= Conformal map from the exterior half-disc to a slit half-plane
{title2=$g(z)=2z/(1-z^2)$}
The <rational function> $g(z)=2z/(1-z^2)$ is a <conformal> bijection from $\{z:\operatorname{Im}z>0,\ |z|>1\}$ onto $\{w:\operatorname{Im}w>0\}\setminus\{iy:y\geq1\}$. 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.
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