Definition:Gamma Function/Integral Form

Definition
The Gamma function $\Gamma: \C \to \C \ $ is defined, for the open right half-plane, as:
 * $\displaystyle \Gamma \left({z}\right) = \mathcal M \left\{ {e^{-t} }\right\} \left({z}\right) = \int_0^{\to \infty} t^{z-1} e^{-t} \ \mathrm d t$

where $\mathcal M$ is the Mellin transform

and for all other values of $z$ except the non-positive integers as:
 * $\Gamma \left({z + 1}\right) = z \Gamma \left({z}\right)$

Also see

 * Equivalence of Definitions of Gamma Function

Historical Note
The symbol $\Gamma \left({x}\right)$ was introduced by Adrien-Marie Legendre.