A conformal map from $\CC\setminus[1,\infty)$ onto $\DD$

The map \[ z\longmapsto 1-z \] sends the slit plane \(\mathbb C\setminus[1,\infty)\) onto \(\mathbb C\setminus(-\infty,0]\). On this latter domain take the principal square root, which maps biholomorphically onto the right half-plane. Finally, the Cayley transform \[ w\longmapsto\frac{w-1}{w+1} \] maps the right half-plane biholomorphically onto \(\mathbb D\). Hence one conformal map is \[ \boxed{ F(z)=\frac{\sqrt{\,1-z\,}-1}{\sqrt{\,1-z\,}+1}, } \] where \(\sqrt{\cdot}\) is the principal square root on \(\mathbb C\setminus(-\infty,0]\).