From 369209c2a674c2167372672f1cb273e02c0ab084 Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Fri, 17 Oct 2025 13:47:03 +0200 Subject: [PATCH] Cleanup --- .vscode/ltex.hiddenFalsePositives.de-DE.txt | 1 + 03-wegintegrale.tex | 269 ++++++++++---------- Notizen/220511-Vorlesung.pdf | Bin 144053 -> 0 bytes 3 files changed, 134 insertions(+), 136 deletions(-) delete mode 100644 Notizen/220511-Vorlesung.pdf diff --git a/.vscode/ltex.hiddenFalsePositives.de-DE.txt b/.vscode/ltex.hiddenFalsePositives.de-DE.txt index 7c7bd1f..5b8fdd5 100644 --- a/.vscode/ltex.hiddenFalsePositives.de-DE.txt +++ b/.vscode/ltex.hiddenFalsePositives.de-DE.txt @@ -10,3 +10,4 @@ {"rule":"GERMAN_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QDie Funktion \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q ist bei \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q differenzierbar und die Ableitungsmatrix ist eine Drehstreckung.\\E$"} {"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QNach der Kettenregel für differenzierbare Funktionen gilt für jedes \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q also die Gleichung \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q-Matrix \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\QVektor \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Qkompl.\\E$"} {"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QZahl \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Qkompl.\\E$"} +{"rule":"DE_SENTENCE_WHITESPACE","sentence":"^\\QAlso ist \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q und damit \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} diff --git a/03-wegintegrale.tex b/03-wegintegrale.tex index 28005bf..2c46680 100644 --- a/03-wegintegrale.tex +++ b/03-wegintegrale.tex @@ -179,11 +179,11 @@ Die folgenden Aussagen sollten Ihnen aus den Analysis-Vorlesungen bekannt sein. \subsection{Elementare Fakten zur Wegintegration} \begin{prop}[Umkehrung des Weges] - In der Situation von Definition~\ref{def:3-2-1} sei $\overline{γ}: [a,b] → U$ - derselbe Weg wie $γ$, nur umgekehrt durchlaufen: $\overline{γ}(t) = γ(b+a-t)$. + In der Situation von Definition~\ref{def:3-2-1} sei $\overlineγ: [a,b] → U$ + derselbe Weg wie $γ$, nur umgekehrt durchlaufen: $\overlineγ(t) = γ(b+a-t)$. Dann ist \[ - \int_{\overline{γ}} f(z) \, dz = - \int_γ f(z) \, dz. \eqno\qed + \int_{\overlineγ} f(z) \, dz = - \int_γ f(z) \, dz. \eqno\qed \] \end{prop} @@ -298,7 +298,7 @@ Fakten der Analysis und Topologie. stetig differenzierbaren Weg $γ: [0,1] → U$ mit $γ(0) = z_1$ und $γ'(0) = z_2$. Dann ist aber \[ - f(z_2) - f(z_1) = \int_γ f'(z) \, dz = \int_{γ} 0 \, dz = 0. + f(z_2) - f(z_1) = \int_γ f'(z) \, dz = \int_γ 0 \, dz = 0. \] Da dies für alle Punkte $z_1, z_2 ∈ U$ gilt, ist die Konstanz der Funktion $f$ gezeigt. @@ -309,18 +309,17 @@ Fakten der Analysis und Topologie. \sideremark{Vorlesung 5} -Wir erinnern uns an die Ergebnisse des letzten Abschnitts: Es sei $U \subset -\bC$ offen und es sei $f: U \to \bC$ stetig. Wenn $f$ eine Stammfunktion $F: U -\to \bC$ besitzt, dann gilt für jeden (stückweise) stetig differenzierbaren Weg -$\gamma: [a,b] \to U$ die Gleichung +Wir erinnern uns an die Ergebnisse des letzten Abschnitts: Es sei $U ⊂ ℂ$ offen +und es sei $f: U → ℂ$ stetig. Wenn $f$ eine Stammfunktion $F: U → ℂ$ besitzt, +dann gilt für jeden (stückweise) stetig differenzierbaren Weg $γ: [a,b] → U$ die +Gleichung \[ -\int_\gamma f(z)\, dz = F(\gamma(b)) - F(\gamma(a)). +\int_γ f(z)\, dz = F(γ(b)) - F(γ(a)). \] -Das Wegintegral hängt also nur von Start- und Endpunkt ab. Insbesondere gilt: -Wenn der Weg $\gamma$ geschlossen ist (das bedeutet: $\gamma(a) = \gamma(b)$), -dann ist +Das Wegintegral hängt also nur von Start- und Endpunkt ab. Insbesondere gilt: +Wenn der Weg $γ$ geschlossen ist (das bedeutet: $γ(a) = γ(b)$), dann ist \[ -\int_\gamma f(z)\, dz = 0. +\int_γ f(z)\, dz = 0. \] Ziel dieses Kapitels ist die Umkehrung dieser Aussage: Wir wollen zeigen, dass die Existenz einer Stammfunktion bereits aus der Tatsache folgt, dass das @@ -328,77 +327,80 @@ Wegintegral nur von Start- und Endpunkt abhängt. Die folgende Beobachtung wird beim Beweis helfen. \begin{beobachtung} - Es sei $U \subseteq \bC$ offen und es sei $f: U \to \bC$ eine Funktion, sodass - für jeden geschlossenen Weg $\gamma$ stets $\int_\gamma f(z)\, dz = 0$ ist. - Gegeben seien zwei (stückweise) stetig differenzierbare Wege + Es sei $U ⊆ ℂ$ offen und es sei $f: U → ℂ$ eine Funktion, sodass für jeden + geschlossenen Weg $γ$ stets $\int_γ f(z)\, dz = 0$ ist. Gegeben seien zwei + (stückweise) stetig differenzierbare Wege \[ - \gamma_1: [a_1, b_1] \to U \qquad \gamma_2: [a_2, b_2] \to U + γ_1: [a_1, b_1] → U \qquad γ_2: [a_2, b_2] → U \] mit identischem Start- und Endpunkt, \[ - \gamma_1(a_1) = \gamma_2(a_2) =: z_a \quad\text{und}\quad \gamma_1(b_1) = \gamma_2(b_2) =: z_b. - \] - Wir wollen zeigen, dass die Wegintegrale über $\gamma_1$ und $\gamma_2$ gleich - sind. Dazu betrachten wir den Weg, der zuerst $\gamma_1$ hin und dann - $\gamma_2$ zurück durchläuft. Genauer: Betrachte den folgenden, stückweise - stetig differenzierbaren Weg + γ_1(a_1) = γ_2(a_2) =: z_a + \quad\text{und}\quad + γ_1(b_1) = γ_2(b_2) =: z_b. + \] + Wir wollen zeigen, dass die Wegintegrale über $γ_1$ und $γ_2$ gleich sind. + Dazu betrachten wir den Weg, der zuerst $γ_1$ hin und dann $γ_2$ zurück + durchläuft. Genauer: Betrachte den folgenden, stückweise stetig + differenzierbaren Weg \[ - \delta: [a_1, (b_1 - a_1), (b_2 - a_2)] \to U, \quad t \mapsto + δ: [a_1, (b_1 - a_1), (b_2 - a_2)] → U, \quad t ↦ \begin{cases} - \gamma_1(t + a_1) & \text{falls } t < b_1 - a_1 \\ - \gamma_2(b_2 + b_2 - a_1 - t) & \text{sonst}. + γ_1(t + a_1) & \text{falls } t < b_1 - a_1 \\ + γ_2(b_2 + b_2 - a_1 - t) & \text{sonst}. \end{cases} \] Dann gilt: \begin{align*} - 0 & = \int_\delta f(z)\, dz && \text{weil $\delta$ geschlossen} \\ - & = \int_{\gamma_1} f(z)\, dz - \int_{\gamma_2} f(z)\, dz. + 0 & = \int_{δ} f(z)\, dz && \text{weil $δ$ geschlossen} \\ + & = \int_{γ_1} f(z)\, dz - \int_{γ_2} f(z)\, dz. \end{align*} Also folgt: \[ - \int_{\gamma_1} f(z)\, dz = \int_{\gamma_2} f(z)\, dz. + \int_{γ_1} f(z)\, dz = \int_{γ_2} f(z)\, dz. \] Zusammenfassung: Wenn das Wegintegral über jeden geschlossenen Weg - verschwindet, dann hängen die Wegintegral $\int_{\bullet} f(d)\, dz$ nur von - Start- und Endpunkt ab. + verschwindet, dann hängen die Wegintegral $\int_• f(d)\, dz$ nur von Start- + und Endpunkt ab. \end{beobachtung} \begin{satz}[Existenz von Stammfunktionen]\label{satz:3-3-9}% - Sei $U \subset \mathbb{C}$ offen, und sei $f: U \to \bC$ stetig. Falls die - Wegintegral $\int_{\bullet} f(d)\, dz$ nur von Start- und Endpunkt abhängen, - dann existiert eine Stammfunktion. + Sei $U ⊂ ℂ$ offen, und sei $f: U → ℂ$ stetig. Falls die Wegintegral $\int_• + f(d)\, dz$ nur von Start- und Endpunkt abhängen, dann existiert eine + Stammfunktion. \end{satz} \begin{proof} Wir können ohne Beschränkung der Allgemeinheit annehmen, dass die offene Menge $U$ zusammenhängend ist -- ansonsten betrachte die Zusammenhangskomponenten - einzeln. Wähle einen Punkt $z_0 \in U$. Die Annahme, dass Wegintegrale nur von + einzeln. Wähle einen Punkt $z_0 ∈ U$. Die Annahme, dass Wegintegrale nur von Start- und Endpunkt abhängen, erlaubt die Definition der folgenden Funktion: \[ - F: U \to \bC, \quad z \mapsto \int_\gamma f(z)\, dz, \text{ wobei $\gamma$ ein Weg ist, der $z_0$ und $z$ verbindet.} + F: U → ℂ, \quad z ↦ \int_γ f(z)\, dz, \text{ wobei $γ$ ein Weg ist, der $z_0$ und $z$ verbindet.} \] Ich behaupte, dass $F$ eine Stammfunktion von $f$ ist. Dazu müssen wir zeigen, dass $F$ an jedem Punkt von $U$ komplex differenzierbar ist und dass - $F' = f$ gilt. Sei also ein Punkt $p \in U$ gegeben. Wir müssen zeigen, dass + $F' = f$ gilt. Sei also ein Punkt $p ∈ U$ gegeben. Wir müssen zeigen, dass \begin{equation}\label{eq:3-3-9-1} - \lim_{h \to 0} \frac{F(p+h) - F(p)}{h} = f(p) + \lim_{h → 0} \frac{F(p+h) - F(p)}{h} = f(p) \end{equation} - ist. Dazu wähle irgendeinen Weg $\delta: [0, 1] \to U$ mit $\delta(0) = z_0$, - und $\delta(1) = p$. Für komplexe Zahlen $h$ von hinreichend kleinem Betrag - ist die Kreisscheibe um $p$ mit Radius $|h|$ komplett in $U$ enthalten. - Gegeben ein solches $h$, betrachten wir den Weg - \[ - \gamma_h: [0, 1] \to U, \quad t \mapsto p + t \cdot h. + ist. Dazu wähle irgendeinen Weg $δ: [0, 1] → U$ mit $δ(0) = z_0$, und $δ(1) = + p$. Für komplexe Zahlen $h$ von hinreichend kleinem Betrag ist die + Kreisscheibe um $p$ mit Radius $|h|$ komplett in $U$ enthalten. Gegeben ein + solches $h$, betrachten wir den Weg + \[ + γ_h: [0, 1] → U, \quad t ↦ p + t · h. \] Dann ist \[ - F(p) = \int_\delta f(z)\, dz \quad\text{und}\quad F(p+h) = \int_\delta f(z)\, dz + \int_{\gamma_h} f(z)\, dz. + F(p) = \int_{δ} f(z)\, dz \quad\text{und}\quad F(p+h) = \int_{δ} f(z)\, dz + \int_{γ_h} f(z)\, dz. \] - Also gilt für jede komplexe Zahl $h$ mit ausreichend kleinem Betrag die Gleichung + Also gilt für jede komplexe Zahl $h$ mit ausreichend kleinem Betrag die + Gleichung \begin{align*} - \frac{F(p+h) - F(h)}{h} & = \frac{\int_{\gamma_h} f(z)\, dz}{h} \\ - & = \frac{1}{h} \int_0^1 f(\gamma_h(t)) \cdot \gamma_h'(t)\, dt \\ - & = \frac{1}{h} \int_0^1 f(p + t \cdot h) \cdot h\, dt \\ - & = \int_0^1 f(p + th)\, dt. + \frac{F(p+h) - F(h)}{h} & = \frac{\int_{γ_h} f(z)\, dz}{h} \\ + & = \frac{1}{h} \int_0¹ f(γ_h(t)) · γ_h'(t)\, dt \\ + & = \frac{1}{h} \int_0¹ f(p + t · h) · h\, dt \\ + & = \int_0¹ f(p + th)\, dt. \end{align*} Gleichung~\eqref{eq:3-3-9-1} folgt sofort, weil $f$ bei $p$ stetig ist! \end{proof} @@ -407,15 +409,15 @@ beim Beweis helfen. \subsection{Rechteckwege} Satz~\ref{satz:3-3-9} gibt es sehr allgemein. Die Voraussetzung ist aber sehr -stark: Das Wegintegral muss über jeden geschlossenen Weg verschwinden. Das ist +stark: Das Wegintegral muss über jeden geschlossenen Weg verschwinden. Das ist in der Praxis schwer zu überprüfen. Wir wollen daher eine Variante des Satzes beweisen, die eine schwächere Voraussetzung hat. Dazu betrachten statt beliebigen nur noch achsenparallele Wege der folgenden Art. \begin{center} \begin{tikzpicture} - \draw[->] (0,0) -- (2,0) node[midway,below] {$\gamma_1$}; - \draw[->] (2,0) -- (2,1.5) node[midway,right] {$\gamma_2$}; + \draw[->] (0,0) -- (2,0) node[midway,below] {$γ_1$}; + \draw[->] (2,0) -- (2,1.5) node[midway,right] {$γ_2$}; \end{tikzpicture} \end{center} @@ -424,103 +426,101 @@ folgenden Art. \begin{center} \begin{tikzpicture} - \draw[->] (0,0) -- (2,0) node[midway,below] {$\gamma_1$}; - \draw[->] (2,0) -- (2,1.5) node[midway,right] {$\gamma_2$}; - \draw[->] (2,1.5) -- (0,1.5) node[midway,above] {$\gamma_3$}; - \draw[->] (0,1.5) -- (0,0) node[midway,left] {$\gamma_4$}; + \draw[->] (0,0) -- (2,0) node[midway,below] {$γ_1$}; + \draw[->] (2,0) -- (2,1.5) node[midway,right] {$γ_2$}; + \draw[->] (2,1.5) -- (0,1.5) node[midway,above] {$γ_3$}; + \draw[->] (0,1.5) -- (0,0) node[midway,left] {$γ_4$}; \end{tikzpicture} \end{center} \begin{notation} - Gegeben eine offene Menge $U \subseteq \bC$, ein achsenparalleles Rechteck - $\mathcal{R} \subset U$ und eine stetige Funktion $f : U \to \bC$, dann nennt - man + Gegeben eine offene Menge $U ⊆ ℂ$, ein achsenparalleles Rechteck $\mathcal{R} + ⊂ U$ und eine stetige Funktion $f : U → ℂ$, dann nennt man \[ - \int_{\gamma_1} f(z)\, dz + \int_{\gamma_2} f(z)\, dz + \int_{\gamma_3} f(z)\, dz + \int_{\gamma_4} f(z)\, dz + \int_{γ_1} f(z)\, dz + \int_{γ_2} f(z)\, dz + \int_{γ_3} f(z)\, dz + \int_{γ_4} f(z)\, dz \] das \emph{Randintegral}\index{Randintegral} über das Rechteck $\mathcal{R}$. \end{notation} \begin{satz}[Stammfunktionen auf der Kreisscheibe]\label{satz:3-3-11}% - Es sei $U = \{z \in \bC \mid |z| < 1\}$ die Kreisscheibe und es sei $f: U \to - \bC$ eine stetige Funktion, sodass das Randintegral über achsenparallele - Rechtecke stets verschwindet. Dann besitzt $f$ eine Stammfunktion. + Es sei $U = \{z ∈ ℂ \mid |z| < 1\}$ die Kreisscheibe und es sei $f: U → ℂ$ + eine stetige Funktion, sodass das Randintegral über achsenparallele Rechtecke + stets verschwindet. Dann besitzt $f$ eine Stammfunktion. \end{satz} \begin{bemerkung} - Die Aussage „Dann besitzt $f$ eine Stammfunktion.“ ist nicht optimal. Es gilt: - „Die Funktion $f$ besitzt genau dann eine Stammfunktion, wenn \ldots“. Die + Die Aussage „Dann besitzt $f$ eine Stammfunktion.“ ist nicht optimal. Es + gilt: „Die Funktion $f$ besitzt genau dann eine Stammfunktion, wenn …“. Die Umkehrrichtung ist ja schon aus dem letzten Kapitel bekannt. \end{bemerkung} \begin{bemerkung} Satz~\ref{satz:3-3-11} zeigt dieselbe Folgerung wie Satz~\ref{satz:3-3-9}, - aber unter schwächeren Voraussetzungen. Also ist der Beweis aufwändiger. Die + aber unter schwächeren Voraussetzungen. Also ist der Beweis aufwändiger. Die Grundidee ist aber dieselbe. \end{bemerkung} \begin{proof}[Beweis von Satz~\ref{satz:3-3-11} als Bildgeschichte] - Gegeben irgendeinen Punkt $p \in U$, dann kann ich den $0$-Punkt wie folgt - durch zwei achsenparallele Wege mit $p$ verbinden, die ganz innerhalb von $U$ - verlaufen. Hier benutze ich natürlich, dass $U$ eine Kreisscheibe ist. + Gegeben irgendeinen Punkt $p ∈ U$, dann kann ich den $0$-Punkt wie folgt durch + zwei achsenparallele Wege mit $p$ verbinden, die ganz innerhalb von $U$ + verlaufen. Hier benutze ich natürlich, dass $U$ eine Kreisscheibe ist. \begin{center} \begin{tikzpicture} \draw (0,0) circle (2cm); - \draw[->] (0,0) -- (1.5,0) node[midway,below] {$\gamma_1$}; - \draw[->] (1.5,0) -- (1.5,1) node[midway,left] {$\gamma_2$}; + \draw[->] (0,0) -- (1.5,0) node[midway,below] {$γ_1$}; + \draw[->] (1.5,0) -- (1.5,1) node[midway,left] {$γ_2$}; \node at (1.5,1) [circle,fill,inner sep=1.5pt,label=left:$p$] {}; \node at (0,0) [circle,fill,inner sep=1.5pt,label=below:$0$] {}; \end{tikzpicture} \end{center} Definiere damit eine Funktion \[ - F : U \to \bC, \quad p \mapsto \int_{\gamma_1} f(z)\, dz + \int_{\gamma_2} f(z)\, dz. + F : U → ℂ, \quad p ↦ \int_{γ_1} f(z)\, dz + \int_{γ_2} f(z)\, dz. \] - Ich behaupte, dass $F$ eine Stammfunktion von $f$ ist. Dazu müssen wir zeigen, - dass $F$ an jedem Punkt von $U$ komplex differenzierbar ist und dass $F' = f$ - gilt. Sei also ein Punkt $p \in U$ gegeben. + Ich behaupte, dass $F$ eine Stammfunktion von $f$ ist. Dazu müssen wir + zeigen, dass $F$ an jedem Punkt von $U$ komplex differenzierbar ist und dass + $F' = f$ gilt. Sei also ein Punkt $p ∈ U$ gegeben. Ich diskutiere erst einmal die partielle Ableitung von $F$ nach $x$. Gegeben eine hinreichend kleine reelle Zahl $h$, betrachte die folgenden Wege, die wieder vollständig innerhalb von $U$ verlaufen. \begin{center} \begin{tikzpicture}[scale=0.8] - \draw[->] (0,0) -- (2,0) node[midway,below] {$\gamma_1$}; - \draw[->] (2,0) -- (3,0) node[midway,below] {$\delta_1$}; - \draw[->] (2,0) -- (2,2) node[midway,left] {$\gamma_2$}; - \draw[->] (3,0) -- (3,2) node[midway,right] {$\delta_2$}; - \draw[->] (3,2) -- (2,2) node[midway,above] {$\delta_3$}; + \draw[->] (0,0) -- (2,0) node[midway,below] {$γ_1$}; + \draw[->] (2,0) -- (3,0) node[midway,below] {$δ_1$}; + \draw[->] (2,0) -- (2,2) node[midway,left] {$γ_2$}; + \draw[->] (3,0) -- (3,2) node[midway,right] {$δ_2$}; + \draw[->] (3,2) -- (2,2) node[midway,above] {$δ_3$}; \node at (2,2) [circle,fill,inner sep=1pt,label=left:$p$] {}; \node at (3,2) [circle,fill,inner sep=1pt,label=right:$p+h$] {}; \node at (0,0) [circle,fill,inner sep=1pt,label=below:$0$] {}; \end{tikzpicture} \end{center} - Weil das Randintegral über das Rechteck verschwindet, ist + Weil das Randintegral über das Rechteck verschwindet, ist \begin{align*} - F(p + h) & = \int_{\gamma_1} f(z)\, dz + \int_{\delta_1} f(z)\, dz + \int_{\delta_2} f(z)\, dz \\ - & = \int_{\gamma_1} f(z)\, dz + \int_{\delta_1} f(z)\, dz + \int_{\delta_2} f(z)\, dz \\ - & \qquad - \bigl(\int_{\delta_1} f(z)\, dz + \int_{\delta_2} f(z)\, dz + \int_{\delta_3} f(z)\, dz - \int_{\gamma_2} f(z)\, dz\bigr) \\ - & = \int_{\gamma_1} f(z)\, dz - \bigl(\int_{\delta_3} f(z)\, dz - \int_{\gamma_2} f(z)\, dz\bigr) \\ - & = F(p) - \int_{\delta_3} f(z)\, dz. + F(p + h) & = \int_{γ_1} f(z)\, dz + \int_{δ_1} f(z)\, dz + \int_{δ_2} f(z)\, dz \\ + & = \int_{γ_1} f(z)\, dz + \int_{δ_1} f(z)\, dz + \int_{δ_2} f(z)\, dz \\ + & \qquad - \bigl(\int_{δ_1} f(z)\, dz + \int_{δ_2} f(z)\, dz + \int_{δ_3} f(z)\, dz - \int_{γ_2} f(z)\, dz\bigr) \\ + & = \int_{γ_1} f(z)\, dz - \bigl(\int_{δ_3} f(z)\, dz - \int_{γ_2} f(z)\, dz\bigr) \\ + & = F(p) - \int_{δ_3} f(z)\, dz. \intertext{Also ist} - F(p+h) - F(p) & = -\int_{\delta_3} f(z)\, dz = \int_0^1 f(p + t \cdot h) \cdot h\, dt \\ + F(p+h) - F(p) & = -\int_{δ_3} f(z)\, dz = \int_0¹ f(p + t · h) · h\, dt \\ \end{align*} und damit \[ - \lim_{h \to 0} \frac{F(p+h) - F(p)}{h} = f(p). + \lim_{h → 0} \frac{F(p+h) - F(p)}{h} = f(p). \] Zusammenfassung: Die Funktion $F$ ist am Punkt $p$ partiell nach $x$ - differenzierbar und es ist $\frac{\partial F}{\partial x}(p) = f(p)$. Analog - beweist man: Die Funktion $F$ ist am Punkt $p$ partiell nach $y$ - differenzierbar und es ist $\frac{\partial F}{\partial y}(p) = i \cdot f(p)$. - Das hat zwei Konsequenzen. + differenzierbar und es ist $\frac{∂ F}{∂ x}(p) = f(p)$. Analog beweist man: + Die Funktion $F$ ist am Punkt $p$ partiell nach $y$ differenzierbar und es ist + $\frac{∂ F}{∂ y}(p) = i · f(p)$. Das hat zwei Konsequenzen. \begin{itemize} \item Weil $f$ stetig ist, haben wir gezeigt, dass $F$ stetig partiell differenzierbar ist, also total differenzierbar. - \item Es ist $\frac{\partial F}{\partial \overline{z}} = \frac{\partial - F}{\partial x} + i \frac{\partial F}{\partial y} = 0$. Also erfüllen die - partiellen Ableitungen von $F$ die Cauchy-Riemann-Differenzialgleichungen. + \item Es ist $\frac{∂ F}{∂ \overline{z}} = \frac{∂ F}{∂ x} + i \frac{∂ F}{∂ + y} = 0$. Also erfüllen die partiellen Ableitungen von $F$ die + Cauchy-Riemann-Differenzialgleichungen. \end{itemize} In der Summe sehen wir, dass $F$ komplex differenzierbar ist und dass $F' = f$ gilt. @@ -529,91 +529,88 @@ folgenden Art. \section{Homotopie von Wegen} -Gegeben eine offene Menge $U \subset \bC$ und Punkte $z_0, z_1 \in U$, so -betrachten wir Wege, die $z_0$ und $z_1$ verbinden. Anschaulich ist klar, dass -manche dieser Wege stetig ineinander übergeführt werden können und andere nicht. +Gegeben eine offene Menge $U ⊂ ℂ$ und Punkte $z_0, z_1 ∈ U$, so betrachten wir +Wege, die $z_0$ und $z_1$ verbinden. Anschaulich ist klar, dass manche dieser +Wege stetig ineinander übergeführt werden können und andere nicht. \begin{definition}[Homotopie von Wegen]\label{def:3-4-1}% - Es sei $U$ ein topologischer Raum, und $[a, b] \subset \bR$ ein kompaktes - Intervall. Zwei stetige Wege + Es sei $U$ ein topologischer Raum, und $[a, b] ⊂ ℝ$ ein kompaktes Intervall. + Zwei stetige Wege \[ - \gamma_0: [a, b] \to U, \quad \gamma_1: [a, b] \to U - \quad\text{mit}\quad - \gamma_0(a) = \gamma_1(a), \quad \gamma_0(b) = \gamma_1(b) + γ_0: [a, b] → U, \quad γ_1: [a, b] → U + \quad\text{mit}\quad + γ_0(a) = γ_1(a), \quad γ_0(b) = γ_1(b) \] heißen \emph{homotop}\index{homotop}, wenn es stetige Abbildung \[ - \Gamma: [a, b] \times [0, 1] \longrightarrow U + Γ: [a, b] ⨯ [0, 1] \longrightarrow U \] gibt, sodass die folgenden Bedingungen erfüllt sind: \begin{itemize} - \item $\forall s \in [0, 1] : \Gamma(a, s) = \gamma_0(a) \text{ und } \Gamma(b, s) = \gamma_0(b)$ - \item $\forall t \in [a, b] : \Gamma(t, 0) = \gamma_0(t) \text{ und } \Gamma(t, 1) = \gamma_1(t)$. + \item $\forall s ∈ [0, 1] : Γ(a, s) = γ_0(a) \text{ und } Γ(b, s) = γ_0(b)$ + \item $\forall t ∈ [a, b] : Γ(t, 0) = γ_0(t) \text{ und } Γ(t, 1) = γ_1(t)$. \end{itemize} - Eine Abbildung $\Gamma$ mit diesen Eigenschaften heißt \emph{Homotopie}\index{Homotopie} - zwischen den Wegen $\gamma_0$ und $\gamma_1$. + Eine Abbildung $Γ$ mit diesen Eigenschaften heißt + \emph{Homotopie}\index{Homotopie} zwischen den Wegen $γ_0$ und $γ_1$. \end{definition} In der Situation von Definition~\ref{def:3-4-1} kann man die Homotopie als -Familie von Wegen auffassen, $\gamma_s := \Gamma(\bullet, s)$, die stetig -zwischen den Wegen $\gamma_0$ und $\gamma_1$ interpoliert. +Familie von Wegen auffassen, $γ_s := Γ(•, s)$, die stetig zwischen den Wegen +$γ_0$ und $γ_1$ interpoliert. \begin{definition}[Zusammenziehbare Wege und einfach zusammenhängende Räume]\label{def:3-4-2}% - Es sei $U$ ein topologischer Raum, und $[a, b] \subset \bR$ ein kompaktes - Intervall. Ein \emph{geschlossener Weg}\index{geschlossener Weg} in $U$ ist - ein stetiger Weg + Es sei $U$ ein topologischer Raum, und $[a, b] ⊂ ℝ$ ein kompaktes Intervall. + Ein \emph{geschlossener Weg}\index{geschlossener Weg} in $U$ ist ein stetiger + Weg \[ - \gamma: [a, b] \to U \quad\text{mit}\quad \gamma(a) = \gamma(b). + γ: [a, b] → U \quad\text{mit}\quad γ(a) = γ(b). \] Ein geschlossener Weg heißt \emph{zusammenziehbar}\index{zusammenziehbar} oder \emph{kontrahierbar}\index{kontrahierbar}, wenn er homotop zu einem konstanten - Weg ist. Der Raum $U$ heißt \emph{wegweise einfach + Weg ist. Der Raum $U$ heißt \emph{wegweise einfach zusammenhängend}\index{wegweise einfach zusammenhängend}, wenn jeder geschlossener Weg zusammenziehbar ist. \end{definition} \begin{bsp}[Zentrierte geschlossene Wege in der Kreisscheibe] - Es sei $U \subset \bC$ die Einheits-Kreisscheibe und es sei $\gamma_0: [a, b] - \to U$ ein Weg mit $\gamma_0(a) = \gamma_0(b) = 0$. Dann ist + Es sei $U ⊂ ℂ$ die Einheits-Kreisscheibe und es sei $γ_0: [a, b] → U$ ein Weg + mit $γ_0(a) = γ_0(b) = 0$. Dann ist \[ - \Gamma: [a, b] \times [0, 1] \to U, \quad (t, s) \mapsto (1 - t) \cdot \gamma_0(t) + Γ: [a, b] ⨯ [0, 1] → U, \quad (t, s) ↦ (1 - t) · γ_0(t) \] - eine Homotopie zwischen dem Weg $\gamma_0$ und dem konstanten Weg $\gamma_1 - \equiv 0$. Also ist $\gamma_0$ zusammenziehbar. + eine Homotopie zwischen dem Weg $γ_0$ und dem konstanten Weg $γ_1 \equiv 0$. + Also ist $γ_0$ zusammenziehbar. \end{bsp} \begin{bsp}[Allgemeine geschlossene Wege in der Kreisscheibe]\label{bsp:3-4-4}% - Es sei $U \subset \bC$ die Einheits-Kreisscheibe und es sei $\gamma_0: [a, b] - \to U$ irgendein geschlossener Weg mit $\gamma_0(a) = \gamma_0(b) =: z$. Dann - ist + Es sei $U ⊂ ℂ$ die Einheits-Kreisscheibe und es sei $γ_0: [a, b] → U$ + irgendein geschlossener Weg mit $γ_0(a) = γ_0(b) =: z$. Dann ist \[ - \Gamma: [a, b] \times [0, 1] \to U, \quad (t, s) \mapsto (1 - s) \cdot (\gamma_0(t) - z) + z + Γ: [a, b] ⨯ [0, 1] → U, \quad (t, s) ↦ (1 - s) · (γ_0(t) - z) + z \] - eine Homotopie zwischen $\gamma_0$ und dem konstanten Weg $\gamma_1 \equiv z$. - Also ist die Kreisscheibe einfach zusammenhängend. + eine Homotopie zwischen $γ_0$ und dem konstanten Weg $γ_1 \equiv z$. Also ist + die Kreisscheibe einfach zusammenhängend. \end{bsp} \begin{bemerkung}[Konvexe Mengen sind einfach zusammenhängend] Der wesentliche Punkt in Beispiel~\ref{bsp:3-4-4} ist, dass die Bildmenge von - $\Gamma$ ganz in der Kreisscheibe enthalten ist. Die zum Beweis notwendige + $Γ$ ganz in der Kreisscheibe enthalten ist. Die zum Beweis notwendige Rechnung verwendet allerdings nur, dass die Kreisscheibe konvex ist. - Tatsächlich sind alle konvexen Mengen wegweise einfach zusammenhängend. - In nicht-konvexen Mengen können geschlossene Wege durchaus nicht - zusammenziehbar sein. + Tatsächlich sind alle konvexen Mengen wegweise einfach zusammenhängend. In + nicht-konvexen Mengen können geschlossene Wege durchaus nicht zusammenziehbar + sein. \end{bemerkung} \begin{bemerkung} Homotopie ist eine Äquivalenzrelation auf der Menge der Wege mit festen - Anfangs- und Endpunkten. Insbesondere ist Homotopie reflexiv, symmetrisch und + Anfangs- und Endpunkten. Insbesondere ist Homotopie reflexiv, symmetrisch und transitiv. \end{bemerkung} \begin{bemerkung} Wir haben noch kein Kriterium dafür, dass eine Menge \emph{nicht} einfach - zusammenhängend ist. Der Integralsatz von Cauchy wird uns aber bald ein solches - Kriterium liefern. + zusammenhängend ist. 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