I've been reviewing conformal maps lately and I feel like I understand them, until I reach a certain point in a problem. I'll give an example to illustrate my confusion:
In the process of constructing a conformal map $\{z\in \mathbb{C}\mid |z|>1 \text{ and } z\notin (0,\infty)\}\to \{z\in\mathbb{C}\mid Re(z)>0\},$ I map the first region successfully to the upper half of the unit disk $\{z\in \mathbb{D}\mid Im(z)>0\}$ (using $z^{1/2}$ and $1/z$) and then I want to map this region onto a quadrant.
I know that the inverse of the Cayley transform (the map the takes $\mathbb{D}\to \text{Upper Half Plane}$ conformally) is $z\mapsto i\frac{1-z}{1+z},$ but I don't have any idea how to figure out the range of this function when restricted to the upper half of the unit disk.
So, my first question is: is there a general way to determine the range of these maps? Especially when restricting a known conformal mapping to a smaller domain?
My second question is: I have often seen linear fractional transformations determined by 3 points. I understand that lines and circles on $\mathbb{C}_\infty$ are determined in this way, but I get confused when it comes to disks and half planes.
For example, if I wanted to determine a linear fractional transformation from the unit disk, I would likely pick $i,-i,$ and $1.$ However, if I wanted to pick 3 more points for the upper half plane, I might pick $-1,1,$ and $\infty.$ How do I know the transformation I design will give me the upper half plane and not the lower half plane? Is there a clearer way to always pick these 3 points? Furthermore, is there an intuitive way to know where to send which point? So far, I've just been assigning them pretty randomly.
Any help and clarification is greatly appreciated!