Find conformal mapping from sector to unit disc

Solution 1:

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.

Solution 2:

Here is a plan: first, apply $z \to z^2$. It will conformally map your sector onto the half-plane $\mathrm{Re}(z) > 0$. Then find a Möbius transformation that will map this half-plane to the unit disk.