Q:

Map $\{ z=x+iy : |x|\geq y \}$ with a branch cut on the negative imaginary axis from $[-i,0]$ to the unit disc

I have a conformal map that I've been having problems with.

Map the set $\{ z=x+iy : |x|\geq y \}$ with a branch cut on the negative imaginary axis from $[-i,0]$ to the unit disc.  

I've tried a few things but the branch cut keeps messing me up.

A:

The interior of your domain consists of all points $z=x+iy$ which satisfies one of the three conditions: (i) $x>0$ and $y<x$, (ii) $x=0$ and $y<-1$, or (iii) $x<0$ and $y<-x$. 
First map $z$ to $w=iz$. This will rotate your picture counterclockwise by $90^\circ$. Then map $w$ to $\tau=w^{4/3}$. This will close the open sector, and your domain will be mapped to the complex plane minus a cut from $\tau=1$ along the real axis to $\tau=-\infty$. Then map $\tau$ to $\zeta=\sqrt{\tau-1}$, which will send your domain to the right half plane $\Re(\zeta)>0$. Finally map $\zeta$ to ${1-\zeta \over 1+\zeta}$, which will map your domain to the unit disc.

