\begin{definition}[Definition:Gamma Function/Euler Form]
The '''{{AuthorRef|Leonhard Paul Euler|Euler}} form''' of the '''gamma function''' is:
:$\ds \map \Gamma z = \frac 1 z \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac 1 n}^z \paren {1 + \frac z n}^{-1} } = \lim_{m \mathop \to \infty} \frac {m^z m!} {z \paren {z + 1} \paren {z + 2} \cdots \paren {z + m} }$
which is valid except for $z \in \set {0, -1, -2, \ldots}$.
\end{definition}