Notation of "defined for all complex numbers except the negative integers and zero"

Question

The Euler and Weierstrass forms of the gamma function are :

$$\mathop{\mathrm{\Gamma}}\left(z\right)=\frac{1}{z}\prod^{\infty}_{n=1}\frac{\left(1+\frac{1}{n}\right)^{z}}{\left(1+\frac{z}{n}\right)}=\frac{e^{-\gamma z}}{z}\prod^{\infty}_{n=1}\frac{e^{\frac{z}{n}}}{\left(1+\frac{z}{n}\right)},\quad\mbox{for}\;z \in \mathbb{C}$$

but how do you note mathematically instead of $z\in\mathbb{C}$ : "for all complex numbers except the negative integers and zero" ?

(+bonus question: can you confirm me that the Euler and Weiestrass forms are striclty equivalent ?)

Answer 1

If your natural numbers include $0,$ you could say $$z\in\Bbb C\setminus\{-n:n\in\Bbb N\},$$ and if not, you could say $$z\in\Bbb C\setminus\bigl\{-n:n\in\Bbb N\cup\{0\}\bigr\}.$$ If you're used to $\Bbb C^\times$ meaning the non-zero complex numbers (those with multiplicative inverses), then regardless of whether your natural numbers include $0,$ you could just say $$z\in\Bbb C^\times\setminus\{-n:n\in\Bbb N\}.$$ The Euler and Weierstrass forms are indeed equivalent.

Answer 2

I suggest, You could also simply say that the three expressions define the same meromorphic function on (all of) $\mathbb C$, since "meromorphic on $D$" already stands for "holomorphic on $D$ except for a set of isolated poles"

Answer 3

You can say

Let $\Omega$ be the set of complex numbers excluding the negative integers. Then the above formulas are valid in $\Omega$.