How to transform a set of complex numbers by applying the function $f(z)=\frac{1}{z^2}$

Question

I have a set of complex numbers, $S = \left\{ z \in C 1 \leq \lvert z \rvert \leq 2, \frac{\pi}{4} \leq \lvert arg(z) \rvert \leq \frac{\pi}{2} \right\}$. How do I determine the subset $ f(S) = \frac{1}{z^2} $?

Answer 1

Hint: Consider what this function does to (1) rays from the origin, and (2) circles around the origin.

To do this, look at $f(r e^{i\theta_0})$ as $r$ varies (with $\theta_0$ fixed), and at $f(r_0e^{i\theta})$ as $\theta$ varies (with $r_0$ fixed).

Note that $f(re^{i\theta}) = \frac{1}{r^2}e^{i(-2\theta)}$.