Conformal map from the exterior half-disc to a slit half-plane ($g(z)=2z/(1-z^2)$)

The rational function $g(z)=2z/(1-z^2)$ is a conformal bijection from $\{z:\mathrm{Im}z>0,|z|>1\}$ onto $\{w:\mathrm{Im}w>0\}\setminus \{iy:y\ge 1\}$. For a proof, the Cayley transform between the half-plane and disk $f(z)=(z-i)/(z+i)$ maps the source onto the right half of the unit disc. Squaring maps this half-disc bijectively onto $\mathbb{D}\setminus (-1,0]$. Finally $T(s)=i(1-s)/(1+s)$ maps the disc to the upper half-plane and $(-1,0]$ to the excluded ray. The composition $T(f(z{)}^2)$ simplifies to $g(z)$. Each factor is holomorphic with nonzero derivative and bijective on its indicated domain.