From b43e393dacfe0b58949a61fd3da9b0d8f426d962 Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Wed, 22 Oct 2025 11:04:51 +0200 Subject: [PATCH] Cleanup --- .vscode/ltex.dictionary.de-DE.txt | 1 + 04-wegintegraleStetig.tex | 79 ++++++++++++++++-------------- Notizen/220518-Vorlesung.pdf | Bin 3975356 -> 0 bytes 3 files changed, 42 insertions(+), 38 deletions(-) delete mode 100644 Notizen/220518-Vorlesung.pdf diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt index 9091dbc..aeb9a62 100644 --- a/.vscode/ltex.dictionary.de-DE.txt +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -16,3 +16,4 @@ Homotopieinvarianz zusammenziehbarer Homotopien Hin- +Demleitner diff --git a/04-wegintegraleStetig.tex b/04-wegintegraleStetig.tex index a8447b9..17155f8 100644 --- a/04-wegintegraleStetig.tex +++ b/04-wegintegraleStetig.tex @@ -204,9 +204,9 @@ dies präzise dar. \] eine Homotopie zwischen den Wegen $γ_0$ und $γ_1$. Weil die Menge $[a, b] ⨯ [0, 1]$ kompakt ist, können wir nach Erinnerung~\ref{eri:3-4-1} die Bildmenge - $γ\bigl([a, b] ⨯ [0, 1]\bigr) ⊂ U$ ist kompakt mit endlich viele Kreisscheiben $Δ_1, …, - Δ_n$ aus $U$ überdecken. Weiter können wir eine Unterteilung der Intervalle - finden, + $γ\bigl([a, b] ⨯ [0, 1]\bigr) ⊂ U$ ist kompakt mit endlich viele Kreisscheiben + $Δ_1, …, Δ_n$ aus $U$ überdecken. Weiter können wir eine Unterteilung der + Intervalle finden, \begin{align*} a & = t_0 < t_1 < t_2 < … < t_m = b && \text{von } [a, b], \\ 0 & = s_0 < s_1 < s_2 < … < s_k = 1 && \text{von } [0, 1], @@ -243,9 +243,9 @@ dies präzise dar. \end{kor} \begin{bsp}[Die Menge $ℂ^*$ ist nicht einfach zusammenhängend]\label{bsp:4-3-3}% - Wir betrachten die offene Menge $U := ℂ^* = ℂ ∖ \{0\}$ und die - holomorphe Funktion $f: U → ℂ$, $z ↦ 1/z$. Weiter betrachten wir den - geschlossenen Kreisweg + Wir betrachten die offene Menge $U := ℂ^* = ℂ ∖ \{0\}$ und die holomorphe + Funktion $f: U → ℂ$, $z ↦ 1/z$. Weiter betrachten wir den geschlossenen + Kreisweg \[ γ: [0, 1] → U, \quad t ↦ e^{2πi t}. \] @@ -255,41 +255,44 @@ dies präzise dar. \int_γ \frac{1}{z} \, dz = 2πi. \] 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 $ℂ^*$ + 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 $f$. + 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 + 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 + 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_1(b_1)$. Betrachte als Nächstes den Weg $γ$, definiert durch \[ - γ : [a_0, b_0 + b_1 - a_1] \to U, \quad t \mapsto := \begin{cases} + γ : [a_0, b_0 + b_1 - a_1] → U, \quad t ↦ := + \begin{cases} γ_0(t) & t ∈ [a_0, b_0), \\ γ_1(b_0 + b_1 - t) & t ∈ [b_0, b_0 + b_1 - a_1]. \end{cases} \] - 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, + Beachte, dass der Weg $γ$ 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. + & = \int_{γ_0} f(z) \, dz - \int_{γ_1} f(z) \, dz && \text{Konstruktion von }γ. \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. +Homotopie zu verwenden. Abbildung~\ref{fig:4-3-1-2} veranschaulicht die +Situation. \begin{figure} \begin{center} @@ -325,20 +328,20 @@ Homotopie zu verwenden. Abbildung~\ref{fig:4-3-1-2} veranschaulicht die Situatio 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. +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 + : [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 + Es sei $Γ : [a,b] ⨯ [0,1] → U$ 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} @@ -348,23 +351,23 @@ sich mit kleinen Änderungen auch auf freie Homotopien von geschlossenen Wegen \label{fig:4-3-3} \end{figure} \[ - \delta : [0, 1] → U, s \mapsto \Gamma(a, s). + δ : [0, 1] → U, s ↦ Γ(a, s). \] 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: + γ_1(b)$ verbindet. Konstruiere nun einen geschlossenen Weg $\wtilde{γ}_1 : + [a,b] → U$, 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 + Ich behaupte, dass die Wege $γ_0$ und $\wtilde{γ}_1$ 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 + $\wtilde{γ}_1$ überführt. Für jeden Parameterwert $s ∈ [0, 1]$ definieren wir + den Weg \[ - \wtilde{\gamma}_s : [a,b] \to U, + \wtilde{γ}_s : [a,b] → U, \] der Folgendes macht: \begin{itemize} @@ -375,13 +378,13 @@ sich mit kleinen Änderungen auch auf freie Homotopien von geschlossenen Wegen \end{itemize} Die Abbildung \[ - \wtilde{\Gamma} : [a,b] ⨯ [0,1] \to U, \quad (t,s) \mapsto \wtilde{\gamma}_s(t) + \wtilde{Γ} : [a,b] ⨯ [0,1] → U, \quad (t,s) ↦ \wtilde{γ}_s(t) \] definiert dann die gesuchte Homotopie zwischen den Wegen $γ_0$ und - $\wtilde{\gamma}_1$. Nach Satz~\ref{satz:4-3-1} gilt daher + $\wtilde{γ}_1$. Nach Satz~\ref{satz:4-3-1} gilt daher \[ - \int_{γ_0} f(z) \, dz = \int_{\wtilde{\gamma}_1} f(z) \, dz = \int_{γ_1} f(z) \, dz. - \] + \int_{γ_0} f(z) \, dz = \int_{\wtilde{γ}_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. @@ -391,9 +394,9 @@ sich mit kleinen Änderungen auch auf freie Homotopien von geschlossenen Wegen \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. +auszurechnen, die uns auch schon in der Vorlesung „Analysis II“ interessiert +hätten. 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