From 9f0ad8c53d4c3b5d5e050e12b1d82ad01564e832 Mon Sep 17 00:00:00 2001 From: flavis Date: Fri, 24 Apr 2020 22:37:09 +0200 Subject: [PATCH] update ana --- sose2020/ana/lectures/ana1.tex | 5 +- sose2020/ana/lectures/analysisII.pdf | Bin 249632 -> 249632 bytes sose2020/ana/uebungen/ana1.pdf | Bin 0 -> 124651 bytes sose2020/ana/uebungen/ana1.tex | 165 +++++++++++++++++++++++++++ 4 files changed, 168 insertions(+), 2 deletions(-) create mode 100644 sose2020/ana/uebungen/ana1.pdf create mode 100644 sose2020/ana/uebungen/ana1.tex diff --git a/sose2020/ana/lectures/ana1.tex b/sose2020/ana/lectures/ana1.tex index 4995832..2644a41 100644 --- a/sose2020/ana/lectures/ana1.tex +++ b/sose2020/ana/lectures/ana1.tex @@ -226,10 +226,11 @@ $\forall n \ge N_0$ und $\forall x \in [a,b]$. Damit gilt $\forall n, m \ge N_0$ und $\forall x \in [a,b]$: + \begin{align*} &|f_n(x) - f_m(x)| \le |f_n(x) - f(x)| + |f(x) - f_m(x)| \le \frac{\epsilon}{4} + \frac{\epsilon}{4} = \frac{\epsilon}{2} \\ - \implies &\max_{x \in [a,b]} |f_n(x) - f_m(x)| = \Vert f_n - f_m \Vert_\infty + \implies &\max_{x \in [a,b]} |f_n(x) - f_m(x)| = \Vert f_n - f_m \Vert_{\infty} \le \frac{\epsilon}{2} < \epsilon \quad \forall n, m \ge N_0 .\end{align*} @@ -262,7 +263,7 @@ Der Funktionenraum $C[a,b]$ definiert durch \[ C[a,b] := \{ f \colon [a,b] \to \R \mid f \text{ stetig }\} - .\] mit $\Vert f \Vert_\infty$ einen normierten Vektorraum. + ,\] ist mit $\Vert f \Vert_\infty$ ein normierter Vektorraum. \end{definition} \begin{satz}[Vollständigkeit] diff --git a/sose2020/ana/lectures/analysisII.pdf b/sose2020/ana/lectures/analysisII.pdf index a8a0ef3ad3c91d6efe2b1f718de4a324d0a583ad..f6002c02a8d092eff041ab7b64c33ba883a93829 100644 GIT binary patch delta 108 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zF*PvJoNEG}1h8RGcWO-cv-L@>X6`5456SC({wy=dd;fSjDLQ&CEHz_TPlBYQCkaYdDS`Tc;+aIa*hZcHd2!$iHwd(9MGl zKAqg_tEXK-%;20sMp^g4wS?d-%M4Y>jIpFzKu&!Is5&|J64t-ASoIBu7 zNZjo(-pYN{n=bvZ@)|)bvH8o0uV~x0w45UzZ|!Pd4or0rP`!b7x)=R~OoKvLy&*eK z+jFdK6AC^DjBT07K&Pq%cE_9#`?|YHB}bbH23uTeqA;4)tyQgrG1G4Hvatz9nIW33 z4*GUgmHyQA%6e-Vg2gEW8U4isY8mp%J@iy*HG?+#7gd81`WLB^Q|NbZ?q0Gl&`L{- g&j0Tv?&50X?CR 0$ und $n \in \N$ mit $n > \frac{\epsilon \cdot 16}{\sqrt{27} }$. Dann wähle + $x_0 = \frac{1}{\sqrt{3} n}$. Damit folgt + \[ + |f_n(x_0) - f(x_0)| = |f_n(x_0)| + = \left| \frac{\sqrt{27} n}{16} \right| + > \left| \frac{\sqrt{27} \epsilon 16}{16\sqrt{27}} \right| + = \epsilon + .\] + \end{proof} +\end{aufgabe} + +\begin{aufgabe}[Uneigentliche Integrale und Funktionenfolgen] + Für $n \in \N$ sei $f_n \colon [0, \infty) \to \R$ definiert durch + \[ + f_n(x) := \frac{1}{n} e^{-\frac{x}{n}}, \quad x \ge 0 + .\] Beh.: $(f_n)_{n\in\N}$ konvergiert gleichmäßig, aber + \[ + \lim_{n \to \infty} \int_{0}^{\infty} f_n(x) \d x \neq \int_{0}^{\infty} \lim_{n \to \infty} f_n(x) \d x + .\] + \begin{proof} + Zu zeigen: $f_n \xrightarrow[\text{gleichmäßig}]{n \to \infty} f(x) := 0$. + + Sei $\epsilon > 0$ und $x \in [0, \infty)$ beliebig. Wähle $n_0 > \frac{1}{\epsilon}$. Wegen + $x \ge 0$ und $n \ge 1$, folgt $\frac{x}{n} \ge 0 \implies e^{-\frac{x}{n}} \le 1$. + Damit folgt direkt $\forall n \in \N$, $n \ge n_0$: + \[ + \left| \frac{1}{n} e^{-\frac{x}{n}} - 0 \right| \le \frac{1}{n} \cdot 1 \le \frac{1}{n_0} + < \frac{1}{\frac{1}{\epsilon}} = \epsilon + .\] $\implies f_n \xrightarrow[\text{gleichmäßig}]{n \to \infty} f(x)$. + + Damit folgt + \[ + \int_{0}^{1} \frac{1}{n} e^{-\frac{x}{n}} \d x + = - e^{-\frac{x}{n}} \Big|_{0}^{\infty} = - \left( \lim_{x \to \infty} e^{-\frac{x}{n}} - 1 \right) + = 1 \neq 0 = \int_{0}^{\infty} f(x) \d x = \int_{0}^{\infty} \lim_{n \to \infty} f_n(x) \d x + .\] + \end{proof} + $[0, \infty)$ ist kein kompaktes Intervall, weshalb Satz 1.3.1 nicht anwendbar ist. +\end{aufgabe} + +\begin{aufgabe}[Stammfunktionen] + Mit Produktintegration folgt sofort + \begin{align*} + &\int \cos(x) \sin(x) \d x = - \cos^2(x) - \int \cos(x) \sin(x) \d x \\ + \implies & 2 \int \cos(x) \sin(x) \d x = - \cos^2(x) \\ + \implies & \int \cos(x) \sin(x) \d x = - \frac{1}{2} \cos^2(x) + .\end{align*} +\end{aufgabe} + +\end{document}