Aritalab:Lecture/Math/Function

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

$\Gamma(1) = 1\,$

$\Gamma(1/2) = \sqrt{\pi}\,$

$\Gamma(3/2) = \sqrt{\pi}/2\,$

$\Gamma(z+1/2) = \frac{(2n)!}{2^{2n}n!}\sqrt{\pi}$

Stirling の近似

$\Gamma(z+1) = z! \sim \sqrt{2\pi z} \Big(\frac{z}{e}\Big)^z$

ベータ関数

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

例

$\Beta(x,y) = \Beta(y,x)\,$

$\Beta(x,y) = \frac{x+y}{y} \Beta(x,y+1)\,$

$\Beta(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x + y)}$

$\textstyle \Beta(\frac{1}{2},\frac{1}{2}) = \pi\,$