Zeroes of the Gamma Function

Theorem

The Gamma function is never equal to $0$.

Proof

Suppose $\exists z$ such that $\Gamma \left({z}\right) = 0$.

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

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 $\dfrac 1 z \neq 0$, 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 $\left\{{0,-1,-2, \ldots}\right\}$, which is discussed at Poles of the Gamma Function.

$\blacksquare$