Gamma function is zero-free

Statement

The gamma function $\Gamma$ has no zeros. In particular, since we also know that all poles of the gamma function are simple poles, the reciprocal of the gamma function is an entire function.

Facts used

• Reflection principle for gamma function

Proof

The reflection principle for the gamma function states that for $z$ not an integer:

$\Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin(\pi z)}}$

Thus, if $z$ is not an integer, $\Gamma (z)\neq 0$. Further, for $z$ a positive integer, $\Gamma (z)=(z-1)!$, which is again nonzero. Finally, when $z$ is a negative integer, $\Gamma$ has a simple pole at $z$.