From f1a1d0dd6630b18c5693cd0b93e95da9327aa351 Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Tue, 18 Nov 2025 16:59:52 +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 | 3 + .vscode/ltex.hiddenFalsePositives.de-DE.txt | 4 + 03-wegintegraleDiffbar.tex | 3 +- 06-potenz.tex | 2 +- 09-singularities.tex | 349 +------------------- 09-annulus.png => 10-annulus.png | Bin 10-laurent.tex | 255 ++++++++++++++ 10-mittagLeffler.png | Bin 0 -> 14266 bytes 11-applications.tex | 292 ++++++++++++++++ 12-res1.png | Bin 0 -> 7497 bytes 12-res2.png | Bin 0 -> 26045 bytes 12-res3.png | Bin 0 -> 15219 bytes 12-residuum.tex | 159 +++++++++ Funktionentheorie.tex | 6 + Notizen/220622-Vorlesung.pdf | Bin 6505840 -> 0 bytes 15 files changed, 739 insertions(+), 334 deletions(-) rename 09-annulus.png => 10-annulus.png (100%) create mode 100644 10-laurent.tex create mode 100644 10-mittagLeffler.png create mode 100644 11-applications.tex create mode 100644 12-res1.png create mode 100644 12-res2.png create mode 100644 12-res3.png create mode 100644 12-residuum.tex delete mode 100644 Notizen/220622-Vorlesung.pdf diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt index d599b1e..1a02216 100644 --- a/.vscode/ltex.dictionary.de-DE.txt +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -59,3 +59,6 @@ Laurentreihen Laurentreihenentwicklung Alphonse funktionentheoretischen +Summendarstellung +Mittag-Leffler +Djursholm diff --git a/.vscode/ltex.hiddenFalsePositives.de-DE.txt b/.vscode/ltex.hiddenFalsePositives.de-DE.txt index 89b4e89..0af5e11 100644 --- a/.vscode/ltex.hiddenFalsePositives.de-DE.txt +++ b/.vscode/ltex.hiddenFalsePositives.de-DE.txt @@ -31,3 +31,7 @@ {"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$"} +{"rule":"DE_CASE","sentence":"^\\QAnwendung der Goldenen Regeln\\E$"} +{"rule":"UNPAIRED_BRACKETS","sentence":"^\\QDie Aussage folgt dann aus Korollar \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q (“Integralsatz von Cauchy”), \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} +{"rule":"DE_CASE","sentence":"^\\QDie zweite Gleichheit gilt nach der Goldenen Regel 1, weil \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q und \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q in derselben Zusammenhangskomponente von \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q ist.\\E$"} +{"rule":"DE_CASE","sentence":"^\\QDamit ist die Goldene Regel 3 zumindest halbwegs gerechtfertigt.\\E$"} diff --git a/03-wegintegraleDiffbar.tex b/03-wegintegraleDiffbar.tex index 4dedcb2..f36f0e8 100644 --- a/03-wegintegraleDiffbar.tex +++ b/03-wegintegraleDiffbar.tex @@ -171,7 +171,8 @@ diese Begriffe nicht verwechseln! \end{bemerkung} \begin{bsp}[Integration von $z^n$ über den Rand des Einheitskreises]\label{bsp:3-2-2}% - Es sei $U = ℂ^*$ und es sei $f(z) = z^n$. Weiter betrachten wir den Weg + Es sei $U = ℂ^*$, es sei $n ∈ ℤ$ und es sei $f(z) = z^n$. Weiter + betrachten wir den Weg \[ γ: [0,2π] → ℂ^*, \quad t ↦ r\exp(it), \] diff --git a/06-potenz.tex b/06-potenz.tex index a5fb2aa..ea86378 100644 --- a/06-potenz.tex +++ b/06-potenz.tex @@ -82,7 +82,7 @@ B_r(ρ)$ der Kreisscheibe liegen. \section{Potenzreihen liefern holomorphe Funktionen} -\begin{proposition}[Kompakt konvergierende Folgen holomorpher Funktionen]\label{prop:potenzreihe-holomorph}% +\begin{proposition}[Lokal gleichmäßig konvergierende Folgen holomorpher Funktionen]\label{prop:potenzreihe-holomorph}% Es sei $U ⊂ ℂ$ offen und es sei $(f_i)_{i ∈ ℕ}$ eine Folge von holomorphen Funktionen, $f_i ∈ 𝒪(U)$, die auf $U$ kompakt gegen Grenzfunktion $f$ konvergiert. Dann gilt: diff --git a/09-singularities.tex b/09-singularities.tex index 2aa3285..cc8b178 100644 --- a/09-singularities.tex +++ b/09-singularities.tex @@ -2,9 +2,9 @@ \selectlanguage{german} \chapter{Singuläre Stellen holomorpher Funktionen} +\label{sec:9}% \section{Isolierte Singularitäten} -\label{sec:9-1}% Wir interessieren uns für die folgende Situation: Sei $U ⊂ ℂ$ ein Gebiet, sei $ρ ∈ U$ ein Punkt. Gegeben eine holomorphe Funktion $f ∈ 𝒪(U ∖ ρ)$. Was kann ich @@ -57,6 +57,9 @@ Wir interessieren uns für die folgende Situation: Sei $U ⊂ ℂ$ ein Gebiet, s \end{enumerate} \end{definition} + +\section{Hebbare Singularitäten} + \begin{satz}[Hebbarkeitsatz von Riemann]\label{satz:9-1-hebbarkeit}% \index{Hebbarkeitsatz von Riemann}In der Situation von Definition~\ref{def:9-1-1} sei $ρ ∈ T$. Falls die Funktion $|f|$ in der Nähe @@ -91,13 +94,13 @@ Wir interessieren uns für die folgende Situation: Sei $U ⊂ ℂ$ ein Gebiet, s $ℝ²$ überhaupt nicht differenzierbar. \end{bemerkung} -\begin{frage} - Was kann ich über Beiträge von Funktionen mit Polstelle sagen? -\end{frage} -Als Antwort eine Beispielrechnung. +\section{Polstellen} -\begin{rem}[Beispielrechnung zu Funktionen mit Polstellen] +Was kann ich über Beiträge von Funktionen mit Polstelle sagen? Als Antwort eine +Beispielrechnung. + +\begin{rem}[Beispielrechnung zu Funktionen mit Polstellen]\label{bsp:9-3-1}% In der Situation von Definition~\ref{def:9-1-1} sei $ρ ∈ T$ eine Polstelle der Ordnung $n > 0$ hat. Also gibt es eine Zahl $ε > 0$ und eine holomorphe Funktion $g ∈ 𝒪(B_ε(ρ))$, sodass folgendes gilt. @@ -128,9 +131,14 @@ Pol, dann gilt für jedes $M ∈ ℝ⁺$ und alle ausreichend kleinen $ε > 0$. \item Für jedes $z ∈ B_ε(ρ) ∖ \{ρ\}$ ist $|f(z)| > M$. \end{enumerate} -Die Funktionswerte von $f$ explodieren also betragsmäßig, wenn ich mich dem -Punkt $ρ$ annähere. Auf jeden Fall sind die Funktionswerte von $0$ weg -beschränkt. \textbf{Das ist bei wesentlichen Singularitäten ganz anders!} + + +\section{Wesentliche Singularitäten} + +Beispielrechnung~\ref{bsp:9-3-1} zeigt, dass die Funktionswerte von $f$ +betragsmäßig explodieren, wenn ich mich einer Polstelle annähere. Auf jeden +Fall sind die Funktionswerte in der Nähe der Polstelle von $0$ weg beschränkt. +\textbf{Das ist bei wesentlichen Singularitäten ganz anders!} \begin{satz}[Satz von Casorati\footnote{Felice Casorati (* 17.~Dezember 1835 in Pavia; † 11.~September 1890 in Casteggio) war ein italienischer @@ -174,327 +182,4 @@ beschränkt. \textbf{Das ist bei wesentlichen Singularitäten ganz anders!} deshalb keine wesentlichen Singularitäten hat. \end{proof} - -\section{Entwicklung in Laurentreihen} -\sideremark{Vorlesung 13} - -In Kapitel~\ref{chap:6} haben wir gesehen, dass holomorphe Funktionen auf -Kreisscheiben Potenzreihenentwicklungen besitzen. Wir wollen jetzt einen -ähnlichen Satz für holomorphe Funktionen auf Kreisringen beweisen. Dazu -verwenden wir denselben Trick wie bei der Potenzreihenentwicklung. - -\begin{bemerkung}[Erinnerung] - Gegeben Kreisscheibe $B_r(0)$ und $f ∈ 𝒪(B_r(0))$, dann wähle $0 < ρ < r$. - Nach Satz~\vref{satz:4-4-1} („Integralformel von Cauchy“) gilt für alle $w ∈ - B_{ρ}(0)$ - \[ - f(w) = \frac{1}{2π i} \int_{∂ B_{ρ}(0)} \frac{f(z)}{z - w} \, dz. - \] - Schreibe - \[ - \frac{1}{z - w} = \frac{1}{z} · \frac{1}{1 - w/z} = \frac{1}{z} · \sum_{i=0}^∞ \left(\frac{w}{z}\right)ⁱ, - \] - und setze dies in das Integral ein. Vertausche Integral und Summe und erhalte - eine Potenzreihendarstellung von $f$. -\end{bemerkung} - -So etwas wollen wir jetzt auch für holomorphe Funktionen auf $B_r(0)$ machen, -die bei $0$ eine Singularität haben. Sogar noch allgemeiner: für Funktionen auf -Kreisringen. - -\begin{notation}[Kreisring] - Seien reelle Zahlen $0 < r < R$ gegeben und sei $ρ ∈ ℂ$. Dann sei - \[ - K_{r,R}(ρ) = \{ z ∈ ℂ \::\: r < |z - ρ| < R \} - \] - der offene \emph{Kreisring}\index{Kreisring} um $p$ mit Radien $r$ und $R$. -\end{notation} - -\begin{situation}[Holomorphe Funktion auf Kreisring]\label{situation:9-2-2}% - Es seien reelle Zahlen $0 < r < R$ und es sei eine holomorphe Funktion auf dem - Kreisring $K_{r,R}(0)$ gegeben, $f ∈ 𝒪(K_{r,R}(0))$. -\end{situation} - -\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 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} - - \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 - \[ - f(w) = \frac{1}{2π i} \int_{∂ B_ε(w)} \frac{f(z)}{z - w} dz. - \] - Jetzt beobachte, dass der Weg $γ$ frei homotop zum Weg rund um $∂ B_ε(w)$ ist. - Also sind nach Satz~\ref{satz:4-3-6} („Invarianz von Wegintegralen unter - freier Homotopie“) die Integrale über diese Wege gleich, - \begin{align*} - f(w) & = \frac{1}{2π i} \int_{∂ B_ε(w)} \frac{f(z)}{z - w} \, dz \\ - & = \frac{1}{2π i} \int_{γ} \frac{f(z)}{z - w} \, dz \\ - & = \frac{1}{2π i} \left[ \int_{∂ B_A(0)} \frac{f(z)}{z - w} \, dz - \int_{∂ B_a(0)} \frac{f(z)}{z - w} \, dz \right]. - \end{align*} - Jetzt wenden wir den Trick mit der geometrischen Reihe zweimal an. - \begin{itemize} - \item Für alle $z ∈ ∂ B_A(0)$ und alle $w ∈ K_{r,A}(0)$ ist - \[ - \frac{f(z)}{z - w} = \frac{f(z)}{z} · \sum \left(\frac{w}{z}\right)ⁱ - \] - und deshalb - \[ - \int_{∂ B_A(0)} \frac{f(z)}{z - w} \, dz - = \sum_{i=0}^∞ \left( \int_{∂ B_A(0)} \frac{f(z)}{z^{i+1}} \, dz \right) · wⁱ. - \] - - \item Für alle $z ∈ ∂ B_a(0)$ und alle $w ∈ K_{a,R}(0)$ ist - \[ - \frac{f(z)}{z - w} = \frac{-f(z)}{w - z} = \frac{-f(z)}{w} · \frac{1}{1 - z/w} = \frac{-f(z)}{w} \sum \left(\frac{z}{w}\right)ⁱ - \] - und deshalb - \[ - \int_{∂ B_a(0)} \frac{f(z)}{z - w} \, dz = \sum_{i=0}^∞ \left( \int_{∂ B_a} (-f(z) \, zⁱ) \, dz \right) · w^{-(i+1)}. - \] - \end{itemize} - Wie zuvor gilt: Der Wert der Integrale hängen nicht von der Wahl von $a$ und - $A$ ab und die Reihen konvergieren auf $K_{r,R}(0)$ lokal gleichmäßig. - Zusammenfassend können wir sagen: Es gibt Familien von komplexen Zahlen - $(α_i)_{i ∈ ℤ}$, sodass für jedes $w ∈ K_{r,R}(0)$ die folgende Gleichung - gilt, - \[ - f(w) = \sum_{i=0}^∞ α_i \, wⁱ + \sum_{i=0}^∞ α_{-(i+1)} \, w^{-(i+1)}. - \] - Beide Reihen konvergieren auf $K_{r,R}(0)$ lokal gleichmäßig, und es gilt - \[ - f(w) = \lim_{n → ∞} \sum_{i=-n}^n α_i \, wⁱ. - \] - Die Konstruktion endet hier. -\end{konstruktion} - -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.} -bezeichnet. - -\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. -\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 - \[ - \sum_{i=-n}^n c_i \, (z - ρ)ⁱ - \] - als die \emph{$n$.te Partialsumme}\index{Partialsumme!einer Laurentreihe} der - Laurentreihe. Man sagt, die Laurentreihe konvergiert für ein $z_0 ∈ ℂ$ gegen - $g ∈ ℂ$, wenn die Folge $\sum_{i=-n}^n c_i \, (z_0 - p)ⁱ$ gegen $g$ - konvergiert. Man sagt, die Laurentreihe konvergiert auf einer Menge $M ⊆ ℂ$ - lokal gleichmäßig, wenn die Folge der Partialsummen auf $M$ lokal gleichmäßig - 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/09-annulus.png b/10-annulus.png similarity index 100% rename from 09-annulus.png rename to 10-annulus.png diff --git a/10-laurent.tex b/10-laurent.tex new file mode 100644 index 0000000..3916644 --- /dev/null +++ b/10-laurent.tex @@ -0,0 +1,255 @@ +% spell checker language +\selectlanguage{german} + +\chapter{Entwicklung in Laurentreihen} +\sideremark{Vorlesung 13} + +\section{Holomorphe Funktionen auf Kreisringen} + +In Kapitel~\ref{chap:6} haben wir gesehen, dass holomorphe Funktionen auf +Kreisscheiben Potenzreihenentwicklungen besitzen. Wir wollen jetzt einen +ähnlichen Satz für holomorphe Funktionen auf Kreisringen beweisen. Dazu +verwenden wir denselben Trick wie bei der Potenzreihenentwicklung. + +\begin{bemerkung}[Erinnerung] + Gegeben Kreisscheibe $B_r(0)$ und $f ∈ 𝒪(B_r(0))$, dann wähle $0 < ρ < r$. + Nach Satz~\vref{satz:4-4-1} („Integralformel von Cauchy“) gilt für alle $w ∈ + B_{ρ}(0)$ + \[ + f(w) = \frac{1}{2π i} \int_{∂ B_{ρ}(0)} \frac{f(z)}{z - w} \, dz. + \] + Schreibe + \[ + \frac{1}{z - w} = \frac{1}{z} · \frac{1}{1 - w/z} = \frac{1}{z} · \sum_{i=0}^∞ \left(\frac{w}{z}\right)ⁱ, + \] + und setze dies in das Integral ein. Vertausche Integral und Summe und erhalte + eine Potenzreihendarstellung von $f$. +\end{bemerkung} + +So etwas wollen wir jetzt auch für holomorphe Funktionen auf $B_r(0)$ machen, +die bei $0$ eine Singularität haben. Sogar noch allgemeiner: für Funktionen auf +Kreisringen. + +\begin{notation}[Kreisring] + Seien reelle Zahlen $0 < r < R$ gegeben und sei $ρ ∈ ℂ$. Dann sei + \[ + K_{r,R}(ρ) = \{ z ∈ ℂ \::\: r < |z - ρ| < R \} + \] + der offene \emph{Kreisring}\index{Kreisring} um $p$ mit Radien $r$ und $R$. +\end{notation} + +\begin{situation}[Holomorphe Funktion auf Kreisring]\label{situation:9-2-2}% + Es seien reelle Zahlen $0 < r < R$ und es sei eine holomorphe Funktion auf dem + Kreisring $K_{r,R}(0)$ gegeben, $f ∈ 𝒪(K_{r,R}(0))$. +\end{situation} + +\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 in Abbildung~\ref{fig:9-2-1} gezeigten Weg $γ$ in $K_{r,R}(0)$. + \begin{figure} + \begin{center} + \includegraphics[width=10cm]{10-annulus.png} + \end{center} + + \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 + \[ + f(w) = \frac{1}{2π i} \int_{∂ B_ε(w)} \frac{f(z)}{z - w} dz. + \] + Jetzt beobachte, dass der Weg $γ$ frei homotop zum Weg rund um $∂ B_ε(w)$ ist. + Also sind nach Satz~\ref{satz:4-3-6} („Invarianz von Wegintegralen unter + freier Homotopie“) die Integrale über diese Wege gleich, + \begin{align*} + f(w) & = \frac{1}{2π i} \int_{∂ B_ε(w)} \frac{f(z)}{z - w} \, dz \\ + & = \frac{1}{2π i} \int_{γ} \frac{f(z)}{z - w} \, dz \\ + & = \frac{1}{2π i} \left[ \int_{∂ B_A(0)} \frac{f(z)}{z - w} \, dz - \int_{∂ B_a(0)} \frac{f(z)}{z - w} \, dz \right]. + \end{align*} + Jetzt wenden wir den Trick mit der geometrischen Reihe zweimal an. + \begin{itemize} + \item Für alle $z ∈ ∂ B_A(0)$ und alle $w ∈ K_{r,A}(0)$ ist + \[ + \frac{f(z)}{z - w} = \frac{f(z)}{z} · \sum \left(\frac{w}{z}\right)ⁱ + \] + und deshalb + \[ + \int_{∂ B_A(0)} \frac{f(z)}{z - w} \, dz + = \sum_{i=0}^∞ \left( \int_{∂ B_A(0)} \frac{f(z)}{z^{i+1}} \, dz \right) · wⁱ. + \] + + \item Für alle $z ∈ ∂ B_a(0)$ und alle $w ∈ K_{a,R}(0)$ ist + \[ + \frac{f(z)}{z - w} = \frac{-f(z)}{w - z} = \frac{-f(z)}{w} · \frac{1}{1 - z/w} = \frac{-f(z)}{w} \sum \left(\frac{z}{w}\right)ⁱ + \] + und deshalb + \[ + \int_{∂ B_a(0)} \frac{f(z)}{z - w} \, dz = \sum_{i=0}^∞ \left( \int_{∂ B_a} (-f(z) \, zⁱ) \, dz \right) · w^{-(i+1)}. + \] + \end{itemize} + Wie zuvor gilt: Der Wert der Integrale hängen nicht von der Wahl von $a$ und + $A$ ab und die Reihen konvergieren auf $K_{r,R}(0)$ lokal gleichmäßig. + Zusammenfassend können wir sagen: Es gibt Familien von komplexen Zahlen + $(α_i)_{i ∈ ℤ}$, sodass für jedes $w ∈ K_{r,R}(0)$ die folgende Gleichung + gilt, + \[ + f(w) = \sum_{i=0}^∞ α_i \, wⁱ + \sum_{i=0}^∞ α_{-(i+1)} \, w^{-(i+1)}. + \] + Beide Reihen konvergieren auf $K_{r,R}(0)$ lokal gleichmäßig, und es gilt + \[ + f(w) = \lim_{n → ∞} \sum_{i=-n}^n α_i \, wⁱ. + \] + Die Konstruktion endet hier. +\end{konstruktion} + + +\section{Laurentreihen} + +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.} +bezeichnet. + +\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. +\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 + \[ + \sum_{i=-n}^n c_i \, (z - ρ)ⁱ + \] + als die \emph{$n$.te Partialsumme}\index{Partialsumme!einer Laurentreihe} der + Laurentreihe. Man sagt, die Laurentreihe konvergiert für ein $z_0 ∈ ℂ$ gegen + $g ∈ ℂ$, wenn die Folge $\sum_{i=-n}^n c_i \, (z_0 - p)ⁱ$ gegen $g$ + konvergiert. Man sagt, die Laurentreihe konvergiert auf einer Menge $M ⊆ ℂ$ + lokal gleichmäßig, wenn die Folge der Partialsummen auf $M$ lokal gleichmäßig + 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 der Laurentreihe konvergieren auf + $K_{r,R}(p)$ ebenfalls jeweils lokal gleichmäßig. \qed +\end{satz} + +\begin{bsp}[Laurentreihenentwicklung]\label{bsp:9-2-10}% + Betrachte die Funktion + \[ + f : ℂ ∖ \{2\} → ℂ, \quad z ↦ \frac{1}{z - 2}. + \] + Das Ziel ist, die Laurentreihenentwicklung von $f$ auf verschiedenen + Kreisringen zu bestimmen. + \begin{enumerate} + \item\label{il:10-2-6-1} Auf dem Kreisring $K_{0,1}(0)$ gilt für jedes $z$ die + Gleichung + \[ + f(z) = -\frac{1}{2} · \frac{1}{1-\frac{z}{2}} + = \sum_{k=0}^∞ -\frac{1}{2} \left(\frac{z}{2}\right)^k = \sum_{k=0}^∞ -\frac{1}{2^{k+1}} z^k. + \] + Die Summendarstellung $\frac{1}{1-\frac{z}{2}}$ gilt wie angegeben, weil + $|z|<2$, also $|\frac{z}{2}|<1$ ist. + + \item Auf dem Kreisring $K_{1, 2}(0)$ gilt für jedes $z$ dieselbe Darstellung + wie in \ref{il:10-2-6-1}, weil immer noch $|z|<2$ ist. + + \item Auf dem Kreisring $K_{2,∞}(0)$ können wir die Reihe aus + \ref{il:10-2-6-1} definitiv \emph{nicht} nehmen. Es gilt aber für jedes $z$ + die Gleichung $|z|>2$ und deshalb $|\frac{2}{z}|<1$. + Also ist + \[ + f(z) = \frac{1}{z-2} = \frac{1}{z} · \frac{1}{1-\frac{2}{z}} = \frac{1}{z} · \sum_{k=0}^∞ \left(\frac{2}{z}\right)^k = \sum_{k=-∞}⁰ \frac{1}{2^k} z^{k-1}. + \] + \end{enumerate} +\end{bsp} + +\begin{kor}[Eindeutigkeit der Laurentreihenentwicklung]\label{kor:10-2-5}% + In der Situation von Satz~\ref{satz:9-2-8} ist die Darstellung von $f$ als + Laurentreihe eindeutig. Genauer: wenn + \[ + \sum_{i ∈ ℤ} c_i (z - ρ)ⁱ + \] + eine auf ganz $K_{r,R}(p)$ lokal gleichmäßig konvergierende Laurentreihe ist, + deren Grenzfunktion mit $f$ übereinstimmt, dann für jede reelle Zahl $a$ mit + $r < a < R$ und jede ganze Zahl $n ∈ ℤ$ die Formel + \[ + c_n = \frac{1}{2π i} \int_{∂ B_a(ρ)} f(z) · (z - ρ)^{-(n+1)} \, dz. + \] +\end{kor} +\begin{proof} + Seien Zahlen $a$ und $m$ gegeben. Betrachte die folgenden Integrale, + \begin{multline*} + \int_{∂ B_a(ρ)} f(z) · (z - ρ)^{-(n+1)} \, dz \\ + \begin{aligned} + & = \int_{∂ B_a(ρ)} \left(\lim_{k → ∞} \sum_{i=-k}^k c_i (z - ρ)ⁱ \right) (z - ρ)^{-(n+1)} \, dz && \text{Laurentreihe von } f \\ + & = \lim_{k → ∞} \sum_{i=-k}^k \int_{∂ B_a(ρ)} c_i (z - ρ)^{i-(n+1)} \, dz && \text{lokal gleichmäßige Konvergenz} \\ + & = \lim_{k → ∞} \sum_{i=-k}^k c_i·\int_{∂ B_a(ρ)} (z - ρ)^{i-(n+1)} \, dz \\ + & = 2π i · c_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} + +\begin{kor}[Laurentreihenentwicklung und lokal gleichmäßige Konvergenz]\label{kor:10-2-7}% + Es sei $ρ ∈ ℂ$ ein Punkt und es seien reelle Zahlen $0 < r < R$ gegeben. + Weiter seien $f$ und $(f_n)_{n ∈ ℕ} ∈ 𝒪(K_{r,R}(p))$ auf dem Kreisring $K_{r,R}(p)$ holomorphe + Funktionen, gegeben durch auf ganz $K_{r,R}(p)$ lokal gleichmäßig + konvergierende Laurentreihen + \[ + f(x) = \sum_{i ∈ ℤ} c_i (z - ρ)ⁱ + \quad\text{und}\quad + f_n(x) = \sum_{i ∈ ℤ} c_{n,i} (z - ρ)ⁱ. + \] + Falls die $f_n$ auf $K_{r,R}(p)$ lokal gleichmäßig gegen $f$ konvergieren, + dann ist für jeden Index $i$ + \[ + c_i = \lim_{n → ∞} c_{n,i}. + \] +\end{kor} +\begin{proof} + Seien ein Index $i$. Wähle eine reelle Zahlen $r < a < R$ und schreibe + \begin{align*} + c_i & = \frac{1}{2π i} \int_{∂ B_a(ρ)} f(z) · (z - ρ)^{-(i+1)} \, dz && \text{Korollar~\ref{kor:10-2-5}} \\ + & = \frac{1}{2π i} \int_{∂ B_a(ρ)} \lim_{n → ∞} f_n(z) · (z - ρ)^{-(i+1)} \, dz && \text{Annahme} \\ + & = \lim_{n → ∞} \frac{1}{2π i} \int_{∂ B_a(ρ)} f_n(z) · (z - ρ)^{-(i+1)} \, dz && \text{lokal glm.~Konvergenz} \\ + & = \lim_{n → ∞} c_{n,i}. && \text{Korollar~\ref{kor:10-2-5}} + \end{align*} + Damit folgt die Behauptung. +\end{proof} + +% !TEX root = Funktionentheorie diff --git a/10-mittagLeffler.png b/10-mittagLeffler.png new file mode 100644 index 0000000000000000000000000000000000000000..98903e77f4f364f80d5516f709538f88b91fecd6 GIT binary patch literal 14266 zcmWk#WmFwa4_^E&6p9pghl>_>cjt1^BE{X^p%izyP~6>$yR^8wYjKC-&d2v-&z?!L zlgZ38Cz)&}Tt!J56`24T002~383`}|z=8k(2J63f0wCD$l=A+9aRy6^!7L3i`oA~e z%|#VO0iY%p<=GJ7y-j8;16Bk8FB$*{_yGWq@1%eO0B~gmfTM2!!1ogXa2+yQRR!J| 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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} + +\begin{bsp}[Funktion mit wesentlicher Singularität]\label{bsp:9-2-11}% + Die Funktion + \[ + f : ℂ ∖ \{0\} → ℂ, \quad z ↦ \exp\left(\frac{1}{z}\right) + \] + ist durch die Laurentreihe + \[ + f(z) = \sum_{k=-∞}⁰ \frac{1}{|k|!} z^k + \] + gegeben und hat deshalb in $0$ eine wesentliche Singularität. +\end{bsp} + + +\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} + + +\section{Funktionen mit vorgeschriebenen Singularitäten} +\sideremark{Vorlesung 14} + +\begin{frage}[Funktionen mit vorgeschriebenen Singularitäten, vereinfachte Fragestellung]\label{fr:11-3-1}% + Sei $P ⊂ ℂ$ eine abgeschlossene und diskrete Teilmenge. Gibt es eine Funktion + $f ∈ 𝒪(ℂ ∖ P)$, sodass $f$ in jedem Punkt von $P$ einen Pol oder eine + wesentliche Singularität hat? +\end{frage} + +\begin{bemerkung}[Naiver Ansatz] + Wenn die Menge $P$ aus Frage~\ref{fr:11-3-1} endlich ist, ist das alles kein + Hexenwerk. Wir können zum Beispiel die Funktion + \[ + f(z) = \sum_{p ∈ P} \frac{1}{z-p} \quad \text{oder} \quad f(z) = \sum_{p ∈ P} \exp\left(\frac{1}{z-p}\right) \quad \text{oder} \quad \dots + \] + nehmen. Aber was, wenn die Menge $P$ unendlich ist? Kann man dann unendliche + Summen nehmen? Wie garantieren wir deren Konvergenz? +\end{bemerkung} + +Der folgende Satz beantwortet Frage~\ref{fr:11-3-1} vollständig. Der Satz +erlaubt sogar, die Hauptteile der $f$ für jeden Punkt $p ∈ P$ einen vorzugeben. + +\begin{satz}[Mittag-Leffler\footnote{Magnus Gösta Mittag-Leffler, genannt Gösta, + (* 16.~März 1846 in Stockholm; † 7.~Juli 1927 in Djursholm[1]) war ein + schwedischer Mathematiker, der sich vor allem mit Analysis + beschäftigte.}]\label{satz:mittag-leffler}% + \index{Satz von Mittag-Leffler}Sei $P ⊂ ℂ$ eine abgeschlossene, diskrete + Teilmenge. Für jedes $p ∈ P$ sei eine Laurentreihe $f_p$ mit + Entwicklungspunkt $p$ gegeben, die auf $K_{0,∞}(p) = ℂ ∖ \{p\}$ konvergiert. + Dann gibt es eine Funktion $f ∈ 𝒪(ℂ ∖ P)$, sodass für jeden Punkt $p ∈ P$ + gilt: Die Laurententwicklung von $f$ am Punkt $p$ hat denselben Hauptteil wie + die Laurentreihe $f_p$. +\end{satz} + +\begin{bemerkung} + Satz~\ref{satz:mittag-leffler} ist nicht optimal. Es gibt allgemeinere + Versionen, bei denen zum Beispiel statt $ℂ$ nur durch eine offene Menge $U ⊆ + ℂ$ betrachtet wird. +\end{bemerkung} + +\begin{proof} + Falls die Menge $P$ endlich ist, so können wir einfach $f(z) = \sum_{p ∈ P} + f_p(z)$ nehmen. Im Folgenden nehmen wir also ohne Beschränkung der + Allgemeinheit an, dass die Menge $P$ unendlich ist. Der Einfachheit halber + nehmen wir zusätzlich noch an, dass die Menge $P$ den Nullpunkt nicht enthält. + Um die Notation zu vereinfachen, schreiben wir für jeden Punkt $p ∈ P$ die + $h_p$ für den Hauptteil der Laurentreihe $f_p$. Dies ist eine Laurentreihe + mit trivialem Nebenteil, die auf ganz $ℂ ∖ \{p\}$ konvergiert. + + + \paragraph*{Schritt 1} + + Um das Problem von Mittag-Leffler auf den endlichen Fall zurückzuführen, + beachte, dass für jede kompakte Menge $K ⊂ ℂ$ nur endlich viele $p ∈ P$ in $K$ + liegen. Dies gilt, weil die Menge $P$ diskret ist. Wir nutzen dies, und + zerlegen die Menge $P$ in Teilmengen der Form wie folgt. + \begin{center} + \includegraphics[width=8cm]{10-mittagLeffler.png} + \end{center} + Gegeben eine natürliche Zahl $n ∈ ℕ$, dann ist die Summe der Hauptteile, + \[ + S_n := \sum_{n ≤ |p| < n+1} h_p, + \] + eine endliche Summe von Laurentreihen\footnote{Wobei $h_p$ jeweils auf ganz $ℂ + ∖ \{p\}$ konvergiert} und deshalb selbst eine Laurentreihe die auf ganz + \[ + ℂ ∖ \{ p ∈ P \::\: n ≤ |p| < n+1\} + \] + konvergiert. Wir erhalten also eine Funktion, die wir (nicht ganz korrekt) + wieder mit $S_n$ bezeichnen, + \[ + S_n : \underbrace{ℂ ∖ \{ p ∈ P \::\: n ≤ |p| < n+1\}}_{\text{enthält die gesamte offene Kreisscheibe }B_n(0)} → ℂ. + \] + + + \paragraph*{Schritt 2} + + Beachte, dass die Funktion $S_n$ auf der Kreisscheibe $B_n(0)$ holomorph ist, + weil dort keine der Singularitäten liegen. Wir können die Funktion $S_n$ + deshalb am Nullpunkt in eine Potenzreihe entwickeln, deren Konvergenzradius + mindestens $n$ ist. Aufgrund der lokal gleichmäßigen Konvergenz dieser + Potenzreihe wird die Folge der Partialsummen auf der kompakten Kreisscheibe + $\overline{B_{n-1}(0)}$ gleichmäßig gegen $S_n$ konvergieren. Wir finden also + für jedes $n ≥ 1$ ein Polynom $Q_n$, sodass + \begin{equation}\label{eq:11-3-4-1}% + \forall z ∈ B_{n-1}(0) : |S_n(z) - Q_n(z)| ≤ 2^{-n} + \end{equation} + gilt. + + + \paragraph*{Schritt 3} + + Definiere jetzt die Funktionenfolge + \[ + f_m : ℂ ∖ P → ℂ, \quad z ↦ \sum_{n=1}^m (S_n(z) - Q_n(z)). + \] + Damit gilt schon einmal Folgendes. + \begin{enumerate} + \item\label{il:11-3-4-1} Für jedes $m$ ist die Funktion $f_m$ auf $ℂ ∖ P$ + holomorph, weil sie als endliche Summe von holomorphen Funktionen + dargestellt ist. + + \item\label{il:11-3-4-2} Für jeden Punkt $p ∈ P$ und jedes $m > |p|$ hat die + Funktion $f_m$ bei $p$ denselben Hauptteil wie die Laurentreihe $f_p$, + weil die Polynome $Q_n$ nichts an den Hauptteilen in $p ∈ P$ ändern. + \end{enumerate} + Falls wir zeigen können, dass die Funktionenfolge $f_m$ auf $ℂ ∖ P$ lokal + gleichmäßig gegen eine Funktion $f$ konvergiert, sind wir fertig. Der Grund + ist der Folgende. + \begin{enumerate} + \item Als Grenzfunktion einer lokal gleichmäßig konvergierenden Folge + holomorpher Funktionen ist $f$ nach + Proposition~\vref{prop:potenzreihe-holomorph} wieder holomorph. + + \item Nach \ref{il:11-3-4-2} und Korollar~\ref{kor:10-2-7} hat die + Laurentreihe von $f$ bei jedem Punkt $p ∈ P$ hat denselben Hauptteil wie + die Laurentreihe $f_p$. + \end{enumerate} + + + \paragraph*{Schritt 4} + + Um die lokal gleichmäßige Konvergenz zu zeigen, sei ein Punkt $z_0 ∈ ℂ ∖ P$ + gegeben. Wir müssen eine Umgebung dieses Punktes finden, auf der die Folge + $f_m$ gleichmäßig konvergiert. Wähle dazu eine Zahl $n ∈ ℕ$ mit $n > |z_0|$ + und betrachte die Umgebung $B_n(z_0) ∖ P$. Für jedes $z ∈ B_n(z_0) ∖ P$ gilt + dann: + \begin{align*} + |f(z) - f_m(z)| &= \left| \sum_{n=m+1}^∞ (S_n(z) - Q_n(z)) \right| \\ + & ≤ \sum_{n=m+1}^∞ |S_n(z) - Q_n(z)| && \text{Dreiecks-Ungleichung} \\ + & ≤ \sum_{n=m+1}^∞ 2^{-n} && \text{\eqref{eq:11-3-4-1}} \\ + & = 2 - \frac{1-(\frac{1}{2})^{m+1}}{1-\frac{1}{2}} = \left(\frac{1}{2}\right)^{m+1}. + \end{align*} + Damit ist die gleichmäßige Konvergenz von $f_m$ auf $B_n(z_0) ∖ P$ gezeigt. +\end{proof} + +% !TEX root = Funktionentheorie diff --git a/12-res1.png b/12-res1.png new file mode 100644 index 0000000000000000000000000000000000000000..08516c9d9aef516033c2003632726fbb759d21eb GIT binary patch literal 7497 zcmWkzWl$At6MgQbySr0bB&6k1QZFgp-Q6H@2}zNX5NV|mq`N_oP6_Gm?)>~_c6aui zGqXST&(55UQdO44LMKND0AR_>Nvi_@`PV|EQ2$kMG9Bmhf8?$%D+!tV%@OnuAX-Z( zNdQn8hw)&F^bgaT%c&~?;KvLAJPd%le-wNN08efJ_Dlc}P6L3%C8J4A^q+v{BB$#P z00zPT69UpRNdJ*29`Z^uD7zT=IInPXcBv5opv;z+meBB--^}!e5z3NuEF0x|v#Nil 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a/12-residuum.tex b/12-residuum.tex new file mode 100644 index 0000000..c5e3b7a --- /dev/null +++ b/12-residuum.tex @@ -0,0 +1,159 @@ +% spell checker language +\selectlanguage{german} + +\chapter{Der Residuensatz} + +In diesem Abschnitt sei $U ⊂ ℂ$ offen, es sei $P ⊂ U$ endlich, und es $f ∈ 𝒪(U +∖ P)$ eine holomorphe Funktion mit isolierten Singularitäten bei $P$. Gegeben +einen geschlossenen Weg $γ: [a,b] → U ∖ P$, der in $U$ zusammenziehbar ist, dann +fragen wir nach Möglichkeiten, das Integral +\[ + \int_{γ} f(z) \, dz +\] +einfach zu berechnen. Einige Fälle kennen wir schon. + +\begin{bemerkung}[Integralsatz von Cauchy] + Falls $P = ∅$ die leere Menge ist, also $f ∈ 𝒪(U)$, dann sagt + Korollar~\ref{kor:4-3-2} („Integralsatz von Cauchy“), dass das gesuchte + Integral verschwindet, + \[ + \int_{γ} f(z) \, dz = 0. + \] +\end{bemerkung} + +\begin{bemerkung}[Umlaufsinn ist wichtig]\label{bem:12-0-2}% + Wenn $\overline{γ}$ derselbe Weg wie $γ$ ist, aber entgegengesetzter Richtung + läuft, dann gilt + \[ + \int_{\overline{γ}} f(z) \, dz = -\int_{γ} f(z) \, dz. + \] + Also muss es eine Rolle spielen, wie $γ$ durchlaufen wird. +\end{bemerkung} + +\begin{bemerkung}[Spezielle Koeffizienten der Laurent-Entwicklung] + Es sei $U = ℂ$ $P = \{rho\}$ sei ein einziger Punkt. Schreibe $f$ als + Laurentreihe, + \[ + f(z) = \sum_{k=-∞}^∞ c_k (z-p)^k. + \] + Dann gilt nach Korollar~\ref{kor:10-2-5} für jede reelle Zahl $r ∈ ℝ⁺$ die + Gleichung + \[ + \frac{1}{2π i} \int_{\mathcal{B}_R(p)} f(z) \, dz = c_{-1}. + \] + Der Koeffizient $c_{-1}$ der Laurententwicklung scheint also eine besondere + Rolle zu spielen. +\end{bemerkung} + + +\section{Die Umlaufzahl} + +Der Begriff „Umlaufzahl“ präzisiert die in \ref{bem:12-0-2} gemachte +Beobachtung. + +\begin{satz}[Umlaufzahl]\label{satz:12-2-1}% + Sei $p ∈ ℂ$ und $γ: [0,1] → ℂ ∖ \{p\}$ ein geschlossener Weg. Dann gibt es + genau eine Zahl $n ∈ ℤ$, sodass der Weg $γ$ frei homotop zum Weg + \[ + [0,1] → ℂ ∖ \{p\}, \quad t ↦ p + \exp(2π int) + \] + ist, nämlich + \[ + n = \frac{1}{2π i} \int_{γ} \frac{1}{z-p} \, dz. + \] +\end{satz} + +Wir beweisen Satz~\ref{satz:12-2-1} später. Zunächst definieren wir die +Umlaufzahl und diskutieren den Inhalt des Satzes geometrisch. + +\begin{definition}[Umlaufzahl]\label{def:12-2-2}% + In der Situation von Satz~\ref{satz:12-2-1} heißt $n$ die + \emph{Umlaufzahl}\index{Umlaufzahl} oder + \emph{Windungszahl}\index{Windungszahl} des Weges $γ$ um und Punkt $p$. Die + Schreibweise $\Um(γ, p)$ ist üblich. +\end{definition} + +\subsection{Anschauliche Bedeutung der Umlaufzahl} + +Anschaulich gesprochen: Die Umlaufzahl zählt, wie oft $γ$ den Punkt $p$ entgegen +dem Uhrzeigersinn umläuft (Umrundungen im Uhrzeigersinn werden mit $-1$ +gezählt). Mit den folgenden goldenen Regeln kann man die Umlaufzahl praktisch +ermitteln, wenn man den Weg $γ$ zeichnerisch gegeben hat und wenn der Weg $γ$ +einfach durchlaufen wird. Dabei bedeutet „einfach durchlaufen“, dass es eine +endliche Menge $Z ⊂ [0,1]$ gibt, sodass $γ|_{[0,1] ∖ Z}$ injektiv ist. + +\begin{description} + \item[Goldene Regel 1] Die Umlaufzahl $\Um(γ, p)$ hängt nur von der + Zusammenhangskomponente von $ℂ ∖ \Bild(γ)$ ab, in der sich der Punkt $p$ + befindet. + + \item[Goldene Regel 2] Auf der unbeschränkten Zusammenhangskomponente von $\bC + ∖ \Bild(γ)$ ist die Umlaufzahl gleich $0$. + + \item[Goldene Regel 3] Die Windungszahl in benachbarten + Zusammenhangskomponenten unterscheidet sich um $± 1$, wobei die größere Zahl + in Fahrtrichtung links liegt. +\end{description} + +Abbildung~\ref{fig:12-1-1} zeigt, was dabei herauskommt. Wir beweisen die +goldenen Regeln nicht wirklich vollständig, sondern geben nur gute Gründe für +deren Gültigkeit an. Details sind Inhalt der Vorlesung „Topologie“. +\begin{figure} + \begin{center} + \includegraphics[width=4cm]{12-res1.png} + \end{center} + + \caption{Anwendung der Goldenen Regeln} + \label{fig:12-1-1} +\end{figure} + +\begin{proof}[Begründung zur Goldenen Regel 1] + Der Satz über parameterabhängige Integral zeigt, dass die Funktion + \[ + \Um(γ, ·): ℂ ∖ \Bild(γ) → ℤ + \] + stetig ist. Die Aussage folgt, weil die Bildmenge $ℤ$ diskret ist. +\end{proof} + +\begin{proof}[Begründung zur Goldenen Regel 2] + In der Topologie zeigt man: Der Weg $γ$ ist in $ℂ ∖ \{p\}$ zusammenziehbar, + falls $p$ in der unbeschränkten Zusammenhangskomponente von $γ$ liegt. Die + Aussage folgt dann aus Korollar~\ref{kor:4-3-2} („Integralsatz von Cauchy“), + \[ + \int_γ \frac{1}{z-p} \, dz = \Um(γ, p) = 0. + \] +\end{proof} + +\begin{proof}[Begründung zur Goldenen Regel 3] + Seien $p_1, p_2 ∈ ℂ ∖ \Bild(γ)$ in benachbarten Zusammenhangskomponenten. Das + sieht dann etwa so aus: + \begin{center} + \includegraphics[width=14cm]{12-res2.png} + \end{center} + Wir ändern den Weg $γ$ wie eingezeichnet ab und vergleichen die Umlaufzahlen + der Wege $γ$ und $\widetilde{γ}_{ε}$. Es ist + \[ + \Um(γ, p_1) = \Um(\widetilde{γ}_{ε}, p_1) = \Um(\widetilde{γ}_{ε}, p_2). + \] + Dabei gilt das erste Gleichheitszeichen, weil der Weg $γ$ in $ℂ ∖ \{p_1\}$ + homotop zu $\widetilde{γ}_{ε}$ ist. Die zweite Gleichheit gilt nach der + Goldenen Regel 1, weil $p_1$ und $p_2$ in derselben Zusammenhangskomponente + von $ℂ ∖ \Bild(\widetilde{γ}_{ε})$ ist. Die Umlaufzahl $\Um(\widetilde{β}, + p_2)$ hängt natürlich nicht von der Größe $ε$ ab. Wenn man $ε$ gegen null + gehen lässt, erhält man folgendes Bild: + \begin{center} + \includegraphics[width=6cm]{12-res3.png} + \end{center} + Dann ist + \begin{align*} + \Um(γ, p_1) & = \Um(\widetilde{γ}_{ε}, p_2) \\ + & = \frac{1}{2π i} \int_{\widetilde{γ}} \frac{1}{z-p_2} \, dz \\ + & = \frac{1}{2π i} \left( \int_{γ} \frac{1}{z-p_2} \, dz + \int_{β} \frac{1}{z-p_2} \, dz \right) \\ + & = \Um(γ, p_2) + \frac{1}{2π i} · \int_{β} \frac{1}{z-p_2} \, dz + \intertext{und für $r$ ausreichend klein ist nach der Cauchy-Integralformel} + \int_{β} \frac{1}{z-p_2} \, dz & = \int_{B_r(p_2)} \frac{1}{z-p_2} \, dz = ± 2π i. + \end{align*} + Damit ist die Goldene Regel 3 zumindest halbwegs gerechtfertigt. +\end{proof} + +% !TEX root = Funktionentheorie diff --git a/Funktionentheorie.tex b/Funktionentheorie.tex index f273460..1a3fb54 100644 --- a/Funktionentheorie.tex +++ b/Funktionentheorie.tex @@ -32,12 +32,15 @@ \DeclareMathOperator{\ad}{ad} \DeclareMathOperator{\Bij}{Bij} +\DeclareMathOperator{\Bild}{Bild} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Hau}{Hau} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\rang}{rang} +\DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\spur}{spur} +\DeclareMathOperator{\Um}{Um} \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)}} @@ -151,6 +154,9 @@ Link in den Text ein. \part{Singularitäten} \input{09-singularities} +\input{10-laurent} +\input{11-applications} +\input{12-residuum} \addchap{Lizenz} diff --git a/Notizen/220622-Vorlesung.pdf b/Notizen/220622-Vorlesung.pdf deleted file mode 100644 index 2f603c311db336c9ddfbea4dfeba0451e72dee35..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 6505840 zcmeFabySq=_cpG_7R43;K~x&08vz9o>FyHgt{Gw|3lZsV5TsLShEk9Yi5YqTl^jZe 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