Definition:Gamma Function/Integral Form

Definition

The Gamma function $\Gamma: \C \to \C$ is defined, for the open right half-plane, as:

$\displaystyle \map \Gamma z = \map {\mathcal M \set {e^{-t} } } z = \int_0^{\to \infty} t^{z - 1} e^{-t} \rd t$

where $\mathcal M$ 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

• Results about the Gamma function can be found here.

Historical Note

The integral form of the gamma function $\Gamma \left({z}\right)$ was discovered by Leonhard Paul Euler.