Good notation to require that z ≠ 0, -1, -2, -3, ...

Question

An engineer in my 50s, I have gradually been trying to improve my mathematical notation, an effort your answers here at Math.SE have partly inspired. Therefore, if $z \in \mathbb C$, may I ask whether I have written this in good style? $$z \notin \{n | n \in \mathbb Z \wedge n \le 0\}$$ Or is this better? $$z \notin \{n \in \mathbb Z | n \le 0\}$$ Or is this better? $$z \notin \{n \in \mathbb Z \le 0\}$$ Or even this? $$z \notin \{\mathbb Z \le 0\}$$ Or this? $$z \notin\mathbb Z \le 0$$ Or something else? If you find a flaw in any or all of my notations (and I do not doubt that you will), then would you illuminate my misconception?

The $z$ is an arbitrary complex number except that $z = 0, -1, -2, -3,\ldots$ [the poles of $\Gamma(z)$] are not allowed.

Answer 1

Since $z\in\mathbb{C}$ we should also provide this information. Complex numbers which are not zero or negative integers are often specified as \begin{align*} z\in\mathbb{C}\setminus\{0,-1,-2,\ldots\} \end{align*}

Answer 2

Personally, I would say $z\not\in \mathbb{Z}^{\leq 0}$.

Answer 3

I don't like any of these. Too much notation makes questions harder to read, and none of these clarify what $z$ is allowed to be. If $z$ (notice! this is a total guess!) is supposed to be a complex number, I would say, "let $z \in \mathbb{C}$ be a complex number that is not a non-positive integer."

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Edit: I've just seen that $z$ is supposed to be an integer. In this case we should just write $z \in \mathbb{N}$. Most authors define the natural numbers $\mathbb{N}$ to be the set of positive integers.

Answer 4

So one silly answer is just to use $\mathrm{dom}(\Gamma)$ .

Another possibility is to use $ \mathbb{C} \setminus \mathbb{Z}_{\le 0} $ , which was suggested in a comment.

In prose I'm partial to the following:

Let $z$ be a complex number that is not a negative integer or zero.

or

Let $z$ be a complex number that is not a negative integer and is not zero.