ガンマ関数 (Gamma function)

Definition

Main definition

Γ(z)=∫0∞tz−1e−tdt\Gamma \left( z \right) = \int^{\infty }_{0} t^{z-1} e^{-t} dt
Γ(z+1)=zΓ(z)Γ(n)=(n−1)!\begin{aligned} \Gamma \left( z + 1 \right) & = z \Gamma \left( z \right) \\ \Gamma \left( n \right) & = \left( n - 1 \right) ! \end{aligned}

Alternative definitions

Γ(z)=limn→∞n!nzz(z+1)⋯(z+n)=1z∏n=1∞(1+1/n)z1+z/nΓ(z)=e−γzz∏n=1∞(1+zn)−1ez/n\begin{aligned} \Gamma \left( z \right) & = \lim_{ n \to \infty} \frac{ n! n^z } { z \left( z+1 \right) \cdots \left( z + n \right) } \\ & = \frac{ 1 }{ z } \prod_{n=1}^{\infty } \frac{ {\left( 1 + 1/n \right)}^{z} }{ 1 + z/n } \\ \Gamma \left( z \right) & = \frac{ e^{ -\gamma z } }{ z } \prod_{n=1}^{\infty } { \left( 1 + \frac{z}{n} \right) }^{-1} e^{z/n} \end{aligned}

where $ \gamma \approx 0.577216 $ is the Euler-Mascheroni constant.

γ=limn→∞(∑k=1n1k−ln(n))=∫1∞(1?x?−1x)dx\begin{aligned} \gamma & = \lim_{ n \to \infty } \left( \sum_{k=1}^{n} \frac{1}{k} - \ln (n) \right) \\ & = \int_{1}^{\infty } \left( \frac{1}{?x?} - \frac{1}{x} \right) dx \end{aligned}

Properties