(\t) +0.12*sin(5*\t)}); -\end{scope} - -% Label γ -\node at (1.2,-0.2) {$\gamma$}; - -\end{tikzpicture} - - \begin{tikzpicture}[scale=1.2] - % Koordinatenachsen - \draw[->] (-2.5,0) -- (2.5,0); - \draw[->] (0,-2.5) -- (0,2.5); - - % Äußerer Kreis (Radius A) - \draw[thick, ->] (1.8,0) arc (0:45:1.8); - \draw[thick] (1.8,0) arc (0:360:1.8); - \draw[thick, ->] (1.8,0) arc (360:315:1.8); - \draw[thick, ->] (-1.8,0) arc (180:225:1.8); - \draw[thick, ->] (0,-1.8) arc (270:315:1.8); - - % Innerer Kreis (Radius a) - \draw[thick, <-] (1.0,0) arc (0:45:1.0); - \draw[thick] (1.0,0) arc (0:360:1.0); - \draw[thick, <-] (1.0,0) arc (360:315:1.0); - \draw[thick, <-] (-1.0,0) arc (180:225:1.0); - \draw[thick, <-] (0,-1.0) arc (270:315:1.0); - - % Kleiner Kreis um w (mit gestrichelter Linie) - \draw[dashed] (0.5,0.8) circle (0.3); - \draw[thick, ->] (0.5,0.8) + (135:0.3) arc (135:225:0.3); - - % Verbindungslinien (gestrichelt) - \draw[dashed] (0.5,0.5) -- (1.0,0); - \draw[dashed] (0.5,1.1) -- (0,1.8); - - % Beschriftungen - \node at (2.0,0) [below right] {$R$}; - \node at (1.1,0) [below right] {$a$}; - \node at (1.8,0) [below left] {$A$}; - \node at (0.5,0.8) [above right] {$\gamma$}; - \end{tikzpicture} - \end{center} - \begin{center} - \textit{[Hier würde Ihre Skizze mit den Kreisen und Wegen eingefügt.]} - \end{center} - Als erstes werde ich kann den Funktionswert $f(w)$ als Integral über den Weg + \caption{Wege im Kreisring $K_{r,R}(0)$} + \label{fig:9-2-1} + \end{figure} + Als Erstes werde ich kann den Funktionswert $f(w)$ als Integral über den Weg $γ$ ausdrücken. Dazu wähle eine Zahl $ε > 0$, sodass die abgeschlossene Kreisscheibe $\overline{B_ε(w)}$ ganz in $K_{a,A}(0)$ liegt. Dann gilt nach der Cauchy Integralformel die Gleichung @@ -350,18 +283,22 @@ Kreisringen. Die Darstellung von $f$ als „Potenzreihe mit negativen Exponenten“ ist natürlich schrecklich wichtig. Sie wird als -\emph{Laurentreihenentwicklung}\footnote{Pierre Alphonse Laurent (* 18.~Juli 1813 -in Paris; † 2.~September 1854 ebenda) war ein französischer Mathematiker.} +\emph{Laurentreihenentwicklung}\footnote{Pierre Alphonse Laurent (* 18.~Juli +1813 in Paris; † 2.~September 1854 ebenda) war ein französischer Mathematiker.} bezeichnet. -\begin{definition}[Laurentreihen] +\begin{definition}[Laurentreihen]\label{def:9-2-5}% Eine \emph{Laurentreihe}\index{Laurentreihe} mit Entwicklungspunkt $ρ$ ist ein Ausdruck der Form \[ \sum_{i ∈ ℤ} c_i \, (z - ρ)ⁱ. \] - Dabei sind die $c_i$ und $ρ$ komplexe Zahlen. Gegeben eine natürliche Zahl - $n$, so bezeichnet man den Ausdruck + Dabei sind die $c_i$ und $ρ$ komplexe Zahlen. +\end{definition} + +\begin{definition}[Konvergenz von Laurentreihen] + Gegeben eine Laurentreihe wie in Definition~\ref{def:9-2-5} und eine + natürliche Zahl $n$, so bezeichnet man den Ausdruck \[ \sum_{i=-n}^n c_i \, (z - ρ)ⁱ \] @@ -373,4 +310,191 @@ bezeichnet. konvergiert. \end{definition} +\begin{definition}[Haupt- und Nebenteil von Laurentreihen] + Gegeben eine Laurentreihe wie in Definition~\ref{def:9-2-5}, so nennt man die + Teilreihen + \[ + \sum_{i=1}^{∞} c_i (z - p)^{-i} + \quad\text{und}\quad + \sum_{i=0}^{∞} c_i (z - p)ⁱ + \] + den \emph{Hauptteil}\index{Hauptteil einer Laurentreihe} und den + \emph{Nebenteil}\index{Nebenteil einer Laurentreihe} der Laurentreihe. +\end{definition} + +Konstruktion~\ref{kons:9-2-4} arbeitet mit Kreisringen um den Nullpunkt. Wie +immer gelten dieselben Aussagen für Kreisringe mit beliebigem Mittelpunkt. Am +Ende des Tages haben wir Folgendes bewiesen. + +\begin{satz}[Laurentreihenentwicklung]\label{satz:9-2-8}% + Es sei $ρ ∈ ℂ$ ein Punkt und es seien reelle Zahlen $0 < r < R$ + gegeben. Weiter sei $f ∈ 𝒪(K_{r,R}(p))$ eine auf dem Kreisring + $K_{r,R}(p)$ holomorphe Funktion. Dann existiert eine auf ganz $K_{r,R}(p)$ + lokal gleichmäßig konvergierende Laurentreihe deren Grenzfunktion mit $f$ + übereinstimmt. Zusätzlich gilt: Haupt- und Nebenteil Laurentreihe konvergieren + auf $K_{r,R}(p)$ ebenfalls jeweils lokal gleichmäßig. \qed +\end{satz} + +\begin{kor}[Eindeutigkeit der Laurentreihenentwicklung] + In der Situation von Satz~\ref{satz:9-2-8} ist die Darstellung von $f$ als + Laurentreihe eindeutig. +\end{kor} +\begin{proof} + Nach Satz~\ref{satz:9-2-8} gibt es eine Laurentreihe + \[ + f(z) = \sum_{i ∈ ℤ} c_i (z - ρ)ⁱ + \] + die auf $K_{r,R}(ρ)$ lokal gleichmäßig gegen $f$ konvergiert. Wir wollen + zeigen, dass die Koeffizienten $c_i$ eindeutig bestimmt sind. Sei dazu eine + Zahl $n ∈ ℤ$ gegeben. Wir wollen $c_n$ bestimmen. Dazu wähle eine Zahl $r < + a < R$ und betrachte die folgenden Integrale, + \begin{multline*} + \int_{∂ B_a(ρ)} f(z) · (z - ρ)^n \, dz \\ + \begin{aligned} + & = \int_{∂ B_a(ρ)} \left(\lim_{k → ∞} \sum_{i=-k}^k c_i (z - ρ)ⁱ \right) (z - ρ)^n \, dz && \text{Laurentreihe von } f \\ + & = \lim_{k → ∞} \sum_{i=-k}^k \int_{∂ B_a(ρ)} c_i (z - ρ)^{i+n} \, dz && \text{lokal gleichmäßige Konvergenz} \\ + & = 2π i · c_{-1-n} && \text{Beispiel~\vref{bsp:3-2-2}.} + \end{aligned} + \end{multline*} + Also sind die Koeffizienten der Laurentreihe durch eine Formel gegeben, in der + nur die Funktion $f$ vorkommt. Also sind die Koeffizienten der Laurentreihe + eindeutig bestimmt. +\end{proof} + +Wir haben im Abschnitt~\ref{sec:9-1} drei Typen von isolierten Singularitäten +definiert. Mithilfe der Laurentreihenentwicklung können wir diese Typen jetzt +charakterisieren. + +\begin{beobachtung}[Holomorphe Funktionen mit Singularitäten]\label{beo:9-2-10}% + Es sei $U ⊆ ℂ$ offen und es sei $f$ eine holomorphe Funktion mit isolierten + Singularitäten auf $U$. Es sei $T ⊂ U$ die Menge der Singularitäten und sei + $ρ ∈ T$ sei eine der isolierte Singularitäten. Um das Verhalten von $f$ bei + $ρ$ zu verstehen, entwickle $f$ bei $ρ$ in eine Laurentreihe. Genauer: wähle + eine Zahl $ε > 0$, sodass $ρ$ die einzige Singularität von $f$ in der + Kreisscheibe $B_{ε}(ρ)$ ist, also + \[ + B_{ε}(p) ∩ T = \{ρ\}. + \] + Betrachte dann den Kreisring + \[ + K_{\frac{ε}{2},ε}(ρ) = \{ z ∈ ℂ \::\: \frac{ε}{2} < |z - ρ| < ε \}. + \] + Nach Satz~\ref{satz:9-2-8} („Laurentreihenentwicklung“) gibt es eine auf + $K_{\frac{ε}{2},ε}(ρ)$ lokal gleichmäßig konvergierende Laurentreihe $\sum c_i + (z - p)ⁱ$, deren Grenzfunktion auf $K_{\frac{ε}{2},ε}(ρ)$ mit $f$ + übereinstimmt. Dann gilt Folgendes. + \begin{enumerate} + \item Der Punkt $ρ$ ist genau dann eine hebbare Singularität von $f$, wenn + der Hauptteil gleich null ist. + \item Der Punkt $ρ$ ist genau dann eine Polstelle von $f$, wenn der + Hauptteil nur endlich viele Summanden hat. + \item Der Punkt $ρ$ ist genau dann eine wesentliche Singularität von $f$, + wenn der Hauptteil unendlich viele Summanden hat. + \end{enumerate} +\end{beobachtung} + + +\section{Automorphismen der komplexen Ebene} + +Ähnlich wie im Abschnitt~\ref{sec:7-3} möchte ich die Stärke der +funktionentheoretischen Methoden demonstrieren, indem ich die Automorphismen der +komplexen Ebene bestimme. + +\begin{frage} + Was sind die biholomorphen Selbstabbildungen $ℂ → ℂ$? +\end{frage} + +Einige Beispiele und Nicht-Beispiele sind und bereits bekannt. + +\begin{bsp}[Affin-lineare Abbildungen] + Für alle $a,b ∈ ℂ$ mit $a ≠ 0$ ist die Abbildung + \[ + f : ℂ → ℂ, \quad z ↦ az + b + \] + biholomorph. +\end{bsp} + +\begin{beobachtung}[Komposition mit Translationen]\label{beob:9-3-3}% + Gegeben irgendeine biholomorphe Abbildung $f : ℂ → ℂ$, so ist die Abbildung + \[ + f - f(0) : ℂ → ℂ + \] + ebenfalls biholomorph und bildet die Null auf null ab. +\end{beobachtung} + +\begin{bsp}[Polynome höheren Grades] + Polynome vom Grad $≥ 2$ sind ganz bestimmt keine Automorphismen, denn jedes + Polynom $f ∈ ℂ[z]$ zerfällt in Linearfaktoren + \[ + f(z) = λ (z - a_1)(z - a_2) ⋯ (z - a_n). + \] + \begin{itemize} + \item Wenn zwei der $a_•$ gleich sind, dann hat $f$ eine mehrfache + Nullstelle. Die Ableitung $f'$ verschwindet dort. Aber biholomorphe + Abbildungen haben nach dem Satz über die lokale Umkehrbarkeit nirgends + eine verschwindende Ableitung. + + \item Wenn alle $a_•$ verschieden sind, dann hat $f$ mindestens zwei + verschiedene Nullstellen. Also ist $f$ nicht injektiv. + \end{itemize} +\end{bsp} + +\begin{bsp}[Potenzreihen mit unendlich vielen Summanden]\label{bsp:9-3-5}% + Ich behaupte, dass Potenzreihen mit unendlich vielen Summanden ebenfalls + \emph{keine} Automorphismen der komplexen Ebene sind. Dazu führe ich einen + Widerspruchsbeweis. Angenommen, es gäbe einen Automorphismus $f : ℂ → ℂ$ der + durch eine Potenzreihe + \[ + f(z) = \sum a_i zⁱ + \] + mit unendlich vielen Summanden gegeben ist. Nach Beobachtung~\ref{beob:9-3-3} + können wir ohne Einschränkung annehmen, dass $a_0 = 0$ ist, also $f(0) = 0$. + Insbesondere bildet $f$ die Menge $ℂ^*$ biholomorph auf sich selbst ab. + + Aber: die Menge $ℂ^*$ hat einen interessanten Automorphismus, nämlich + \[ + j : ℂ^* → ℂ^*, \quad z ↦ z^{-1}. + \] + Die Abbildung $f ◦ j : ℂ^* → ℂ^*$ ist dann auch holomorph und bijektiv, und + kann als holomorphe Funktion mit Pol bei $0$ aufgefasst werden. Die + Laurentreihe von $f ◦ j$ ergibt sich sehr einfach aus der Laurentreihe von + $f$, + \[ + f ◦ j = \sum a_i z^{-i}. + \] + Ich stelle fest: Der Hauptteil dieser Laurentreihe hat unendlich viele + Summanden. Wie ist in Beobachtung~\ref{beo:9-2-10} gesehen haben, bedeutet + das Folgendes: wenn ich $f ◦ j$ als Funktion mit isolierter Singularität + auffasse, dann hat $f ◦ j$ hat bei $0$ eine wesentliche Singularität. + + Ich behaupte, dass $f ◦ j$ dann aber nicht bijektiv sein kann. Betrachte dazu + die Kreisscheibe $B_{1/2}(1) ⊂ ℂ^*$. Wir wissen, dass die Bildmenge $(f ◦ j) + \left(B_{1/2}(0)\right)$ eine offene Teilmenge von $ℂ^*$ ist. + + Es gilt aber auch noch Satz~\ref{satz:9-1-casorati-weierstrass} + („Casorati-Weierstraß“). Demnach ist die Bildmenge + \[ + (f ◦ j) \left(B_{1/2}(0) ∖ \{0\}\right) ⊆ ℂ + \] + dicht, schneidet also $(f ◦ j) \left(B_{1/2}(0)\right)$ in mindestens einem + Punkt. Dieser Punkt hat demnach zwei Urbilder, nämlich eines in $B_{1/2}(1)$ + und eines $B_{1/2}(0) ∖ \{0\}$. Also ist $f ◦ j$ tatsächlich nicht injektiv! +\end{bsp} + +\begin{bemerkung} + Die Argumentation aus Beispiel~\ref{bsp:9-3-5} zeigt in Wirklichkeit, dass + holomorphe Funktionen mit essenziellen Singularitäten niemals injektiv sind. +\end{bemerkung} + +In der Summe haben wir folgenden Satz bewiesen. + +\begin{satz}[Automorphismen der komplexen Ebene]\label{satz:9-3-6}% + Jede biholomorphe Abbildung $f : ℂ → ℂ$ ist von der Form + \[ + f(z) = az + b + \] + für gewisse $a,b ∈ ℂ$ mit $a ≠ 0$. \qed +\end{satz} + + % !TEX root = Funktionentheorie diff --git a/Notizen/220617-Vorlesung.pdf b/Notizen/220617-Vorlesung.pdf deleted file mode 100644 index e330e1c222e6c66ad0409ff96007a89c9f4df00d..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 281445 zcmbrl19W9uvoIRlc6Myrw$-t1+ji2i)3NQOW7}p29d(?J_0s1%-}&yn|3BWlW4t}G z_F7e|X3d(jFso{0lPQXd(=pL=z>xJ9_ty_p!oV;A7y&Mr=- zhPE)g0v-;A#@41T03%a#OFKT|%kMqJ080}-Vs$n-MmYx&QwvK;FDFwKFL_mCFKc5i 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