From 0b524bc085d4d1cee0e7ff56f2c9b687759ccdf8 Mon Sep 17 00:00:00 2001 From: flavis Date: Fri, 1 May 2020 21:53:47 +0200 Subject: [PATCH] finish ana --- sose2020/ana/lectures | 2 +- sose2020/ana/uebungen/ana2.pdf | Bin 0 -> 137675 bytes sose2020/ana/uebungen/ana2.tex | 295 +++++++++++++++++++++++++++++++++ 3 files changed, 296 insertions(+), 1 deletion(-) create mode 100644 sose2020/ana/uebungen/ana2.pdf create mode 100644 sose2020/ana/uebungen/ana2.tex diff --git a/sose2020/ana/lectures b/sose2020/ana/lectures index a729cf4..0ee655c 160000 --- a/sose2020/ana/lectures +++ b/sose2020/ana/lectures @@ -1 +1 @@ -Subproject commit a729cf48c63d6939f77bfccdf3a8e9490317cebf +Subproject commit 0ee655c856d37c6cf66ac432352429d31e4bf214 diff --git a/sose2020/ana/uebungen/ana2.pdf b/sose2020/ana/uebungen/ana2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aca3a7a5827c12ca33c682b5115356c4e138b2f8 GIT binary patch literal 137675 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Dann gilt + \begin{align*} + \int_{-1}^{1} P_n(x) P_m(x) \d x + &= \frac{1}{2^{n} 2^{m} n! m!} + \int_{-1}^{1} \frac{\mathrm{d}}{\d x^{n}}(x^2-1)^{n} + \frac{\mathrm{d}}{\d x^{m}}(x^2-1)^{m} \d x + .\end{align*} + Zu zeigen: + \[ + \int_{-1}^{1} \frac{\mathrm{d}}{\d x^{n}}(x^2-1)^{n} + \frac{\mathrm{d}}{\d x^{m}}(x^2-1)^{m} \d x = 0 + .\] Mit $(*)$ und $k = m+1 \le n$ folgt + \begin{align*} + \int_{-1}^{1} \frac{\mathrm{d}}{\d x^{n}}(x^2-1)^{n} + \frac{\mathrm{d}}{\d x^{m}}(x^2-1)^{m} \d x + &= (-1)^{m+1} \int_{-1}^{1} \frac{\mathrm{d}^{n-m-1}}{\d x^{n-m-1}} + (x^2-1)^{n} \frac{\mathrm{d}^{2m+1}}{\d x^{2m+1}}(x^2-1)^{m}\d x + .\end{align*} + Wegen $\text{deg } (x^2-1)^{m} = 2^{m}$ folgt + $\frac{\mathrm{d}^{2m+1}}{\d x^{2m+1}}(x^2-1)^{m} = 0$. Damit folgt + \begin{align*} + \int_{-1}^{1} \frac{\mathrm{d}}{\d x^{n}}(x^2-1)^{n} + \frac{\mathrm{d}}{\d x^{m}}(x^2-1)^{m} \d x + &= 0 + .\end{align*} + \end{proof} + \item Beh.: Für ein festes $n \in \N_0$ gilt $\forall k \in \N_0$ mit $0 \le k \le n$: + \[ + \int_{-1}^{1} (1-x)^{n}(1+x)^{n} \d x + = \frac{(n!)^2}{(n-k)!(n+k)!} + \int_{-1}^{1} (1-x)^{n-k}(1+x)^{n+k} \d x + .\] + \begin{proof} + Sei $n \in \N_0$ beliebig. Beweis per Induktion nach $k$. $k = 0$: trivial. + + Es existiere ein $k \in \N_0$ mit $k < n$ mit Beh. Dann folgt für $k+1$ mit + partieller Integration: + \begin{align*} + \frac{(n!)^2}{(n-k-1)!(n+k+1)!}& + \int_{-1}^{1} (1-x)^{n-k-1}(1+x)^{n+k+1} \d x \\ + =& \frac{(n!)^2}{(n-k-1)!(n+k+1)!} + \Big(\underbrace{- \frac{1}{n-k} (1-x)^{n-k} (1+x)^{n+k+1} \Big|_{-1}^{1}}_{=0} \\ &+\int_{-1}^{1} \frac{n+k+1}{n-k} (1-x)^{n-k}(1+x)^{n+k} \d x \Big) \\ + =& \frac{(n!)^2(n-k)}{(n-k)!(n+k)!(n+k+1)} \int_{-1}^{1} \frac{n+k+1}{n-k} + (1-x)^{n-k}(1+x)^{n+k}\d x \\ + =& \frac{(n!)^2}{(n-k)!(n+k)!} \int_{-1}^{1} (1-x)^{n-k}(1+x)^{n+k} \d x \\ + \stackrel{\text{I.V.}}{=}& + \int_{-1}^{1} (1-x)^{n}(1+x)^{n} \d x + .\end{align*} + \end{proof} + \item Beh.: Es gilt $\forall n \in \N_0$: + \[ + \int_{-1}^{1} P_n(x)P_n(x) \d x = \frac{2}{2n+1} + .\] + \begin{proof} + Sei $n \in \N_0$ beliebig. Dann sind wegen $\text{deg } (x^2-1)^{n} = 2n$ + in der $2n$-ten Ableitung alle Terme bis auf den $x^{2n}$ Term null. Damit folgt + \begin{align*} + \frac{\mathrm{d}^{2n}}{\d x^{2n}} (x^2-1)^{n} + = \frac{\mathrm{d}^{2n}}{\d x^{2n}} x^{2n} = (2n)! \qquad (**) + .\end{align*} + + Dann folgt mit $(*)$ und $k = n$: + \begin{align*} + \int_{-1}^{1} P_n(x)P_n(x) \d x + &= \frac{1}{2^{2n} (n!)^2} \int_{-1}^{1} \frac{\mathrm{d}^{n}}{\d x^{n}} (x^2-1)^{n} + \frac{\mathrm{d}^{n}}{\d x^{n}} (x^2-1)^{n} \d x \\ + &\stackrel{(*)}{=} + \frac{(-1)^{n}}{2^{2n}(n!)^2} + \int_{-1}^{1} (x^2-1)^{n} \cdot + \underbrace{\frac{\mathrm{d}^{2n}}{\d x^{2n}} (x^2-1)^{n}}_{= (2n)! \quad (**)} \d x \\ + &= \frac{(2n)!(-1)^{n}}{2^{2n}(n!)^2} + \int_{-1}^{1} ((x-1)(x+1))^{n} \d x \\ + &= \frac{(2n)!(-1)^{n}}{2^{2n}(n!)^2} + \int_{-1}^{1} ((-1)(1-x)(1+x))^{n} \d x \\ + &= \frac{(2n)!}{2^{2n}(n!)^2} + \int_{-1}^{1} (1-x)^{n}(1+x)^{n} \d x \\ + &\stackrel{\text{(b), } k=n}{=} \quad + \frac{1}{2^{2n}} \int_{-1}^{1} (1+x)^{2n} \d x \\ + &= \frac{1}{2^{2n}} \frac{1}{2n+1} (1+x)^{2n+1} \Big|_{-1}^{1}\\ + &= \frac{2^{2n+1}}{2^{2n}} \frac{1}{2n+1} \\ + &= \frac{2}{2n+1} + .\end{align*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Die Funktion $f\colon [0, 2\pi] \to \R$ sei definiert durch: + \[ + f(x) = \min\left( x - \frac{\pi}{2}, \frac{3\pi}{2} - x \right) + .\] Beh.: Für die Fourier Entwicklung $F_{\infty}^{f}$ gilt: + \[ + F_{\infty}^{f} = - \sum_{k=1}^{\infty} \frac{4}{\pi(2k-1)^2} \cos((2k-1)x) + .\] + \begin{proof} + Es gilt zunächst: + \[ + f(x) = \begin{cases} + x - \frac{\pi}{2} & 0 \le x \le \pi \\ + \frac{3\pi}{2} - x & \pi < x \le 2\pi + \end{cases} + .\] + Für die Koeffizienten $a_k$ mit $k > 0$ folgt nach Definition: + \begin{align*} + a_k &= \frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos(kx) \d x \\ + &= \frac{1}{\pi} \left[ \int_{0}^{\pi} \left( x - \frac{\pi}{2} \right) \cos(kx) \d x + + \int_{\pi}^{2\pi} \left( \frac{3\pi}{2} - x) \cos(kx) \right) \d x \right] \\ + &= \frac{1}{\pi} \left[ \frac{1}{k} \sin(kx) \left( x - \frac{\pi}{2} \right)\Big|_{0}^{\pi} + - \int_{0}^{\pi} \frac{1}{k} \sin(kx) \d x + + \frac{1}{k} \sin(kx) \left( \frac{3\pi}{2} -x\right) \Big|_{\pi}^{2\pi} + + \int_{\pi}^{2\pi} \frac{1}{k} \sin(kx) \d x \right] \\ + &= \frac{2}{\pi k^2} (\cos(k\pi) - 1) + .\end{align*} + Für $a_0$ gilt + \begin{align*} + a_0 &= \frac{1}{\pi} \int_{0}^{2\pi} f(x) \d x \\ + &= \frac{1}{\pi} \left[ \int_{0}^{\pi} \left( x - \frac{\pi}{2} \right) \d x + + \int_{\pi}^{2\pi} \left( \frac{3\pi}{2} - x \right) \d x \right] \\ + &= 0 + .\end{align*} + + Für $b_k$ folgt nach Definition mit analoger Rechnung: + \begin{align*} + b_k = \frac{1}{\pi} \int_{0}^{2\pi} f(x) \sin(kx) \d x + = 0 + .\end{align*} + Damit folgt die Fourrier-Entwicklung von $f$ mit: + \begin{align*} + F_{\infty}^{f}(x) + &= \sum_{k=1}^{\infty} \frac{2}{\pi k^2} (\underbrace{\cos(k\pi)}_{= (-1)^{k}} - 1)\cos(kx) \\ + &= \sum_{k=1}^{\infty} \frac{2}{\pi(2k-1)^2} \left[ \underbrace{((-1)^{2k-1} - 1)}_{=-2} + \cos((2k-1)x) + + \underbrace{((-1)^{2k} - 1)}_{= 0} \cos(2kx) \right] \\ + &= -\sum_{k=1}^{\infty} \frac{4}{\pi(2k-1)^2} \cos((2k-1)x) + .\end{align*} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + Beh.: + \[ + \sum_{k=1}^{\infty} \frac{1}{(2k-1)^{4}} = \frac{\pi^{4}}{96} + .\] + \begin{proof} + Nach Vorlesung gilt $a_k = c_k + c_{-k}$ und $b_k = i (c_k - c_{-k})$. Mit + $b_k = 0$ folgt $c_k = c_{-k}$ und mit Aufg. 2.2 damit: + \[ + c_k = c_{-k} = \frac{a_k}{2} = \frac{1}{\pi k^2}((-1)^{k} - 1) + .\] Damit folgt + \begin{align*} + 2 \pi \sum_{-\infty}^{\infty} |c_k|^2 = 2\pi + \sum_{-\infty}^{\infty} \left| \frac{2}{\pi(2k-1)^2} \right|^2 + \qquad \stackrel{c_k = c_{-k}}{=} \qquad + \frac{16}{\pi} \sum_{k=1}^{\infty} \frac{1}{(2k-1)^{4}} + .\end{align*} + Außerdem gilt + \begin{align*} + \int_{0}^{2\pi} |f(x)|^2 \d x &= \int_{0}^{\pi} \left( x - \frac{\pi}{2} \right)^2 \d x + + \int_{\pi}^{2\pi} \left( \frac{3\pi}{2} - x)^2 \right) \d x = \frac{\pi^{3}}{6} + .\end{align*} + Zusammen mit der Parsevalschen Gleichung (PG) folgt dann + \begin{align*} + &\frac{\pi^{3}}{6} \quad \stackrel{\text{(PG)}}{=} \quad \frac{16}{\pi} \sum_{k=1}^{\infty} \frac{1}{(2k-1)^{4}}\\ + \implies &\frac{\pi^{4}}{96} \quad = \quad \sum_{k=1}^{\infty} \frac{1}{(2k-1)^{4}} + .\end{align*} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + Beh.: Für $2\pi$-periodische Funktionen $f\colon \R \to \R$ gilt + \begin{align*} + \int_{0}^{2\pi} f(x)\sin(kx) \d x &= \int_{-\pi}^{\pi} f(x)\sin(kx) \d x \quad (\text{A}) \\ + \int_{0}^{2\pi} f(x)\cos(kx) \d x &= \int_{-\pi}^{\pi} f(x)\cos(kx) \d x \quad (\text{B}) + .\end{align*} + \begin{proof} + Sei $f\colon \R \to \R$ $2\pi$-periodisch, außerdem ist $\sin(x)$ $2\pi$ periodisch $(*)$. + Dann gilt: + \begin{align*} + \int_{0}^{2\pi} f(x)\sin(kx) \d x + &= \int_{0}^{\pi} f(x)\sin(kx) \d x + \int_{\pi}^{2\pi} f(x)\sin(kx) \d x \\ + &= \int_{0}^{\pi} f(x) \sin(kx) \d x + \int_{-\pi}^{0} f(x + 2\pi) \sin(k(x + 2\pi)) \d x \\ + &\stackrel{(*)}{=} \int_{0}^{\pi} f(x) \sin(kx) \d x + \int_{-\pi}^{\pi} f(x) \sin(kx) \d x \\ + &= \int_{-\pi}^{\pi} f(x) \sin(kx) \d x + .\end{align*} + Für $\cos(kx)$ analog. + \end{proof} + Beh.: Für $2\pi$-periodische gerade $f_g$ bzw. ungerade $f_u$ Funktionen $f\colon \R \to \R$ gilt + für ihre Fourier-Entwicklung: + \[ + F_{\infty}^{f_g}(x) = \frac{a_0}{2} + \sum_{k=1}^{\infty} a_k \cos(kx) \quad \text{bzw.} + \quad F_{\infty}^{f_u}(x) = \sum_{k=1}^{\infty} b_k \sin(kx) + .\] + \begin{proof} + Sei $f_g \colon \R \to \R$ gerade $(*)$ und $2\pi$-periodisch. Dann bleibt zu zeigen: + $b_k = 0$. Es gilt $\sin(-x) = -\sin(x)$, also ist $\sin(x)$ ungerade $(**)$. Damit folgt + \begin{align*} + b_k &= \frac{1}{\pi} \int_{0}^{2\pi} f(x) \sin(kx) \d x \\ + &\stackrel{\text{(A)}}{=} \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(kx) \d x \\ + &= \frac{1}{\pi} \int_{-\pi}^{0} f(x) \sin(kx) \d x + + \frac{1}{\pi} \int_{0}^{\pi} f(x) \sin(kx) \d x \\ + &\stackrel{x \to -x}{=} \quad \frac{1}{\pi} \int_{\pi}^{0} - f(-x)\sin(-kx) \d x + + \frac{1}{\pi} \int_{0}^{\pi} f(x) \sin(kx) \d x \\ + &= \frac{1}{\pi} \int_{0}^{\pi} f(-x) \sin(-kx) \d x + + \frac{1}{\pi} \int_{0}^{\pi} f(x) \sin(kx) \d x \\ + &\stackrel{(*),(**)}{=} \quad - \frac{1}{\pi} \int_{0}^{\pi} f(x) \sin(kx) \d x + + \frac{1}{\pi} \int_{0}^{\pi} f(x) \sin(kx) \d x \\ + &= 0 + .\end{align*} + + Sei $f_u \R \to \R$ ungerade und $2\pi$-periodisch. Dann bleibt zu zeigen + $a_k = 0$. Es gilt $\cos(-x) = \cos(x)$, also ist $\cos(x)$ gerade. Dann folgt die Behauptung + analog. + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + Beh.: $\forall n, m \in \N_0$, $n \ge m$, $0 \le k \le n$ gilt zudem nach Aufgabenstellung: + \begin{align*} + \int_{-1}^{1} \frac{\mathrm{d}}{\d x^{n}}(x^2-1)^{n} + \frac{\mathrm{d}}{\d x^{m}}(x^2-1)^{m} \d x + &= (-1)^{m+1} \int_{-1}^{1} \frac{\mathrm{d}^{n-k}}{\d x^{n-k}} + (x^2-1)^{n} \frac{\mathrm{d}^{m+k}}{\d x^{m+k}}(x^2-1)^{m}\d x + .\end{align*} + Es gelte + \[ + \frac{\mathrm{d}^{k}}{\d x^{k}}(x^2- 1)^{n} = (x^2-1)^{n-k} \cdot p_k(x) \quad (*) + .\] + \begin{proof} + Seien $n, m \in \N_0$ beliebig. Beweis per Induktion nach $k$. + + Für $k = 0$ ist Beh. trivialerweise erfüllt. + + Sei nun $k \in \N_0$ mit $k < n$ beliebig mit gegebener Beh. Dann gilt für $k+1$: + \begin{align*} + &(-1)^{k+1} \int_{-1}^{1} \frac{\mathrm{d}^{n-k-1}}{\d x^{n-k-1}}(x^2 - 1)^{n} + \frac{\mathrm{d}^{m+k+1}}{\d x^{m+k+1}}(x^2 -1)^{m} \d x \\ + &= (-1)^{k+1} \left[ \left( \frac{\mathrm{d}^{n-k}}{\d x^{n-k}} (x^2 - 1)^{n} + \cdot \frac{\mathrm{d}^{m+k+1}}{\d x^{m+k+1}}(x^2-1)^{m} \right) \Big|_{-1}^{1} + - \int_{-1}^{1} \frac{\mathrm{d}^{n-k}}{\d x^{n-k}}(x^2-1)^{n} + \cdot \frac{\mathrm{d}^{m+k}}{\d x^{m+k}}(x^2-1)^{m} \d x \right] \\ + &\stackrel{\text{I.V.}}{=} + (-1)^{k+1} \left( \frac{\mathrm{d}^{n-k}}{\d x^{n-k}} (x^2 - 1)^{n} + \cdot \frac{\mathrm{d}^{m+k+1}}{\d x^{m+k+1}}(x^2-1)^{m} \right) \Big|_{-1}^{1} + + \int_{-1}^{1} \frac{\mathrm{d}^{n}}{\d x^{n}} (x^2-1)^{n} + \cdot \frac{\mathrm{d}^{m}}{\d x^{m}}(x^2-1)^{m}\d x + .\end{align*} + Bleibt zu zeigen: + \[ + \left.\left[\frac{\mathrm{d}^{n-k}}{\d x^{n-k}} (x^2 - 1)^{n} + \cdot \frac{\mathrm{d}^{m+k+1}}{\d x^{m+k+1}}(x^2-1)^{m} \right] \right\rvert_{-1}^{1} = 0 + .\] + \begin{align*} + \left.\left[\frac{\mathrm{d}^{n-k}}{\d x^{n-k}} (x^2 - 1)^{n} + \cdot \frac{\mathrm{d}^{m+k+1}}{\d x^{m+k+1}}(x^2-1)^{m} \right] \right\rvert_{-1}^{1} + \stackrel{(*)}{=}& + \left.\left[(x^2 - 1)^{k} \cdot p_{n-k}(x) \cdot + \frac{\mathrm{d}^{m+k+1}}{\d x^{m+k+1}}(x^2-1)^{m} \right] \right\rvert_{-1}^{1} \\ + =&\; (1 - 1)^{k} \cdot p_{n-k}(1) \cdot + \left(\frac{\mathrm{d}^{m+k+1}}{\d x^{m+k+1}}(x^2-1)^{m}\right)(1) \\ + &- (1-1)^{k} \cdot p_{n-k}(-1) \cdot + \left(\frac{\mathrm{d}^{m+k+1}}{\d x^{m+k+1}}(x^2-1)^{m}\right)(-1) \\ + =& \;0 + .\end{align*} + Damit folgt die Induktionsbehauptung für $k+1$. + \end{proof} +\end{aufgabe} + +\end{document}