From 263ff9991f84905b913ca478fff82390506e6efe Mon Sep 17 00:00:00 2001 From: christian Date: Fri, 31 Jan 2020 12:13:09 +0100 Subject: [PATCH] add ana26, LAST LECTURE --- lecture.cls | 5 ++ ws2019/ana/lectures/analysis26.pdf | Bin 0 -> 132247 bytes ws2019/ana/lectures/analysis26.tex | 133 +++++++++++++++++++++++++++++ 3 files changed, 138 insertions(+) create mode 100644 ws2019/ana/lectures/analysis26.pdf create mode 100644 ws2019/ana/lectures/analysis26.tex diff --git a/lecture.cls b/lecture.cls index cf7c3de..e0b2c34 100644 --- a/lecture.cls +++ b/lecture.cls @@ -111,3 +111,8 @@ \renewcommand{\stackrel}[2]{% \oldstackrel{\mathclap{#1}}{#2} }% + +% integral d sign +\makeatletter \renewcommand\d[1]{\ensuremath{% + \;\mathrm{d}#1\@ifnextchar\d{\!}{}}} +\makeatother diff --git a/ws2019/ana/lectures/analysis26.pdf b/ws2019/ana/lectures/analysis26.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fb346840f4c119da96a8f95c3f5b5ae89ce2f7a5 GIT binary patch literal 132247 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zMq=IlDs}1KWa)pKdH=&HW`-yyMv$&267>{dDG88LkhW$78My|b{!AG`rT|MRfHXjg zvS^6G_)-=nDL7^L=Y$i668-U+Kam!|QdwOFA`O+2g2+qCszbH4WT8-5NwAhYR8AJG z309B;s{sE07Ro(vl&7<=3qTek^$+6!{wNRHm%>gUD4JqhS|~>h67>g(e0(Va=E3-s zAUJN09!7A|{qX5EeEC@1WR-JB95>2^QTUSRT9^lP9V|%IZ z(KT6SEWJ7$7<&{UFBqmq@=Nlf5|9foLVdW!V%4?UKI;Nz5Hd1Ov9;$fW-U^ob*)i9 z6HXFw+LvM1s1itn3rqLI!SPYavjVS%@3P%DFTR()L#q&)WJ!B47(NH(jUf3ixzf?H z7`V`D2Z!UBINvp}ONfL}KU?pbe<-=flXbx^w#3IY)Wx6IhNk(|+(c3^Be98{GGrY1 z1YSA4dY?C(t8FwrKn=xhHbqsj+2tnpzJ$K&?d2^CH9v7P{I2_bwF16L8!B#SS8ssb zL~U~e%t4qXy#<$5ww~(ILDguX9?5||42Lo`Uu{&qtn`cb(aDF$CDz-aC$eKC+HJpx z5YO692A(FQf4q>Cz?0%N2_cpPTGp!(Uv7J4r6 0$ $\exists a < b_{\epsilon} < b$ s.d. + $\forall b_{\epsilon} < b_1 < b_2 < b$ gilt + \begin{align*} + \left| \int_{b_1}^{b_2} f(x) dx \right| < \epsilon + .\end{align*} +\end{satz} + +\begin{satz}[Majoranten-Minoranten Kriterium] + Seien $f, g, h \colon [a, b) \to \R (b \le \infty)$ + integrierbar $\forall [a,c] \subset [a,b)$ und + $ 0 \le h(x) \le |f(x)| \le g(x)$ $\forall x \in [a,b)$. + Dann gilt: + \begin{align*} + \int_{a}^{b} |f(x)| dx \begin{cases} + \text{konvergent, falls } \int_{a}^{b} g(x) \d x \text{ konvergent} \\ + \text{divergent, falls } \int_{a}^{b} h(x) \d t \text{ divergent} + \end{cases} + .\end{align*} +\end{satz} + +\begin{satz}[Grenzwertkriterium] + Seien $f, g\colon [a, b) \to \R$ $(b \le \infty)$ integrierbar + $\forall [a,c] \subset [a,b)$ und es ex. der Grenzwert + $\lim_{t \nearrow b} \frac{f(t)}{g(t)} \in (0, +\infty) $ Dann + sind die Integrale $\int_{a}^{b} f(x) \d x $ und + $\int_{a}^{b} g(x) \d x $ entweder beide konvergent oder beide + divergent. +\end{satz} + +\begin{satz} + Seien $a_n, b_n$ positiv und $\frac{a_n}{b_n} \xrightarrow{n \to \infty} q \in (0, \infty)$. Dann sind $\sum_{k=1}^{\infty} a_n$ und + $\sum_{k=1}^{\infty} b_n$ entweder beide konvergent oder beide divergent. +\end{satz} + +\begin{satz}[Dirichlet-Kriterium] + Sei $f\colon [a, \infty) \to \R$ in $[a, \infty)$ integrierbar und + $\sup_{x \ge a} \left| \int_{a}^{x} f(t) \d t \right| = M < \infty$. + + Sei $g\colon [a, \infty) \to \R_{+}$ differenzierbar und + monoton gegen Null fallend, dann ex. das uneigentliche Integral + \begin{align*} + \int_{a}^{\infty} f(t) g(t) \d t = \lim_{x \to \infty} \int_{a}^{x} f(t)g(t) \d t + .\end{align*} +\end{satz} + +\begin{bsp} + $\int_{1}^{\infty} \frac{\sin x}{x} \d x$ mit + $f(x) = \sin x$ und $g(x) = \frac{1}{x}$. +\end{bsp} + +\begin{proof} + $f, g$ sind integrierbar, $f\cdot g$ auch + integrierbar auf $[a,x] \subset [a, \infty)$ $\forall x$. + Das Integral $F(x) = \int_{a}^{x} f(t) \d t $ ex. und + ist Stammfunktion von $f$ nach HDI. + + Es gilt (partielle Integration) + \begin{align*} + \int_{a}^{x} f(t) g(t) \d t = F(t) g(t) \Big|_{a}^{x} - + \int_{a}^{x} f(t) g'(t) \d t + .\end{align*} + Sei $\epsilon > 0$ beliebig. Dann ex. $\beta_{\epsilon} > a$ s.d. + \begin{align*} + g(x) < \frac{\epsilon}{2M} \text{ für } x \ge \beta_{\epsilon} + \quad g \text{ (monoton gegen Null fallend)} + .\end{align*} und $g'(x) \le 0$. + + Sei $\beta > \alpha \ge \beta_{\epsilon}$ + \begin{align*} + \left| \int_{\alpha}^{\beta} F(t) g'(t)\d t \right| + &\le M \int_{\alpha}^{\beta} |g'(t)| \d t \\ + &= - M \int_{\alpha}^{\beta} g'(t) \d t \\ + &= - M g(t) \Big|_{\alpha}^{\beta} \\ + &= - M (g(\beta) - g(\alpha)) = M (g(\alpha) - g(\beta)) \\ + &\le 2 M g(\alpha) \le \epsilon \quad + \forall \alpha \ge \beta_{\epsilon} + .\end{align*} + + Nach Cauchy-Kriterium existiert + \begin{align*} + \lim_{x \to \infty} \int_{a}^{x} F(t) g'(t) \d t + = \int_{a}^{\infty} F(t) g'(t) \d t + .\end{align*} + + Dann gilt + \begin{align*} + \lim_{x \to \infty} \int_{a}^{x} f(t) g(t) \d t + = \lim_{x \to \infty} \underbrace{F(x)}_{\text{beschränkt}} + \underbrace{g(x)}_{\xrightarrow{x \to \infty} 0} + - \lim_{x \to \infty} \underbrace{F(a)}_{= 0} g(a) - + \underbrace{\int_{a}^{\infty} F(t) g'(t) \d t}_{\text{existiert}} + .\end{align*} + $\implies \lim_{x \to \infty} \int_{a}^{x} f(t) g(t) \d t $ existiert. +\end{proof} + +\begin{satz}[Integralkriterium für Reihen] + Sei $f\colon [n_0, \infty) \to \R$ eine stetige + monton fallende Funktion. Dann gilt: + \[ + \sum_{k=n_0}^{\infty} f(k) < \infty \iff + \int_{n_0}^{\infty} f(x) \d x < \infty + .\] +\end{satz} + +\begin{proof} + ,,$\implies$'' Die Reihe ist konvergent. Sei $n > n_0$, $n \in \N$ + \[ + \int_{n_0}^{n+1} f(x) \d x = \sum_{k=n_0}^{n} \int_{k}^{k+1} f(x) \d x + \quad \qquad \stackrel{f \text{ monoton fallend}}{\le } \qquad \quad + \sum_{k=n_0}^{n} f(k) \cdot 1 \le \sum_{k=n_0}^{\infty} f(k) < \infty + .\] $\implies \int_{n_0}^{\infty} f(x) \d x $ existiert. + + ,,$\impliedby$'' $\int_{n_0}^{\infty} f(x) \d x $ existiert. Dann gilt + \begin{align*} + \sum_{k=n_0}^{n} f(k) &= f(n_0) + \sum_{k=n_0}^{n-1} f(k+1) \\ + &\le f(n_0) + \sum_{k=n_0}^{n-1} \int_{k}^{k+1} f(t) \d t \\ + &\le f(n_0) + \int_{n_0}^{\infty} f(t) \d t + < \infty \quad \forall n + .\end{align*} + $\implies$ die Reihe ist konvergent. +\end{proof} + +\end{document}