Definition:Gamma Function/Integral Form

Definition
The gamma function $\Gamma: \C \setminus \Z_{\le 0} \to \C$ is defined, for the open right half-plane, as:
 * $\ds \map \Gamma z = \map {\MM \set {e^{-t} } } z = \int_0^{\to \infty} t^{z - 1} e^{-t} \rd t$

where $\MM$ is the Mellin transform.

For all other values of $z$ except the non-positive integers, $\map \Gamma z$ is defined as:
 * $\map \Gamma {z + 1} = z \map \Gamma z$

Also see

 * Equivalence of Definitions of Gamma Function