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zVH$@DyezY87W;lpGpefE_k8^6Ye$@Q&;Aq0I_8; g{{R2HoLvl^Ts)k<-_tkv%E8J6Lq;YlF9!2}0Ga+>L;wH) literal 0 HcmV?d00001 diff --git a/ws2019/ana/lectures/analysis25.tex b/ws2019/ana/lectures/analysis25.tex new file mode 100644 index 0000000..6c50960 --- /dev/null +++ b/ws2019/ana/lectures/analysis25.tex @@ -0,0 +1,263 @@ +\documentclass{../../../lecture} + +\begin{document} + +\begin{korrolar}[2. Mittelwertsatz] + Seien $f\colon I \to \R$ monoton, $g\colon I \to \R$ R.-integrierbar. Dann + ex. $\xi \in [a,b]$ s.d. + \[ + \int_{a}^{b} f(x) g(x) dx = f(a) \int_{a}^{\xi} g(x) dx + f(b) \int_{\xi}^{b} g(x) dx + .\] +\end{korrolar} + +\begin{proof} + o.B.d.A: $f$ monoton fallend. + + Definiere $\phi(t) := f(a) \int_{a}^{t} g(x) dx + f(b) \int_{t}^{b} g(x) dx $, $a \le t \le b$. + Nach HDI $\phi(t)$ stetig. + \[ + \varphi(a) = f(b) \int_{a}^{b} g(x) dx \qquad \quad + \stackrel{f \text{ monoton fallend}}{\le} \qquad \quad + \int_{a}^{b} f(x) g(x) dx + \le f(a) \int_{a}^{b} g(x) dx = \varphi(b) + .\] Nach ZWS $\exists \xi \in [a,b]$ s.d. + $\varphi(\xi) = \int_{a}^{b} f(x) g(x) dx $. +\end{proof} + +\begin{bem} + Monotonie unverzichtbar. $f(x) = x^2$, $g(x) = 1$, $I = [-1,1]$. + \begin{align*} + &f(-1) \int_{-1}^{\xi} g(x) dx + f(1) \int_{\xi}^{1} g(x) dx + = \int_{-1}^{1} 1 dx = 2 \quad \forall \xi \in I\\ + & \int_{-1}^{1} x^2 dx = \frac{x^{3}}{3} \Big|_{-1}^{1} = \frac{1}{3} - \left( -\frac{1}{3} \right) = \frac{2}{3} \neq 2 + .\end{align*} +\end{bem} + +\subsection{Integrationsformeln} + +\begin{lemma}[Partielle Integration] + $f, g\colon [a,b] \to \R$ stetig differenzierbar. Dann gilt + \begin{align*} + \int_{a}^{b} f'(x) g(x) dx = \left[ f(x) \cdot g(x) \right] \Big|_{a}^{b} - \int_{a}^{b} f(x) g'(x) dx + .\end{align*} +\end{lemma} + +\begin{proof} + $(f \cdot g)'(x) = f' \cdot g + f\cdot g' \implies$ + \begin{align*} + \int_{a}^{b} (f' \cdot g + f\cdot g')(x) dx = + \int_{a}^{b} (f \cdot g)'(x) dx + \stackrel{\text{HDI}}{=} (f \cdot g)(x) \Big|_{a}^{b} + .\end{align*} +\end{proof} + +\begin{bsp} + \begin{align*} + \int_{a}^{b} \cos^2(x) &= \int_{a}^{b} \cos x \cdot \cos x dx \\ + &= \cos x \cdot \sin x \Big|_{a}^{b} - \int_{a}^{b} (- \sin x) \sin x dx \\ + &= \cos x \cdot \sin x |Big_{a}^{b} + \int_{a}^{b} (1 - \cos^2(x))dx \\ + \implies 2 \int_{a}^{b} \cos^2(x) dx &= \cos x \cdot \sin x + \Big|_{a}^{b} + \int_{a}^{b} dx + .\end{align*} +\end{bsp} + +\begin{lemma} + Seien $[a,b], [\alpha, \beta] \subset \R$, $f\colon [a,b] \to \R$ stetig, + $\varphi\colon [\alpha, \beta] \to [a,b]$ stetig differenzierbar + mit $a = \varphi(\alpha)$, $b= \varphi(\beta)$. Dann gilt + \begin{align*} + \int_{a}^{b} f(\varphi(t)) \varphi'(t) dt = + \int_{a = \varphi(\alpha)}^{b = \varphi(\beta)} f(x) dx + .\end{align*} +\end{lemma} + +\begin{proof} + Sei $F$ eine Stammfunktion von $f$. Dann + $F \circ \varphi \colon [\alpha, \beta] \to \R$ stetig differenzierbar + und + \begin{align*} + (F \circ \varphi)' = (F'(\varphi(t))) \cdot \varphi'(t) + = f(\varphi(t)) \cdot \varphi'(t) + .\end{align*} + \begin{align*} + \int_{\alpha}^{\beta} f(\varphi(t)) \varphi'(t) dt = + \int_{\alpha}^{\beta} (F \circ \varphi)'(t) dt + = (F \circ \varphi)(t) \Big|_{\alpha}^{\beta} + = F(\varphi(\beta)) - F(\varphi(\alpha)) + = \int_{\varphi(\alpha)}^{\varphi(\beta)} f(x) dx + .\end{align*} +\end{proof} + +\begin{bem} + Formal: + $x = \varphi(t)$ + \begin{align*} + \frac{dx}{dt} = \varphi'(t) \implies dx = \varphi'(t) dt \\ + \int_{\varphi(\alpha) = a}^{\varphi(\beta) = b} + f(x) dx = \int_{a}^{b} f(\varphi(t)) \varphi'(t) dt + .\end{align*} +\end{bem} + +\begin{bsp} + \begin{align*} + \int_{0}^{2} t \cdot \cos(\underbrace{t^2 + t}_{x}) dt + = \frac{1}{2} \int_{0}^{2} \cos(\underbrace{t^2 + t}_{\varphi(t)}) + \cdot 2 t dt + = \frac{1}{2} \int_{\varphi(0) = 1}^{\varphi(2) = 5} \cos x dx + .\end{align*} +\end{bsp} + +\subsection{Uneigentliche Integrale} + +\begin{satz}[Uneigentliches R.-Integral Typ 1] + Sei $f\colon (a, b] \to \R$ auf $(a, b]$ + R.-integrierbar, d.h. $f$ R.-integrierbar auf + $\forall [a', b] \subset (a, b]$, aber nicht auf + $[a,b]$. + + Falls für alle Folgen $a_n \in (a,b]$ ex. + \begin{align*} + \lim_{a_n \searrow a} \int_{a_n}^{b} f(x) dx =: \int_{a}^{b} f(x) dx + .\end{align*} + Dann gilt: Dieser Limes ist von der Wahl der Folge $a_n$ unabhängig und + heißt das uneigentliche Integral von $f$ über $[a,b]$. +\end{satz} + +\begin{figure}[h!] + \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=10, + ymin=0, + width=.5\textwidth + ] + \addplot[domain=0:2,samples=100,smooth,red] {1/x}; + \end{axis} + \end{tikzpicture} +\end{figure} + +\begin{proof} + Sei $(a_n')_{n \in \N}$ eine weitere Folge mit + \[ + \lim_{a_n \searrow a} \int_{a'}^{b} f(x) dx = A' + .\] Konstruiere Folge $\{a_1, a_1', a_2, a_2', \ldots\} = (a''_n)_{n \in \N}$. Nach Voraussetzungen + \begin{align*} + \exists \lim_{a_n'' \searrow a} \int_{a''}^{b} f(x) dx = A'' + .\end{align*} + Alle Teilfolgen konvergenter Folgen, konvergieren gegen denselben + Limes wie die Gesamtfolge $\implies A''= A'$. +\end{proof} + +\begin{lemma} + Sei $f\colon (a,b] \to \R$ auf $(a, b]$ aber nicht auf $[a,b]$ + integrierbar. + + Falls das uneigentliche Integral von $|f|$ auf $[a,b]$ ex., dann + ex. das uneigentliche Integral von $f$ über $[a,b]$ und es gilt + \begin{align*} + \left| \int_{a}^{b} f(x) dx \right| \le \int_{a}^{b} |f(x)| dx + .\end{align*} +\end{lemma} + +\begin{proof} + Sei $\epsilon > 0, \epsilon < b - a$. Betrachte + \begin{align*} + \int_{a + \epsilon}^{b} f(x) dx = + \int_{a + \epsilon}^{b} \frac{|f(x)| + f(x)}{2} dx + - \int_{a + \epsilon}^{b} \frac{|f(x)| - f(x)}{2} dx + .\end{align*} + Integrale sind gleichmäßig beschränkt. + \begin{align*} + \frac{|f(x)| + f(x)}{2} > 0 \quad \forall x \text{ und } + \frac{|f(x)| - f(x)}{2} > 0 \quad \forall x + .\end{align*} + $\implies \int_{a+ \epsilon}^{b} \ldots dx $ monoton wachsend für + $\epsilon \to 0$ und + \begin{align*} + \left| \int_{a+\epsilon}^{b} \frac{|f(x)| + f(x)}{2} dx \right| + + \left| \int_{a+\epsilon}^{b} \frac{|f(x)| - f(x)}{dx} \right| + \le \frac{4}{2} \int_{a+\epsilon}^{b} |f(x)| dx + \le 2 \int_{a}^{b} |f(x)|dx + .\end{align*} + $\implies$ Für $\epsilon \to 0$: + \begin{align*} + \exists \lim_{\epsilon \to 0} + \int_{a + \epsilon}^{b} f(x) dx =: \int_{a}^{b} f(x) dx + .\end{align*} +\end{proof} + +\begin{bem} + \begin{enumerate} + \item Umkehrung der Aussage + (d.h. $f$ uneigentlich integrierbar $\implies |f|$ uneigentlich + integrierbar) ist i.A. nicht richtig. + + ,,einfache'' Konvergenz, d.h. + $\exists \lim_{\epsilon \to 0} \int_{a+ \epsilon}^{b} f(x) dx$. + + ,,absolute'' Konvergenz / absolut uneigentlich integrierbar, d.h. + $\exists \lim_{\epsilon \to 0} \int_{a + \epsilon}^{b} |f(x)| dx$. +\item + Sei $\int_{a}^{b} f(x) dx $ bei $b$ uneigentlich und + bei $a $ nicht uneigentlich, dann definiert man das uneigentliche Integral + \begin{align*} + \int_{a}^{b} f(x) dx := &\lim_{x \to b} \int_{a}^{x} f(t) dt \quad \text{oder }\\ + &\lim_{\epsilon \to 0} \int_{a}^{b - \epsilon} f(x) dx + .\end{align*} + falls der Limes existiert! +\item Sei $\int_{a}^{b} f(x) dx $ bei $a$ und $b$ uneigentlich, dann + \begin{align*} + \int_{a}^{b} f(x) dx := \lim_{\epsilon \to 0} \int_{c}^{b - \epsilon} f(x) dx + \lim_{\epsilon \to 0} \int_{a+\epsilon}^{c} f(x) dx + .\end{align*} mit $c \in (a,b)$, falls beide Grenzwerte + existieren, ist der Wert unabhängig von der Wahl von $c \in (a,b)$. +\item Uneigentliches Integral existiert $\iff$ uneigentliches Integral + konvergiert. + \end{enumerate} +\end{bem} + +\begin{lemma}[wie bei Reihen] + Absolute Konvergenz $\implies$ Einfache Konvergenz +\end{lemma} + +\begin{bsp} + \begin{align*} + \int_{a + \epsilon}^{b} \frac{dx}{x - a} = \ln(b-a) - \ln(\epsilon) + \xrightarrow{\epsilon \to 0} \infty + .\end{align*} + \begin{align*} + \int_{a + \epsilon}^{b} \frac{dx}{(x-a)^{\mu}} + = \frac{1}{1 - \mu} \frac{1}{(x-a)^{\mu -1}} \Big|_{a + \epsilon}^{b} + = \frac{1}{1 - \mu} \left( \frac{1}{(b-a)^{\mu - 1}} - \frac{1}{\epsilon^{\mu - 1}} \right) + .\end{align*} + $\implies$ Integral ex. für $0 < \mu < 1$, ex. nicht für $\mu \ge 1$. +\end{bsp} + +\begin{satz}[Uneigentliche R.-Integrale Typ 2] + Sei $f\colon [a, \infty] \to \R$ eine lokal integrierbare + Funktion, d.h. $f$ ist auf $[a,b'] \subset [a, \infty)$ integrierbar + $\forall b'$. + + Falls für alle Folgen $b_n \in [a, +\infty)$ der Limes + \begin{align*} + \lim_{b_n \to \infty} \int_{a}^{b_n} f(x) dx =: + \int_{a}^{\infty} f(x) dx + .\end{align*} existiert, dann ist dieser unabhängig von der Wahl + der Folge $(b_n)_{n\in\N}$ und heißt uneigentliches Integral von + $f$ über $[a, \infty)$. +\end{satz} + +\begin{lemma} + Sei $f\colon [a, \infty) \to \R$ lokal integrierbar und es existiere + $\int_{a}^{\infty} |f(x)| dx $. Dann ex. $\int_{a}^{\infty} f(x) dx $ + und es gilt + \begin{align*} + \left| \int_{a}^{\infty} f(x) dx \right| \le \int_{a}^{\infty} |f(x)| dx + .\end{align*} +\end{lemma} + +\end{document}