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

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.

• What's wrong with just writing "[$z\in \mathbb{C}$ with] $z\not = 0, -1, -2, \dots$"? – anomaly Sep 14 at 20:39
• @anomaly Only that I did not know that that is an accepted notation. – thb Sep 14 at 20:41
• There's also an option of "$-z\notin\mathbb N$", assuming your readers know that $0\in\mathbb N$. – Oscar Cunningham Sep 16 at 12:28

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*}

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

• I assume that some of my other notations are overelaborate, awkward and/or wrong. If you elaborated, I would be interested. – thb Sep 14 at 20:32
• @thb The second one you gave is the only one that I think I would find in a text. $z\not\in \{n\in \mathbb{Z}\mid n\leq 0\}$, but this assume that the total domain is $\mathbb{Z}$. I now realize you might be talking about the complex plane. In which case, you might write it as, $z\in \mathbb{C}\setminus \mathbb{Z}^{\leq 0} = \{z\in\mathbb{C}| z\ne n, n\in \mathbb{Z}^{\leq 0}\}$. – Laarz Sep 14 at 20:39
• Yes, that is my trouble, isn't it? My notation was poor, so you did not know what I was talking about. I am indeed talking about the complex plane. Your answer shows me how to improve my notation. – thb Sep 14 at 20:45

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."

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.

• Yes, that is how I usually write. Good advice, but today I am explicitly trying to master the formalism. The $z$ is indeed complex: it's the same $z$ that appears in Abramowitz & Stegun. – thb Sep 14 at 20:33
• I have edited my question to clarify. – thb Sep 14 at 20:35
• +1 for minimizing notation. If necessary the OP can clarify that $0 \not\in \mathbb{N}$. – Ethan Bolker Sep 14 at 21:51

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.

• Besides wishing to learn the notation, I am most interested to learn that professional mathematicians may prefer plain English to symbolism. Titchmarsh's Theory of Functions uses plain English, but that was in the 1930s and I had somehow got the idea that mathematicians just didn't write that way any longer, except maybe when mathematicians condescend to communicate with engineers like me! Thanks for the notational instruction and also for the contrary advice. – thb Sep 15 at 14:32