= Conformal map of a segment complement to a punctured disk
{title2=$w=\frac{\sqrt{(z-1)/(z+1)}-1}{\sqrt{(z-1)/(z+1)}+1}$}
On $\mathbb C\setminus[-1,1]$, take the <principal square root> with positive real part. The <Möbius transformation> $(z-1)/(z+1)$ has image $\bigl(\mathbb C\setminus(-\infty,0]\bigr)\setminus\{1\}$. The <principal square root> therefore maps to the right half-plane with $1$ omitted, and the final <Möbius transformation> gives exactly $0<|w|<1$. The inverse is $z=-(w+w^{-1})/2$. Adding $\infty$ to the source on the <Riemann sphere> fills the missing value $w=0$ and gives a <conformal map> onto the full disk. Without $\infty$, a loop surrounding the segment has nonzero <winding number>, so the source is not a <simply connected space> and cannot be <conformally equivalent> to a disk.
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