From 8fe976ff88bfc9ac40a3fd049967e5749f6b9e69 Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 22 Apr 2020 12:45:39 +0200 Subject: [PATCH] update ana 1 scripts --- lecture.cls | 2 +- ws2019/ana/lectures/analysis.pdf | Bin 620606 -> 620163 bytes ws2019/ana/lectures/analysis10.tex | 8 +++--- ws2019/ana/lectures/analysis11.tex | 14 +++++---- ws2019/ana/lectures/analysis13.tex | 4 +-- ws2019/ana/lectures/analysis14.tex | 10 +++---- ws2019/ana/lectures/analysis15.tex | 9 +++--- ws2019/ana/lectures/analysis16.tex | 10 +++---- ws2019/ana/lectures/analysis18.tex | 4 +-- ws2019/ana/lectures/analysis19.tex | 11 ++++---- ws2019/ana/lectures/analysis20.tex | 10 +++---- ws2019/ana/lectures/analysis21.tex | 2 +- ws2019/ana/lectures/analysis22.tex | 20 ++++++------- ws2019/ana/lectures/analysis23.tex | 8 +++--- ws2019/ana/lectures/analysis24.tex | 12 ++++---- ws2019/ana/lectures/analysis3.tex | 25 ++++++++-------- ws2019/ana/lectures/analysis4.tex | 44 ++++++++++++++--------------- ws2019/ana/lectures/analysis5.tex | 24 ++++++++-------- 18 files changed, 109 insertions(+), 108 deletions(-) diff --git a/lecture.cls b/lecture.cls index cf7c3de..5f51916 100644 --- a/lecture.cls +++ b/lecture.cls @@ -14,7 +14,7 @@ \RequirePackage{transparent} \RequirePackage{xcolor} \RequirePackage{array} -\RequirePackage{enumerate} +\RequirePackage[shortlabels]{enumitem} \RequirePackage{tikz} \RequirePackage{pgfplots} \RequirePackage[nobottomtitles]{titlesec} diff --git a/ws2019/ana/lectures/analysis.pdf b/ws2019/ana/lectures/analysis.pdf index e9945432e11fd71e70041fa4ded29e487c32b069..c028b00ebe544c5c1439732db5b727afdd811433 100644 GIT binary patch delta 266613 zcmV)1K+V6t@Fj!oC9sh?0yr>}@Cht`bK6D|e%G&n-@uAFp8HVB5B6GioK3kLTiM!* zl?N(>X{<;nOH%fhe*Jc1KmY>{!BbvuDla6CK~GN~Umr71F5kUn$z*XxlSx98S7*;J z&z`(gDsjYVu9M4;iR47_#4TE-ughea2ueny?jZ3bRp}{60JT z@Ec9W@ao5e65(v}r5rd*q;NzSxSS^M&wj4C4Y<~Hjca2R;X-=0fk~ZCm)X1+4TdaT z4p^Gw#b>y<9w?S(U$T5L7z$;7(^RAV9PC#c|EJ+EqA@{i={OH!YN1*+ 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= 1,2 \text{ gilt} + \] \begin{enumerate}[(a)] \item $\overline{z} = r \cdot e^{-i\varphi}$ \item $z_1 \cdot z_2 = r_1\cdot r_2 \cdot e^{i(\varphi_1 + \varphi_2)}$ @@ -101,8 +101,8 @@ tolle Grafik von der Kostina, die gibt's nur in der Vollversion. \begin{align*} &z^{n} = \rho^{n} e^{i n \psi} \stackrel{!}{=} r e^{i \varphi} = w \\ - \iff &\rho^{n} = r \text{ und } n \psi = \varphi + 2 \pi k, k \in \Z \\ - \iff &\varrho = \sqrt[n]{r} \text{ und } \psi = \frac{1}{n}(\varphi + 2 \pi k) k \in \Z + \iff &\rho^{n} = r \text{ und } n \psi = \varphi + 2 \pi k \quad k \in \Z \\ + \iff &\rho = \sqrt[n]{r} \text{ und } \psi = \frac{1}{n}(\varphi + 2 \pi k) \quad k \in \Z .\end{align*} Betrachte: $z_k = \sqrt[n]{r} e^{i\psi_k}, \psi_k = \frac{1}{n}\left( \varphi + 2 \pi k \right), k = 0, 1, \ldots, n-1$ diff --git a/ws2019/ana/lectures/analysis11.tex b/ws2019/ana/lectures/analysis11.tex index aaba798..378e353 100644 --- a/ws2019/ana/lectures/analysis11.tex +++ b/ws2019/ana/lectures/analysis11.tex @@ -4,17 +4,19 @@ \section{Folgen und Reihen} \subsection{Folgen} -Eine Zahlenfolge ist eine Abbildung von $\N$ nach $\R$, d.h. $n \mapsto a_n \in \R$. +\begin{definition}[Folgen] + Eine Zahlenfolge ist eine Abbildung von $\N$ nach $\R$, d.h. $n \mapsto a_n \in \R$. -Teilfolge: $(a_{n_k})_{k \in\N}$ von $(a_n)_{n\in\N}$ wobei $(n_k)_{k \in \N}$ eine Folge natürlicher -Zahlen, weile streng monoton wächst, d.h. $n_1 < n_2 < \ldots.$ bzw. $n_k < n_{k+1} \forall k \in \N$ + Die Folge $(a_{n_k})_{k \in\N}$ ist eine Teilfolge von $(a_n)_{n\in\N}$ + wobei $(n_k)_{k \in \N}$ eine Folge natürlicher + Zahlen ist, die streng monoton wächst, d.h. $n_1 < n_2 < \ldots.$ bzw. $n_k < n_{k+1} \forall k \in \N$. +\end{definition} \begin{bsp} $(-1)^{n}$ hat zwei Teilfolgen: $(-1)^{2n} = 1$ und $(-1)^{2n+1} = -1$. \end{bsp} \begin{definition}[Konvergenz, Beschränktheit, Monotonie von Folgen] - \begin{enumerate} \item Eine Folge $(a_n)_{n\in\N} \in \R$ heißt \textit{beschränkt}, wenn es eine Konstante $c \in \R$ gibt mit $|a_n| \le C$. @@ -50,10 +52,10 @@ Zahlen, weile streng monoton wächst, d.h. $n_1 < n_2 < \ldots.$ bzw. $n_k < n_{ \begin{proof} Angenommen $a \neq a'$. Definiere $\epsilon := \frac{|a - a'|}{2} > 0$. - Dann $\exists n_1,n 2 \in \N$ mit $|a_n - a| < \epsilon \quad \forall n \ge n_1$ + Dann $\exists n_1,n_2 \in \N$ mit $|a_n - a| < \epsilon \quad \forall n \ge n_1$ und $|a_n - a'| < \epsilon \quad \forall n \ge n_2$. - Dann $\forall n \ge \text{max}\{n_1, n_2\}$ gilt: + Dann gilt $\forall n \ge \text{max}\{n_1, n_2\}$: \begin{align*} |a - a'| = |a - a_n + a_n -a'| \le |a-a_n| + |a_n - a'| < \epsilon + \epsilon = | a - a'| .\end{align*} diff --git a/ws2019/ana/lectures/analysis13.tex b/ws2019/ana/lectures/analysis13.tex index 5c1058b..1067946 100644 --- a/ws2019/ana/lectures/analysis13.tex +++ b/ws2019/ana/lectures/analysis13.tex @@ -14,7 +14,7 @@ Aus Definitionen: |z| := \sqrt{(\text{Re}(z))^{2} + (\text{Im}(z))^{2}} .\] und der Ungleichung: \[ - max(|x|, |y|) \le \sqrt{x^2 + y^2} \le |x| + |y| \quad \forall x,y \in \R + \max(|x|, |y|) \le \sqrt{x^2 + y^2} \le |x| + |y| \quad \forall x,y \in \R .\] folgt: \begin{enumerate} @@ -73,7 +73,7 @@ Aus Definitionen: Folge der Partialsummen \[ s_n = \sum_{k=0}^{n} q^{k} = \begin{cases} - \frac{1 - q^{n+1}}{1 - q} \quad q \neq 1 & q \neq 1 \\ + \frac{1 - q^{n+1}}{1 - q} & q \neq 1 \\ n + 1 & q = 1 \end{cases} .\] diff --git a/ws2019/ana/lectures/analysis14.tex b/ws2019/ana/lectures/analysis14.tex index 76e8576..6a720c0 100644 --- a/ws2019/ana/lectures/analysis14.tex +++ b/ws2019/ana/lectures/analysis14.tex @@ -163,7 +163,7 @@ Kostina glaubt, dass das so stimmt, aber offensichtlich ist sie sich nicht siche \subsection{Umordnen von Reihen} \begin{definition}[Umordnung] - Sei $\sum_{n=1}^{\infty} a_n$ eine Reihe und $\tau: N \to N$ eine + Sei $\sum_{n=1}^{\infty} a_n$ eine Reihe und $\tau: \N \to \N$ eine bijektive Abbildung. Dann heißt $\sum_{n=1}^{\infty} a_{\tau(n)} = @@ -214,10 +214,10 @@ Kostina glaubt, dass das so stimmt, aber offensichtlich ist sie sich nicht siche absolut konvergente Reihen. Für $n \in \N_0$ sei $c_n$ definiert durch \[ - c_n := \sum_{k=0}^{\infty} a_k b_{n-k} = a_0b_n + a_1b_{n-1} + c_n := \sum_{k=0}^{n} a_k b_{n-k} = a_0b_n + a_1b_{n-1} + \ldots + a_nb_0 - .\] Dann ist die Reihe $\sum_{n=0}^{\infty} c_n$ absolut konvergent - mit $\sum_{k=0}^{\infty} c_n = \left( \sum_{n=0}^{\infty} a_n \right) + .\] Dann ist die Reihe $\sum_{k=0}^{\infty} c_k$ absolut konvergent + mit $\sum_{k=0}^{\infty} c_k = \left( \sum_{n=0}^{\infty} a_n \right) \left( \sum_{n=0}^{\infty} b_n \right)$ \end{satz} @@ -317,7 +317,7 @@ Kostina glaubt, dass das so stimmt, aber offensichtlich ist sie sich nicht siche Zur Potenzreihe $\sum_{k=1}^{\infty} a_k(z-z_0)^{k}$ definiere den Konvergenzradius $\rho$ durch \[ - \rho := \frac{1}{\lim_{n \to \infty} \text{sup } \sqrt[n]{|a_n|} } + \rho := \frac{1}{\lim \sup_{n \to \infty} \sqrt[n]{|a_n|} } .\] \end{definition} diff --git a/ws2019/ana/lectures/analysis15.tex b/ws2019/ana/lectures/analysis15.tex index 9b3859c..dad1fa6 100644 --- a/ws2019/ana/lectures/analysis15.tex +++ b/ws2019/ana/lectures/analysis15.tex @@ -15,7 +15,7 @@ (f+g)(x) &:= f(x) + g(x) \\ (f-g)(x) &:= f(x) - g(x) \\ (f\cdot g)(x) &:= f(x) \cdot g(x) \\ - \left(\frac{f}{g}\right)(x) &:= \frac{f(x)}{g(x)}\\ + \left(\frac{f}{g}\right)(x) &:= \frac{f(x)}{g(x)} .\end{align*} \end{definition} @@ -98,9 +98,10 @@ H(x) := \begin{cases} 1 & x > 0 \\ \frac{1}{2} & x = 0 \\ - 0, x < 0 + 0 & x < 0 \end{cases} - .\] Für $x_0 > 0$ gilt $\lim_{x \to x_0} H(x) = 1$. + .\] + Für $x_0 > 0$ gilt $\lim_{x \to x_0} H(x) = 1$. \begin{proof} Sei $\epsilon > 0$, wähle $\delta := \frac{x_0}{2} > 0$. Dann gilt \[ @@ -116,7 +117,7 @@ Dann wählen wir $\epsilon = \frac{1}{4}$, dann ex. $\delta > 0$ mit \[ |H(x) - y_0| < \frac{1}{4} \quad \forall x \text{ mit } x \in \;]-\delta, \delta[ - .\] $\implies 1 = |H(-\delta)| - H(\delta)| \le |H(-\delta) - y_0| + |y_0 - H(\delta)| < \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ Widerspruch! \\ + .\] $\implies 1 = |H(-\delta) - H(\delta)| \le |H(-\delta) - y_0| + |y_0 - H(\delta)| < \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ Widerspruch! \\ $\implies \lim_{x \to 0} H(x) $ existiert nicht. \end{proof} diff --git a/ws2019/ana/lectures/analysis16.tex b/ws2019/ana/lectures/analysis16.tex index 6f849b3..27e19d8 100644 --- a/ws2019/ana/lectures/analysis16.tex +++ b/ws2019/ana/lectures/analysis16.tex @@ -116,7 +116,7 @@ in dieser Umgebung. 0 \le x_n \le 1 \; \forall n \stackrel{\text{Sandwich}}{\implies} 0 \le x_0 \le 1 \implies x_0 \in [0,1] \quad \text{abgeschlossen} .\] - $[0,1]$ ist nicht offen, da $0 \in [0,1]$, aber $\underbrace{]-r, r[}_{B_r(0) \subset [0,1]} \quad \forall r > 0$ + $[0,1]$ ist nicht offen, da $0 \in [0,1]$, aber $\underbrace{]-r, r[}_{B_r(0) \not\subset [0,1]} \quad \forall r > 0$ \item $\R$ ist offen, abgeschlossen aber nicht kompakt. \end{enumerate} \end{bsp} @@ -167,15 +167,15 @@ in dieser Umgebung. Sei $f \colon D \to \R$, $D \subset \R$. \begin{align*} - \operatorname{sup}_{x \in D} f(x) \quad &\text{kleinste obere Grenze der Bildmenge } B_f := \{f(x) \mid x \in D\} \\ - & := \text{sup } B_f := \text{min}\{\beta \in \R \mid y \le \beta \; \forall y \in B_f \} + \sup_{x \in D} f(x) \quad &\text{kleinste obere Grenze der Bildmenge } B_f := \{f(x) \mid x \in D\} \\ + & := \sup B_f := \text{min}\{\beta \in \R \mid y \le \beta \; \forall y \in B_f \} .\end{align*} \begin{align*} - \operatorname{inf}_{x \in D} f(x) \; := \text{inf } B_f := \text{min}\{\beta \in \R \mid y \le \beta \; \forall y \in B_f\} + \inf_{x \in D} f(x) \; := \text{inf } B_f := \text{min}\{\beta \in \R \mid y \le \beta \; \forall y \in B_f\} .\end{align*} Falls $B_f := f(D)$ beschränkt ist, dann $\exists $ inf und sup. - $x_min \in D$ heißt Minimum, $x_max$ Maximum von $f$, falls + $x_{\text{min}} \in D$ heißt Minimum, $x_{\text{max}}$ Maximum von $f$, falls \[ \begin{cases} \text{inf } f(x) = f(x_{min}) =: \text{min } f(x) \\ diff --git a/ws2019/ana/lectures/analysis18.tex b/ws2019/ana/lectures/analysis18.tex index 2f91bec..2dbe9f0 100644 --- a/ws2019/ana/lectures/analysis18.tex +++ b/ws2019/ana/lectures/analysis18.tex @@ -83,7 +83,7 @@ $k \to \infty$. Dann konvergiert auch $(y_{n_k})_{k\in\N}$ gegen $p$, d.h. $y_{n_k} \to p, k \to \infty$ (weil $|x_{n_k} - y_{n_k}| < \frac{1}{n_k}$ \\ - $\implies \epsilon_0 \le |f(x_{n_k} - f(y_{n_k})| \to |f(p) - f(p)| = 0$. + $\implies \epsilon_0 \le |f(x_{n_k}) - f(y_{n_k})| \to |f(p) - f(p)| = 0$. Widerspruch \end{proof} @@ -113,7 +113,7 @@ \subsection{Trigonometrische Funktionen} \begin{satz} - Für $x \in \R$ definiere $\cos(x) := \text{Re}(e^{-x})$ und + Für $x \in \R$ definiere $\cos(x) := \text{Re}(e^{ix})$ und $\sin(x) := \text{Im}(e^{ix})$. Dann gilt $\forall x \in \R$. \begin{enumerate} diff --git a/ws2019/ana/lectures/analysis19.tex b/ws2019/ana/lectures/analysis19.tex index a01dffb..0852b02 100644 --- a/ws2019/ana/lectures/analysis19.tex +++ b/ws2019/ana/lectures/analysis19.tex @@ -202,21 +202,21 @@ Ziel: Analytische Definition von $\pi \in \R$. \begin{enumerate}[(i)] \item Die Tangensfunktion \begin{align*} - &\tan: \R \setminus \{x = (k + \frac{1}{2}) \pi | k \in \Z\} + &\tan: \R \setminus \left\{x = \left(k + \frac{1}{2}\right) \pi \mid k \in \Z\right\} \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 + &\cot: \R \setminus \{x = k \pi \mid 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] +\begin{figure}[h!] \centering \begin{tikzpicture} \begin{axis}% @@ -235,7 +235,7 @@ Ziel: Analytische Definition von $\pi \in \R$. \caption{$\tan(x)$} \end{figure} -\begin{figure}[htpb] +\begin{figure}[h!] \centering \begin{tikzpicture} \begin{axis}% @@ -254,8 +254,7 @@ Ziel: Analytische Definition von $\pi \in \R$. \caption{$\cot(x)$} \end{figure} -\begin{definition}[Arcusfunktionen (Umkehrfunktionen der Trigonometrischen - Funktionen)] +\begin{definition}[Arcusfunktionen] \begin{enumerate}[(i)] \item $\cos\colon [0, \pi] \to [-1, 1]$ ist streng monoton fallend und bijektiv. Die Umkehrfunktion heißt diff --git a/ws2019/ana/lectures/analysis20.tex b/ws2019/ana/lectures/analysis20.tex index fa3948c..9bdc254 100644 --- a/ws2019/ana/lectures/analysis20.tex +++ b/ws2019/ana/lectures/analysis20.tex @@ -152,7 +152,7 @@ \[ f(x) - f(x_0) = f'(x_0)\underbrace{(x-x_0)}_{\to 0} + \underbrace{\frac{R(x)}{x -x_0}}_{\to 0}\underbrace{(x-x_0)}_{\to 0} - .\] d.h. $f(x) \to f(x_0) \stackrel{\text{Def. Stetigkeit}}{\implies} f$ stetig. + .\] d.h. $f(x) \to f(x_0) \qquad \stackrel{\text{Def. Stetigkeit}}{\implies} \qquad f$ stetig. \end{proof} \begin{bem} @@ -173,8 +173,8 @@ stetig differenzierbar mit $f'(x) = n x^{n-1}$ $\forall x$, weil \begin{align*} \lim_{h \to 0} \frac{(x+h)^{n} - x^{n}}{h} - &\stackrel{a^{n} - b^{n} = \ldots}{=} - \lim_{h \to 0} \frac{(x+h-x)(x+h)^{n-1}+(x+h)^{n-2}\cdot x + \ldots + (x-h)x^{n-2} + x^{n-1}}{h} \\ + &= + \lim_{h \to 0} \frac{(x+h-x)(x+h)^{n-1}+(x+h)^{n-2}\cdot x + \ldots + (x+h)x^{n-2} + x^{n-1}}{h} \\ &= \underbrace{x^{n-1} + x^{n-2} \cdot x + \ldots + x^{n-1}}_{n\text{-mal}} \\ &= n x^{n-1} .\end{align*} @@ -222,7 +222,7 @@ mit \begin{align*} e^{h} &= \sum_{k=0}^{\infty} \frac{h^{k}}{k!} = 1 + h + \frac{h^2}{2} + \frac{h^{3}}{3!} + \ldots \\ - \frac{e^{h} - 1}{h} &= 1 + \frac{h}{2} + \frac{h^2}{3!} + \ldots \xrightarrow{h \to 0} \to 1 + \frac{e^{h} - 1}{h} &= 1 + \frac{h}{2} + \frac{h^2}{3!} + \ldots \xrightarrow{h \to 0} 1 .\end{align*} \item Sinus / Cosinus. mit $\sin(x) - \sin(y) = 2 \cos \frac{1}{2} (x+y) \cdot \sin \frac{1}{2}(x-y)$ folgt @@ -335,7 +335,7 @@ \begin{satz}[Kettenregel] Seien $f\colon D_f \to \R$, $g\colon D_g \to \R$ stetige Funktionen. - $f \in x_0 \in D_f$ differenzierbar, $g \in y_0 = f(x_0) \in D_g$ + $f$ in $x_0 \in D_f$ differenzierbar, $g$ in $y_0 = f(x_0) \in D_g$ differenzierbar. Dann ist $(g \circ f): D_f \to \R$ differenzierbar in $x_0$ und es gilt die Kettenregel \[ diff --git a/ws2019/ana/lectures/analysis21.tex b/ws2019/ana/lectures/analysis21.tex index b5b55b7..13834d5 100644 --- a/ws2019/ana/lectures/analysis21.tex +++ b/ws2019/ana/lectures/analysis21.tex @@ -7,7 +7,7 @@ \begin{definition}[globales / lokales Extremum] Die Funktion $f\colon D \to \R$ hat in $x_0 \in D$ ein globales Extremum (Maximum oder Minimum), falls - gilt $f(x_0) \ge f(x)$ bzw. $f(x_0 \le f(x)$ $\forall x \in D$. + gilt $f(x_0) \ge f(x)$ bzw. $f(x_0) \le f(x)$ $\forall x \in D$. Die Funktion $f$ hat in $x_0 \in D$ ein lokales Extremum (Maximum oder Minimum), falls diff --git a/ws2019/ana/lectures/analysis22.tex b/ws2019/ana/lectures/analysis22.tex index e016c56..52873ab 100644 --- a/ws2019/ana/lectures/analysis22.tex +++ b/ws2019/ana/lectures/analysis22.tex @@ -26,18 +26,19 @@ Für $x = x_0$: klar. Sei $x \neq x_0$. Betrachte $R = R(x, x_0)$ definiert durch - $f(x) = T_N(f, x, x_0) + \frac{(x-x_0)^{n+1}}{(n+1)!}\cdot R$. + $f(x) = T_n(f, x, x_0) + \frac{(x-x_0)^{n+1}}{(n+1)!}\cdot R$. Für $y \in D$ definiere \[ \varphi(y) := f(x) - \sum_{k=0}^{n} \frac{f^{(k)}(y)}{k!}(x-y)^{k} - \frac{(x-y)^{n+1}}{(n+1)!}R .\] Dann folgt $\varphi(x_0) = 0 = \varphi(x)$, $\varphi \in C^{1}$.\\ - $\stackrel{\text{Satz von Rolle}}{\implies}$ $\exists \xi$ zwischen $x$ und $x_0$ mit $\varphi'(\xi) = 0$. + $\stackrel{\text{Satz von Rolle}}{\implies} \qquad$ + $\exists \xi$ zwischen $x$ und $x_0$ mit $\varphi'(\xi) = 0$. \begin{align*} 0 = \varphi'(\xi) &= - \sum_{k=0}^{n} \frac{f^{(k+1)}(\xi)}{k!} (x-\xi)^{k} - \sum_{k=0}^{n} \frac{f^{(k)}(\xi)}{k!}k(x-\xi)^{k-1}(-1) - - \frac{(n+1)(x-\xi)^{n}}{(n+1)!} R \\ + + \frac{(n+1)(x-\xi)^{n}}{(n+1)!} R \\ &\stackrel{\mathclap{\text{Teleskop}}}{=} \quad - \frac{f^{n+1}(\xi)}{n!}(x-\xi)^{n} + \frac{(x-\xi)^{n}}{n!} R \\ &= \frac{(x-\xi)^{n}}{n!} \left( - f^{(n+1)}(\xi) + R \right) .\end{align*} $\implies R = f^{(n+1)}(\xi)$, $\xi$ zwischen $x$ und $x_0$. @@ -59,7 +60,7 @@ \begin{proof} \[ |f(x) - T_n(f, x, x_0)| \le \frac{\left| f^{(n+1)}(\xi)\right|}{(n+1)!}|x-x_0|^{n+1} - \le \frac{M}{(n+1)!(b-a)^{n+1}} + \le \frac{M}{(n+1)!}(b-a)^{n+1} .\] $\implies \forall \epsilon > 0$ $\exists n_{\epsilon}$, s.d. $\frac{M}{(n+1)!}(b-a)^{n+1} < \epsilon$ \end{proof} @@ -78,7 +79,7 @@ Sei $f \in C^{n}(D, \R)$, $D = (a,b)$ und für $x_0 \in (a,b)$ gelte \[ - f'(x_0) = f''(x_0) = \ldots = f^{(n-1)}(x_0) \quad \text{und} \quad f^{(n)}(x_0) \neq 0 + f'(x_0) = f''(x_0) = \ldots = f^{(n-1)}(x_0) = 0 \quad \text{und} \quad f^{(n)}(x_0) \neq 0 .\] Dann gilt \begin{enumerate} \item Ist $n$ gerade und $f^{(n)}(x_0) < 0$ bzw. $f^{(n)}(x_0) > 0$, dann ist @@ -88,7 +89,7 @@ \end{enumerate} \end{korrolar} -\begin{figure} +\begin{figure}[h!] \centering \begin{subfigure}{.4\textwidth} \caption{$f(x) = x^3$} @@ -129,7 +130,7 @@ \end{figure} \begin{proof} - Taylor Satz $\implies$ $f(x) = f(x_0) + \frac{f^{(n)}(\xi)}{n!} (x-x_0)^{n}$ + Satz von Taylor $\implies$ $f(x) = f(x_0) + \frac{f^{(n)}(\xi)}{n!} (x-x_0)^{n}$ $f^{(n)}$ stetig in $x_0$, $f^{(n)} \neq 0$ in $x_0$ \\ $\implies \exists \delta > 0$, s.d. $f^{(n)}(x) \neq 0$ für $x \in \; ]x_0 - \delta , x_0 + \delta [$ und hat das gleiche @@ -139,9 +140,8 @@ $x \neq x_0$. \[ f(x) = f(x_0) + \frac{f^{(n)}(\xi)}{n!}\underbrace{(x-x_0)^n}_{> 0} - .\] $\implies$ $f(x) > f(x_0)$, falls $f^{(n)}(\xi) > 0$, dann - $f^{(n)}(x_0) > 0$. - $f(x) < f(x_0)$, falls $f^{(n)}(x_0) < 0$. + .\] $\implies$ Wegen $f^{(n)}(x_0) = f^{(n)}(\xi)$ folgt $f(x) > f(x_0)$, falls $f^{(n)}(x_0) > 0$ + und $f(x) < f(x_0)$, falls $f^{(n)}(x_0) < 0$. \item $n$ ungerade, wechselt $(x-x_0)^{n}$ das Vorzeichen. \end{enumerate} \end{proof} diff --git a/ws2019/ana/lectures/analysis23.tex b/ws2019/ana/lectures/analysis23.tex index f6e8398..1ce0e06 100644 --- a/ws2019/ana/lectures/analysis23.tex +++ b/ws2019/ana/lectures/analysis23.tex @@ -16,7 +16,7 @@ berechnen. \subsection{Riemannintegral} -\begin{definition}[Zerlegungen] +\begin{definition}[Zerlegungen, Stützpunkte] Eine endliche Zerlegung $Z$ von einem (beschränkten) Intervall $[a,b]$ ist eine endliche Folge $z = (x_0, x_1, \ldots, x_n)$ mit $x_0 = a < x_1 < \ldots < x_n = b$. $x_k$ heißen Teilungs- oder @@ -45,7 +45,7 @@ berechnen. \begin{bem} Eine Verfeinerung der Zerlegung $Z$ ist eine - Zerlegung $Z'' = (x_0'', \ldots, x''_{n''}$ + Zerlegung $Z'' = (x_0'', \ldots, x''_{n''})$ s.d. $(x_0, \ldots, x_n) \subset (x_0'', \ldots, x''_{n''})$ und $h'' \le h$. Zu Zerlegungen $Z = (x_0, \ldots, x_n)$ und $Z' = (x_0', \ldots, x'_{n'})$ gibt es eine @@ -83,10 +83,10 @@ berechnen. Ober- und Unterintegral und für jede Folge von Zerlegungen $Z_n \in \mathcal{Z}(a,b)$, $n \in \N$ mit $h_n \xrightarrow{n \to \infty} 0$ gilt \[ - \lim_{n \to \infty} \underline{S}_{Z_n} + \lim_{n \to \infty} \underline{S}_{Z_n}(f) = \underline{\int_{a}^{b} } f(x) dx \le \overline{\int_{a}^{b}} f(x) dx - = \lim_{n \to \infty} f(x) \overline{S}_{Z_n} + = \lim_{n \to \infty} \overline{S}_{Z_n}(f) .\] \end{lemma} diff --git a/ws2019/ana/lectures/analysis24.tex b/ws2019/ana/lectures/analysis24.tex index 2880d84..c752a36 100644 --- a/ws2019/ana/lectures/analysis24.tex +++ b/ws2019/ana/lectures/analysis24.tex @@ -57,7 +57,7 @@ \lim_{h \to 0} (\beta \cdot RS_Z(g)) \\ &= \lim_{h \to 0} RS_Z(\alpha f) + \lim_{h \to 0} RS_Z(\beta g) \\ &= \lim_{h \to 0} RS_Z(\alpha f + \beta g) \\ - &= \int_{a}^{b} (\alpha f + \beta g)(x)dx \\ + &= \int_{a}^{b} (\alpha f + \beta g)(x)dx .\end{align*} \end{proof} @@ -91,7 +91,7 @@ $g \equiv 1 \implies \int_{a}^{b} 1 dx = (b - a)$ Damit folgt \begin{align*} m (b-a) = \int_{a}^{b} m \cdot dx - \stackrel{\text{Satz \ref{satz:riemann-monoton}}}{\le} + \quad \stackrel{\text{Satz \ref{satz:riemann-monoton}}}{\le} \quad \int_{a}^{b} f(x) dx \le \int_{a}^{b} M dx = M (b-a) @@ -296,17 +296,17 @@ Für $h \neq 0$, $x \in [a,b]$ \begin{align*} \frac{F_0(x+h) - F_0(x)}{h} - = \frac{1}{h} \left( \int_{a}^{x+h} f(t) dt + &= \frac{1}{h} \left( \int_{a}^{x+h} f(t) dt - \int_{a}^{x} f(t) dt \right) = \frac{1}{h} \int_{x}^{x+h} f(t) dt \quad \stackrel{\text{1. MWS}}{=} \quad \frac{1}{h} f(\xi_h) \cdot h \xrightarrow[h \to 0]{f\text{ stetig}} f(x) \\ - \implies F_0'(x) = \lim_{h \to 0} \frac{F_0(x+h) - F_0(x)}{h} = f(x) + \implies F_0'(x) &= \lim_{h \to 0} \frac{F_0(x+h) - F_0(x)}{h} = f(x) .\end{align*} - \item Sei $F$ eine Stammfunktion ovn $f$, dann gilt + \item Sei $F$ eine Stammfunktion von $f$, dann gilt $(F - F_0)' = f - f = 0 - \stackrel{\text{MWS Diff.}}{\implies} F - F_0 \equiv + \quad \stackrel{\text{MWS Diff.}}{\implies} \quad F - F_0 \equiv \text{konstant} = C \implies F = F_0 + C$ für ein $C \in \R$. \item $F(b) - F(a) \stackrel{\text{(b)}}{=} F_0(b) - F_0(a) = \int_{a}^{b} f(t) dt $ diff --git a/ws2019/ana/lectures/analysis3.tex b/ws2019/ana/lectures/analysis3.tex index 092842f..9e9921f 100644 --- a/ws2019/ana/lectures/analysis3.tex +++ b/ws2019/ana/lectures/analysis3.tex @@ -60,10 +60,11 @@ \begin{proof}[Alternativbeweis mit Teleskop-Summe] \begin{align*} - 1-x^{n+1} &= 1 - x + x - x^2 + x^2 - \ldots - x^n + x^n - x^n+1 \\ + 1-x^{n+1} &= 1 - x + x - x^2 + x^2 - \ldots - x^n + x^n - x^{n+1} \\ &= \sum_{k=0}^{n} x^{k} - \sum_{k=1}^{n+1} x^{k} \\ &= \sum_{k=0}^{n} x^{k} - \sum_{k=0}^{n} x^{k+1} \\ - &= \sum_{k=0}^{n} x^{k} - x \\ + &= \sum_{k=0}^{n} x^{k} - x \sum_{k=0}^{n} x^{k} \\ + &= (1-x) \sum_{k=0}^{n} x^{k} .\end{align*} \end{proof} @@ -79,13 +80,11 @@ $ \forall a, b \in \R$ und $n \in \N$ gilt: \begin{proof} Für $a=0$ und $a = b$ stimmt die Formel offenbar.\\ Betrachte geometrische Reihe mit $x := \frac{b}{a} \neq 1$ - \[ - 1 - \left(\frac{b}{a}\right)^n = 1 - x^n = (1 - x) \sum_{k=0}^{n-1} x^k - = (1 - \frac{b}{a}) \sum_{k=0}^{n-1} \left(\frac{b}{a}\right)^k - \] - \[ - a^n - b^n = (a-b) \sum_{k=0}^{n-1} b^k a^{-k} a^{n-1} = (a - b) \sum_{k=0}^{n-1} a^{n-1-k} b^k - \] + \begin{align*} + &1 - \left(\frac{b}{a}\right)^n = 1 - x^n = (1 - x) \sum_{k=0}^{n-1} x^k + = (1 - \frac{b}{a}) \sum_{k=0}^{n-1} \left(\frac{b}{a}\right)^k \quad \Big| \cdot a^{n}\\ + \implies &a^n - b^n = (a-b) \sum_{k=0}^{n-1} b^k a^{-k} a^{n-1} = (a - b) \sum_{k=0}^{n-1} a^{n-1-k} b^k + .\end{align*} \end{proof} \subsection{Elemente der Kombinatorik} @@ -113,16 +112,16 @@ Für $n \in \N$ ist die Fakultät $n!$ rekursiv definiert durch: \end{proof} \begin{definition}[Binomialkoeffizient] - Für $n, k \in \N$ definieren wir:\\ + Für $n, k \in \N_0$ definieren wir:\\ \begin{align*} - n \ge k \ge 1:& \binom{n}{k} := \frac{n(n-1) \ldots (n -k + 1)}{k!} \\ - k = 0:& \binom{n}{0} := 1 + n \ge k \ge 1\colon & \binom{n}{k} := \frac{n(n-1) \ldots (n -k + 1)}{k!} \\ + k = 0\colon & \binom{n}{0} := 1 \end{align*} $\binom{n}{k}$ ist die Anzahl der k-Elementigen Teilmengen einer n-Elementigen Menge, z.B.: Lotto $\binom{49}{6} = 13.983.816$. \begin{align*} \binom{n}{k} &= \frac{n(n-1) \ldots (n - k +1)}{k!}\\ &= \frac{n(n-1) \ldots (n-k+1)(n-k)!}{k!(n-k)!}\\ - &= \frac{n!}{k!(n-k)!} = \binom{n}{n-k}\\ + &= \frac{n!}{k!(n-k)!} = \binom{n}{n-k} .\end{align*} Es folgt $\binom{n}{0} = 1$, $\binom{n}{n} = 1$, $\binom{n}{1} = \binom{n}{n-1} = n, \forall n \in \N.$ diff --git a/ws2019/ana/lectures/analysis4.tex b/ws2019/ana/lectures/analysis4.tex index bd02473..5a349df 100644 --- a/ws2019/ana/lectures/analysis4.tex +++ b/ws2019/ana/lectures/analysis4.tex @@ -56,7 +56,7 @@ \binom{n}{n-1}a b^{n-1} + \binom{n}{n}b^{n}\right) \\ &= \binom{n}{0}a^{n+1}+\binom{n}{1}a^{n}b + \ldots + \binom{n}{n-1} a^{2} b^{n-1} + \binom{n}{n} a b^{n} \\ - &+ \binom{n}{0} a^{n}b + \binom{n}{1} a^{n-1}b^{2} + \ldots + + & \quad \; + \binom{n}{0} a^{n}b + \binom{n}{1} a^{n-1}b^{2} + \ldots + \binom{n}{n-1}a b^{n}+\binom{n}{n}b^{n+1} \\ &= \binom{n+1}{0}a^{n+1}+\binom{n+1}{1}a^{n}b+\ldots+ \binom{n+1}{n}ab^{n} + \binom{n+1}{n+1}b^{n+1} @@ -66,13 +66,13 @@ \subsection{Grundlegendes über Zahlenmengen} \[ \N = \{1, 2, 3, \ldots\} \text{ natürliche Zahlen} -.\] auf $\N$ sind die arithmetischen Operationen ,,$+$'' (Addition) und ,,$\cdot$'' +.\] Auf $\N$ sind die arithmetischen Operationen ,,$+$'' (Addition) und ,,$\cdot$'' (Multiplikation) definiert. Für diese gelten u.a. die Regeln: -\begin{align*} - n + m = m + n &\text{ bzw. } n \cdot m = m \cdot n \text{ Kommutativität} \\ - (n + m) + k = n + (m + k) &\text{ bzw. } (n \cdot m) \cdot k = n \cdot (m \cdot k) \text{ Assoziativität}\\ - (n + m) \cdot k &= n \cdot k + m \cdot k \text{ Distributivität} -\end{align*} +\begin{flalign*} + &n + m = m + n \text{ bzw. } n \cdot m = m \cdot n &\text{ Kommutativität} \\ + &(n + m) + k = n + (m + k) \text{ bzw. } (n \cdot m) \cdot k = n \cdot (m \cdot k) &\text{ Assoziativität}\\ + &(n + m) \cdot k = n \cdot k + m \cdot k &\text{ Distributivität} +\end{flalign*} Subtraktion und Division sind nicht für alle Paare der natürlichen Zahlen definiert ($1 - 2 \not\in \N, \frac{1}{2} \not\in \N$), d.h. die natürlichen Zahlen @@ -89,8 +89,8 @@ Deshalb werden die natürlichen Zahlen zu den ganzen Zahlen erweitert. $x = m - n \in \Z$. $\Z$ ist vollständig bezüglich der Subtraktion aber unvollständig bezüglich -der Division, d.h. für beliebige $b, y \in \Z$, $b \neq 0$, die ,,lineare'' -Gleichung $b \cdot x = y$ nicht immer durch ein $x \in \Z$ lösbar ist. +der Division, d.h. für beliebige $b, y \in \Z$, $b \neq 0$, ist die ,,lineare'' +Gleichung $b \cdot x = y$ nicht immer durch ein $x \in \Z$ lösbar. Diese Beschränkung wird durch die Einführung der rationalen Zahlen behoben: \[ @@ -105,29 +105,29 @@ a = \frac{r}{s}, b = \frac{u}{v} \in \Q = \begin{cases} a \cdot b &:= \frac{r \cdot u}{s \cdot v} \\ \frac{a}{b} &:= \frac{r \cdot v}{s \cdot u} \\ \end{cases} -.\] $\Q$ bildet mit der Operation ,,+'' und ,,-'' einen ,,Körper'' bildet. +.\] $\Q$ bildet mit der Operation ,,$+$'' und ,,$\cdot$'' einen ,,Körper''. \subsection{Was ist ein Körper?} Sei $K$ eine Menge mit Operationen ,,$+$'' und ,,$\cdot$''. Operation ,,$+$'' erfüllt die Axiome der Addition -\begin{enumerate} - \item Kommutativität $\forall a,b \in K: a + b = b + a$ - \item Assoziativität $\forall a, b, c \in K: (a+b)+c = a+(b+c)$ - \item Neutrales Element $\exists 0 \in K: \forall a \in K: a + 0 = a$ - \item Additives Inverses $\forall a \in K: \exists -a \in K: a + (-a) = 0$ +\begin{enumerate}[label=(A\arabic*)] + \item Kommutativität $\forall a,b \in K\colon a + b = b + a$ + \item Assoziativität $\forall a, b, c \in K\colon (a+b)+c = a+(b+c)$ + \item Neutrales Element $\exists 0 \in K\colon \forall a \in K\colon a + 0 = a$ + \item Additives Inverses $\forall a \in K\colon \exists -a \in K\colon a + (-a) = 0$ \end{enumerate} -Operation ,,$\cdot$'' erfüllt Axiome der Multiplikation -\begin{enumerate} - \item Kommutativität $\forall a,b \in K: a \cdot b = b \cdot a$ - \item Assoziativität $\forall a, b, c \in K: (a\cdot b)\cdot c = a\cdot (b\cdot c)$ - \item Neutrales Element $\exists 1 \in K \setminus\{0\} =: K^{*}: \forall a \in K: a \cdot 1 = a$ - \item Additives Inverses $\forall a \in K: \exists a^{-1} \in K: a \cdot a^{-1} = 1$ +Operation ,,$\cdot$'' erfüllt die Axiome der Multiplikation +\begin{enumerate}[label=(M\arabic*)] + \item Kommutativität $\forall a,b \in K\colon a \cdot b = b \cdot a$ + \item Assoziativität $\forall a, b, c \in K\colon (a\cdot b)\cdot c = a\cdot (b\cdot c)$ + \item Neutrales Element $\exists 1 \in K \setminus\{0\} =\colon K^{*}\colon \forall a \in K\colon a \cdot 1 = a$ + \item Additives Inverses $\forall a \in K\colon \exists a^{-1} \in K\colon a \cdot a^{-1} = 1$ \end{enumerate} -Zusätzlich gilt ,,$+$'' und ,,$\cdot$'' erfüllen die Distributivität (D): +Zusätzlich erfüllen ,,$+$'' und ,,$\cdot$'' die Distributivität (D): \[ \forall a,b,c \in K: a \cdot (b + c) = a \cdot b + a \cdot c .\] diff --git a/ws2019/ana/lectures/analysis5.tex b/ws2019/ana/lectures/analysis5.tex index 5443414..cdff478 100644 --- a/ws2019/ana/lectures/analysis5.tex +++ b/ws2019/ana/lectures/analysis5.tex @@ -26,7 +26,7 @@ $ab$. \end{definition} -Es gelten Regeln: +Es gelten folgende Regeln: \begin{itemize} \item $a < b, b < c \implies a < c$ Transitivität \item $a < b \implies a + c < b +c, c \in K$ @@ -43,7 +43,7 @@ Es gelten Regeln: \begin{definition}[Absolutbetrag] Sei $(K, +, \cdot, >$ ein angeordneter Körper - Dann + Dann ist \[ |a| := \begin{cases} a & \text{für } a > 0 \\ @@ -52,9 +52,9 @@ Es gelten Regeln: \end{cases} .\] - Abbildung $|\cdot|$: $K \to K$ mit Eigenschaften: + eine Abbildung $|\cdot|$: $K \to K$ mit den Eigenschaften: \begin{itemize} - \item $|a| = 0 \iff a = 0$ (Definiertheit) + \item $|a| = 0 \iff a = 0$ (Definitheit) \item $|ab| = |a| |b|$ (Multiplikativität) \item $|a+b| \le |a| + |b|$ (Dreiecksungleichung) \end{itemize} @@ -80,7 +80,7 @@ Es folgt aus den Eigenschaften: oder periodische Dezimalbruchdarstellung der Form: \[ a = \pm (a_0 + 0,d_1\ldots d_s): \iff - a = \pm \left(a_0 + \sum_{k=1}^{s} ds \cdot 10^{-k}\right) + a = \pm \left(a_0 + \sum_{k=1}^{s} d_k \cdot 10^{-k}\right) .\] bzw. \[ a = \pm (a_0 + 0,d_1\ldots d_s \overline{d_{s+1} \ldots d_{s+t}}) @@ -142,7 +142,7 @@ ein $x \in \Q$ lösbar. E \implies E_1 \implies E_2 \implies \ldots \implies E_k \implies V .\] Indirekter Beweis: Zeigen $E$ und $\neg V$ immer falsch. Da $E$ immer wahr ist, muss $\neg V$ falsch sein. - Da $\neg V$ falsch ist, dann ist $V$ wahr. + Da $\neg V$ falsch ist, ist $V$ wahr. \end{bem} \textbf{Ziel}: Konstruiere rationale Zahlen, welche die @@ -162,22 +162,22 @@ $a_1 < b_1, a_1^2 = 1,96 < 2 < 2,25 = b_1^2$ Fall a) Es liege für ein $n \in \N$ eine Einschließung vor: \[ - a_n = 1,d_1d_2, \ldots d_{n-1}d_n < b_n = 1,d_1d_2\ldots d_{n-1}(d_{n+1}) + a_n = 1,d_1d_2, \ldots d_{n-1}d_n < b_n = 1,d_1d_2\ldots d_{n-1}(d_n + 1) .\] \[ a_n^2 < 2 < b_n^2 -.\] $d_k \in \{0,1, \ldots, 9\}, k=1 \ldots n-1, dn \le 8 $ +.\] $d_k \in \{0,1, \ldots, 9\}, k=1 \ldots n-1, d_n \le 8 $ Die nächste Einschließung ist \[ - a_{n+1} := 1,d_1\ldots d_n, d_{n+1}, d_{n+1} \in \{0,1,\ldots_,9\} + a_{n+1} := 1,d_1\ldots d_n, d_{n+1} \qquad d_{n+1} \in \{0,1,\ldots_,9\} .\] $a_{n+1}$ möglichst groß aber $a_{n+1}^2 < 2$. und \[ b_{n+1} := \begin{cases} 1,d_1, \ldots d_n (d_{n+1} + 1) & \text{für } d_{n+1} \le 8 \\ - 1,d_1, \ldots (d_n + 1) 0 & für d_{n+1} = 9 \\ + 1,d_1, \ldots (d_n + 1) 0 & \text{für } d_{n+1} = 9 \\ \end{cases} .\] Nach Konstruktion: \[ @@ -192,7 +192,7 @@ Fall b) Für ein $n \in \N$ liegt eine Einschließung vor a_1 = 1, d_1\ldots d_{n-1} d_{n} < b_{n} = 1, d_1 d_2 \ldots d_{n-1} (d_n + 1) 0 \ldots 0 .\] \[ - a_n^2 < 2 < b_n^2 \text{ mit } d_k \in \{0, 1, \ldots ,9\}, k=1, n = 1 + a_n^2 < 2 < b_n^2 \text{ mit } d_k \in \{0, 1, \ldots ,9\}, k=1 \ldots n-1 .\] \[ d_n \le 8, d_{n+1} = \ldots = d_n = 9 @@ -241,7 +241,7 @@ $\implies$ wir sollen die Zahl $\sqrt{2}$ eventuell erfassen! Offenbar, $1 + \frac{1}{n} \to 1, n \to \infty$ bzw. \[ - \lim_{n \to \infty} (1+ \frac{1}{n}) = 1 + \lim_{n \to \infty} \left(1+ \frac{1}{n}\right) = 1 .\] d.h. Folge konvergiert gegen 1 \begin{definition}[Konvergenz]