Plot of a domain in the complex plane

Question

I am trying to plot the following domain in the complex plane:

$\lbrace x\in\mathbb{C}|\: |x^{2}-1|<r\rbrace$

for some $r>1$.

I know that in general to take a square root of a complex number $z$ one would have to go to polar coordinates and the get $\sqrt{z}=r^{1/2}\exp (i\theta/2)$.

How do I proceed in taking a "square root of a disk"? in the sense that I can also rewrite it as $\lbrace \sqrt{w}|\: |w-1|<r\rbrace$. What kind of geometrical object is it? Still a disk?

(it's not a homework question, I'm just curious)

Answer 1

Let be $x=u+iv:$ $$ \sqrt{(u^2-v^2-1)^2+(2uv)^2}=|u^2-v^2-1+2uvi|=|x^2-1|<r. $$