From ff6be5e50eb5c06ab7da37f6ae640460accca7a1 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Thu, 9 Jul 2026 08:09:39 +0200 Subject: [PATCH] =?UTF-8?q?letzte=20=C3=84nderungen=20auf=20Grundlage=20de?= =?UTF-8?q?r=20Konsultation=20von=20Luca?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Mathematischer_Hintergrund.tex | 68 +++++++++---------------- galton.pdf | Bin 458000 -> 453107 bytes 2 files changed, 23 insertions(+), 45 deletions(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 3832ecf..d6bc685 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -265,6 +265,25 @@ Folgende Proposition ist leicht aus dem oberen Beweis ableitbar (aufgrund der Ab wobei $\beta = \sqrt{\frac{q}{np}}$ und $\gamma = \sqrt{\frac{p}{nq}}$ definiert sind. \end{proposition} +\subsection{Gamma-Funktion} +% wird für Beweis der Stirling-Approximation gebraucht +Zur nachfolgenden Betrachtung und des Beweises der Stirling-Approximation benötigen wir die Gamma-Funktion, auch \textbf{Eulersches Integral zweiter Gattung} genannt. Sie erweitert die Fakultätsfunktion von den natürlichen Zahlen $\mathbb{N}$ auf reelle und komplexe Zahlen (mit einigen Ausnahmen). Hier betrachten wir der Einfachheit halber nur die Gamma-Funktion in $\mathbb{R}$. +\begin{definition}[Gamma-Funktion]{gammafunktion}{\cite[Kap.~XI, S.~142]{nielsen1906}} + Sei $x>0\in\mathbb{R}$, dann ist die \textbf{Gamma-Funktion} definiert durch + \begin{align*} + \Gamma(x):\mathbb{R^+}&\rightarrow\mathbb{R}\\ + x&\mapsto\int_{0}^{\infty}t^{x-1}e^{-t} dt + \end{align*} +\end{definition} + +Die Gamma-Funktion berechnet wie folgt die Fakultät in $\mathbb{N}$: +\begin{satz}[Vergleich mit der Fakultät]{fakultaetgammafunktion} + Für $n\in\mathbb{N}$ gilt: + \[ + \Gamma(n+1)=n! + \] +\end{satz} + \subsection{Stirling-Approximation} Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der Fakultät einer Zahl. Die Approximation ist vorallem in der Stochastik und der statistischen Physik ein unverzichtbares Werkzeug. Wir schauen sie daher im Folgenden im Detail an. \begin{satz}[Stirlingformel]{stirlingformel}{\cite{freitag1995-oj}} @@ -332,33 +351,10 @@ Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der F \end{proof} \subsection{Taylor-Approximation} -Um eine Taylorpproximation für eine Funktion zu finden, müssen wir zunächst nachweisen, dass die Funktion die wir approximieren wollen, $(n+1)$-mal differenzierbar ist. Da wir unsere Approximation nach dem quadratischen Term abbrechen werden genügt es zu zeigen, dass $\ln(1-x)$ dreimal differenzierbar ist. -\begin{lemma}{dreimalableiten} +Um eine Taylorpproximation für eine Funktion zu finden, müssen wir zunächst nachweisen, dass die Funktion die wir approximieren wollen, $(n+1)$-mal differenzierbar ist. Da wir unsere Approximation nach dem quadratischen Term abbrechen werden genügt es zu zeigen, dass $\ln(1-x)$ dreimal differenzierbar ist. Da $\ln(1-x)$ sogar unendlich oft differenzierbar ist, ist diese Aussage trivial. +\begin{proposition}{dreimalableiten} $f(x)=\ln(1-x)$ ist mindestens dreimal differenzierbar. -\end{lemma} -\begin{proof} - Zum Beweis berechnen wir die drei Ableitungen. - \begin{align*} - f'(x)&=(\ln{(1-x)})'\\ - &=\frac{1}{1-x}\cdot(1-x)'&&\text{Kettenregel}\\ - &=-\frac{1}{1-x}\\\\ - f''(x)&=(f'(x))'\\ - &=\left(-\frac{1}{1-x}\right)'\\ - &=\frac{(-1)'\cdot(1-x)-(-1)\cdot(1-x)'}{(1-x)^2}&&\text{Quotientenregel}\\ - &=\frac{(1-x)'}{(1-x)^2}&&\text{da } (-1)'\cdot(1-x)=0\\ - &=\frac{1}{(1-x)^2}\\\\ - f'''(x)&=(f''(x))'\\ - &=(\frac{1}{(1-x)^2})'\\ - &=\frac{(-1)'\cdot(1-x)^2-(-1)\cdot\left((1-x)^2\right)'}{\left((1-x)^2\right)^2}&&\text{Quotientenregel}\\ - &=\frac{\left((1-x)^2\right)'}{(1-x)^4}&&\text{da } (-1)'\cdot(1-x)^2=0\\ - &=\frac{-2\cdot(1-x)}{(1-x)^4}&&\text{Kettenregel}\\ - &=\frac{-2}{(1-x)^3} - \end{align*} -\end{proof} -Nun haben wir nachgewiesen, dass die Funktion $\ln(1-x)$ dreimal differenzierbar ist. -\begin{bemerkung} - Mit einem Blick auf die Funktion und wie ihre Ableitungen gebildet werden, ist leicht zu erkennen, dass $\ln(x-1)$ sogar unendlich oft differenzierbar ist. -\end{bemerkung} +\end{proposition} Bevor wir die Approximation nachweisen, ist eine formale Definition des Satzes von Taylor und dem Taylorpolynom nötig, wobei letzteres aus dem Satz von Taylor folgt. \begin{satz}[Satz von Taylor]{taylor}{\cite{enders_analysis_2024}} Seien $n\in\mathbb{N}_0, f:[a,b]\rightarrow\mathbb{R}$ eine $(n+1)$-mal differenzierbare Funktion und $x_0\in[a,b]$. @@ -402,7 +398,7 @@ Nun können wir unsere Approximation von $\ln(1-x)=-x-\frac{x^2}{2}+O(x^3)$ zeig Das zweite Taylor-Polynom lautet \begin{align*} T_{\ln(1-x),0,2}(x)&=\sum_{k=0}^{2}\frac{1}{k!}\cdot (\ln(1-x_0))^{(k)}(x-x_0)^k\\ - &=\frac{1}{0!}\cdot\ln(1)\cdot 1+\frac{1}{1!}\cdot(\ln(1))^{(1)}\cdot x+\frac{1}{2!}\cdot(\ln(1))^{(2)}\cdot x^2&&\text{\cref{lem:dreimalableiten}}\\ + &=\frac{1}{0!}\cdot\ln(1)\cdot 1+\frac{1}{1!}\cdot(\ln(1))^{(1)}\cdot x+\frac{1}{2!}\cdot(\ln(1))^{(2)}\cdot x^2&&\text{\cref{prop:dreimalableiten}}\\ &=0+1\cdot(-1)\cdot x+\frac{1}{2}\cdot(-1)\cdot x^2\\ &=-x-\frac{x^2}{2} \end{align*} @@ -459,23 +455,5 @@ Im Plot können wir sehen, dass beide Funktionen um unsere Entwicklungsstelle $x \caption{Darstellung der Funktionen $\ln(1-x)$ und $-x-\frac{x^2}{2}$} \end{figure} Würden wir die Entwicklung weiterführen, approximiert die violette Funktion (unser Taylorpolynom) die orangene Funktion $(\ln(1-x))$ immer weiter. Graphisch lässt sich erkennen, dass für Werte im Intervall $I(-0.5,0.5)$ die Funktion $\ln(1-x) \sim -x\frac{-x}{2}$ plus ein kleiner Fehler ist. -\subsection{Gamma-Funktion} -% wird für Beweis der Stirling-Approximation gebraucht -Zum nachfolgenden Betrachtung und des Beweises der Stirling-Approximation betrachten wir nun die Gamma-Funktion, auch \textbf{Eulersches Integral zweiter Gattung} genannt. Sie erweitert die Fakultätsfunktion von den natürlichen Zahlen $\mathbb{N}$ auf reelle und komplexe Zahlen (mit einigen Ausnahmen). Hier betrachten wir der Einfachheit halber nur die Gamma-Funktion in $\mathbb{R}$. -\begin{definition}[Gamma-Funktion]{gammafunktion}{\cite[Kap.~XI, S.~142]{nielsen1906}} - Sei $x>0\in\mathbb{R}$, dann ist die \textbf{Gamma-Funktion} definiert durch - \begin{align*} - \Gamma(x):\mathbb{R^+}&\rightarrow\mathbb{R}\\ - x&\mapsto\int_{0}^{\infty}t^{x-1}e^{-t} dt - \end{align*} -\end{definition} - -Die Gamma-Funktion berechnet wie folgt die Fakultät in $\mathbb{N}$: -\begin{satz}[Vergleich mit der Fakultät]{fakultaetgammafunktion} - Für $n\in\mathbb{N}$ gilt: - \[ - \Gamma(n+1)=n! - \] -\end{satz} diff --git a/galton.pdf b/galton.pdf index 1efcf823fc6099397cb2cb2ee1b69e0f06594a9d..5656d6ffe01308e882f185f622e73e285e5f1b08 100644 GIT binary patch delta 101959 zcmaI7bzD^I*FG#mgMiX4ARPlULw7ewcXv0^jI?w!fRr>yNJ~n0w}gO7gQRr)#^X89 zH=gHx|CrC-_rBM$Qixp zs5q$9sMw`#J*W(+goQCIoh+Qq>@Z+m5lBo(U@$KaY-$id0M(Oow%~f!v8~-E!qE>r znr2k{>OtCtl|uF{v^Pr{?costcE}7{t%iM+L&KNa^{#C1U~c*r; 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