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

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

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.