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--- a/ws2019/ana/lectures/analysis23.tex +++ b/ws2019/ana/lectures/analysis23.tex @@ -27,14 +27,14 @@ berechnen. Eine Zerlegung mit $|x_k - x_{k-1}| = h$ $\forall k$ heißt äquidistant. - $Z(a,b) = $ Menge aller Zerlegungen des Intervalls $[a,b]$ + $\mathcal{Z}(a,b) = $ Menge aller Zerlegungen des Intervalls $[a,b]$ \end{definition} \begin{definition}[Ober- und Untersumme] Sei $f\colon [a,b] \to \R$ beschränkt, d.h. $\exists M \in \R$, s.d. $|f(x)| \le M$ $\forall x \in [a,b]$. - Die Riemannsche Ober- / Untersummen sind + Die Riemannschen Ober- / Untersummen sind \[ \overline{S}_Z(f) := \sum_{k=1}^{n} \sup_{x \in I_k} f(x) \cdot (x_k - x_{k-1}) .\] bzw. @@ -81,12 +81,12 @@ berechnen. \begin{lemma} Sei $f\colon [a,b] \to \R$ beschränkt. Dann ex. für $f$ das Ober- und Unterintegral und für jede Folge von Zerlegungen - $z_n \in Z(a,b)$, $n \in \N$ mit $h_n \xrightarrow{n \to \infty} 0$ gilt + $Z_n \in \mathcal{Z}(a,b)$, $n \in \N$ mit $h_n \xrightarrow{n \to \infty} 0$ gilt \[ - \lim_{n \to \infty} \underline{S}_{z_n} + \lim_{n \to \infty} \underline{S}_{Z_n} = \underline{\int_{a}^{b} } f(x) dx - \le \overline{\int_{a}^{b}} dx - = \lim_{n \to \infty} f(x) \overline{S}_{z_n} + \le \overline{\int_{a}^{b}} f(x) dx + = \lim_{n \to \infty} f(x) \overline{S}_{Z_n} .\] \end{lemma} @@ -110,7 +110,7 @@ berechnen. Eine beschränkte Funktion $f\colon [a,b] \to \R$ ist genau dann auf $I = [a,b]$ integrierbar, falls $\forall \epsilon > 0$ $\exists $ Zerlegung - $z \in Z(a,b)$, s.d. + $Z \in \mathcal{Z}(a,b)$, s.d. $|\overline{S}_Z(f) - \underline{S}_Z(f)| < \epsilon$. \end{satz} @@ -119,7 +119,7 @@ berechnen. \end{proof} \begin{definition}[Riemann-Summen] - Sei $z = (x_0, x_1, \ldots, x_n)$ eine Zerlegung von + Sei $Z = (x_0, x_1, \ldots, x_n)$ eine Zerlegung von $[a,b]$ und $x_{i-1} \le \xi_i \le x_i$, $i = 1 \ldots n$. \[ RS_Z(f) = \sum_{k=1}^{n} f(\xi_i) (x_i - x_{i-1}) @@ -157,17 +157,17 @@ berechnen. \begin{satz} Eine beschränkte Funktion $f\colon I = [a,b] \to \R$ ist genau - dann R-integrierbar wenn $\forall $ Folgen $z_n \in Z(a,b)$ mit + dann R.-integrierbar wenn $\forall $ Folgen $Z_n \in \mathcal{Z}(a,b)$ mit $h_n \xrightarrow{n \to \infty} 0$ alle zugehörigen R.-Summen zu dem selben Limes konvergieren. \[ - RS_{z_n}(f) \xrightarrow{n \to \infty} \int_{a}^{b} f(x) dx + RS_{Z_n}(f) \xrightarrow{n \to \infty} \int_{a}^{b} f(x) dx .\] \end{satz} \begin{proof} - ,,$\implies$'': Sei $f$ R.-integrierbar. Sei $ z \in Z(a,b)$ mit - Feinheit $h$. Dann + ,,$\implies$'': Sei $f$ R.-integrierbar. Sei $ Z \in \mathcal{Z}(a,b)$ mit + Feinheit $h$. Dann gilt \[ \underline{S}_Z(f) \le \underbrace{RS_Z(f)}_{\forall \xi = (\xi_1, \ldots, \xi_n)} \le \overline{S}_Z(f) .\] Aus der Konvergenz @@ -176,7 +176,7 @@ berechnen. \int_{a}^{b} f(x) dx$. ,,$\impliedby$'' Seien alle R.-Summen konvergent gegen denselben - Limes. Sei $ z \in \mathcal{Z}(a,b)$, $\epsilon > 0$ beliebig. + Limes. Sei $ Z \in \mathcal{Z}(a,b)$, $\epsilon > 0$ beliebig. Offenbar $\exists $ R.-S. $\underline{RS}_Z(f)$, $\overline{RS}_Z(f)$ s.d. $\underline{RS}_Z(f) - \epsilon \le \underline{S}_Z(f)$ und