1. ## Domain of definition

The question

Describe the domain of definition that is understood:

$f(z)=\frac{1}{1-|z|^2}$

My attempt:
I noticed that $|z|^2=z\bar{z}$ and that $z=re^{i\theta}$ in polar form. So,

$z\bar{z}=re^{i\theta}re^{-i\theta}=r^2e^{0}=r^2$

Thus the denominator becomes 0 if $r=\pm1$, so there is no domain at these points. Is this correct? Any assistance would be truly appreciated!

2. |z|^2 is a real number, so 1/(1 - |z|^2) is also a real number. For what values will this real number be defined?

3. When (1 - |z|^2) != 0

Or, x^2 + y^2 != 1

4. Correct.

So the domain is all z such that |z| =/= 1.