Domain of definition

The question

Describe the domain of definition that is understood:

http://latex.codecogs.com/png.latex?...ac{1}{1-|z|^2}

My attempt:
I noticed that http://latex.codecogs.com/png.latex?|z|^2=z\bar{z} and that http://latex.codecogs.com/png.latex?z=re^{i\theta} in polar form. So,

http://latex.codecogs.com/png.latex?...}=r^2e^{0}=r^2

Thus the denominator becomes 0 if http://latex.codecogs.com/png.latex?r=\pm1, so there is no domain at these points. Is this correct? Any assistance would be truly appreciated!

|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?

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

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

Correct.

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