From c394e31a70ce5f137f3f576b988c4b214e2f4f33 Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Mon, 3 Nov 2025 11:02:14 +0100 Subject: [PATCH] Done. --- .vscode/ltex.dictionary.de-DE.txt | 1 + 03-wegintegraleDiffbar.tex | 4 +- 05-cauchy.tex | 2 +- 06-potenz.tex | 119 +++++++++++++++--------------- 07-nullstelle.tex | 71 +++++++++--------- Notizen/220527-Vorlesung.pdf | Bin 294619 -> 0 bytes 6 files changed, 102 insertions(+), 95 deletions(-) delete mode 100644 Notizen/220527-Vorlesung.pdf diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt index e557cda..9095b20 100644 --- a/.vscode/ltex.dictionary.de-DE.txt +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -42,3 +42,4 @@ Cauchy Brook Middlesex Somerset +Maximumsprinzip diff --git a/03-wegintegraleDiffbar.tex b/03-wegintegraleDiffbar.tex index ba6b392..4dedcb2 100644 --- a/03-wegintegraleDiffbar.tex +++ b/03-wegintegraleDiffbar.tex @@ -583,7 +583,7 @@ einer Viertelungsargumentation basiert. \begin{satz}[Satz von Goursat]\label{satz:3-4-6}% \index{Satz von Goursat!über Wegintegrale}Es sei $U ⊆ ℂ$ offen, es sei $f ∈ - 𝒪(U)$ holomorph und sei $\mathcal{R} ⊂ U$ ein achsenparalleles Rechteck. Dann + 𝒪(U)$ holomorph und sei $\mathcal{R} ⊂ U$ ein achsenparalleles Rechteck. Dann gilt: \begin{equation}\label{eq:3-4-6-1} \int_{∂ R} f(z) \, dz = 0 @@ -693,7 +693,7 @@ einer Viertelungsargumentation basiert. \paragraph{Schritt 6: Integralabschätzung} Für hinreichend großes $n$ liegt $R^{(n)}$ ganz in der $δ$-Um\-gebung von - $z_0$. Für jedes $z ∈ ∂ R^{(n)}$ mit $|z - z_0| ≤ d_n = 2^{-n} d$ gilt dann + $z_0$. Für jedes $z ∈ ∂ R^{(n)}$ mit $|z - z_0| ≤ d_n = 2^{-n} d$ gilt dann nach~\eqref{eq:3-4-6-7} die Ungleichung \begin{equation}\label{eq:3-4-6-8} |r(z)| ≤ ε · 2^{-n} d. diff --git a/05-cauchy.tex b/05-cauchy.tex index a1be1a0..c953d4a 100644 --- a/05-cauchy.tex +++ b/05-cauchy.tex @@ -321,7 +321,7 @@ Es gilt noch mehr, aber dafür müssen wir etwas mehr arbeiten. \begin{kor}[Satz von Morera\footnote{Giacinto Morera (* 18.~Juli 1856 in Novara, Italien; † 8.~Februar 1909 in Turin, Italien) war ein italienischer Ingenieur - und Mathematiker. Er ist für den Satz von Morera in der Funktionentheorie und + und Mathematiker. Er ist für den Satz von Morera in der Funktionentheorie und für seine Arbeiten über lineare Elastizität bekannt.}, Charakterisierung holomorpher Funktionen]\label{kor:5-2-6}% \index{Satz von Morera}Sei $U ⊆ ℂ$ offen und $f : U → ℂ$ sei stetig. Dann diff --git a/06-potenz.tex b/06-potenz.tex index 87bd1e2..2802a71 100644 --- a/06-potenz.tex +++ b/06-potenz.tex @@ -54,7 +54,7 @@ Alle Aussagen gelten mit denselben Beweisen auf für komplexe Zahlen. \item\label{il:6-0-2-4} Es sei $\sum_{i=0}^∞ a_i (x - ρ)ⁱ$ eine komplexe Potenzreihe mit Konvergenzradius $R > 0$. Dann gilt: die Folge der - Partialsummen konvergiert auf der offenen Kreisscheibe $B_r(ρ)$ kompakt. Das + Partialsummen konvergiert auf der offenen Kreisscheibe $B_r(ρ)$ kompakt. Das bedeutet: auf jeder kompakten Teilmenge $K$ von $B_r(ρ)$ konvergiert die Folge der Partialsummen gleichmäßig. \end{enumerate} @@ -64,18 +64,20 @@ Man beachte: Genau wie in der reellen Situation machen wir in \ref{il:6-0-2-4} keinerlei Aussagen über Konvergenz der Reihe für Punkte $z$, die auf dem Rand $∂ B_r(ρ)$ der Kreisscheibe liegen. -\begin{bsp}[Potenzreihen mit Entwicklungspunkt 0] - --- - \begin{enumerate} - \item $\sum_{i=0}^∞ \frac{zⁱ}{i!}$ --- diese Potenzreihe kennen wir schon, das - ist die Exponentialfunktion. Der Konvergenzradius ist $∞$. - - \item $\sum_{i=0}^∞ zⁱ$ --- diese Potenzreihe kennen wir auch schon. Aus - Analysis I wissen wir: Reihe konvergiert für reelle $z$ mit $|z| < 1$. Für - $z = 1$ konvergiert die Reihe nicht. Also ist der Konvergenzradius $= 1$. - Wie in der Vorlesung Analysis~I rechnet man nach: für jede Zahl $z ∈ B_1(0)$ - konvergiert die Reihe konvergiert gegen $\frac{1}{1-z}$. - \end{enumerate} +\sideremark{Vorlesung 10} + +\begin{bsp}[Exponentialreihe] + \index{Exponentialreihe}Die Potenzreihe $\sum_{i=0}^∞ \frac{zⁱ}{i!}$ kennen + wir schon, das ist die Exponentialfunktion. Der Konvergenzradius ist $∞$. +\end{bsp} + +\begin{bsp}[Geometrische Reihe] + \index{Geometrische Reihe}Die Potenzreihe $\sum_{i=0}^∞ zⁱ$ kennen wir auch + schon. Aus Analysis I wissen wir: die Reihe konvergiert für reelle $z$ mit + $|z| < 1$. Für $z = 1$ konvergiert die Reihe nicht. Also ist der + Konvergenzradius $= 1$. Wie in der Vorlesung Analysis~I rechnet man nach: für + jede Zahl $z ∈ B_1(0)$ konvergiert die Reihe konvergiert gegen + $\frac{1}{1-z}$. \end{bsp} @@ -83,10 +85,10 @@ B_r(ρ)$ der Kreisscheibe liegen. \begin{proposition}[Kompakt 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, $f_i ∈ \sO(U)$, die auf $U$ kompakt gegen Grenzfunktion $f$ - konvergiert. Dann gilt: + Funktionen, $f_i ∈ 𝒪(U)$, die auf $U$ kompakt gegen Grenzfunktion $f$ + konvergiert. Dann gilt: \begin{enumerate} - \item\label{il:6-0-4-1} Die Grenzfunktion ist holomorph, $f ∈ \sO(U)$. + \item\label{il:6-0-4-1} Die Grenzfunktion ist holomorph, $f ∈ 𝒪(U)$. \item\label{il:6-0-4-2} Die Folge $(f_i')_{i ∈ ℕ}$ der komplexen Ableitungen konvergiert kompakt gegen $f'$. @@ -95,52 +97,52 @@ B_r(ρ)$ der Kreisscheibe liegen. \begin{proof}[Beweis von \ref{il:6-0-4-1}] Die kompakte Konvergenz (= lokal gleichmäßige Konvergenz) garantiert schon - einmal, dass $f$ stetig ist. Um zu zeigen, dass $f$ holomorph ist, verwenden - wir das Kriterium aus Korollar~\ref{kor:5-2-6}. Wenn $R ⊂ U$ ein + einmal, dass $f$ stetig ist. Um zu zeigen, dass $f$ holomorph ist, verwenden + wir das Kriterium aus Korollar~\ref{kor:5-2-6}. Wenn $R ⊂ U$ ein Achsenparalleles Rechteck ist, dann ist \[ \int_{∂R} f(z)\,dz = \lim_{i→∞} \int_{∂R} f_i(z)\,dz = 0. \] Die erste Gleichheit gilt, weil $∂R$ kompakt ist und $f_i$ kompakt gegen $f$ - konvergiert. Die Integrale $\int_{∂R} f_i(z)\,dz$ verschwinden nach + konvergiert. Die Integrale $\int_{∂R} f_i(z)\,dz$ verschwinden nach Korollar~\ref{kor:5-2-6}, weil die Funktionen $f_i$ holomorph sind. \end{proof} \begin{proof}[Beweis von \ref{il:6-0-4-2}] - Es sei ein Punkt $z_0 ∈ U$ gegeben. Wir müssen zeigen, dass die Folge + Es sei ein Punkt $z_0 ∈ U$ gegeben. Wir müssen zeigen, dass die Folge $(f_i')_{i ∈ ℕ}$ in der Nähe von $z_0$ gleichmäßig gegen $f'$ konvergiert. - Wähle dazu einen Radius $r ∈ ℝ^+$, sodass die abgeschlossene Kreisscheibe - $\overline{B_r(z_0)}$ ganz in $U$ liegt. Dann wissen wir nach dem + Wähle dazu einen Radius $r ∈ ℝ⁺$, sodass die abgeschlossene Kreisscheibe + $\overline{B_r(z_0)}$ ganz in $U$ liegt. Dann wissen wir nach dem Satz~\ref{satz:4-4-2} von Goursat schon: für jede Zahl $w$ aus dem Inneren der Kreisscheibe, $w ∈ B_r(z_0)$ gelten die Gleichungen \begin{align*} - f'(w) & = \frac{1}{2πi} \int_{∂B_r(z_0)} \frac{f(z)}{(z-w)^2}\,dz \\ - f_i'(w) & = \frac{1}{2πi} \int_{∂B_r(z_0)} \frac{f_i(z)}{(z-w)^2}\,dz + f'(w) & = \frac{1}{2πi} \int_{∂B_r(z_0)} \frac{f(z)}{(z-w)²}\,dz \\ + f_i'(w) & = \frac{1}{2πi} \int_{∂B_r(z_0)} \frac{f_i(z)}{(z-w)²}\,dz \end{align*} Betrachte als Nächstes die Hilfsabbildungen \begin{align*} - φ: ∂B_r(z) \times B_r(z_0) &\to ℂ & (z, w) & \mapsto \frac{f(z)}{(z-w)^2} \\ - φ_i: ∂B_r(z) \times B_r(z_0) &\to ℂ & (z, w) & \mapsto \frac{f_i(z)}{(z-w)^2}. + φ: ∂B_r(z) ⨯ B_r(z_0) &→ ℂ & (z, w) & ↦ \frac{f(z)}{(z-w)²} \\ + φ_i: ∂B_r(z) ⨯ B_r(z_0) &→ ℂ & (z, w) & ↦ \frac{f_i(z)}{(z-w)²}. \end{align*} - Dann ist klar: auf der kompakten Menge $∂B_r(z_0) \times - \overline{B_{r/2}(z_0)}$ konvergiert die Funktionenfolge $φ_i$ gleichmäßig - gegen $φ$. Also konvergiert die Funktionenfolge $f_i'$ auf - $\overline{B_{r/2}(z_0)}$ gleichmäßig gegen $f'$. + Dann ist klar: auf der kompakten Menge $∂B_r(z_0) ⨯ \overline{B_{r/2}(z_0)}$ + konvergiert die Funktionenfolge $φ_i$ gleichmäßig gegen $φ$. Also konvergiert + die Funktionenfolge $f_i'$ auf $\overline{B_{r/2}(z_0)}$ gleichmäßig gegen + $f'$. Insgesamt sehen wir: jedes Kompaktum $K ⊂ U$ ist von endlich vielen abgeschlossenen Kreisscheiben überdeckt, auf denen $f_i'$ gleichmäßig gegen $f'$ konvergiert, also liegt kompakte Konvergenz vor. \end{proof} -\begin{kor}[Potenzreihen und Holomorphie]\label{kor:6-1-2} +\begin{kor}[Potenzreihen und Holomorphie]\label{kor:6-1-2}% Es sei $\sum_{i=0}^∞ a_i (z-ρ)ⁱ$ eine Potenzreihe mit Entwicklungspunkt $ρ$ - und Konvergenzradius $r > 0$. Weiter sei $f: B_r(ρ) → ℂ$ die zugehörige + und Konvergenzradius $r > 0$. Weiter sei $f: B_r(ρ) → ℂ$ die zugehörige Funktion. Dann gilt: \begin{enumerate} - \item Die Funktion $f$ ist holomorph, $f \in \sO(B_r(ρ))$. + \item Die Funktion $f$ ist holomorph, $f ∈ 𝒪(B_r(ρ))$. \item Der Konvergenzradius der Potenzreihe $\sum_{i=1}^∞ i · a_i (z-ρ)^{i-1}$ - ist mindestens $r$. Auf der Kreisscheibe $B_r(ρ)$ ist die zugehörige + ist mindestens gleich $r$. Auf der Kreisscheibe $B_r(ρ)$ ist die zugehörige Funktion exakt die Ableitung $f'$. \qed \end{enumerate} \end{kor} @@ -148,8 +150,8 @@ B_r(ρ)$ der Kreisscheibe liegen. \section{Holomorphe Funktionen liefern Potenzreihen} -\begin{proberechnung}[Konstruktion von Potenzeihen]\label{prorech:6-1-2} - Es sei $r > 0$ und es sei $f ∈ \sO(B_r(0))$. Wenn $ρ < r$ eine positive Zahl +\begin{proberechnung}[Konstruktion von Potenzeihen]\label{prorech:6-1-2}% + Es sei $r > 0$ und es sei $f ∈ 𝒪(B_r(0))$. Wenn $ρ < r$ eine positive Zahl ist, dann nach der Integralformel von Cauchy, Satz~\ref{satz:4-4-1}, für jede Zahl $w ∈ B_ρ(0)$ die Gleichung \begin{equation}\label{eq:6-0-6-1} @@ -166,26 +168,26 @@ B_r(ρ)$ der Kreisscheibe liegen. f(w) & = \frac{1}{2πi} \int_{∂B_ρ(0)} \frac{f(z)}{z - w}\,dz && \eqref{eq:6-0-6-1} \\ & = \frac{1}{2πi} \int_{∂B_ρ(0)} \frac{f(z)}{z} \sum_{i=0}^∞ \left(\frac{w}{z}\right)ⁱ\,dz && \eqref{eq:6-0-6-2} \\ & = \frac{1}{2πi} \sum_{i=0}^∞ \int_{∂B_ρ(0)} \frac{f(z)}{z^{i+1}} · wⁱ\,dz && \text{kompakte Konvergenz} \\ - & = \sum_{i=0}^∞ \underbrace{\left( \frac{1}{2πi}·\int_{∂B_ρ(0)} \frac{f(z)}{z^{i+1}}\,dz \right)}_{:= a_i} · wⁱ. + & = \sum_{i=0}^∞ \underbrace{\left( \frac{1}{2πi}·\int_{∂B_ρ(0)} \frac{f(z)}{z^{i+1}}\,dz \right)}_{:= a_i} · wⁱ. \end{align*} - Nach dem Integralsatz von Cauchy hängen die Zahlen $a_i$ aber gar nicht von der - Wahl von $ρ$ ab! Die Gleichung + Nach dem Integralsatz von Cauchy hängen die Zahlen $a_i$ aber gar nicht von + der Wahl von $ρ$ ab! Die Gleichung \begin{equation}\label{eq:6-0-6-3} f(w) = \sum_{i=0}^∞ a_i wⁱ \end{equation} - gilt also für alle $w ∈ B_r(0)$. Insbesondere ist der Konvergenzradius der + gilt also für alle $w ∈ B_r(0)$. Insbesondere ist der Konvergenzradius der Potenzreihe \eqref{eq:6-0-6-3} mindestens gleich $r$. \end{proberechnung} Die Proberechnung funktioniert natürlich nicht nur bei Kreisscheiben um den Nullpunkt. In der Summe haben wir folgenden Satz bewiesen. -\begin{satz}[Potenzreihendarstellung holomorpher Funktionen]\label{satz:6-0-7} - Es sei $B_r(ρ) ⊂ ℂ$ eine Kreisscheibe und $f ∈ \sO(B_r(ρ))$ eine holomorphe - Funktion. Dann kann $f$ auf ganz $B_r(ρ)$ als Potenzreihe dargestellt werden. +\begin{satz}[Potenzreihendarstellung holomorpher Funktionen]\label{satz:6-0-7}% + Es sei $B_r(ρ) ⊂ ℂ$ eine Kreisscheibe und $f ∈ 𝒪(B_r(ρ))$ eine holomorphe + Funktion. Dann kann $f$ auf ganz $B_r(ρ)$ als Potenzreihe dargestellt werden. Genauer: Es existiert Potenzreihe $\sum_{i=0}^∞ a_i (z-ρ)ⁱ$ mit Entwicklungspunkt $ρ$ und Konvergenzradius $≥ r$, sodass für jede Zahl $z ∈ - B_r(ρ)$ die Gleichung $f(z) = \sum_{i=0}^∞ a_i (z-ρ)ⁱ$ gilt. \qed + B_r(ρ)$ die Gleichung $f(z) = \sum_{i=0}^∞ a_i (z-ρ)ⁱ$ gilt. \qed \end{satz} Umkehrung die Umkehrung von Satz~\ref{satz:6-0-7} gilt natürlich auch: @@ -195,22 +197,23 @@ Korollar~\ref{kor:6-1-2} holomorph. \subsection{Praktische Fragen} -\begin{frage} +\begin{frage}[Muss ich Integrale ausrechnen?!] Gegeben eine Kreisscheibe $B_r(ρ) ⊂ ℂ$ und eine holomorphe Funktion $f ∈ - \sO(B_r(ρ))$. Wie komme ich an die Darstellung von $f$ als Potenzreihe? Muss + 𝒪(B_r(ρ))$. Wie komme ich an die Darstellung von $f$ als Potenzreihe? Muss ich die Integrale aus Proberechnung~\ref{prorech:6-1-2} wirklich ausrechnen? \end{frage} -Die beruhigende Antwort ist ein klares „Nein!“ Wenn ich bereits weiß, dass es +Die beruhigende Antwort ist ein klares „Nein!“ Wenn ich bereits weiß, dass es eine Darstellung von $f$ als Potenzreihe gibt, \begin{equation}\label{eq:6-2-3-1} - f(z) = \sum a_i (z-ρ)ⁱ, + f(z) = \sum a_i (z-ρ)ⁱ, \end{equation} -dann ist $a_0 = f(ρ)$. Außerdem haben wir in Korollar~\ref{prop:potenzreihe-holomorph} bewiesen, dass +dann ist $a_0 = f(ρ)$. Außerdem haben wir in +Korollar~\ref{prop:potenzreihe-holomorph} bewiesen, dass \[ f'(z) = \sum_{i=1}^∞ a_i · i (z-ρ)^{i-1} \] -ist. Insgesamt gilt damit für jeden Index $i$, +ist. Insgesamt gilt damit für jeden Index $i$, \[ a_i = \frac{f^{(i)}(ρ)}{i!}. \] @@ -219,26 +222,26 @@ Diese Formel hat zwei interessante Konsequenzen. \item Die Darstellung \eqref{eq:6-2-3-1} ist also einfach die aus der Vorlesung „Analysis~I“ bekannte Taylor\footnote{Brook Taylor (* 18.~August 1685 in Edmonton, Middlesex; † 29.~Dezember 1731 in Somerset House, London) - war ein britischer Mathematiker und Mitglied der Royal Society. Nach ihm wurde - unter anderem die Taylorreihe benannt.}-Reihe. + war ein britischer Mathematiker und Mitglied der Royal Society. Nach ihm + wurde unter anderem die Taylorreihe benannt.}-Reihe. - \item Die Darstellung von $f$ als Potenzreihe mit Entwicklungspunkt $\rho$ ist + \item Die Darstellung von $f$ als Potenzreihe mit Entwicklungspunkt $ρ$ ist eindeutig! \end{enumerate} \begin{bemerkung} Manchmal tritt folgende Situation auf: gegeben eine Kreisscheibe $B_r(ρ)$ und - eine holomorphe Funktion $f ∈ \sO(B_r(ρ))$. Dann schreibe ich $f$ als + eine holomorphe Funktion $f ∈ 𝒪(B_r(ρ))$. Dann schreibe ich $f$ als Potenzreihe mit Entwicklungspunkt $ρ$ und stelle fest, dass der - Konvergenzradius $\tilde R > r$ ist. In dieser Situation ist \emph{sofort} + Konvergenzradius $\tilde R > r$ ist. In dieser Situation ist \emph{sofort} klar, dass ich $f$ zu einer holomorphen Funktion auf der größeren Kreisscheibe $B_{\tilde R}(ρ)$ fortsetzen kann. Falls $r = \tilde R$ ist, dann ist klar, dass ich $f$ niemals auf einer - größeren Kreisscheibe fortsetzen kann. Es könnte aber sein, dass ich $f$ zu + größeren Kreisscheibe fortsetzen kann. Es könnte aber sein, dass ich $f$ zu einer holomorphen Funktion auf einer offenen Menge $U$ fortsetzen kann, die - zwar größer als $B_r(ρ)$ ist, aber keine Kreisschreiben um $\rho$ mit Radius - $> r$ enthält. + zwar größer als $B_r(ρ)$ ist, aber keine Kreisschreiben um $ρ$ mit Radius $> + r$ enthält. \end{bemerkung} % !TEX root = Funktionentheorie diff --git a/07-nullstelle.tex b/07-nullstelle.tex index c382e0e..b73a195 100644 --- a/07-nullstelle.tex +++ b/07-nullstelle.tex @@ -5,8 +5,8 @@ Wir betrachten die folgende Situation. -\begin{situation}[Nullstellen von holomorphen Funktionen]\label{set:7-0-1} - Es sei $U ⊂ ℂ$ offen, es sei $f ∈ \sO(U)$ holomorph und es sei +\begin{situation}[Nullstellen von holomorphen Funktionen]\label{set:7-0-1}% + Es sei $U ⊂ ℂ$ offen, es sei $f ∈ 𝒪(U)$ holomorph und es sei \[ Z = \{ z ∈ U \mid f(z) = 0 \} \] @@ -14,21 +14,21 @@ Wir betrachten die folgende Situation. \end{situation} Weil holomorphe Funktionen stetig ist, ist klar, dass die Menge $Z$ eine -abgeschlossene Teilmenge von $U$ ist. Aber was können wir sonst noch sagen? +abgeschlossene Teilmenge von $U$ ist. Aber was können wir sonst noch sagen? \section{Zwei Typen von Nullstellen} In Situation~\ref{set:7-0-1} sei $ρ ∈ Z$ eine Nullstelle von $f$ und sei \begin{equation}\label{eq:7-2-0-1} - \sum_{i=0}^∞ a_i(z-ρ)ⁱ + \sum_{i=0}^∞ a_i(z-ρ)ⁱ \end{equation} - die Potenzreihenentwicklung von $f$ im Punkt $ρ$. Der Konvergenzradius sei - $r$. Dann gibt es zwei Möglichkeiten: + die Potenzreihenentwicklung von $f$ im Punkt $ρ$. Der Konvergenzradius sei + $r$. Dann gibt es zwei Möglichkeiten: \begin{description} \item[Nullstelle vom Typ 1] Alle Koeffizienten $a_i$ der - Potenzreihe~\eqref{eq:7-2-0-1} sind - gleich 0. Dann ist $f$ in einer Umgebung von $f$ konstant Null. + Potenzreihe~\eqref{eq:7-2-0-1} sind gleich 0. Dann ist $f$ bereits in einer + ganzen Umgebung von $f$ konstant Null. \item[Nullstelle vom Typ 2] Nicht alle Koeffizienten $a_i$ der Potenzreihe~\eqref{eq:7-2-0-1} sind gleich 0. Betrachte in diesem Fall den @@ -36,43 +36,45 @@ abgeschlossene Teilmenge von $U$ ist. Aber was können wir sonst noch sagen? \[ n := \min \{ i \::\: a_i ≠ 0 \} \] - und nenne $n$ die ``Ordnung der Nullstelle $ρ$ von $f$''. Schreibe weiter + und nenne $n$ die „Ordnung der Nullstelle $ρ$ von $f$“. Schreibe weiter \begin{align*} f(z) & = \sum_{i=n}^∞ a_i (z-ρ)ⁱ \\ & = (z-ρ)^n · \sum_{i=n}^∞ a_i (z-ρ)^{i-n} \\ & = (z-ρ)^n · \sum_{i=0}^∞ b_i (z-ρ)^{i-n}, \end{align*} - wobei $b_i := a_{n+i}$ ist. Man rechne nach, dass die Potenzreihe + wobei $b_i := a_{n+i}$ ist. Man rechne nach, dass die Potenzreihe \[ \sum_{i=0}^∞ b_i (z-ρ)^{i-n} \] - ebenfalls Konvergenzradius $r$ hat, also eine Funktion $g ∈ \sO(B_r(ρ))$ - definiert, die aber bei $ρ$ \emph{keine} Nullstelle hat (… und in einer - ausreichend kleinen Umgebung von $ρ$ ebenfalls nicht). Auf $B_r(ρ)$ gilt die - Gleichung + ebenfalls Konvergenzradius $r$ hat, also eine Funktion $g ∈ 𝒪(B_r(ρ))$ + definiert, die aber bei $ρ$ \emph{keine} Nullstelle hat (und deshalb in + einer ausreichend kleinen Umgebung von $ρ$ ebenfalls nicht). Auf $B_r(ρ)$ + gilt die Gleichung \[ f(z) = (z-ρ)^n · g(z) \] - und es gibt ein $ε > 0$, sodass $Z ∩ B_ε(ρ) = \{ρ\}$ ist. Man sagt: $ρ$ - ist eine isolierte Nullstelle von $f$. + und es gibt ein $ε > 0$, sodass $Z ∩ B_ε(ρ) = \{ρ\}$ ist. Man sagt: $ρ$ ist + eine isolierte Nullstelle von $f$. \end{description} Zusammenfassung: Ich kann die Nullstellenmenge $Z$ aufteilen \[ - Z = \text{Typ 1} \: \cup \text{Typ 2} + Z = \text{Typ 1} \: ∪ \text{Typ 2} \] Dabei gilt Folgendes: \begin{itemize} \item Die Menge der Nullstellen vom Typ~2 ist eine abgeschlossene, diskrete - Teilmenge von $U$. + Teilmenge von $U$. + \item Die Menge der Nullstellen von Typ~1 ist offen. + \item Die Menge der Punkte von $U$, an denen \emph{keine} Nullstelle vom Typ~1 - vorliegt, ist ebenfalls offen. Grund: Wenn $f$ bei $ρ$ überhaupt keine - Nullstelle hat, dann hat $f$ als stetige Funktion auch in einer Umgebung von - $\rho$ keine Nullstelle. Wenn $f$ bei $ρ$ eine Nullstelle vom Typ~2 hat, dann - haben wir gesehen, dass $f$ in einer Umgebung von $\rho$ keine weitere - Nullstelle hat. -\end{itemize} + vorliegt, ist ebenfalls offen. Grund: Wenn $f$ bei $ρ$ überhaupt keine + Nullstelle hat, dann hat $f$ als stetige Funktion auch in einer Umgebung von + $ρ$ keine Nullstelle. Wenn $f$ bei $ρ$ eine Nullstelle vom Typ~2 hat, dann + haben wir gesehen, dass $f$ in einer Umgebung von $ρ$ keine weitere + Nullstelle hat. +\end{itemize} In der Summe sehen wir: Die Menge Nullstellen vom Typ 1 ist offen \emph{und} abgeschlossen, also eine ganze Zusammenhangskomponente von $U$! @@ -80,43 +82,44 @@ abgeschlossen, also eine ganze Zusammenhangskomponente von $U$! \section{Identitätssatz und Maximumsprinzip} \begin{satz} - In Situation~\ref{set:7-0-1} sei $U$ zusammenhängend. Dann sind die folgenden + In Situation~\ref{set:7-0-1} sei $U$ zusammenhängend. Dann sind die folgenden Aussagen äquivalent. \begin{enumerate} \item Die Nullstellenmenge $Z$ ist \emph{nicht} diskret. Äquivalent: die Nullstellenmenge $Z$ hat einen Häufungspunkt. - \item Die Funktion $f$ ist konstant gleich $0$. \qed + + \item Die Funktion $f$ ist konstant gleich $0$. \qed \end{enumerate} \end{satz} -\begin{kor}[Identitätssatz für holomorphe Funktionen]\label{kor:7-2-2} +\begin{kor}[Identitätssatz für holomorphe Funktionen]\label{kor:7-2-2}% \index{Identitätssatz}In Situation~\ref{set:7-0-1} sei $U$ zusammenhängend und - es sei $g ∈ \sO(U)$ eine weitere holomorphe Funktion auf $U$. Dann sind die + es sei $g ∈ 𝒪(U)$ eine weitere holomorphe Funktion auf $U$. Dann sind die folgenden Aussagen äquivalent. \begin{enumerate} \item Die Menge \[ \{ z ∈ U \mid f(z) = g(z) \} \] - ist \emph{nicht} diskret. Äquivalent: die Menge hat einen Häufungspunkt. + ist \emph{nicht} diskret. Äquivalent: die Menge hat einen Häufungspunkt. \item Die Funktionen $f$ und $g$ sind gleich. \end{enumerate} \end{kor} \begin{bsp}[Eindeutigkeit der Exponentialfunktion] - Die Funktion $\exp: ℂ → ℂ$ ist die einzige holomorphe Funktion auf $\bC$, die + Die Funktion $\exp: ℂ → ℂ$ ist die einzige holomorphe Funktion auf $ℂ$, die für reelle Zahlen mit der bekannten Exponentialfunktion übereinstimmt. \end{bsp} \begin{kor}[Maximumsprinzip] - In Situation~\ref{set:7-0-1} sei $U$ zusammenhängend. Falls $|f|$ ein Maximum + In Situation~\ref{set:7-0-1} sei $U$ zusammenhängend. Falls $|f|$ ein Maximum hat, dann ist $f$ konstant. \end{kor} \begin{proof} - \index{Maximumprinzip}Es sei $ρ ∈ U$ ein Maximum von $|f|$. Nach - Satz~\ref{satz:5-2-3} (``Starkes Maximumprinzip'') wissen wir schon: es gibt - ein $ε > 0$, sodass $f|_{B_ε(ρ)}$ konstant ist. Nach dem Identitätssatz, + \index{Maximumprinzip}Es sei $ρ ∈ U$ ein Maximum von $|f|$. Nach + Satz~\ref{satz:5-2-3} („Starkes Maximumsprinzip“) wissen wir schon: es gibt + ein $ε > 0$, sodass $f|_{B_ε(ρ)}$ konstant ist. Nach dem Identitätssatz, Korollar~\ref{kor:7-2-2}, ist $f$ dann aber auf ganz $U$ konstant. \end{proof} diff --git a/Notizen/220527-Vorlesung.pdf b/Notizen/220527-Vorlesung.pdf deleted file mode 100644 index 44638b00d483ec71c83ab42f00766cbbbcf76fd0..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 294619 zcmbrl1yp3)t~LrZ?(XjH?(Wuj;|_(pySqcACXUr zkJ#MF$&Qv2q&c>1CQ^V(c{?Q?7 z;%MXmuyX>~+7SMs4Ge9ao%kpy{!wiAshHv)#Ws%gf7wBAWNXdv`2_J)z-n-k^gUrf0KXAV)!3+{cmmk zoVEYEt!@T(pSCjo+tyD(ZaG_HfT_Evfs+Y8Gvnv{FtX7xbEq(KaWk@Uv$1IMGW-vr zzfJ$c6#4Y6lgaacFqn~e>v6Ih{43l#QM)-|9mPY1`0xBBW_b$ z2Wx{*zZuxsSpkd;{rzDr+kn-Q*;$Omlq33f6 z{{a~xr6~6wgfg-*b1-r-vVDT{kJx`=_Mg!HCuZ~V{S;_!L4`V%)yjBJff{>vue23$GwWqw@~ zHDt#%M|-~6dac=-;x40q^{=b7=BmO+rS765Oq{+e2t`5txH;V_r#e#S(?yxv4=&Vr z;M}rH$ep7Rb$-6rI%i4}!Cpz2=_x!`s|(626;IDlny&8x6cGlF$An}+ooVi^CnthY zrflVC9G%#GqSx<3kG58z*x*4M@J>(7i^$c`h$KB|$Cj(LxaO9pj2pO;GZO*>0FTB6 z+Ye6PE|U_uC#FuUux*zW*tuW+HuxoM0$zBQG5CfpxIp`h-s#xI6ko9THwpY|;qSgQ 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