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ws2019/ana/lectures/analysis19.tex
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| \documentclass{../../../lecture} | |||
| \begin{document} | |||
| \begin{satz}[Reihenentwicklung Sinus / Cosinus] | |||
| Für alle $x \in \R$ gilt (absolut konvergente | |||
| Potenzreihendarstellung) | |||
| \[ | |||
| \cos(x) = \sum_{k=0}^{\infty} (-1)^{k}\frac{x^{2k}}{(2k)!} = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \ldots | |||
| .\] und | |||
| \[ | |||
| \sin(x) = \sum_{k=0}^{\infty} (-1)^{k}\frac{x^{2k+1}}{(2k+1)!} = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \ldots | |||
| .\] | |||
| \end{satz} | |||
| \begin{proof} | |||
| Die absolute Konvergenz folgt als Teilreihe der Exponentialreihe (als Majorante) | |||
| Es gilt für $m \in \N_0$ | |||
| \[ | |||
| i^{n} = \begin{cases} | |||
| 1 & n = 4m \\ | |||
| i & n = 4m+1 \\ | |||
| -1 & n = 4m+2 \\ | |||
| -i & n = 4m+3 | |||
| \end{cases} | |||
| .\] Es folgt | |||
| \begin{align*} | |||
| e^{ix} &= \sum_{n=0}^{\infty} \frac{(ix)^{n}}{n!} = \sum_{n=0}^{\infty} i^{n} \frac{x^{n}}{n!} \\ | |||
| &= \underbrace{\sum_{k=0}^{\infty} (-1)^{k} \frac{x^{2k}}{(2k)!}}_{\cos(x)} + i \underbrace{\sum_{k=0}^{\infty} (-1)^{k} \frac{x^{2k+1}}{(2k+1)!}}_{\sin(x)} | |||
| .\end{align*} | |||
| \end{proof} | |||
| \begin{satz}[Restgliedabschätzung Sinus / Cosinus] | |||
| Für $n \in \N_0$ gilt | |||
| \[ | |||
| \cos(x) = \sum_{k=0}^{n} (-1)^{k} \frac{x^{2k}}{(2k)!} + R_{2n+2}(x) | |||
| .\] und | |||
| \[ | |||
| \sin(x)= \sum_{k=0}^{n} (-1)^{k} \frac{x^{2k+1}}{(2k+1)!} + R_{2n+3}(x) | |||
| .\] | |||
| mit | |||
| \[ | |||
| |R_{2n+2}(x)| \le \frac{|x|^{2n+2}}{(2n+2)!} \text{ für } |x| \le 2n+3 | |||
| .\] bzw. | |||
| \[ | |||
| |R_{2n+3}(x)| \le \frac{|x|^{2n+3}}{(2n+3)!} \text{ für } |x| \le 2n+4 | |||
| .\] | |||
| \end{satz} | |||
| \begin{proof} | |||
| Es gilt | |||
| \begin{align*} | |||
| R_{2n+2}(x) &= \sum_{k=n+1}^{\infty} (-1)^{k} \frac{x^{2k}}{(2k)!} \\ | |||
| &= (-1)^{n+1} \frac{x^{2n+2}}{(2n+2)!} | |||
| \left( \sum_{k=n+1}^{\infty} (-1)^{k-(n+1)} | |||
| \frac{x^{2(k - (n+1))}}{(2k)! \frac{1}{(2n+2)!}}\right) \\ | |||
| &= (-1)^{n+1} \frac{x^{2n+2}}{(2n+2)!} | |||
| \left( \sum_{k=0}^{\infty} (-1)^{k} | |||
| \frac{x^{2k}(2n+2)!}{(2k+2n+2)!} \right) | |||
| .\end{align*} | |||
| Für $k \in \N$ setze | |||
| \begin{align*} | |||
| a_k :&= \frac{x^{2k}(2n+2)!}{(2k+2n+2)!} | |||
| = \frac{x^{2k}}{(2n+3)(2n+4) \ldots (2k + 2n + 2)} \\ | |||
| a_{k-1} &= \frac{x^{2k-2}(2n+2)!}{(2k+2n)!} | |||
| \intertext{damit} | |||
| a_k &= a_{k-1} \cdot \frac{x^{2}}{(2k+2n+1)(2k+2n+2)} | |||
| .\end{align*} | |||
| Es gilt für $|x| \le 2n+3, k\ge 1$ | |||
| \[ | |||
| \frac{x^{2}}{(2k+2n+1)(2k+2n+2)} \le \frac{(2n+3)^{2}}{(2n+3)(2n+4)} < 1 | |||
| .\] $\implies$ | |||
| \[ | |||
| a_k \le \frac{(2n+3)^{k}}{(2n+4)^{k}} a_0 \quad a_0 = \frac{1}{(2n+2)!} | |||
| .\] $\stackrel{\text{Leibniz}}{\implies}$ | |||
| \[ | |||
| \sum_{k=0}^{\infty} (-1)^{k} a_k | |||
| .\] konvergent mit | |||
| \[ | |||
| 0 < \underbrace{\underbrace{1 - a_1}_{> 0} + \underbrace{a_2 - a_3}_{> 0} | |||
| + \underbrace{a_4 - \ldots}_{> 0}}_{< 1} < 1 | |||
| .\] $\implies$ | |||
| \[ | |||
| |R_{2n+2}(x)| \le \frac{|x|^{2n+2}}{(2n+2)!} \text{ für } |x| \le 2n+3 | |||
| .\] Genauso für $R_{2n+3}(x)$ (Sinus). | |||
| \end{proof} | |||
| \begin{lemma} | |||
| Sinus und Cosinus Funktionen haben das folgende Verhalten | |||
| \[ | |||
| \lim_{x \to 0} \frac{\sin(x)}{x} = 1 | |||
| .\] | |||
| \[ | |||
| \lim_{x \to 0} \frac{\cos(x)-1}{x} = 0 | |||
| .\] | |||
| \end{lemma} | |||
| \begin{proof} | |||
| \begin{align*} | |||
| \left| \frac{\sin(x)}{x} - 1 \right| | |||
| &= \left| \underbrace{1 - \frac{x^{2}}{3!} + \frac{x^{4}}{5!}}_{\frac{\sin(x)}{x}} - \ldots - 1\right| \\ | |||
| &= \left| x \sum_{k=1}^{\infty} (-1)^{k}\frac{x^{2k-1}}{(2k+1)!} \right| \\ | |||
| &\stackrel{|x| < 1}{\le |x|} \cdot \left| \sum_{k=1}^{\infty} \frac{1}{(2k+1)!} \right| | |||
| \le |x| \cdot e | |||
| .\end{align*} $\implies$ | |||
| \[ | |||
| \underbrace{\left| \frac{\sin(x)}{x} -1 \right|}_{\to 0} | |||
| \le \underbrace{|x| \cdot e}_{\to 0} | |||
| .\] | |||
| genauso für $\lim_{x \to 0} \frac{\cos(x) - 1}{x}$. | |||
| \end{proof} | |||
| \subsection{Die Zahl $\pi$} | |||
| Ziel: Analytische Definition von $\pi \in \R$. | |||
| \begin{satz}[und Definition] | |||
| Die Funktion $\cos\colon [0,2] \to \R$ hat genau eine Nullstelle | |||
| im Intervall $[0,2]$, welche mit $\frac{\pi}{2}$ bezeichnet | |||
| wird ($\pi := 2 \frac{\pi}{2}$ ). | |||
| \end{satz} | |||
| \begin{proof} | |||
| in 4 Schritten. | |||
| Schritt 1 / Lemma 1: $\cos(2) \le -\frac{1}{3}$. \\ | |||
| Restgliedabschätzung liefert ($|x| \le 5$ ). | |||
| \[ | |||
| \cos(x) = 1 - \frac{x^2}{2} + R_4(x) \text{ mit } |R_4(x)| \le \frac{|x|^{4}}{24} | |||
| .\] $\implies$ | |||
| \[ | |||
| \cos(2) = 1 - 2 + \underbrace{R_4(2)}_{\le \frac{16}{24} = \frac{2}{3}} \le -1 + \frac{2}{3} = -\frac{1}{3} | |||
| .\] | |||
| Schritt 2 / Lemma 2: $\sin(x) > 0$ $\forall x \in \; ]0, 2[$\\ | |||
| Es gilt | |||
| \begin{align*} | |||
| \sin(x) = x + R_3(x) = x (1 + \frac{R_3(x)}{x}) | |||
| \left| \frac{R_3(x)}{x} \right| \le \frac{|x|^2}{6} | |||
| \stackrel{0 < x \le 2}{\le} \frac{4}{6} = \frac{2}{3} | |||
| \intertext{$\implies$} | |||
| 1 + \frac{R_3(x)}{x} \ge \frac{1}{3} | |||
| .\end{align*} | |||
| Schritt 3 / Lemma 3: $\cos: [0,2] \to \R$ ist streng monoton fallend.\\ | |||
| Sei $0 \le y < x \le 2$. Dann gilt | |||
| \begin{align*} | |||
| \cos(x) - \cos(y) \stackrel{\text{Additionstheorem}}{=} | |||
| - 2 \underbrace{\sin\left( \frac{x+y}{2} \right)}_{> 0} | |||
| \underbrace{\sin\left( \frac{x-y}{2} \right)}_{> 0} < 0 | |||
| .\end{align*} | |||
| Schritt 4 (Beweis der Definition von $\pi$ ) | |||
| $\cos(0) = 1$ (nach Definition). | |||
| \[ | |||
| \cos(2) \le - \frac{1}{3} \stackrel{\text{Zwischenwertsatz}}{\implies} | |||
| \exists x_0 \in [0,2] \text{ mit } \cos(x_0) = 0 | |||
| .\] Nach Lemma 3 ist $x_0$ eindeutig. | |||
| \end{proof} | |||
| \begin{korrolar}[Spezielle Werte von $\exp$] | |||
| Es gilt: $e^{i \frac{\pi}{2}} = i$, | |||
| $e^{i \pi} = -1$, $e^{i \frac{3\pi}{2}} = -i$, $e^{2\pi i} = 1$ | |||
| \end{korrolar} | |||
| \begin{proof} | |||
| Übung. | |||
| \end{proof} | |||
| \begin{korrolar}[Eigenschaften Sinus / Cosinus] | |||
| $\forall x \in \R$ gilt: | |||
| \begin{enumerate}[(i)] | |||
| \item $\cos(x + 2\pi) = \cos(x) \quad \sin(x+2\pi) = \sin(x)$ \\ | |||
| $2 \pi$: Periodizität | |||
| \item $\cos(x + \pi) = - \cos(x) \quad \sin(x+ \pi) = - \sin(x)$ | |||
| \item $\cos(x) = \sin(\frac{\pi}{2} - x) \quad \sin(x) = \cos(\frac{\pi}{2} - x)$ | |||
| \item Nullstellen von $\sin / \cos$.\\ | |||
| $\{x \in \R | \sin x = 0\} = \{x = k\pi | k \in \Z\} $ \\ | |||
| $\{x \in \R | \cos x = 0\} = \{x = \left(k+\frac{1}{2}\right)\pi | k \in \Z\} $ \\ | |||
| \end{enumerate} | |||
| \end{korrolar} | |||
| \begin{proof} | |||
| folgt aus den Additionstheoremen, der Definition von $\frac{\pi}{2}$, | |||
| den speziellen Werten von $\exp$ und folgender Tabelle | |||
| \begin{tabular}{l|l|l|l|l|l} | |||
| x & 0 & $\frac{\pi}{2}$ & $\pi$ & $\frac{3}{2} \pi$ & $2 \pi$ \\ \hline | |||
| $\cos x$ & 1 & 0 & $-1$ & 0 & 1 \\ \hline | |||
| $\sin x$ & 0 & 1 & 0 & $-1$ & 0 \\ | |||
| \end{tabular}. | |||
| \end{proof} | |||
| \begin{korrolar}[$e^{z} = 1$] | |||
| Es gilt $\{z \in \mathbb{C} | e^{z} = 1\} = \{i 2 \pi k | k \in \Z\} $ | |||
| \end{korrolar} | |||
| \begin{proof} | |||
| ohne Beweis. | |||
| \end{proof} | |||
| \begin{definition}[Tangens, Cotangens] | |||
| \begin{enumerate}[(i)] | |||
| \item Die Tangensfunktion | |||
| \begin{align*} | |||
| &\tan: \R \setminus \{x = (k + \frac{1}{2}) \pi | k \in \Z\} | |||
| \to \R | |||
| \intertext{ist definiert durch} | |||
| &\tan x := \frac{\sin x}{\cos x} | |||
| .\end{align*} | |||
| \item Die Cotangensfunktion | |||
| \begin{align*} | |||
| &\cot: \R \setminus \{x = k \pi | k \in \Z\} \to \R | |||
| \intertext{ist definiert durch} | |||
| &\cot x := \frac{\cos(x)}{\sin(x)} | |||
| .\end{align*} | |||
| \end{enumerate} | |||
| \end{definition} | |||
| \begin{figure}[htpb] | |||
| \centering | |||
| \begin{tikzpicture} | |||
| \begin{axis}% | |||
| [grid=both, | |||
| minor tick num=4, | |||
| grid style={line width=.1pt, draw=gray!10}, | |||
| major grid style={line width=.2pt,draw=gray!50}, | |||
| axis lines=middle, | |||
| enlargelimits={abs=0.2}, | |||
| ymax=5, | |||
| ymin=-5 | |||
| ] | |||
| \addplot[domain=-3:3,samples=50,smooth,red] {tan(deg(x))}; | |||
| \end{axis} | |||
| \end{tikzpicture} | |||
| \caption{$\tan(x)$} | |||
| \end{figure} | |||
| \begin{figure}[htpb] | |||
| \centering | |||
| \begin{tikzpicture} | |||
| \begin{axis}% | |||
| [grid=both, | |||
| minor tick num=4, | |||
| grid style={line width=.1pt, draw=gray!10}, | |||
| major grid style={line width=.2pt,draw=gray!50}, | |||
| axis lines=middle, | |||
| enlargelimits={abs=0.2}, | |||
| ymax=5, | |||
| ymin=-5 | |||
| ] | |||
| \addplot[domain=-3:3,samples=50,smooth,red] {cot(deg(x))}; | |||
| \end{axis} | |||
| \end{tikzpicture} | |||
| \caption{$\cot(x)$} | |||
| \end{figure} | |||
| \begin{definition}[Arcusfunktionen (Umkehrfunktionen der Trigonometrischen | |||
| Funktionen)] | |||
| \begin{enumerate}[(i)] | |||
| \item $\cos\colon [0, \pi] \to [-1, 1]$ ist streng monoton fallend | |||
| und bijektiv. Die Umkehrfunktion heißt | |||
| Arcus-Cosinus. | |||
| \[ | |||
| \arccos: [-1,1] \to [0, \pi] | |||
| .\] | |||
| \item $\sin\colon \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \to [-1, 1]$ | |||
| ist streng monoton wachsend und bijektiv. Die Umkehrfunktion | |||
| heißt Arcus-Sinus. | |||
| \[ | |||
| \arcsin: [-1,1] \to \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] | |||
| .\] | |||
| \item $\tan\colon \; ] - \frac{\pi}{2}, \frac{\pi}{2} [ \to \R$ ist streng | |||
| monoton wachsend und bijektiv. Die Umkehrfunktion heißt | |||
| Arcus-Tangens. | |||
| \[ | |||
| \arctan: \R \to ] - \frac{\pi}{2}, \frac{\pi}{2} [ | |||
| .\] | |||
| \end{enumerate} | |||
| \end{definition} | |||
| \begin{satz}[Polarkoordinaten] | |||
| Jedes $z \in \mathbb{C}$ lässt sich schreiben als | |||
| $z = r\cdot e^{i \varphi}$, $\varphi \in \R$ und | |||
| $r = |z| \in [0, \infty[$. | |||
| Für $z \neq 0$ ist $\varphi$ bis auf ein ganzzahliges Vielfaches von | |||
| $2\pi$ eindeutig bestimmt. | |||
| \end{satz} | |||
| \begin{proof} | |||
| Rannacher. | |||
| \end{proof} | |||
| \end{document} | |||