From bfc450f229ca8a4eb4c7f8bf72f21f2665190fc3 Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Wed, 22 Oct 2025 10:48:07 +0200 Subject: [PATCH] Working --- .vscode/ltex.dictionary.de-DE.txt | 2 + 04-freeHomotopy-1.png | Bin 0 -> 45769 bytes 04-freeHomotopy-2.png | Bin 0 -> 29720 bytes 04-integration-1.png | Bin 0 -> 58573 bytes 04-integration-2.png | Bin 0 -> 52886 bytes 04-integration-3.png | Bin 0 -> 23398 bytes 04-wegintegraleStetig.tex | 168 +++++++++++++++++++++++++++--- 7 files changed, 158 insertions(+), 12 deletions(-) create mode 100644 04-freeHomotopy-1.png create mode 100644 04-freeHomotopy-2.png create mode 100644 04-integration-1.png create mode 100644 04-integration-2.png create mode 100644 04-integration-3.png diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt index f474313..9091dbc 100644 --- a/.vscode/ltex.dictionary.de-DE.txt +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -14,3 +14,5 @@ Lebesgue-Zahl homotope Homotopieinvarianz 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z%G}8gHqY4M3yQcJ3#k-(ypN+R}DrT6fTmDWnVZJT>6e_eTSjM2CTKHsv~=fzkc zO@c%2_Gml1iUO6Tum6yiPS%>r-1OrYpM{PxP(5WzBL6Ud(JyLaaR0Fr>02TW@|>^4 z-!S#?@~V|=LnF~si2-G%hM4oCOYwhQ=~{wjkVoPn2ML^C;tQWafO{<7|Lz{ O0000vg literal 0 HcmV?d00001 diff --git a/04-wegintegraleStetig.tex b/04-wegintegraleStetig.tex index 9186392..a8447b9 100644 --- a/04-wegintegraleStetig.tex +++ b/04-wegintegraleStetig.tex @@ -181,7 +181,7 @@ $γ_0$ und $γ_1$ interpoliert. \end{bemerkung} -\section{Homotopieinvarianz von Wegintegralen und der Integralsatz von Cauchy} +\section{Der Integralsatz von Cauchy} \sideremark{Vorlesung 6} @@ -189,7 +189,7 @@ Nach den vorhergehenden Abschnitten werden Sie sich schon denken, was jetzt kommt: Wegintegrale über homotope Wege sind gleich. Der folgende Satz stellt dies präzise dar. -\begin{satz}[Homotopieinvarianz von Wegintegralen] +\begin{satz}[Homotopieinvarianz von Wegintegralen]\label{satz:4-3-1}% Es sei $U ⊂ ℂ$ offen und es sei $f : U → ℂ$ holomorph. Weiter seien $γ_0, γ_1 : [a, b] → U$ zwei stetige Wege mit $γ_0(a) = γ_1(a)$ und $γ_0(b) = γ_1(b)$. Wenn $γ_0$ und $γ_1$ zueinander homotop sind, dann gilt die Gleichheit @@ -222,7 +222,7 @@ dies präzise dar. \int_{β_{j,l}} f(z) \, dz = 0. \] Gleichung~\eqref{eq:4-3-1-1} folgt nun durch Addition über alle Indizes $i$ - und $j$. Man beachte dabei, dass sich Wegeintegrale über entgegengesetzt + und $j$. Man beachte dabei, dass sich Wegintegrale über entgegengesetzt orientierte Wege aufheben. \begin{figure} \begin{center} @@ -234,9 +234,9 @@ dies präzise dar. \end{figure} \end{proof} -\begin{kor}[Integralsatz von Cauchy] - Es sei $U ⊂ ℂ$ offen und es sei $f : U → ℂ$ holomorph. Weiter seien $γ : [a, - b] → U$ ein zusammenziehbarer, stetiger Weg. Dann ist +\begin{kor}[Integralsatz von Cauchy]\label{kor:4-3-2}% + Es sei $U ⊂ ℂ$ offen und es sei $f : U → ℂ$ holomorph. Weiter sei $γ : [a, b] + → U$ ein zusammenziehbarer, stetiger Weg. Dann ist \[ \int_{γ} f(z) \, dz = 0. \] @@ -249,14 +249,158 @@ dies präzise dar. \[ γ: [0, 1] → U, \quad t ↦ e^{2πi t}. \] - Wir haben haben in Beispiel~\vref{bsp:3-2-2} aber bereits ausgerechnet, dass - das Wegintegral gleich + Wir haben in Beispiel~\vref{bsp:3-2-2} aber bereits ausgerechnet, was das + Wegintegral ist: \[ - \int_γ \frac{1}{z} \, dz = 2πi + \int_γ \frac{1}{z} \, dz = 2πi. \] - ist. Nach dem Integralsatz von Cauchy ist der Weg $γ$ also nicht - zusammenziehbar. Wir haben damit gezeigt, dass die Menge $ℂ^*$ nicht einfach - zusammenhängend ist. + Nach Korollar~\ref{kor:4-3-2}, dem Integralsatz von Cauchy, ist der Weg $γ$ + also nicht zusammenziehbar. Wir haben damit gezeigt, dass die Menge $ℂ^*$ + nicht einfach zusammenhängend ist. \end{bsp} +\sideremark{Vorlesung 7} + +\begin{kor}[Existenz von Stammfunktionen in einfach zusammenhängenden Mengen]% + Es sei $U ⊂ ℂ$ offen und wegweise einfach zusammenhängend, und es sei $f : U + → ℂ$ holomorph. Dann gibt es eine Stammfunktion $F: U → ℂ$ von $f$. +\end{kor} +\begin{proof} + Sei eine holomorphe Funktion $f$ gegeben. Um die Existenz einer Stammfunktion + zu beweisen, genügt es nach Satz~\vref{satz:3-3-9}zu zeigen, dass die + Wegintegrale $\int_• f(z)\, dz$ nur von Start- und Endpunkt des jeweiligen + Weges abhängen. Seien also $γ_0 : [a_0, b_0] → U$ $γ_1 : [a_1, b_1] → U$ zwei + stetige Wege gleichem Start- und Endpunkt, $γ_0(a_0) = γ_1(a_1)$ und $γ_0(b_0) + = γ_1(b_1)$. Betrachte als Nächstes den Weg $γ$, definiert durch + \[ + γ : [a_0, b_0 + b_1 - a_1] \to U, \quad t \mapsto := \begin{cases} + γ_0(t) & t ∈ [a_0, b_0), \\ + γ_1(b_0 + b_1 - t) & t ∈ [b_0, b_0 + b_1 - a_1]. + \end{cases} + \] + Beachte, dass der Weg $\gamma$ stetig ist und $γ(a_0) = γ(b_0 + b_1 - a_1)$ gilt, also + dass $γ$ ein geschlossener Weg ist. Da $U$ wegweise einfach zusammenhängend ist, ist der Weg $γ$ zusammenziehbar. + Nach dem gilt daher nach dem Integralsatz von Cauchy, + \begin{align*} + 0 & = \int_{γ} f(z) \, dz && \text{Korollar~\ref{kor:4-3-2}}\\ + & = \int_{γ_0} f(z) \, dz - \int_{γ_1} f(z) \, dz && \text{Konstruktion von }\gamma. + \end{align*} + Damit folgt die Behauptung. +\end{proof} + +Gelegentlich ist es nützlich, die folgende schwächere Version des Begriffs der +Homotopie zu verwenden. Abbildung~\ref{fig:4-3-1-2} veranschaulicht die Situation. + +\begin{figure} + \begin{center} + \includegraphics[width=13cm]{04-freeHomotopy-1.png} + \end{center} + + \caption{Freie Homotopie von geschlossenen Wegen} + \label{fig:4-3-1-2} +\end{figure} + +\begin{definition}[Freie Homotopie von geschlossenen Wegen]\label{def:4-3-5}% + Es sei $U$ ein topologischer Raum, und $[a, b] ⊂ ℝ$ ein kompaktes Intervall. + Zwei geschlossene Wege + \[ + γ_0: [a, b] → U, \quad γ_1: [a, b] → U + \quad\text{mit}\quad + γ_0(a) = γ_0(b), \quad γ_1(a) = γ_1(b) + \] + heißen \emph{frei homotop}\index{freie Homotopie von Wegen}, wenn es eine + stetige Abbildung + \[ + Γ: [a, b] ⨯ [0, 1] \longrightarrow U + \] + gibt, sodass die folgenden Bedingungen erfüllt sind: + \begin{itemize} + \item $\forall s ∈ [0, 1] : Γ(a, s) = γ_0(b)$ + \item $\forall t ∈ [a, b] : Γ(t, 0) = γ_0(t) \text{ und } Γ(t, 1) = γ_1(t)$. + \end{itemize} + Eine Abbildung $Γ$ mit diesen Eigenschaften heißt \emph{freie + Homotopie}\index{freie Homotopie von Wegen} zwischen den Wegen $γ_0$ und + $γ_1$. +\end{definition} + +Der Satz~\ref{satz:4-3-1} über die Homotopieinvarianz von Wegintegralen lässt +sich mit kleinen Änderungen auch auf freie Homotopien von geschlossenen Wegen +übertragen. + +\begin{satz}[Invarianz von Wegintegralen unter freier Homotopie]\label{satz:4-3-6}% + Es sei $U ⊂ ℂ$ offen und es sei $f : U → ℂ$ holomorph. Weiter seien $γ_0, γ_1 + : [a, b] → U$ zwei stetige. Wenn $γ_0$ und $γ_1$ zueinander frei homotop sind, + dann gilt die Gleichheit + \begin{equation}\label{eq:4-3-6-1} + \int_{γ_0} f(z) \, dz = \int_{γ_1} f(z) \, dz. + \end{equation} +\end{satz} +\begin{proof}[Beweisskizze mit Bild] + Es sei $\Gamma : [a,b] ⨯ [0,1] \to U$ eine freie Homotopie zwischen den + geschlossenen Wegen $γ_0$ und $γ_1$. Betrachte weiter den in + Abbildung~\ref{fig:4-3-3} gezeigten Weg + \begin{figure} + \begin{center} + \includegraphics[width=5cm]{04-freeHomotopy-2.png} + \end{center} + + \caption{Beweis: Invarianz von Wegintegralen unter freier Homotopie} + \label{fig:4-3-3} + \end{figure} + \[ + \delta : [0, 1] → U, s \mapsto \Gamma(a, s). + \] + Dies ist ein Weg, der den Punkt $γ_0(a) = γ_0(b)$ mit dem Punkt $γ_1(a) = + γ_1(b)$ verbindet. Konstruiere nun einen geschlossenen Weg $\wtilde{\gamma}_1 : + [a,b] \to U$, der Folgendes macht: + \begin{itemize} + \item Er beginnt bei $γ_0(a)$, + \item folgt dem Weg $δ$ bis zu $γ_1(a)$, + \item folgt dann dem Weg $γ_1$ bis zu $γ_1(b)$, + \item folgt dann dem Weg $δ$ zurück bis zu $γ_0(a)$. + \end{itemize} + Ich behaupte, dass die Wege $γ_0$ und $\wtilde{\gamma}_1$ homotop sind. Dazu + konstruiere eine Homotopie, also eine stetige Familie von Wegen, die $γ_0$ in + $\wtilde{\gamma}_1$ überführt. Für jeden Parameterwert $s ∈ [0, 1]$ + definieren wir den Weg + \[ + \wtilde{\gamma}_s : [a,b] \to U, + \] + der Folgendes macht: + \begin{itemize} + \item Er beginnt bei $γ_0(a)$, + \item folgt dem Weg $δ$ bis zu $Γ(a,s)$, + \item folgt dann dem Weg $Γ(•, s)$ bis zu $Γ(b,s)$, + \item folgt dann dem Weg $δ$ zurück bis zu $γ_0(a)$. + \end{itemize} + Die Abbildung + \[ + \wtilde{\Gamma} : [a,b] ⨯ [0,1] \to U, \quad (t,s) \mapsto \wtilde{\gamma}_s(t) + \] + definiert dann die gesuchte Homotopie zwischen den Wegen $γ_0$ und + $\wtilde{\gamma}_1$. Nach Satz~\ref{satz:4-3-1} gilt daher + \[ + \int_{γ_0} f(z) \, dz = \int_{\wtilde{\gamma}_1} f(z) \, dz = \int_{γ_1} f(z) \, dz. + \] + Die letzte Gleichung folgt unmittelbar aus der Definition des Wegintegrals + über stetige Wege (Definition~\ref{def:4-1-5}) und der Tatsache, dass die + Wegintegrale über den Hin- und Rückweg entlang von $δ$ sich aufheben. +\end{proof} + + +\section{Anwendung: Reelle Integration} + +Der Integralsatz von Cauchy kann verwendet werden, um rein reelle Integrale +auszurechnen, die uns auch schon in der Vorlesung ``Analysis II'' interessiert +hätten. Die Notizen von Andreas Demleitner zur letzten Vorlesung +``Funktionentheorie'' sind so gut, dass ich die Seiten hier einfach wiedergebe. + +\begin{center} + \noindent\includegraphics[width=14cm]{04-integration-1.png} + + \noindent\includegraphics[width=14cm]{04-integration-2.png} + + \noindent\includegraphics[width=14cm]{04-integration-3.png} +\end{center} + % !TEX root = Funktionentheorie

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