From 61a1663529680960fab7f1119631a045f17fb117 Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Wed, 10 Dec 2025 11:07:08 +0100 Subject: [PATCH] Cleanup --- 13-applResiduum.tex | 89 +++++++++++++++++++++++++++++--------------- 13-integration.png | Bin 0 -> 56020 bytes 2 files changed, 59 insertions(+), 30 deletions(-) create mode 100644 13-integration.png diff --git a/13-applResiduum.tex b/13-applResiduum.tex index 5e3832b..b34a378 100644 --- a/13-applResiduum.tex +++ b/13-applResiduum.tex @@ -55,6 +55,8 @@ Damit ist Satz~\ref{satz:12-5-1} bewiesen. \end{proof} +\sideremark{Vorlesung 18} + \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}% @@ -139,10 +141,10 @@ sodass für $z$ in einer Umgebung von $p$ folgendes gilt. \subsection{Beweis des Satzes von Weierstraß} -Die Vorüberlegung legt nahe, den Satz von Mittag–Leffler zu verwenden. Sei also -$g ∈ 𝒪(ℂ ∖ P)$ eine Funktion, deren Hauptteil an jeder Stelle $p ∈ P$ exakt -gleich $\frac{n(p)}{z-p}$ ist. Das Ziel ist jetzt, aus der Funktion $g$ die -gesuchte Funktion $f$ zu konstruieren. +Die Vorüberlegung legt nahe, den Satz~\ref{satz:mittag-leffler} von +Mittag–Leffler zu verwenden. Sei also $g ∈ 𝒪(ℂ ∖ P)$ eine Funktion, deren +Hauptteil an jeder Stelle $p ∈ P$ exakt gleich $\frac{n(p)}{z-p}$ ist. Das Ziel +ist jetzt, aus der Funktion $g$ die gesuchte Funktion $f$ zu konstruieren. \begin{erinnerung} Der Residuensatz besagt, dass für jede geschlossene Kurve $γ$ in $ℂ ∖ P$ gilt: @@ -206,34 +208,61 @@ Satz von Weierstraß bewiesen. \qed \subsection{Uneigentliche Integrale rationaler Funktionen} Der Residuensatz erlaubt die Berechnung gewisser uneigentlichen Integrale. Ich -skizziere das Vorgehen am Beispiel rationaler Funktionen. Sei $f(z) = -\frac{a(z)}{b(z)}$ eine rationale Funktion, wobei folgendes gilt. -\begin{enumerate} - \item Die Funktion $f$ habe auf der reellen Achse keine Polstellen. +skizziere das Vorgehen am Beispiel rationaler Funktionen. - \item Die Grade der Polynome $a(z)$ und $b(z)$ erfüllen die Ungleichung $\deg - b \ge \deg a + 2$. -\end{enumerate} -und $f$ habe auf der reellen Achse keine Pole. Dann kann man den Residuensatz -anwenden, um das uneigentliche Integral -\[ - \int_{-∞}^{∞} f(x) \, dx -\] -zu berechnen. Beachte dazu folgendes: Wenn $r ∈ ℝ⁺$ hinreichend groß ist, dann -liegen alle Polstellen von $f$ im Inneren der Kreisscheibe $B_r(0)$. Sei $γ_r$ -der folgende geschlossene Weg: -\begin{itemize} - \item Gehe entlang der reellen Achse von $-r$ nach $r$. +\begin{beobachtung} + Sei $f(z) = \frac{a(z)}{b(z)}$ eine rationale Funktion, wobei folgendes gilt. + \begin{enumerate} + \item Die Funktion $f$ habe auf der reellen Achse keine Polstellen. + + \item Die Grade der Polynome $a(z)$ und $b(z)$ erfüllen die Ungleichung + $\deg b \ge \deg a + 2$. + \end{enumerate} + und $f$ habe auf der reellen Achse keine Pole. Dann kann man den Residuensatz + anwenden, um das uneigentliche Integral + \[ + \int_{-∞}^{∞} f(x) \, dx + \] + zu berechnen. Beachte dazu folgendes: Wenn $r ∈ ℝ⁺$ hinreichend groß ist, + dann liegen alle Polstellen von $f$ im Inneren der Kreisscheibe $B_r(0)$. Sei + $γ_r$ der in Abbildung~\ref{fig:13-1-1} skizzierte geschlossene Weg: + \begin{itemize} + \item Gehe entlang der reellen Achse von $-r$ nach $r$. + + \item Gehe entlang des Halbkreises $\{r·exp(i·t) \::\: 0 ≤ t ≤ π\}$ von $r$ + nach $-r$ zurück. + \end{itemize} + \begin{figure} + \begin{center} + \includegraphics[width=8cm]{13-integration.png} + \end{center} + + \caption{Integrationsweg} + \label{fig:13-1-1} +\end{figure} + + Dann hängt das Integral über $γ_r$ nicht von $r$ ab, und es gilt nach dem + Residuensatz + \[ + \int_{\gamma_r} f(z) \, dz = 2π i \sum_{\Im z > 0} \Res(f,z). + \] + Auf der anderen Seite ist + \[ + \int_{\gamma_r} f(z) \, dz = \int_{-r}^r f(x) dx + \int_{\text{oberer Halbkreis}} f(z) \, dz. + \] + Jetzt rechne man nach, dass die Bedingung $\deg b \ge \deg a + 2$ impliziert, + dass es eine Konstante $\operatorname{const}^+ ∈ ℝ⁺$ gibt, sodass für alle + hinreichend großen $r$ gilt: + \[ + \max_{|z| = r} |f(z)| ≤ \operatorname{const}^+·{r^{-2}}. + \] + Das bedeutet, dass das Integral über den oberen Halbkreis für $r → ∞$ gegen + null geht. Insgesamt ergibt sich also die folgende Formel. + \[ + \int_{-∞}^{∞} f(x) \, dx = 2π i \sum_{\Im z > 0} \Res(f,z). + \] +\end{beobachtung} - \item Gehe entlang des Halbkreises $\{r·exp(i·t) \::\: 0 ≤ t ≤ π\}$ von $r$ - nach $-r$ zurück. -\end{itemize} -Dann hängt das Integral über $γ_r$ nicht von $r$ ab, und es gilt nach dem -Residuensatz -\[ - \int_{-r}^{r} f(z) \, dz = 2π i \sum_{\Im z > 0} \Res(f,z), -\] -da das Integral über den oberen Halbkreis für $r → ∞$ verschwindet. \begin{rem}[Variante] Sei $f(z) = \dfrac{a(z)}{b(z)}$ rational ohne Pole auf der reellen Achse und diff --git a/13-integration.png b/13-integration.png new file mode 100644 index 0000000000000000000000000000000000000000..97806824d885cb20dbc6c7cd948638ef27bbc352 GIT binary patch literal 56020 zcmZ6zcRbba|37|^5ostR3FRcEBqcK%)`=*zGeSv2cFHWNPI5tPThv@je=mh{S)4-g zyhfo+;Ey~zC=^F&3Z=`GLQ%X&q43*8WbRkSU(C1BI$}?ua4jYOW1>8W5WpX@JJ9y3 zvv)4w66*-;F8o7<9wa? 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