Zeroes of Gamma Function

Theorem

The Gamma function is never equal to $0$.

Proof

Suppose $\exists z$ such that $\map \Gamma z = 0$.

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

The Euler form, equated with zero, yields

$\ds 0 = \frac 1 z \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac 1 n}^z \paren {1 + \frac z n}^{-1} }$

It is clear that $\dfrac 1 z \ne 0$, 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 $\set {0, -1, -2, \ldots}$, which is discussed at Poles of Gamma Function.

$\blacksquare$