User:Gamma

$\ds \map \Gamma z = \int_0^\infty t^{z-1} e^{-t}\,dt$

$\ds \begin {align} \map \Gamma z & = \lim_{n \mathop \to \infty} \frac {n! \; n^z} {z \; \paren {z + 1} \cdots \paren {z + n} } = \frac 1 z \prod_{n \mathop = 1}^\infty \frac {\paren {1 + \frac 1 n}^z} {1 + \frac z n} \\ \map \Gamma z & = \frac {e^{-\gamma z} } z \prod_{n \mathop = 1}^\infty \paren {1 + \frac z n}^{-1} e^{z / n} \\ \end{align}$