Holomorphy of the gamma function

Remarks

On the half-plane $\operatorname{Re} z > 0$, Euler's integral $\int_0^\infty t^{z-1} e^{-t}\,dt$ is holomorphic, see Euler integral for the gamma function. On $\operatorname{Re} z > -n$, with $z \notin \set{0, \ldots, -n+1}$, the formula $\Gamma(z + n)/\big(z (z + 1) \cdots (z + n - 1)\big)$ is holomorphic and agrees with the previous one on the overlap; these pieces glue to a holomorphic function on $U$. Uniqueness follows from the identity theorem, because $U$ is open and connected and $(0, \infty) \subset U$ has accumulation points. The function cannot be extended holomorphically to any point of $\set{0, -1, -2, \ldots}$, since it has poles there. In Mathlib the statement is Complex.differentiableAt_Gamma.

Going deeper

For a student

Holomorphy is what lets one prove identities for $\Gamma$ on the real line and then transfer them to complex arguments, by the identity theorem. The reflection and duplication formulas are usually proved in this way. Together with the description of the poles, it says that $\Gamma$ is a meromorphic function on $\C$.