概要
階乗の一般化
$$\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}dt$$
> gamma(1)
[1] 1
> gamma(2)
[1] 1
> gamma(3)
[1] 2
> gamma(4)
[1] 6
> gamma(5)
[1] 24
> gamma(1.5)
[1] 0.886227
> gamma(0.5)
[1] 1.772454
> gamma(0.1)
> curve(gamma,0.1,4)
$z=n+1$ で整数のとき・・・
$$\Gamma(n+1)=n!$$
不完全ガンマ関数として次の２種類がある。
$$\gamma(z,x)=\int_0^x t^{z-1}e^{-t}dt$$
$$\Gamma(z,x)=\int_x^\infty t^{z-1}e^{-t}dt$$
ガンマ関数の性質
$$\Gamma(x+1)=x\Gamma(x)$$
一階の微分
$$\Gamma'(x)=\Gamma(x)\frac{\Gamma'(x)}{\Gamma(x)}=\Gamma(x)\varphi(x)$$
二階の微分
$$\Gamma”(x)=\Gamma(x)\left(\frac{\Gamma'(x)}{\Gamma(x)}\right)^2+\Gamma(x)\left(\frac{\Gamma'(x)}{\Gamma(x)}\right)’=\Gamma(x)\varphi(x)^2+\Gamma(x)\varphi'(x)$$
$\varphi(n)$は・・・
$$\varphi(1)=\frac{\Gamma'(1)}{\Gamma(1)}=-\gamma$$
$$\varphi(n)=\frac{\Gamma'(n)}{\Gamma(n)}=-\gamma+\sum_{k=1}^{n-1}\frac{1}{k}, \ n\gt 2$$
> psigamma(1)
[1] -0.5772157
> psigamma(2)
[1] 0.4227843
> psigamma(3)
[1] 0.9227843
> curve(psigamma,0.2,4)
$\varphi'(n)$は・・・
$$\varphi'(1)=\left(\frac{\Gamma'(1)}{\Gamma(1)}\right)’=\frac{\pi^2}{6}$$
$$\varphi'(n)=\left(\frac{\Gamma'(n)}{\Gamma(n)}\right)’=\frac{\pi^2}{6}-\sum_{k=1}^{n-1}\frac{1}{k^2}, \ n\gt 2$$
> psigamma(1,1)
[1] 1.644934
> pi*pi/6
[1] 1.644934
> psigamma(3,1)
[1] 0.3949341
> psigamma(1,1)-1-1/4
[1] 0.3949341