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+\end{definition} + +\begin{lemma}[Eigenschaften von $a^{x}$] + Sei $a > 0$ : + \begin{enumerate} + \item $\exp_a\colon \R \to \R$ ist stetig + \item $\exp_a(x+y) = \exp_a(x) \cdot \exp_a(y) \quad \forall x,y \in \R$ + \item $\exp_a(n) = a^{n} = \underbrace{a \cdot a \cdot a \cdot \ldots \cdot a}_{n-\text{mal}} \quad n \in \N$ + \item $\exp_a(n) = a^{n} \quad n \in \Z$ + \item $\exp_a\left( \frac{p}{q} \right) = \sqrt[q]{a^{p}} \quad \forall p \in \Z, q \in \N$ + \item $a^{x}\cdot a^{y} = a^{x + y}$ + \item $(a^{x})^{y} = a^{x\cdot y}$ + \item $a^{x}b^{x} = (ab)^{x} \quad b > 0, x \in \R$ + \item $\frac{1}{a^{x}} = a^{-x} \quad \forall x \in \R$ + \end{enumerate} +\end{lemma} + +\begin{proof} + trivial. +\end{proof} + +\subsection{Gleichmäßige Stetigkeit} + +\begin{definition}[gleichmäßige Stetigkeit] + Eine Funktion $f\colon D \to \R$, $D \subset \R$ heißt + gleichmäßig stetig auf $D$, falls gilt: + \[ + \forall \epsilon > 0 \exists \delta > 0 \text{ mit } |f(x) - f(y)| < \epsilon \quad \forall x, y \in D + \text{ mit } |x-y| < \delta + .\] +\end{definition} + +\begin{bem} \begin{enumerate} + \item Jede gleichmäßige stetige Funktion auf $D$ ist auch stetig + \item Unterschied zwischen stetig und gleichmäßig stetig: + \begin{itemize} + \item stetig: $\delta$ hängt von $\epsilon$ und $x$ ab + \item gleichmäßig stetig: $\delta$ hängt nur von $\epsilon$ ab + \end{itemize} + \end{enumerate} +\end{bem} + +\begin{bsp} + $f\colon ]0,1] \to \R$ mit $f(x) = \frac{1}{x}$ + + $f$ stetig, aber nicht gleichmäßig stetig. +\end{bsp} + +\begin{proof} + Wähle $\epsilon = 1$. Angenommen: $\exists \delta > 0$ mit + $|f(x) - f(y)| < 1$ $\forall x, y \in ]0,1]$ + mit $|x - y| < \delta$. + + $\exists n \in \N$ mit $\frac{1}{n} < \delta$. Für $x := \frac{1}{n}$ und + $y := \frac{1}{2n}$ gilt $|x-y| = |\frac{1}{n} - \frac{1}{2n}| = |\frac{1}{2n}| < \delta$, aber + $|f(x) - f(y)| = |n-2n| = n \ge 1$. Widerspruch +\end{proof} + +\begin{satz} + Auf kompakten Mengen (Intervallen) gilt: stetig $\iff$ gleichmäßig stetig + + Sei $f\colon D \to \R$ und $D \subset \R$ kompakt. Dann ist + $f$ gleichmäßig stetig. +\end{satz} + +\begin{proof} + Ang. $f$ ist nicht gleichmäßig stetig. Dann $\exists \epsilon_0 > 0$ mit + $\forall n \in \N$ $\exists x_n, y_n \in D$, s.d. $|x_n - y_n| < \frac{1}{n}$ und + $|f(x_n) - f(y_n)| \ge \epsilon_0$. + + Folgenkompakt $\implies$ $\exists$ konvergente Teilfolge + $(x_{n_k})_{k \in \N}$, $x_{n_k} \to p \in D$. + + $k \to \infty$. Dann konvergiert auch $(y_{n_k})_{k\in\N}$ gegen $p$, + d.h. $y_{n_k} \to p, k \to \infty$ (weil $|x_{n_k} - y_{n_k}| < \frac{1}{n_k}$ \\ + $\implies \epsilon_0 \le |f(x_{n_k} - f(y_{n_k})| \to |f(p) - f(p)| = 0$. + Widerspruch +\end{proof} + +\begin{definition}[Lipschitzstetigkeit] + Eine Funktion $f\colon D \to \R$ heißt lipschitz stetig auf $D$, falls + $\exists $ Konstante $L > 0$ (sog. Lipschitzkonstante), s.d. + \[ + |f(x) - f(y)| \le L |x - y| \quad \forall x, y \in D + .\] +\end{definition} + +\begin{bsp}[] + für $x = 3$ nicht lipschitzstetig. +\end{bsp} + +\begin{tikzpicture} + \begin{axis} + \addplot[samples=100, domain=0:6]{-abs(1/(5*(x - 3)))+6}; + \end{axis} +\end{tikzpicture} + +\begin{bem} + Lipschitzstetige Funktionen sind gleichmäßig stetig (stärker + als gleichmäßige Stetigkeit) +\end{bem} + +\subsection{Trigonometrische Funktionen} + +\begin{satz} + Für $x \in \R$ definiere $\cos(x) := \text{Re}(e^{-x})$ und + $\sin(x) := \text{Im}(e^{ix})$. Dann gilt + $\forall x \in \R$. + \begin{enumerate} + \item $e^{ix} = \cos(x) + i \sin(x)$ (Eulersche Formel) + \item $\cos(x) = \frac{1}{2}\left( e^{ix} + e^{-ix} \right) $ \\ + $\sin(x) = \frac{1}{2i}\left( e^{ix} - e^{-ix} \right) $ + \item $\cos(-x) = \cos(x)$ \\ + $\sin(-x) = - \sin(x)$ + \item $\cos^2(x) + \sin^2(x) = 1$ + \end{enumerate} +\end{satz} + +\begin{proof} + trivial. +\end{proof} + +\begin{satz}[$\cos$ und $\sin$ sind stetig] + Restgliedabschätzung von $\exp(x)$ gilt auch für komplexe $z \in \mathbb{C}$ + \[ + (|R_{n+1}(z)| \le 2 \frac{|z|^{N+1}}{(N+1)!} + .\] Damit folgt für eine Nullfolge in $\mathbb{C}$ + ($z_n \to 0, n \to \infty, z_n \in \mathbb{C}$ ) \\ + $\implies \exp(z_n) \to \exp(0) = 1, n \to \infty$ + + Mit Funktionalgleichung $\exp(x\cdot y) = \exp(x) + \exp(y)$ gilt + für eine Folge $(z_n)_{n\in\N}, z_n \to a, n \to \infty$ in $\mathbb{C}$ + $\implies \exp(z_n) \to \exp(a)$. + ($z_n - a \to 0, \exp(z_n - a) \to 1 \implies + \lim_{n \to \infty} \exp(z_n) = \lim_{n \to \infty} + \left(\exp(a) \cdot \exp(z_n - a) \right) = \exp(a)) $ + + Sei $a \in \R$ und $x_n \to a, x_n \in \R$. Dann + $\exp(ix_n) \to \exp(ia)$ mit Re / Im + ($\text{Re}(z_n) \to \text{Re}(a)$, $\text{Im}(z_n) \to \text{Im}(a)$ + , $z_n \to a$ in $\mathbb{C}$. + + $\implies \cos(x_n) \to \cos(a)$ und $\sin(x_n) \to \sin(a)$ \\ + $\implies$ Stetigkeit +\end{satz} + +\begin{satz}[Additionstheoreme] + $\forall x, y \in \R$ gilt: + \begin{enumerate} + \item $\cos(x+y) = \cos x \cdot \cos y - \sin x \cdot \sin y$ \\ + $\sin(x+y) = \sin x \cdot \cos y + \cos x \cdot \sin y$ + \item $\sin x - \sin y = 2 \cos\left( \frac{x+y}{2} \right) + \cdot \sin\left( \frac{x - y}{2} \right) $ \\ + $\cos x - \cos y = - 2 \cdot \sin \left( \frac{x+y}{2} \right) \cdot + \sin \left( \frac{x - \frac{y}{2}}{} \right)$ + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate} + \item \begin{align*} + \cos (x+y) + i \sin (x+y) &= e^{i(x+y)} = e^{ix} \cdot e^{iy} \\ + &= (\cos x + i \sin x)(\cos y + i \sin y) \\ + &= \underbrace{\cos x \cos y - \sin x \sin y}_{\text{Re}} + i \underbrace{(\sin x \cos y + \cos x \sin y)}_{\text{Im}} + .\end{align*} + \item Setze $u := \frac{x+y}{2}, v := \frac{x - y}{2}$. + + $x = u + v, y = u-v$.\\ + \begin{align*} + \sin x - \sin y &= \sin (u+v) - \sin (u - v) \\ + &= \sin u \cdot \cos v + \cos u \cdot \sin v + - (\sin u \underbrace{\cos(-v)}_{= \cos v} + + \cos u \cdot \underbrace{\sin(-v)}_{- \sin v}) \\ + &= 2 \cos u \sin v + = 2 \cos \frac{x+y}{2} \cdot \sin \frac{x - y}{2} + .\end{align*} + \end{enumerate} +\end{proof} + +\end{document} diff --git a/ws2019/la/uebungen/la9.pdf b/ws2019/la/uebungen/la9.pdf index 296a2cbff28a91518f96b0f76bedc9f4c00cc87c..6221aa4f421efcb173d40b33678b178986be7881 100644 GIT binary patch delta 8268 zcmai%Ran&xv&Gq@bc1wA_olmZQhFwc`dY)@6Rv&|x6DFJzZcY)GjT z)6CJ^t%+&IPL`LcW%7FJx0LK z7^p?|QSDW2 z|Na38`uPXl6ud)rk>%0t0(r>_<_CTgdAuN-tlgN#CJ8rPK4GTv%}@f#U<7=dT&H+x zTBvyJVV~AhB7goADO)R;REtXz&6T1Iq5!^`OmHNWt*Y&*nNw`<|&&Wbe?2x+20BF_>?ImBp~ZF$($sVM(21hG{7tt z!SrP&xZ{HRA+?8^M;P^|Po|p#Iuy=sDai26m!V$qrzL|l5ow|*=9?g+K)-_}v0{WW 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