TITLE: Find conformal mapping from sector to unit disc
QUESTION [6 upvotes]: Find a conformal mapping between the sector $\{z\in\mathbb{C}  : -\pi/4<\arg(z) <\pi/4\}$ and the open unit disc $D$.
I know that it should be a Möbius transformation, but other than that I am very stuck, any help would be much appreciated.

REPLY [8 votes]: You know that there is a conformal mapping from the unit disk to the upper half plane given by:
$$z\mapsto -i\frac{z-i}{z+i}$$
Which sends

to

But then you know that the transformation $z\mapsto \sqrt z$ taking the principal value sends the upper half plane to the region you are desiring. This gives:

Reversing these mappings gives:
$$w \mapsto \frac{iw^2+1}{-w^2-i}$$
Which you will see is a conformal mapping sending the first quadrant to the unit disk.