Complex Mapping of a Region Problem
Hello, above is the problem that I'm working on. From what I can tell, in part a the result the region whose upper bound in polar coordinates is bounded by $\pi/3$ and lower bound is the x-axis, which extends to infinity, gets mapped to the upper half of the coordinate plane.
Graph 1
In part b, the results obtained in part a are then mapped to everything above $y=-1$ ? 
Overall, I'm quite confused, it would be nice if I could figure out the math behind what is going on, my understanding is that the $z^3$ transform causes any angle to be tripled( using the knowledge that $z^2$ doubles angles), but I am not certain how to extend this knowledge to the rest of the problem.
Many thanks in advance.
 A: The only mathematics involved in the first part is that you’re taking a point $z=re^{i\theta}$, where $r$ is the distance from the origin, so $r\ge0$, and where $\theta$ is the angle between the positive real axis and the ray that proceeds from the origin through $z$ out infinitely far. And you’re cubing $z$ to get $r^3e^{3i\theta}$, so getting another point whose distance is (of course) still positive, and whose angle is three times the previous one, so your image point may (depending on your choice of the original $z$) be any point in the upper half plane.
For the second part, you’re starting with the upper half plane, and mapping it somehow by means of a “fractional linear transformation”. You need to know the properties of these special maps. It sounds from your question as if you haven’t seen these, and I can only give you a crude sketch of what’s going on. You take a matrix of complex numbers, $M=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ with the property that $\det M\ne0$, and associate to it the mapping
$$
T_M: z\mapsto\frac{az+b}{cz+d}\,.
$$
You need to verify that $T_{MN}=T_M\circ T_N$, in other words that you have a good homomorphism; and that these transformations are one-to-one, onto on the extended complex plane $\Bbb C\cup\{\infty\}$; and that the kernel of the homomorphism consists of just the scalar matrices $\begin{pmatrix}a&0\\0&a\end{pmatrix}$ with $a$ being  nonzero complex numbers. It’s not so easy to verify, but it can be proved that one of these transformations maps generalized circles to generalized circles; these are the circles and the straight lines. So, for instance, $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ gves $z\mapsto1/z$, which I hope you know maps each circle centered $0$ to a circle centered at $0$; you should be able to see that the vertical line of points of the form $1+ti$ goes to the citcle centered at $1/2$, radius $1/2$.
In most cases, you can get a good idea of what a f.l.t. does by seeing what it does to the points $0$, $1$, and $\infty$ of the extended complex plane, and you may get more help by seeing what points of the domain plane go to these three, as well. In your specific case, I think you’ll understand very well by evaluating at $-1$, $0$, $1$, $\infty$, and $i$.
