From 3ce8b986eefd2611784d40b4c4e121aec201b81a Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Tue, 2 Dec 2025 14:34:15 +0100 Subject: [PATCH] Working --- .vscode/ltex.dictionary.de-DE.txt | 9 ++ .vscode/ltex.hiddenFalsePositives.de-DE.txt | 2 + 14-harmonic.tex | 4 +- 15-RiemannMapping.tex | 129 ++++++++++---------- Notizen/220706-Vorlesung.pdf | Bin 162661 -> 0 bytes 5 files changed, 75 insertions(+), 69 deletions(-) delete mode 100644 Notizen/220706-Vorlesung.pdf diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt index 19d996e..ea439fb 100644 --- a/.vscode/ltex.dictionary.de-DE.txt +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -78,3 +78,12 @@ Baptiste Auxerre Beaumont-en-Auge Funktionenfolgen +Montel +Aristide +Félix +Saint-Affrique +Aveyron +Émile +Midi-Pyrénées +Borel +Beschränktheitssatz diff --git a/.vscode/ltex.hiddenFalsePositives.de-DE.txt b/.vscode/ltex.hiddenFalsePositives.de-DE.txt index f4baebd..50c42b3 100644 --- a/.vscode/ltex.hiddenFalsePositives.de-DE.txt +++ b/.vscode/ltex.hiddenFalsePositives.de-DE.txt @@ -43,3 +43,5 @@ {"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QGenauer: Wenn \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q eine Stammfunktion von \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q ist, dann sind alle Lösungen der Differenzialgleichung auf \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q gegeben durch \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Qconst \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} {"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QWir beweisen Satz \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q auf Seite pf:14-3-2.\\E$"} {"rule":"LEERZEICHEN_HINTER_DOPPELPUNKT","sentence":"^\\QWir beweisen Satz \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q auf Seite pf:14-3-2.\\E$"} +{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qvon \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} +{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qkonvergiert bei allen \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q!\\E$"} diff --git a/14-harmonic.tex b/14-harmonic.tex index b5295ae..81f0fc9 100644 --- a/14-harmonic.tex +++ b/14-harmonic.tex @@ -70,7 +70,7 @@ Mittelwert-Eigenschaft per Definition erfüllen. \section{Konstruktion von harmonischen Funktionen} \begin{konstruktion}[Funktionen vom Rand der Kreisscheibe in das Innere Fortsetzen]\label{konst:14-2-1}% - Es sei $S¹ ⊂ ℂ$ der Einheitskreis und $h : S¹ → ℝ$ sei stetig. Dann betrachte + Es sei $S¹ ⊂ ℂ$ der Einheitskreis und $h : S¹ → ℝ$ sei stetig. Dann betrachte \[ \bar{h} : \overline{B_1(0)} → ℝ, \quad @@ -186,7 +186,7 @@ Wir beweisen Satz~\ref{satz:14-3-2} auf Seite~\vpageref{pf:14-3-2}. \begin{notation}[Laplace\footnote{Pierre-Simon Laplace, seit 1817 Marquis de Laplace (* 23.~März 1749 in Beaumont-en-Auge in der Normandie; † 5.~März - 1827 in Paris) war ein französischer Mathematiker, Physiker und Astronom. Er + 1827 in Paris) war ein französischer Mathematiker, Physiker und Astronom. Er beschäftigte sich unter anderem mit der Wahrscheinlichkeitstheorie und mit Differenzialgleichungen.}-Operator]% Statt $\left(\frac{∂²}{∂x²} + \frac{∂²}{∂y²}\right) f$ schreibt man oft $Δf$. diff --git a/15-RiemannMapping.tex b/15-RiemannMapping.tex index 183cd19..c587f6e 100644 --- a/15-RiemannMapping.tex +++ b/15-RiemannMapping.tex @@ -4,39 +4,39 @@ \chapter{Der Riemannsche Abbildungssatz} \begin{satz}[Riemannscher Abbildungssatz]\label{satz:14-5-1}% - \index{Riemannscher Abbildungssatz}% - Es sei $U \subsetneq \bC$ offen, zusammenhängend und einfach zusammenhängend. - Dann ist $U$ biholomorph zur Kreisscheibe $B_1(0)$. + \index{Riemannscher Abbildungssatz}Es sei $U ⊊ ℂ$ offen, zusammenhängend und + einfach zusammenhängend. Dann ist $U$ biholomorph zur Kreisscheibe $B_1(0)$. \end{satz} \begin{bem} - Die Annahme $U \subsetneq \mathbb{C}$ ist wichtig, denn $U = \mathbb{C}$ ist - nicht biholomorph zu $B_1(0)$. + Die Annahme $U ⊊ \bC$ ist wichtig, denn $U = \bC$ ist nicht biholomorph zu + $B_1(0)$. \end{bem} \section{Der Satz von Montel} Ein wesentlicher technisches Hilfsmittel im Beweis ist der folgende Satz über -gleichmäßig beschränkte Funktionenfolgen. In der Literatur findet man statt +gleichmäßig beschränkte Funktionenfolgen. In der Literatur findet man statt „gleichmäßig beschränkt“ manchmal auch den Begriff „betragsmäßig simultan beschränkt“. \begin{defn}[Gleichmäßig beschränkte Funktionenfolgen]\label{def:15-0-3}% - Sei $U \subset \bC$ offen. Eine Funktionenfolge $f_n : U \to \bC$ ist - \emph{gleichmäßig beschränkt}\index{gleichmäßig beschränkte Funktionenfolge}, - wenn eine Zahl $R \in \bR$ existiert, sodass für jedes $p \in U$ und jedes $n - \in \bN$ die Ungleichung $|f_n(p)| < R$ gilt. Die Funktionenfolge ist - \emph{lokal gleichmäßig beschränkt}\index{local gleichmäßig beschränkte - Funktionenfolge}, wenn jeder Punkt $p \in U$ eine Umgebung $V = V(p) \subset - U$ hat, sodass $f_n|_V : V \to \bC$ gleichmäßig beschränkt ist. + Sei $U ⊂ ℂ$ offen. Eine Funktionenfolge $f_n : U → ℂ$ ist \emph{gleichmäßig + beschränkt}\index{gleichmäßig beschränkte Funktionenfolge}, wenn eine Zahl $R + ∈ ℝ$ existiert, sodass für jedes $p ∈ U$ und jedes $n ∈ ℕ$ die Ungleichung + $|f_n(p)| < R$ gilt. Die Funktionenfolge ist \emph{lokal gleichmäßig + beschränkt}\index{local gleichmäßig beschränkte Funktionenfolge}, wenn jeder + Punkt $p ∈ U$ eine Umgebung $V = V(p) ⊂ U$ hat, sodass $f_n|_V : V → ℂ$ + gleichmäßig beschränkt ist. \end{defn} -\begin{satz}[Satz von Montel\footnote{Paul Antoine Aristide Montel (* 29.~April - 1876 in Nizza; † 22.~Januar 1975 in Paris) war ein französischer Mathematiker.}]\label{satz:15-0-4}% - Es sei $U \subset \bC$ offen und $f_n \in \sO(U)$ eine lokal gleichmäßig - beschränkte Folge von holomorphen Funktionen. Dann gibt es eine Teilfolge, - die lokal gleichmäßig konvergiert. +\begin{satz}[Satz von Montel\footnote{Paul Antoine Aristide Montel (* 29.~April + 1876 in Nizza; † 22.~Januar 1975 in Paris) war ein französischer + Mathematiker.}]\label{satz:15-0-4}% + Es sei $U ⊂ ℂ$ offen und $f_n ∈ 𝒪(U)$ eine lokal gleichmäßig beschränkte + Folge von holomorphen Funktionen. Dann gibt es eine Teilfolge, die lokal + gleichmäßig konvergiert. \end{satz} \begin{erinnerung}[Satz von Heine\footnote{Heinrich Eduard Heine (* 18.~März @@ -44,45 +44,45 @@ beschränkt“. Mathematiker und Hochschullehrer.}--Borel\footnote{Félix Édouard Justin Émile Borel (* 7.~Januar 1871 in Saint-Affrique, Département Aveyron, Region Midi-Pyrénées; † 3.~Februar 1956 in Paris) war ein französischer - Mathematiker und Politiker. }]% + Mathematiker und Politiker.}]% Es sei $a_n$ eine beschränkte Folge von komplexen Zahlen. Dann gibt es eine konvergente Teilfolge. Im Kontext von Satz~\ref{satz:15-0-4} bedeutet das: - wenn ein Punkt $p \in U$ gegeben ist, dann gibt es eine Teilfolge $f_{n_1}, - f_{n_2}, \ldots$, sodass $f_{n_k}(p)$ konvergiert. + wenn ein Punkt $p ∈ U$ gegeben ist, dann gibt es eine Teilfolge $f_{n_1}, + f_{n_2}, …$, sodass $f_{n_k}(p)$ konvergiert. \end{erinnerung} \begin{vorueberlegung}[Vorbereitung 1 zum Satz von Montel]\label{vor:15-0-4}% Wir wissen, dass es eine abzählbare Basis der Topologie gibt. Insbesondere - gibt es eine abzählbare, dichte Teilmenge $p_1, p_2, p_3, \ldots$ von $U$. + gibt es eine abzählbare, dichte Teilmenge $p_1, p_2, p_3, …$ von $U$. \begin{itemize} - \item Es gibt Teilfolgen $f_{n_1'}, f_{n_2'}, f_{n_3'}, \ldots$, die bei - $p_1$ konvergiert. + \item Es gibt Teilfolgen $f_{n_1'}, f_{n_2'}, f_{n_3'}, …$, die bei $p_1$ + konvergiert. - \item Davon gibt es Teilfolgen $f_{n_1''}, f_{n_2''}, f_{n_3''}, \ldots$, - die bei $p_1$ und $p_2$ konvergiert. + \item Davon gibt es Teilfolgen $f_{n_1''}, f_{n_2''}, f_{n_3''}, …$, die bei + $p_1$ und $p_2$ konvergiert. \item … \end{itemize} - Am Ende gilt: Die Teilfolge $f_{n_1'}, f_{n_2'}, f_{n_3'}, \ldots$ konvergiert bei + Am Ende gilt: die Teilfolge $f_{n_1'}, f_{n_2'}, f_{n_3'}, …$ konvergiert bei allen $p_k$! \end{vorueberlegung} \begin{vorueberlegung}[Vorbereitung 2 zum Satz von Montel]\label{vor:15-0-5}% - Wenn eine Kreisscheibe $\overline{B_r(p)} \subset U$ gegeben ist, dann gilt - für jede Zahl $n \in \bN$ und jeden Punkt $w \in B_r(p)$ nach der - Integralformel für Ableitungen die Gleichung + Wenn eine Kreisscheibe $\overline{B_r(p)} ⊂ U$ gegeben ist, dann gilt für jede + Zahl $n ∈ ℕ$ und jeden Punkt $w ∈ B_r(p)$ nach der Integralformel für + Ableitungen die Gleichung \[ - f_n'(w) = \frac{1}{2\pi} \int_{\partial B_r(p)} \frac{f_n(z)}{(z-w)^2}\,dz. + f_n'(w) = \frac{1}{2π} \int_{∂ B_r(p)} \frac{f_n(z)}{(z-w)²}\,dz. \] Beachte: \begin{itemize} \item Die Funktionswerte $f_n(z)$ ist per Annahme betragsmäßig beschränkt. - \item Auf $B_{r/2}(p)$ ist die Funktion $\frac{1}{(z-w)^2}$ ebenfalls + \item Auf $B_{r/2}(p)$ ist die Funktion $\frac{1}{(z-w)²}$ ebenfalls betragsmäßig beschränkt. \end{itemize} - Wir erhalten: Die Funktion $f_n'(w)$ ist lokal beschränkt. Genauer: es - existiert $M \in \bR$, sodass für jede Zahl $n \in \bN$ und jeden Punkt $w \in + Wir erhalten: Die Funktion $f_n'(w)$ ist lokal beschränkt. Genauer: es + existiert $M ∈ ℝ$, sodass für jede Zahl $n ∈ ℕ$ und jeden Punkt $w ∈ B_{r/2}(p)$ die Ungleichung \[ |f_n'(w)| < M @@ -90,67 +90,62 @@ beschränkt“. gilt. \end{vorueberlegung} -Der folgenden Mittelwert- und Beschränktheitssätz sing nun eine direkte +Der folgenden Mittelwert- und Beschränktheitssatz sing nun eine direkte Konsequenz der Vorüberlegung~\ref{vor:15-0-5}. \begin{konsequenz}[Mittelwertsatz] In der Situation von Satz~\ref{satz:15-0-4} sei eine Kreisscheibe - $\overline{B_r(p)} \subset U$ gegeben. Dann existiert eine Zahl $M \in \bR$, - sodass für jede Zahl $n \in \bN$ und jeden Punkt $w \in B_{r/2}(p)$ die - Ungleichung + $\overline{B_r(p)} ⊂ U$ gegeben. Dann existiert eine Zahl $M ∈ ℝ$, sodass für + jede Zahl $n ∈ ℕ$ und jeden Punkt $w ∈ B_{r/2}(p)$ die Ungleichung \[ - |f_n(p) - f_n(w)| < M \cdot |p-w| + |f_n(p) - f_n(w)| < M · |p-w| \] - gilt. \qed + gilt. \qed \end{konsequenz} \begin{konsequenz}[Beschränktheitssatz]\label{kon:15-0-6}% - In der Situation von Satz~\ref{satz:15-0-4} sei ein Punkt $p \in U$ und eine - Zahl $\varepsilon > 0$ gegeben. Dann existiert eine positive Zahl $\delta_{p, - \varepsilon} > 0$, sodass die Kreisscheibe $B_{\delta_{p, \varepsilon}}(p)$ - ganz in $U$ liegt und - für jede Zahl $n \in \bN$ und jeden Punkt $w \in B_{\delta_{p, - \varepsilon}}(p)$ die Ungleichung + In der Situation von Satz~\ref{satz:15-0-4} sei ein Punkt $p ∈ U$ und eine + Zahl $ε > 0$ gegeben. Dann existiert eine positive Zahl $δ_{p, ε} > 0$, + sodass die Kreisscheibe $B_{δ_{p, ε}}(p)$ ganz in $U$ liegt und für jede Zahl + $n ∈ ℕ$ und jeden Punkt $w ∈ B_{δ_{p, ε}}(p)$ die Ungleichung \[ - |f_n(p) - f_n(w)| < \varepsilon/3 + |f_n(p) - f_n(w)| < ε/3 \] - gilt. \qed + gilt. \qed \end{konsequenz} \begin{proof}[Beweis des Satzes~\ref{satz:15-0-4} von Montel] Nach Vorüberlegung~\ref{vor:15-0-4} können wir die Folge $f_n$ (falls nötig) durch eine Teilfolge ersetzen und ohne Beschränkung der Allgemeinheit - annehmen, dass eine abzählbare, dichte Teilmenge $p_1, p_2, \ldots \in U$ - existiert, sodass $f_n(p_k)$ konvergiert. + annehmen, dass eine abzählbare, dichte Teilmenge $p_1, p_2, … ∈ U$ existiert, + sodass $f_n(p_k)$ konvergiert. Wir werden zeigen, dass die Folge $f_n$ dann bereits lokal gleichmäßig - konvergiert: gegeben ein Kompaktum $K \subset U$, so gibt es für jedes - $\varepsilon > 0$ einen Index $N$, sodass für alle $n, m > N$ und jedes $p \in - K$ die Ungleichung + konvergiert: gegeben ein Kompaktum $K ⊂ U$, so gibt es für jedes $ε > 0$ einen + Index $N$, sodass für alle $n, m > N$ und jedes $p ∈ K$ die Ungleichung \[ - |f_n(p) - f_m(p)| < \varepsilon + |f_n(p) - f_m(p)| < ε \] - gilt. Seien also $K$ und $\varepsilon$ gegeben. + gilt. Seien also $K$ und $ε$ gegeben. - Man beachte: Die Kreisscheiben $B_{\delta(p_i, \varepsilon/3)}(p_i)$ aus - Konsequenz~\ref{kon:15-0-6} (``Beschränktheitssatz'') bilden eine offene - Überdeckung von $U$. Weil $K$ kompakt ist, überdecken endlich viele dieser + Man beachte: Die Kreisscheiben $B_{δ(p_i, ε/3)}(p_i)$ aus + Konsequenz~\ref{kon:15-0-6} („Beschränktheitssatz“) bilden eine offene + Überdeckung von $U$. Weil $K$ kompakt ist, überdecken endlich viele dieser Kreisscheiben die Menge $K$. Nach Umnummerierung seien dies \[ - B_{\delta_{p_1, \varepsilon/3}}(p_1), \ldots, B_{\delta_{p_\ell, - \varepsilon/3}}(p_\ell). + B_{δ_{p_1, ε/3}}(p_1), …, B_{δ_{p_{ℓ}, ε/3}}(p_{ℓ}). \] - Jetzt kann ich nach Annahme $N \in \bN$ wählen, sodass alle $n, m > N$ und jeden Index $1 \leq i \leq \ell$ die Ungleichung + Jetzt kann ich nach Annahme $N ∈ ℕ$ wählen, sodass alle $n, m > N$ und jeden + Index $1 ≤ i ≤ ℓ$ die Ungleichung \[ - |f_n(p_i) - f_m(p_i)| < \varepsilon/3. + |f_n(p_i) - f_m(p_i)| < ε/3 \] - gilt. Gegeben irgendeinen $p \in K$, so gibt es nun ein $p_i$, mit $1 \leq i \leq - \ell$, sodass $p \in B_{\delta_{p_i, \varepsilon/3}}(p_i)$ ist und für alls $n, m > - N$ gilt: + gilt. Gegeben irgendeinen $p ∈ K$, so gibt es nun ein $p_i$, mit $1 ≤ i ≤ ℓ$, + sodass $p ∈ B_{δ_{p_i, ε/3}}(p_i)$ ist und für alle $n, m > N$ gilt: \begin{align*} |f_n(p_i) - f_m(p)| & = |f_n(p) - f_n(p_i) + f_n(p_i) - f_m(p_i) + f_m(p_i) - f_m(p)| \\ - & \leq |f_n(p) - f_n(p_i)| + |f_n(p_i) - f_m(p_i)| + |f_m(p_i) - f_m(p)| \\ - & \leq \varepsilon. + & ≤ |f_n(p) - f_n(p_i)| + |f_n(p_i) - f_m(p_i)| + |f_m(p_i) - f_m(p)| \\ + & ≤ ε. \end{align*} Damit ist der Satz bewiesen. \end{proof} diff --git a/Notizen/220706-Vorlesung.pdf b/Notizen/220706-Vorlesung.pdf deleted file mode 100644 index 7a72e91e36afa8bbfb97bbb0a275406627aa6ee3..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 162661 zcmbrm19WCvvo;#rwr$(Ct&VNmwrzB5b&QVfbZon0-}L_Wx4(1F|Brjm8224n?^;!> zX04j@sj4-rk`Yn`5iwduI(8`1{*wOspOsKhj06k>c7~Qv1Oz-h1oSE%_9g`M!UoO; z)^=uJysr1d+tDW_uOX=TJ| zOvKLz#pBN9ZewrrWh8;SjkT>4mpdx<@6 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