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zk*+llyNNWri$8+(_*th2E*ryS^*(UcBBFmt&cC+{TELTWZvs^rD)({q=Zn^IM(_n) zkpSv`HZL*cSvwi5I0(cU$w1;Q#?M<_(|qJHI$e z5I;Jlu}qUx%m~K%Yi2(6v%4RDpp-I<`{GY48_B3EPb2*G8QJ`E&cC= 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}