Aritalab:Lecture/Math/Function

From Metabolomics.JP
< Aritalab:Lecture | Math(Difference between revisions)
Jump to: navigation, search
(Created page with "==ガンマ関数== <math> \Gamma(z) = \int^\infty_0 e^{-t} t^{z-1} dt </math> は階乗の一般化で <math> \Gamma(z+1) = z \Gamma(z)\, </math> を満たす。''z'' が正...")

Revision as of 21:05, 20 July 2011

ガンマ関数

\Gamma(z) = \int^\infty_0 e^{-t} t^{z-1} dt

は階乗の一般化で \Gamma(z+1) = z \Gamma(z)\, を満たす。z が正の整数の場合は \Gamma(z + 1) = z!\,。

例
\Gamma(1) = 1\,
\Gamma(1/2) = \sqrt{\pi}\,
\Gamma(3/2) = \sqrt{\pi}/2\,
\Gamma(z+1/2) = \frac{(2n)!}{2^{2n}n!}\sqrt{\pi}
近似
\Gamma(z+1) = z! \sim \sqrt{2\pi z} e^{-z}z^z Stirling の公式


ベータ関数

\Beta(x,y) = \int^1_0 t^{x-1} (1-t)^{y-1} dt

例
\Beta(x,y) = \Beta(y,x)\,
\Beta(x,y) = \frac{x+y}{y} \Beta(x,y+1)\,
\Beta(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x + y)}
\Beta(\frac{1}{2},\frac{1}{2}) = \pi\,
Retrieved from "http://metabolomics.jp/mediawiki/index.php?title=Aritalab:Lecture/Math/Function&oldid=254911"
Personal tools
Namespaces

Variants
Actions
Navigation
metabolites
Toolbox