Denoting the domain of a complex function

Question

If I have a function defined as so: $$f(z)=z^2$$ Can I limit its domain based on its argument (angle in its $re^{i\theta}$ form) like this? $$f(re^{i\theta}) = r^2e^{2i\theta}, 0 \leq \theta < \pi$$ Is this the correct notation and does this function become bijective if defined like this?

Answer 1

It is a bijection from upper half plane to (including non-negative real axis) to $\mathbb C $.