From 441b2276ea12a0feb3ea12bd746132a04a271682 Mon Sep 17 00:00:00 2001 From: Elias Fierke Date: Sat, 11 Jul 2026 12:12:51 +0200 Subject: [PATCH] =?UTF-8?q?fix:=20Beweis=20Stirling-Apprixomation=20=C3=BC?= =?UTF-8?q?berarbeitet?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Mathematischer_Hintergrund.tex | 198 ++++++++++++++++++++----- galton.pdf | Bin 453107 -> 493249 bytes 2 files changed, 165 insertions(+), 33 deletions(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index d479566..5cdeade 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -298,58 +298,190 @@ Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der F \] \end{satz} \begin{proof} - Der Ausgangspunkt ist die nach \cref{def:gammafunktion} definierte \textbf{Gamma-Funktion} + Der Ausdruck aus dem \cref{satz:stirlingformel} ist äquivalent zu \[ - \Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t} dt + \ln {n!}\sim \ln{\sqrt{2\pi}}+\left(n+\frac{1}{2}\right)\ln{n-n} \] - Wir schreiben den Integranden als Exponentialfunktion, dann haben wir + Sei nun \[ - t^{x-1}e^{-t} = \exp((x-1)\ln t-t) + d_n := \ln{n!}-\left(n+\frac{1}{2}\right)\ln{n+n} \] - also ist + Wir zeigen nun, dass \[ - \Gamma(x)=\int_{0}^{\infty}\exp((x-1)\ln t-t) dt + \lim_{n\rightarrow\infty}d_n =\ln\sqrt{2\pi} \] - Setze $t=xu$ und $f(u)=\ln u - u$, dann erhalten wir - \begin{align} - \Gamma(x)&=x^x\cdot\int_{0}^{\infty}u^{x-1}e^{-xu} du\\ - &=x^x\cdot\int_{0}^{\infty}\exp(x(\ln u-u)) du\\ - &=x^x\cdot \int_{0}^{\infty} e^{xf(u)}du - \end{align} - - Für große \(x\) wird das Integral durch die Umgebung des kritischen Punktes von \(f\) dominiert. Diese erhält man aus + Es gilt + \begin{align*} + d_n-d_{n+1}=&\ln n!-\ln{(n+1)!}-\left(n+\frac{1}{2}\right)\ln n \\&+\left(n+1+\frac{1}{2}\right)\ln{(n+1)}+n-(n+1)\\ + =&\ln\frac{n!}{(n+1)!}+\left(n+\frac{1}{2}\right)\left(\ln(n+1)-\ln (n)\right)+\ln(n+1)-1\\ + =&-\ln(n+1)+\left(n+\frac{1}{2}\right)\ln\frac{n+1}{n}+\ln(n+1)-1\\ + =&\frac{2n+1}{2}\ln(\frac{n+1}{n})-1\\ + =&(2n+1)\cdot\frac{1}{2}\ln\frac{1+\frac{1}{2n+1}}{1-\frac{1}{2n1}}-1 + \end{align*} + Für $-1\frac{12}{12\cdot12n^2+12\cdot12n+24n+13} \\ + &= \frac{12}{12\cdot12n^2+12\cdot14n+13} + = \frac{12}{(12n+1)(12n+13)}\\ + &= \frac{1}{12n+1}-\frac{1}{12(n+1)+1} + \end{align*} + Setzen wir beide Ungleichungen zusammen, so erhalten wir \[ - f(u)\sim f(1)+\frac{f''(1)}{2}(u-1)^2 - = -1 - \frac{(u-1)^2}{2}. + \frac{1}{12n+1}-\frac{1}{12(n+1)+1} < d_n-d_{n+1} < \frac{1}{12n}-\frac{1}{12(n+1)} \] - - Einsetzen liefert die lokale Approximation + Aus + \begin{align*} + &\left(d_n-\frac{1}{12n}\right)-\left(d_{n+1}-\frac{1}{12(n+1)}\right)\\ + <\quad0\quad<\quad&\left(d_n-\frac{1}{12n+1}\right)-\left(d_{n+1}-\frac{1}{12(n+1)+1}\right) + \end{align*} + folgt, dass die Folge $\left\{d_n-\frac{1}{12n+1}\right\}$ monton fallend, die Folge $\left\{d_n-\frac{1}{12n}\right\}$ monoton wachsend ist. Beide Folgen haben dabei den Grenzwert $c:=\lim d_n$. + \begin{align*} + d_n-\frac{1}{12n}\quad&<\quad c\quad<\quad d_n-\frac{1}{12n+1}\\ + c+\frac{1}{12n+1}\quad&<\quad d_n\quad<\quad c+\frac{1}{12n} + \end{align*} + Aus unserem Start \[ - \Gamma(x)\sim x^x e^{-x}\cdot\int_0^\infty \exp\!\left(-\frac{x}{2}(u-1)^2\right)\,du. + d_n = \ln{n!}-\left(n+\frac{1}{2}\right)\ln{n+n} \] - - Da der Hauptbeitrag aus einer Umgebung von $u=1$ stammt, kann das Integral asymptotisch auf $\mathbb{R}$ erweitert werden. + folgt nun \[ - \int_{-\infty}^{\infty} \exp\!\left(-\frac{x}{2}(u-1)^2\right)\,du - = \sqrt{\frac{2\pi}{x}}. + e^{d_n} = {n!}\left(\frac{n}{e}\right)^{-1}n^{-\frac{1}{2}} \] - - Damit erhält man + An dieser Stelle notieren wir + \begin{align*} + e^{c+\frac{1}{12n+1}} &< e^{d_n} < e^{c+\frac{1}{12n}} \\ + e^c e^{\frac{1}{12n+1}} &< n! \left(\frac{n}{e}\right)^{-n} n^{-\frac{1}{2}} < e^c e^{\frac{1}{12n}} \\ + e^c \sqrt{n} \left(\frac{n}{e}\right)^n e^{\frac{1}{12n+1}} &< n! < e^c \sqrt{n} \left(\frac{n}{e}\right)^n e^{\frac{1}{12n}} + \end{align*} + Es bleibt zu zeigen, dass \[ - \Gamma(x)\sim \sqrt{2\pi}\, x^{x-\frac12} e^{-x}. + e^c=\sqrt{2\pi} + \] + Wir führen eine Hilfsrechnung über + \[ + I_n:=\int_{0}^{\pi/2}{sin^n x dx} + \] + Es gilt + \begin{align*} + I_n &= \int_{0}^{\pi/2} \sin^{n-1} x \cdot \sin x \, dx \\ + &= \left. \sin^{n-1} x \cdot (-\cos x) \right|_{0}^{\pi/2} - \\ + &\quad \int_{0}^{\pi/2} (n-1) \sin^{n-2} \cos x \cdot (-\cos x) \, dx \\ + &= (n-1) \int_{0}^{\pi/2} \sin^{n-2} x (1 - \sin^2 x) \, dx \\ + &= (n-1)(I_{n-2} - I_n) \\ + I_n &= \frac{n-1}{n} I_{n-2} \\ + I_0 &= \frac{\pi}{2} \\ + I_1 &= 1 \\ + I_2 &= \frac{1}{2} I_0 = \frac{1}{2} \cdot \frac{\pi}{2}\\ + I_{2n} &= \frac{1\cdot3\cdot5\dots(2n-1)}{2\cdot4\cdot6\dots(2n)}\frac{\pi}{2}\\ + I_{2n} &= \frac{2\cdot4\cdot6\dots(2n)}{1\cdot3\cdot5\dots(2n+1)} + \end{align*} + Weiterhin folgt + \begin{align*} + && 0 &< x < \frac{\pi}{2} \\ + &\Rightarrow & 0 &< \sin x < 1 \\ + &\Rightarrow & \sin^{2n-1} x &> \sin^{2n} x > \sin^{2n+1} x \\ + &\Rightarrow & I_{2n-1} &> I_{2n} > I_{2n+1} \\ + &\Rightarrow & \frac{I_{2n-1}}{I_{2n+1}} &> \frac{I_{2n}}{I_{2n+1}} > 1 \\ + &\Rightarrow & \frac{2n+1}{2n} &> \frac{1 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdots (2n-1) \cdot (2n+1)}{2 \cdot 2 \cdot 4 \cdot 4 \cdot 6 \cdot 6 \cdots (2n) \cdot (2n)} \cdot \frac{\pi}{2} > 1 \\ + &\Rightarrow & \lim \ & \frac{1 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdots (2n-1) \cdot (2n+1)}{2 \cdot 2 \cdot 4 \cdot 4 \cdot 6 \cdot 6 \cdots (2n) \cdot (2n)} \cdot \frac{\pi}{2} = 1 \\ + &\Rightarrow & \frac{\pi}{2} &= \lim \left( \frac{2 \cdot 4 \cdot 6 \cdots (2n)}{1 \cdot 3 \cdot 5 \cdots (2n-1)} \right)^2 \cdot \frac{1}{2n+1} \\ + && &= \lim \frac{2^{4n} (n!)^4}{((2n)!)^2 (2n+1)} + \end{align*} + Es ist also für $\lim_{n\rightarrow\infty}{\alpha_n}=\lim_{n\rightarrow\infty}{\beta_n}=0$: + \begin{align*} + n!&=e^c\sqrt{n}n^ne^{-n}e^{\alpha_n}\\ + (2n)!&=e^c\sqrt{2n}2^{2n}n^{2n}e^{-2n}e^{\beta_n} + \end{align*} + Einsetzen liefert schlussendlich + \[ + e^c=\sqrt{2\pi} \] - \end{proof} +Der Beweis entspricht im wesentlichen dem 2006 von Wolfgang Kössler geführten Beweis, siehe \cite{koessler-2006}. + +% Nicht vollständig / zu schwach +%\begin{proof} +% Der Ausgangspunkt ist die nach \cref{def:gammafunktion} definierte \textbf{Gamma-Funktion} +% \[ +% \Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t} dt +% \] +% Wir schreiben den Integranden als Exponentialfunktion, dann haben wir +% \[ +% t^{x-1}e^{-t} = \exp((x-1)\ln t-t) +% \] +% also ist +% \[ +% \Gamma(x)=\int_{0}^{\infty}\exp((x-1)\ln t-t) dt +% \] +% Setze $t=xu$ und $f(u)=\ln u - u$, dann erhalten wir +% \begin{align} +% \Gamma(x)&=x^x\cdot\int_{0}^{\infty}u^{x-1}e^{-xu} du\\ +% &=x^x\cdot\int_{0}^{\infty}\exp(x(\ln u-u)) du\\ +% &=x^x\cdot \int_{0}^{\infty} e^{xf(u)}du +% \end{align} +% +% Für große \(x\) wird das Integral durch die Umgebung des kritischen Punktes von \(f\) dominiert. Diese erhält man aus +% \[ +% f'(u)=\frac{1}{u}-1=0 \quad \Rightarrow \quad u=1. +% \] +% +% Weiter gilt +% \[ +% f''(u)=-\frac{1}{u^2}, \quad \text{also } f''(1)=-1. +% \] +% +% Damit besitzt \(f\) bei \(u=1\) einen stationären Punkt, und wir entwickeln \(f\) dort bis zur zweiten Ordnung: +% \[ +% f(u)\sim f(1)+\frac{f''(1)}{2}(u-1)^2 +% = -1 - \frac{(u-1)^2}{2}. +% \] +% +% Einsetzen liefert die lokale Approximation +% \[ +% \Gamma(x)\sim x^x e^{-x}\cdot\int_0^\infty \exp\!\left(-\frac{x}{2}(u-1)^2\right)\,du. +% \] +% +% Da der Hauptbeitrag aus einer Umgebung von $u=1$ stammt, kann das Integral asymptotisch auf $\mathbb{R}$ erweitert werden. +% \[ +% \int_{-\infty}^{\infty} \exp\!\left(-\frac{x}{2}(u-1)^2\right)\,du +% = \sqrt{\frac{2\pi}{x}}. +% \] +% +% Damit erhält man +% \[ +% \Gamma(x)\sim \sqrt{2\pi}\, x^{x-\frac12} e^{-x}. +% \] +% +%\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. 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