From 3b9193fe9166671badac2651869be82b2c8f940d Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Wed, 26 Nov 2025 13:03:33 +0100 Subject: [PATCH] =?UTF-8?q?Working=E2=80=A6?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .vscode/ltex.dictionary.de-DE.txt | 6 + .vscode/ltex.hiddenFalsePositives.de-DE.txt | 4 + 09-singularities.tex | 4 +- 10-laurent.tex | 18 +- 12-residuum.tex | 325 +++++++++++++++----- Notizen/220624-Vorlesung.pdf | Bin 9264564 -> 0 bytes 6 files changed, 277 insertions(+), 80 deletions(-) delete mode 100644 Notizen/220624-Vorlesung.pdf diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt index 87fec19..7270128 100644 --- a/.vscode/ltex.dictionary.de-DE.txt +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -63,3 +63,9 @@ Summendarstellung Mittag-Leffler Djursholm Konvergenzverhaltens +Eugène +Rouché +Sommières +Hérault +Lunel +Vielfachheit diff --git a/.vscode/ltex.hiddenFalsePositives.de-DE.txt b/.vscode/ltex.hiddenFalsePositives.de-DE.txt index 0af5e11..834728f 100644 --- a/.vscode/ltex.hiddenFalsePositives.de-DE.txt +++ b/.vscode/ltex.hiddenFalsePositives.de-DE.txt @@ -35,3 +35,7 @@ {"rule":"UNPAIRED_BRACKETS","sentence":"^\\QDie Aussage folgt dann aus Korollar \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q (“Integralsatz von Cauchy”), \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} {"rule":"DE_CASE","sentence":"^\\QDie zweite Gleichheit gilt nach der Goldenen Regel 1, weil \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q und \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q in derselben Zusammenhangskomponente von \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q ist.\\E$"} {"rule":"DE_CASE","sentence":"^\\QDamit ist die Goldene Regel 3 zumindest halbwegs gerechtfertigt.\\E$"} +{"rule":"TEST_F_ANSTATT_PH","sentence":"^\\QPierre Alphonse Laurent (* 18. Juli 1813 in Paris; † 2. September 1854 ebenda) war ein französischer Mathematiker.\\E$"} +{"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QWähle eine reelle Zahl \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q und schreibe \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Korollar \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Annahme \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q lokal glm.\\E$"} +{"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QAlso ist \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, weil Stammfkt.\\E$"} +{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qexistiert \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} diff --git a/09-singularities.tex b/09-singularities.tex index c2f98c0..1f374e7 100644 --- a/09-singularities.tex +++ b/09-singularities.tex @@ -34,7 +34,7 @@ Wir interessieren uns für die folgende Situation: Sei $U ⊂ ℂ$ ein Gebiet, s \begin{definition}[Funktionen mit isolierten Singularitäten]\label{def:9-1-1}% Sei $U ⊂ ℂ$ offen. Eine \emph{holomorphe Funktion mit isolierten - Singularitäten}\index{holomorph!mit isolieren Singularitäten} ist eine + Singularitäten}\index{holomorph!mit isolierten Singularitäten} ist eine holomorphe Funktion $f ∈ 𝒪(U ∖ T)$ wobei $T ⊂ U$ eine diskrete Menge ist. \end{definition} @@ -89,7 +89,7 @@ Wir interessieren uns für die folgende Situation: Sei $U ⊂ ℂ$ ein Gebiet, s \end{proof} \begin{bemerkung} - Die Aussage ist in der reellen Analysis überhaupt nicht richtig! Betrachte die + Die Aussage ist in der reellen Analysis überhaupt nicht richtig! Betrachte die Funktion $f : ℝ² ∖ \{0\} → ℝ$, $\vec{v} ↦ |\vec{v}|$. Die Abbildung ist stetig und außerhalb von $0$ differenzierbar. Aber auf ganz $ℝ²$ überhaupt nicht differenzierbar. diff --git a/10-laurent.tex b/10-laurent.tex index 8f46d21..b4e1fcf 100644 --- a/10-laurent.tex +++ b/10-laurent.tex @@ -144,7 +144,7 @@ bezeichnet. Gegeben eine Laurentreihe wie in Definition~\ref{def:9-2-5}, so nennt man die Teilreihen \[ - \sum_{i=1}^{∞} c_i (z - p)^{-i} + \sum_{i=1}^{∞} c_{-i} (z - p)^{-i} \quad\text{und}\quad \sum_{i=0}^{∞} c_i (z - p)ⁱ \] @@ -255,13 +255,13 @@ Ende des Tages haben wir Folgendes bewiesen. \section{Konvergenz von Hauptteilen} Ich erinnere an die Beschreibung des Konvergenzverhaltens von komplexen -Potenzreihen, Punkt~\ref{il:6-0-2-2} aus Fakt~\vref{fact:6-0-2}. Gegeben sei -eine komplexe Potenzreihe $\sum_{i=0}^∞ a_i (z - ρ)ⁱ$ mit Entwicklungspunkt $ρ$ +Potenzreihen, Punkt~\ref{il:6-0-2-2} aus Fakt~\vref{fact:6-0-2}. Gegeben sei +eine komplexe Potenzreihe $\sum_{i=0}^∞ a_i (z - ρ)ⁱ$ mit Entwicklungspunkt $ρ$ und ein $z_0 ∈ ℂ$, sodass $\sum a_i (z_0 - ρ)ⁱ$ konvergiert. Dann gilt für alle $z$ mit $|z - ρ| < |z_0 - ρ|$, dass die Reihe $\sum a_i (z - ρ)ⁱ$ absolut -konvergiert. Die zugehörige Funktionenfolge konvergiert auf $B_{|z_0 - ρ|}(ρ)$ -sogar lokal gleichmäßig. Die Aussage gilt analog für den Nebenteil von -Laurentreihen. +konvergiert. Die zugehörige Funktionenfolge konvergiert auf $B_{|z_0 - ρ|}(ρ)$ +sogar lokal gleichmäßig. Die Aussage gilt analog für den Nebenteil von +Laurentreihen. \begin{fakt}[Konvergenz von Hauptteilen]\label{fakt:10-3-1}% Gegeben sei eine komplexe Laurentreihe ohne Nebenteil und mit @@ -269,10 +269,10 @@ Laurentreihen. \[ \sum_{i=1}^∞ a_i (z - ρ)^{-i}. \] - Gegeben sei weiter ein $z_0 ∈ ℂ$, sodass die Reihe bei $z_0$ konvergiert. Dann + Gegeben sei weiter ein $z_0 ∈ ℂ$, sodass die Reihe bei $z_0$ konvergiert. Dann gilt für alle $z$ mit $|z - ρ| > |z_0 - ρ|$, dass die Reihe bei $z$ absolut - konvergiert. Die zugehörige Funktionenfolge konvergiert auf $\bC \setminus - \overline{B_{|z_0 - ρ|}(ρ)}$ sogar lokal gleichmäßig. \qed + konvergiert. Die zugehörige Funktionenfolge konvergiert auf $ℂ ∖ + \overline{B_{|z_0 - ρ|}(ρ)}$ sogar lokal gleichmäßig. \qed \end{fakt} % !TEX root = Funktionentheorie diff --git a/12-residuum.tex b/12-residuum.tex index 2a2c32e..ec25141 100644 --- a/12-residuum.tex +++ b/12-residuum.tex @@ -69,7 +69,7 @@ Umlaufzahl und diskutieren den Inhalt des Satzes geometrisch. \begin{definition}[Umlaufzahl]\label{def:12-2-2}% In der Situation von Satz~\ref{satz:12-2-1} heißt $n$ die \emph{Umlaufzahl}\index{Umlaufzahl} oder - \emph{Windungszahl}\index{Windungszahl} des Weges $γ$ um und Punkt $p$. Die + \emph{Windungszahl}\index{Windungszahl} des Weges $γ$ um und Punkt $p$. Die Schreibweise $\Um(γ, p)$ ist üblich. \end{definition} @@ -87,8 +87,8 @@ endliche Menge $Z ⊂ [0,1]$ gibt, sodass $γ|_{[0,1] ∖ Z}$ injektiv ist. Zusammenhangskomponente von $ℂ ∖ \Bild(γ)$ ab, in der sich der Punkt $p$ befindet. - \item[Goldene Regel 2] Auf der unbeschränkten Zusammenhangskomponente von $\bC - ∖ \Bild(γ)$ ist die Umlaufzahl gleich $0$. + \item[Goldene Regel 2] Auf der unbeschränkten Zusammenhangskomponente von $ℂ ∖ + \Bild(γ)$ ist die Umlaufzahl gleich $0$. \item[Goldene Regel 3] Die Windungszahl in benachbarten Zusammenhangskomponenten unterscheidet sich um $± 1$, wobei die größere Zahl @@ -123,7 +123,6 @@ deren Gültigkeit an. Details sind Inhalt der Vorlesung „Topologie“. \int_γ \frac{1}{z-p} \, dz = \Um(γ, p) = 0. \] \end{proof} - \begin{proof}[Begründung zur Goldenen Regel 3] Seien $p_1, p_2 ∈ ℂ ∖ \Bild(γ)$ in benachbarten Zusammenhangskomponenten. Das sieht dann etwa so aus: @@ -131,14 +130,14 @@ deren Gültigkeit an. Details sind Inhalt der Vorlesung „Topologie“. \includegraphics[width=14cm]{12-res2.png} \end{center} Wir ändern den Weg $γ$ wie eingezeichnet ab und vergleichen die Umlaufzahlen - der Wege $γ$ und $\widetilde{γ}_{ε}$. Es ist + der Wege $γ$ und $\widetilde{γ}_ε$. Es ist \[ - \Um(γ, p_1) = \Um(\widetilde{γ}_{ε}, p_1) = \Um(\widetilde{γ}_{ε}, p_2). + \Um(γ, p_1) = \Um(\widetilde{γ}_ε, p_1) = \Um(\widetilde{γ}_ε, p_2). \] Dabei gilt das erste Gleichheitszeichen, weil der Weg $γ$ in $ℂ ∖ \{p_1\}$ - homotop zu $\widetilde{γ}_{ε}$ ist. Die zweite Gleichheit gilt nach der + homotop zu $\widetilde{γ}_ε$ ist. Die zweite Gleichheit gilt nach der Goldenen Regel 1, weil $p_1$ und $p_2$ in derselben Zusammenhangskomponente - von $ℂ ∖ \Bild(\widetilde{γ}_{ε})$ ist. Die Umlaufzahl $\Um(\widetilde{β}, + von $ℂ ∖ \Bild(\widetilde{γ}_ε)$ ist. Die Umlaufzahl $\Um(\widetilde{β}, p_2)$ hängt natürlich nicht von der Größe $ε$ ab. Wenn man $ε$ gegen null gehen lässt, erhält man folgendes Bild: \begin{center} @@ -146,7 +145,7 @@ deren Gültigkeit an. Details sind Inhalt der Vorlesung „Topologie“. \end{center} Dann ist \begin{align*} - \Um(γ, p_1) & = \Um(\widetilde{γ}_{ε}, p_2) \\ + \Um(γ, p_1) & = \Um(\widetilde{γ}_ε, p_2) \\ & = \frac{1}{2π i} \int_{\widetilde{γ}} \frac{1}{z-p_2} \, dz \\ & = \frac{1}{2π i} \left( \int_{γ} \frac{1}{z-p_2} \, dz + \int_{β} \frac{1}{z-p_2} \, dz \right) \\ & = \Um(γ, p_2) + \frac{1}{2π i} · \int_{β} \frac{1}{z-p_2} \, dz @@ -168,8 +167,8 @@ Wir bewiesen Satz~\ref{satz:12-2-1} in mehreren Schritten. \begin{behauptung}[Existenz eines Logarithmus-Wegs]\label{beh:12-2-3}% Sei $a ∈ ℂ$ ein Logarithmus von $γ(0)$ (das bedeutet: $\exp(a) = γ(0)$). Dann - gibt es einen stetigen Weg $\widetilde{γ}: [0,1] → ℂ$ mit $\widetilde{γ}(0) = - a$, sodass für jedes $t \in [0,1]$ die Gleichung + gibt es einen stetigen Weg $\widetilde{γ}: [0,1] → ℂ$ mit $\widetilde{γ}(0) = + a$, sodass für jedes $t ∈ [0,1]$ die Gleichung \[ γ(t) = exp \bigl( \widetilde{γ}(t) \bigr) \] @@ -178,30 +177,29 @@ Wir bewiesen Satz~\ref{satz:12-2-1} in mehreren Schritten. \begin{proof}[Beweis von Behauptung~\ref{beh:12-2-3}] Einige Fälle sind einfach. \begin{itemize} - \item Falls $\Bild(γ)$ ganz in der geschlitzten Ebene $S \subseteq \bC$ liegt, - nehmen wir + \item Falls $\Bild(γ)$ ganz in der geschlitzten Ebene $S ⊆ ℂ$ liegt, nehmen + wir \[ \widetilde{γ}(t) = \log (γ(t)) + 2π i k. \] Dabei ist $\log$ der Hauptzweig des Logarithmus und $k ∈ ℤ$ ist so gewählt, dass $\widetilde{γ}(0) = a$ ist. - \item Falls $\Bild(γ)$ ganz in $-S \subseteq \bC$ liegt, können wir analog - vorgehen. + \item Falls $\Bild(γ)$ ganz in $-S ⊆ ℂ$ liegt, können wir analog vorgehen. \end{itemize} - Im Allgemeinen wird das Bild von $\gamma$ weder in $S$ noch in $-S$ liegen. - Dann zerlegen wir das Intervall $[0,1]$ durch Zwischenpunkte, $0 = t_0 < t_1 < - … < t_r = 1$ so, dass $γ([t_{i-1}, t_i])$ jeweils in einem $S$ oder in $-S$ + Im Allgemeinen wird das Bild von $γ$ weder in $S$ noch in $-S$ liegen. Dann + zerlegen wir das Intervall $[0,1]$ durch Zwischenpunkte, $0 = t_0 < t_1 < … < + t_r = 1$ so, dass $γ([t_{i-1}, t_i])$ jeweils in einem $S$ oder in $-S$ enthalten ist. Wähle dann induktiv Logarithmus-Wege \[ \widetilde{γ}_i: [t_{i-1}, t_i] → ℂ \quad\text{mit}\quad - \widetilde{γ}_i(t_{i-1}) = + \widetilde{γ}_i(t_{i-1}) = \begin{cases} a & \text{falls } i = 1 \\ \widetilde{γ}_{i-1}(t_{i-1}) & \text{sonst} \end{cases} - \] + \] und erhalte durch Zusammenfügen der Wege $\widetilde{γ}_i$ den gewünschten Weg $\widetilde{γ}$. \end{proof} @@ -209,7 +207,7 @@ Wir bewiesen Satz~\ref{satz:12-2-1} in mehreren Schritten. \begin{behauptung}[Existenz von $n$]\label{beh:12-2-4}% Es gibt eine Zahl $n ∈ ℤ$, sodass der Weg $γ$ frei homotop zum Weg \[ - \delta_n : [0,1] → ℂ ∖ \{p\}, \quad t ↦ \exp(2πin·t) + δ_n : [0,1] → ℂ ∖ \{p\}, \quad t ↦ \exp(2πin·t) \] ist. \end{behauptung} @@ -218,42 +216,41 @@ Wir bewiesen Satz~\ref{satz:12-2-1} in mehreren Schritten. \[ \exp(\widetilde{γ}(0)) = γ(0) = γ(1) = \exp(\widetilde{γ}(1)). \] - Also ist $\widetilde{γ}(1) - \widetilde{γ}(0) ∈ \ker(\exp) = 2π i ℤ$. Demnach + Also ist $\widetilde{γ}(1) - \widetilde{γ}(0) ∈ \ker(\exp) = 2π i ℤ$. Demnach gibt es eine Zahl $n ∈ ℤ$ mit \[ \widetilde{γ}(1) = \widetilde{γ}(0) + 2π i·n. \] - Wir erinnern uns daran, dass der Topologische Raum $\bC$ einfach - zusammenhängend ist. Daher ist der $\widetilde{γ}$ in $ℂ$ homotop zu jedem - anderen Weg mit denselben Anfangs- und Endpunkt. Insbesondere ist - $\widetilde{γ}$ homotop zu + Wir erinnern uns daran, dass der Topologische Raum $ℂ$ einfach zusammenhängend + ist. Daher ist der $\widetilde{γ}$ in $ℂ$ homotop zu jedem anderen Weg mit + denselben Anfangs- und Endpunkt. Insbesondere ist $\widetilde{γ}$ homotop zu \[ β: [0,1] → ℂ, \quad t ↦ \widetilde{γ}(0) + 2πin·t. \] - Also ist der Weg $γ = \exp \circ \widetilde{γ}$ in $\bC^*$ homotop zu + Also ist der Weg $γ = \exp ◦ \widetilde{γ}$ in $ℂ^*$ homotop zu \[ - \exp \circ β: [0,1] → ℂ, \quad t ↦ γ(0)·\exp(2πin·t). + \exp ◦ β: [0,1] → ℂ, \quad t ↦ γ(0)·\exp(2πin·t). \] - Dieser Weg wiederum ist in $ℂ^*$ frei homotop zu $\delta_n$. Damit ist die + Dieser Weg wiederum ist in $ℂ^*$ frei homotop zu $δ_n$. Damit ist die Existenz von $n$ gezeigt. \end{proof} \begin{behauptung}[Eindeutigkeit von $n$]\label{beh:12-2-5}% Es gibt genau Zahl $n ∈ ℤ$, sodass der Weg $γ$ frei homotop zum Weg \[ - \delta_n : [0,1] → ℂ ∖ \{p\}, \quad t ↦ \exp(2πin·t) + δ_n : [0,1] → ℂ ∖ \{p\}, \quad t ↦ \exp(2πin·t) \] - ist, nämlich + ist, nämlich \[ n = \frac{1}{2π i} \int_{γ} \frac{1}{z} \, dz. \] \end{behauptung} \begin{proof}[Beweis von Behauptung~\ref{beh:12-2-5}] - Sei $n$ eine Zahl mit der Eigenschaft, dass $γ$ frei homotop zu $\delta_n$ - ist. Dann ist + Sei $n$ eine Zahl mit der Eigenschaft, dass $γ$ frei homotop zu $δ_n$ ist. + Dann ist \begin{align*} - n & = \frac{1}{2π i} \int_{\delta_n} \frac{1}{z} \, dz && \text{Beispiel~\ref{bsp:3-2-2}} \\ - & = \frac{1}{2π i} \int_{β} \frac{1}{z} \, dz. && \text{Satz~\ref{satz:4-3-6}.} + n & = \frac{1}{2π i} \int_{δ_n} \frac{1}{z} \, dz && \text{Beispiel~\ref{bsp:3-2-2}} \\ + & = \frac{1}{2π i} \int_{β} \frac{1}{z} \, dz. && \text{Satz~\ref{satz:4-3-6}.} \end{align*} Damit ist die Zahl $n$ eindeutig bestimmt. \end{proof} @@ -262,87 +259,277 @@ Wir bewiesen Satz~\ref{satz:12-2-1} in mehreren Schritten. \section{Das Residuum} \begin{definition}[Residuensatz]\label{def:12-3-1}% - Es sei $f \in \sO(U \setminus p)$ und es sei $R > 0$, sodass - $\overline{K_{0,R}(p)} \subset U$ ist. Der Koeffizient von $(z-p)^{-1}$ in - der Laurententwicklung von $f$ auf $K_{0,R}(p)$ wird - \emph{Residuum}\index{Residuum} von $f$ in $p$ genannt. Die Bezeichnung - $\Res_p(f)$ ist üblich. + Es sei $f ∈ 𝒪(U ∖ p)$ und es sei $R > 0$, sodass $\overline{K_{0,R}(p)} ⊂ U$ + ist. Der Koeffizient von $(z-p)^{-1}$ in der Laurententwicklung von $f$ auf + $K_{0,R}(p)$ wird \emph{Residuum}\index{Residuum} von $f$ in $p$ genannt. Die + Bezeichnung $\Res_p(f)$ ist üblich. \end{definition} \begin{bsp}[Hebbare Singularität] - Wenn $f$ in $p$ eine hebbare Singularität hat, dann ist $\Res_p(f) = 0$. + Wenn $f$ in $p$ eine hebbare Singularität hat, dann ist $\Res_p(f) = 0$. \end{bsp} \begin{bsp}[Polstelle] - Es sei $U = \bC$, $p = 0$ und $f(z) = \frac{1}{z}$. Dann ist $\Res_p(f) = 1$. + Es sei $U = ℂ$, $p = 0$ und $f(z) = \frac{1}{z}$. Dann ist $\Res_p(f) = 1$. \end{bsp} \begin{bsp}[Polstelle] - Es sei $U = \bC^*$, $p \ne 0$ und $f(z) = \frac{1}{z}$. Dann ist $\Res_p(f) = + Es sei $U = ℂ^*$, $p \ne 0$ und $f(z) = \frac{1}{z}$. Dann ist $\Res_p(f) = 0$. \end{bsp} \begin{erinnerung} In der Situation von Definition~\ref{def:12-3-1} ist \[ - \Res_p(f) = \frac{1}{2\pi i} \int_{\partial K_R(p)} f(z) \, dz. - \] + \Res_p(f) = \frac{1}{2π i} \int_{∂ K_R(p)} f(z) \, dz. + \] \end{erinnerung} \section{Der Residuensatz} - \begin{satz}[Residuensatz]\label{satz:12-3-2}% - Es sei $U \subset \bC$ offen, es sei $P \subset U$ eine endliche Teilmenge, - und $f \in \sO(U \setminus P)$. Sei weiter $\gamma: [a,b] \to U \setminus P$ - ein geschlossener und stetiger Weg, der in $U$ zusammenziehbar ist. Dann gilt: + Es sei $U ⊂ ℂ$ offen, es sei $P ⊂ U$ eine endliche Teilmenge, und $f ∈ 𝒪(U ∖ + P)$. Sei weiter $γ: [a,b] → U ∖ P$ ein geschlossener und stetiger Weg, der in + $U$ zusammenziehbar ist. Dann gilt: \[ - \int_\gamma f(z) \, dz = 2\pi i \cdot \sum_{p \in P} \Um(\gamma, p) \cdot \Res_p(f). + \int_{γ} f(z) \, dz = 2π i · \sum_{p ∈ P} \Um(γ, p) · \Res_p(f). \] \end{satz} \begin{proof}[Beweis] - Betrachte zunächst einen Punkt $p \in P$. Entwickle $f$ in eine Laurentreihe - in $p$: + Betrachte zunächst einen Punkt $p ∈ P$. Entwickle $f$ in eine Laurentreihe in + $p$: \[ - f(z) = \sum_{k=-\infty}^\infty c_k \cdot (z-p)^k + f(z) = \sum_{k=-∞}^∞ c_k · (z-p)^k \] und definiere \[ - h_p(z) := \sum_{k=-\infty}^{-2} c_k \cdot (z-p)^k. + h_p(z) := \sum_{k=-∞}^{-2} c_k · (z-p)^k. \] Die Funktion $h_p$ hat folgende Eigenschaften. \begin{enumerate} - \item Nach Fakt~\ref{fakt:10-3-1} definiert die Reihe $h_p$ eine auf ganz - $\bC \setminus \{p\}$ holomorphe Funktion. + \item Nach Fakt~\ref{fakt:10-3-1} definiert die Reihe $h_p$ eine auf ganz $ℂ + ∖ \{p\}$ holomorphe Funktion. \item Die Reihe \[ - \sum_{k=-\infty}^{-2} c_k \cdot \frac{(z-p)^{k+1}}{k+1} + \sum_{k=-∞}^{-2} c_k · \frac{(z-p)^{k+1}}{k+1} \] - definiert eine auf ganz $\bC \setminus \{p\}$ holomorphe Stammfunktion von - $h_p$. + definiert eine auf ganz $ℂ ∖ \{p\}$ holomorphe Stammfunktion von $h_p$. - \item Die auf $U \setminus P$ definierte Funktion + \item Die auf $U ∖ P$ definierte Funktion \[ - f(z) - h_p(z) - \Res_p(f) \cdot (z-p)^{-1} = \sum_{k=0}^\infty c_k \cdot (z-p)^k + f(z) - h_p(z) - \Res_p(f) · (z-p)^{-1} = \sum_{k=0}^∞ c_k · (z-p)^k \] hat in $p$ eine hebbare Singularität! - \end{enumerate} - Das machen wir jetzt für alle $p \in P$ und erhalten: Die Funktion + \end{enumerate} + Das machen wir jetzt für alle $p ∈ P$ und erhalten: Die Funktion \[ - g(z) := f(z) - \sum_{p \in P} h_p(z) - \sum_{p \in P} \Res_p(f) \cdot (z-p)^{-1} + g(z) := f(z) - \sum_{p ∈ P} h_p(z) - \sum_{p ∈ P} \Res_p(f) · (z-p)^{-1} \] - hat in allen Punkte $p \in P$ hebbare Singularitäten. Da die Kurve $\gamma$ in $U$ zusammenziehbar ist, folgt aus dem - Cauchy-Integralsatz + hat in allen Punkte $p ∈ P$ hebbare Singularitäten. Da die Kurve $γ$ in $U$ + zusammenziehbar ist, folgt aus dem Cauchy-Integralsatz \[ - \int_\gamma g(z) \, dz = 0. + \int_{γ} g(z) \, dz = 0. \] Also ist \[ - \int_\gamma f(z) \, dz = \sum_{p \in P} \underbrace{\int_\gamma h_p(z) \, dz}_{=0\text{, weil Stammfkt.~existiert}} - + \sum_{p \in P} \underbrace{\int_\gamma \frac{\Res_p(f)}{z-p} \, dz}_{= 2\pi i \cdot \Um(\gamma, p) \cdot \Res_p(f)}. + \int_{γ} f(z) \, dz = \sum_{p ∈ P} \underbrace{\int_{γ} h_p(z) \, dz}_{=0\text{, weil Stammfkt.~existiert}} + + \sum_{p ∈ P} \underbrace{\int_{γ} \frac{\Res_p(f)}{z-p} \, dz}_{= 2π i · \Um(γ, p) · \Res_p(f)}. \] Damit ist der Satz bewiesen. \end{proof} + + +\section{Berechnung von Residuen} + +Wir geben nun praktische Methoden zur Berechnung von Residuen an. Die +Berechnung ist auf jeden Fall dann einfach, wenn die Reihenentwicklung der +Funktion bekannt ist. Das gilt zum Beispiel für Funktionen wie $\frac{1}{1-z}$, +$e^z$, $\sin z$, … + +\begin{bsp}\label{bsp:12-4-5}% + Betrachte die Funktion $f ∈ 𝒪(ℂ ∖ \{0\})$, gegeben durch + \[ + f(z) = z² \sin\left(\frac{1}{z}\right) = z² · \left(\frac{1}{z} - \frac{1}{6 z³} + \frac{1}{120 z⁵} - … \right). + \] + Dann ist $\Res_0(f) = -\frac{1}{6}$. +\end{bsp} + +Für etwas kompliziertere Funktionen sind die folgenden Resultate nützlich. + +\begin{situation}[Situation für Residuenberechnung]\label{sit:12-4-0}% + Es sei $U ⊂ ℂ$ offen, $p ∈ U$ und $f ∈ 𝒪(U ∖ \{p\})$. +\end{situation} + +\begin{lemma}\label{lem:12-4-0}% + Betrachte Situation~\ref{sit:12-4-0}. Wenn + \[ + \lim_{z → p} (z-p)·f(z) =: α ∈ ℂ + \] + existiert, so gilt $\Res_p(f) = α$. +\end{lemma} +\begin{proof} + Die Existenz des Grenzwertes stellt sicher, dass $f$ in $p$ höchstens einen + Pol der Ordnung 1 hat. Also ist + \[ + f(z) = c_{-1} (z-p)^{-1} + \sum_{k=0}^∞ c_k (z-p)^k. + \] + Das zeigt dann aber sofort, dass $c_{-1} = α$ ist. +\end{proof} + +\begin{bemerkung}[Spezialfall von Lemma~\ref{lem:12-4-0}]\label{bem:12-4-1}% + Lemma~\ref{lem:12-4-0} ist insbesondere in Situationen anwendbar, wo eine + Funktion der Form $f = \frac{g}{h}$ gegeben ist, mit $g,h ∈ 𝒪(U)$ und $h(p) = + 0$, $h'(p) ≠ 0$. Dann ist + \[ + (z-p)·f(z) = \frac{g(z)}{\frac{h(z)-h(p)}{z-p}} + \] + und für $z → p$ strebt dieser Wert gegen $\frac{g(p)}{h'(p)}$. +\end{bemerkung} + +\begin{bemerkung}[Spezialfall von Lemma~\ref{lem:12-4-0}]\label{bem:12-4-2}% + Wenn $f$ in $p$ keine essenzielle Singularität hat, so gibt es $ν_p(f) ∈ ℤ$ + und $g ∈ 𝒪(U)$ mit $g(p) ≠ 0$, sodass für jedes $z ∈ U$ gilt + \[ + f(z) = (z-p)^{ν_p(f)} · g(z). + \] + Dann ist + \begin{align*} + \frac{f'(z)}{f(z)} & = \frac{ν_p(f) (z-p)^{ν_p(f)-1} g(z) + (z-p)^{ν_p(f)} · g'(z)}{(z-p)^{ν_p(f)} · g(z)} \\ + & = \frac{ν_p(f) g(z) + (z-p) · g'(z)}{(z-p) · g(z)}. + \end{align*} + Betrachte die Funktion im Nenner, $h(z) := (z-p)·g(z)$, dann ist $h ∈ 𝒪(U)$ + und $h(p) = 0$, $h'(p) = g(p) ≠ 0$. Nach Bemerkung~\ref{bem:12-4-1} gilt + daher + \[ + \Res_p\left(\frac{f'}{f}\right) = \frac{ν_p(f) · g(p)}{g(p)} = ν_p(f). + \] +\end{bemerkung} + +\begin{lemma}\label{bem:12-4-3}% + In Situation~\ref{sit:12-4-0} sei $f$ von der Form $f(z) = + \frac{g(z)}{(z-p)^ν}$, mit $g ∈ 𝒪(U)$ und $ν ≥ 2$. Dann gilt + \[ + \Res_p(f) = \frac{1}{(ν-1)!} g^{(ν-1)}(p). + \] +\end{lemma} +\begin{proof} + Schreibe $g$ als Potenzreihe, $g(z) = \sum_{k=0}^∞ a_k (z-p)^k$, wobei die + Koeffizienten durch die Taylor-Formel gegeben sind, $a_k = + \frac{g^{(k)}(p)}{k!}$. Dann ist der Koeffizient vor $(z-p)^{-1}$ in der + Laurententwicklung von $f$ gleich + \[ + \frac{1}{(ν-1)!}·g^{(ν-1)}(p). + \] + Die Behauptung folgt. +\end{proof} + + +\section{Anwendungen des Residuensatzes} + +\begin{situation}\label{sit:12-5-1}% + --- + \begin{itemize} + \item Es sei $U ⊂ ℂ$ Gebiet und es sei $P ⊂ U$ eine abgeschlossene und + diskrete Teilmenge. + + \item Es sei $f ∈ 𝒪(U ∖ P)$ eine holomorphe Funktion mit isolierten + Singularitäten. Wir nehmen an, dass $f$ nicht die Nullfunktion ist und + keine essenziellen Singularitäten hat. Für jeden Punkt $z ∈ U$ sei + $ν_z(f)$ die Polstellenordnung von $f$ in $z$; diese ist positiv, wenn $f$ + bei $z$ eine Polstelle hat und negativ bei Nullstellen. + + \item Es sei $N = \{z ∈ U \mid f(z) = 0\}$ die Menge der Nullstellen von + $f$. + + \item Es sei $γ: [a,b] → U ∖ (N ∪ P)$ sei ein in $U$ zusammenziehbarer Weg. + \end{itemize} +\end{situation} + +\begin{bemerkung} + In Situation~\ref{sit:12-5-1} sind die Zahlen $ν_z(f)$ für fast alle $z ∈ U$ + gleich null, und höchstens für $z ∈ P$ positiv. +\end{bemerkung} + +\begin{bemerkung}\label{bem:12-5-3}% + In Situation~\ref{sit:12-5-1} sagt die „Goldene Regel 2“, dass die + Umlaufzahlen $\Um(γ, p)$ höchstens auf einer kompakten Teilmenge von $U$ + ungleich null sind. Da der Schnitt einer diskreten Menge mit einer kompakten + Menge endlich ist, gibt es nur endlich viele Punkte $z ∈ U$, für die das + Produkt $\Um(γ, p) · ν_p(f)$ ungleich null ist. +\end{bemerkung} + +\begin{satz}[Zählen von Null- und Polstellen]\label{satz:12-5-1}% + In Situation~\ref{sit:12-5-1} betrachte den Weg $f ◦ γ : [a,b] → ℂ$. Dann gilt + \[ + \Um(f ◦ γ, 0) = \sum_{p ∈ U} \Um(γ, p) · ν_p(f). + \] + Beachte, dass nach Bemerkung~\ref{bem:12-5-3} nur endlich viele der Summanden + auf der rechten Seite ungleich null sind. +\end{satz} +\begin{proof} + Es gilt + \begin{align*} + \Um(f ◦ γ, 0) & = \frac{1}{2π i} \int_{f◦ γ} \frac{1}{z} \, dz && \text{Definition Umlaufzahl} \\ + & = \frac{1}{2π i} \int_γ \frac{f'(z)}{f(z)} \, dz && \text{Definition Wegintegral, Kettenregel} \\ + & = \sum_{p ∈ U} \Um(γ, p) · ν_p(f) && \text{Residuensatz, Bemerkung~\ref{bem:12-4-2}.} + \end{align*} + Damit ist Satz~\ref{satz:12-5-1} bewiesen. +\end{proof} + +\begin{kor}[Satz von Rouché\footnote{Eugène Rouché (* 18.~August 1832 in + Sommières, Département Hérault; † 19.~August 1910 in Lunel) war ein + französischer Mathematiker. }]\label{kor:12-5-2}% + \index{Satz von Rouché}Sei $U ⊂ ℂ$ offen, $f ∈ 𝒪(U)$ und $K ⊂ U$ eine + abgeschlossene Kreisscheibe. Sei außerdem eine holomorphe Funktion $g ∈ 𝒪(U)$ + gegeben, sodass für jedes $z ∈ δ K$ die Ungleichung + \begin{equation}\label{eq:12-5-5}% + |f(z)| > |g(z)| + \end{equation} + gilt. Dann gilt Folgendes. + \begin{enumerate} + \item\label{il:12-5-5-1} Alle Nullstellen von $f$ und $f+g$ auf $K$ liegen + im Inneren von $K$. + + \item\label{il:12-5-5-2} Mit Vielfachheit gezählt haben $f$ und $f+g$ die + gleiche Anzahl an Nullstellen auf $K$. + \end{enumerate} +\end{kor} +\begin{proof} + Für $t ∈ [0,1]$ betrachte die Familie von Funktionen $h_t(z) := f(z) + + t·g(z)$. Ungleichung~\eqref{eq:12-5-5} zeigt sofort, dass für jedes $t ∈ + [0,1]$ und jedes $z ∈ ∂K$ die Ungleichung + \[ + h_t(z) + \] + gilt. Damit ist~\ref{il:12-5-5-1} bewiesen. Als Nächstes betrachte + \[ + N : [0,1] → ℂ, \quad t ↦ \frac{1}{2π i} \int_{∂K} \frac{h_t'(z)}{h_t(z)} \, dz. + \] + Auf der einen Seite sagt der Satz über parameterabhängige Integrale, dass $N$ + stetig ist. Auf der anderen Seite gilt nach Satz~\ref{satz:12-5-1} für jedes + $t ∈ [0,1]$ die Gleichung + \[ + N(t) = \text{Anzahl der Nullstellen von $h_t$ in $K$}. + \] + Da $N(t)$ also ganzzahlig ist, ist $N$ konstant. Somit gilt insbesondere + $N(0) = N(1)$. +\end{proof} + +\begin{bsp}\label{bsp:12-5-3}% + Wir behaupten, dass die Funktion + \[ + \frac{1}{10} z⁷ + 1 + 5 z² + \] + in $B_1(0)$ genau $2$ Nullstellen hat. Schreibe dazu $f(z) = 5z²$, $g(z) = + \frac{1}{10} z⁷ + 1$ und beobachte, dass für jedes $z$ mit $|z| = 1$ die + Ungleichung + \[ + |f(z)| = 5 > 1 + \frac{1}{10} ≥ |g(z)| + \] + gilt. 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