Is there an interval notation for complex numbers?

Question

Just as $$\{x \in \mathbb{R}: a \leq x \leq b\}$$ can be written in the more-compact form $[a,b],$ is there an analogous notation for $$\{z \in \mathbb{C}:z=x+yi, x \in[a,b], y \in[c,d]\} \quad ?$$

Pictorially, the set of all $z \in \mathbb{C}$ lying in the green area is the set that I'd like to express in a more concise form:

Answer 1

Maybe just define something as $[a,b]+[c,d]i$ ?

Answer 2

Perhaps $\{z \in \mathbb{C}: \operatorname{Re}(z) \in [a,b], \; \operatorname{Im}(z) \in [c,d]\}$.

The complex numbers have no inherent order, so unless you invent something like $[[a+ci, b+di]]$ I know no more compact way to write this.

Answer 3

Well, Cartesian product could work here, $\{z\in\mathbb{C}:\Re(z)\in[a,b],\Im(z)\in[c,d]\}$ as $[a,b]\times[ic,id]$.

But I've never seen a standardized notation for something like this, except for circular regions, such as $|z+1|<4$.

Answer 4

As far as I know there is no widely recognized standard notation. Define one in your paper/book/essay if you need it often.

Answer 5

During the Complex Analysis course I took we used $$[z,w]:=\{(1-t)z+tw:t\in[0,1]\},\quad z,w\in\mathbb{C},$$ which coincides with normal intervals when $z,w\in\mathbb{R}$. But you should define this yourself, because people won't know what you mean if you don't.

Answer 6

You could write it like this:

$\{(x + yi) \in \mathbb{C}: x \in [a,b], \; y \in [c,d]\}$.

Alternatively, choose one of these:

$[a,b] \times i[c,d]$

$[a,b] \times [ic,id]$

Answer 7

A teacher of mine once used an "interval notation" for boxes in $\Bbb R^n$. If $\vec{a} = (a_1, \ldots, a_n)$ and $\vec{b} = (b_1, \cdots, b_n)$, then: $$]\vec{a}, \vec{b}[ = ]a_1, b_1[ \times \cdots \times ]a_n, b_n[ = \prod_{i = 1}^n ]a_i, b_i[$$ Maybe you can adapt it for what you need.

Answer 8

Maybe not the answer you're looking for, but you can include the picture and then write:

Let $X$ be a rectangle in the complex plane, as depicted.

Answer 9

Since $\mathbb C$ is just $\mathbb R^2$ with specific operations, the notation $[a,b]\times[c,d]$ obviously do the trick.