\begin{definition}[Definition:Gamma Function/Integral Form]
The '''gamma function''' $\Gamma: \C \to \C$ 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:
:$\map \Gamma {z + 1} = z \map \Gamma z$
{{MissingLinks|open right half-plane}}
{{explain|How does the last definition translate into $\map \Gamma z$ being defined on $\C \setminus -\N$?}}
\end{definition}