Let $Z : = \{ (x, y) \in \mathbb{C} : x > 0 \}$ be the D3740: Complex open right half-plane.
The gamma function is the D4881: Complex function
\begin{equation}
\Gamma : Z \to \mathbb{C}, \quad
\Gamma(z) = \int^{\infty}_0 t^{z - 1} e^{-t} \, d t
\end{equation}