Let W be the domain ${Im(z) < 0, Re(z) > 0}$. Sketch and describe the image of W under the conformal map $f(z) = z^3$.
I have absolutely no idea how to tackle this practice exam question. I understand how to sketch W, but not sure how to describe it under $z^3$. I was trying to figure it out with a friend who completed this unit prior, and he mentioned that you can view W as $e^{ix}, 0<x<{\pi/2}$. But this doesn't really help.
Cubing is a multiplicative operation, and multiplication of complex numbers is best understood by considering the polar representation of complex numbers. Your set $W$ consists of the nonzero complex numbers with arguments between $-\pi/2$ and $0$, viz., those $z=re^{it}$ with $r>0$ and $-\pi/2<t<0$. Then $z^3$ will have argument between $-3\pi/2$ and $0$. Alternatively, consider $z^3=r^3e^{i3t}$ and ask: what values can $r^3$ take, and what values can $3t$ take.