Definition:Gamma Function/Euler Form

Definition
The Euler form of the Gamma function is:
 * $\displaystyle \Gamma \left({z}\right) = \frac 1 z \prod_{n \mathop = 1}^\infty \left({\left({1 + \frac 1 n}\right)^z \left({1 + \frac z n}\right)^{-1}}\right) = \lim_{m \to \infty} \frac {m^z m!} {z \left({z+1}\right) \left({z+2}\right) \ldots \left({z+m}\right)}$

which is valid except for $z \in \left\{{0, -1, -2, \ldots}\right\}$.

Also see

 * Equivalence of Definitions of Gamma Function