The question here is not so much to find the mapping, but exactly how one goes about to find the correct map.
I have $\{z\in\mathbb{C}: -\pi/4<\arg(z)<\pi/2\}$, which requires to be turned into a quarter disk in the lower left half plane.
Now most of the steps I understand, however what I do not understand is how to get the sector into an easier plane to work with (half plane, or whatever else).
As such the process in order to get through this first map is of much more interest to me than the actual solution. I would be very grateful if someone could explain to me how this particular type of mapping works.
Ronan
Best Answer
The easiest ways to deal with stuff like this (corners) is to use logarithms or fractional powers of $z$ (which use logarithms). So, for instance, taking a log here gives you a rectangle stretching to infinity on the left, which you can play around with, while squaring this gives you three quarters of the unit circle.
If you can get the shape to be a quadrant, you can square it to get a half plane.