I am supposed to represent the following sets:
\begin{align}A&=\{z\in\mathbb{C}:\Re(2z+iz)<0<\Im(z^2)\}, \\ B&=\{w\in\mathbb{C}:w=z^2, z\in A\}, \\ C&=\{u\in\mathbb{C}:u=1/z, z\in A\}.\end{align}
I think I have handled the first one. Writing $z$ in algebraic form, I derived $$A=\{(x,y)\in\mathbb{R^2}:y>2x, xy>0\},$$ so it should individuate the portion of the complex plane enclosed between the line $y=2x$, the positive $y$-semiaxes and the negative $x$-semiaxis.
I am lost with the other two. Sticking to $z=x+iy$ led me to $B=\{(s,t)\in\mathbb{R^2}:s+it=x^2-y^2+2ixy\}.$ Here's my problem: I can see that the ordinates of the points of $B$ lie on the upper half-plane, but how can the abscissas describe a function of two variables? I guess I have no clue about the third set, after using again $z=x+iy$.
I tried writing $z$ in the other forms but it didn't help me, what am I missing?
Hint The key observation is that the set $A$ is a union of two open sectors with vertex at $0$, which suggests describing them in polar form. For example, the sector in the first quadrant is $$\left\{ r e^{i \theta} : r > 0, \arctan 2 < \theta < \tfrac{\pi}{2} \right\} .$$
This is useful for describing $B$, $C$, as those sets are described in terms of powers of elements of $A$ and the expression of (integer) powers is especially simple in polar form: $(r e^{i \theta})^k = r^k e^{i (k \theta)}$.