Zeroes of Gamma Function

Theorem
The Gamma function is never equal to $0$.

Proof
Suppose $\exists z$ such that $\Gamma(z) =0$.

We examine the Euler form of the gamma function, which is defined for $\C - \left\{{0,-1,-2, \dots }\right\}$.

The Euler form, equated with zero, yields

$\displaystyle 0 = \frac{1}{z} \prod_{n=1}^\infty \left({ \left({ 1+\frac{1}{n} }\right)^z \left({1+\frac{z}{n} }\right)^{-1} }\right)$

It is clear that $\frac{1}{z} \neq 0$, so we may divide this out for $z$ in the area of definition.

Now it is clear that as $n \to \infty$, each of the two halves of the term in the product will tend to $1$ for any $z$, and there is no $z$ which yields zero for any $n$ in either of the product terms.

Hence this product will not equal $0$ anywhere.

This leaves only the question of the behavior on $\left\{{0,-1,-2, \dots }\right\}$, which is discussed at Poles of the Gamma Function.