= Joukowski map of the upper exterior of a unit disk
{c}
{title2=$g(z)=z+z^{-1}$}
The <Joukowski map> restricts to a <conformal map> from $\{\operatorname{Im}z>0,\ |z|>1\}$ onto the <upper half-plane>. Indeed, $\operatorname{Im}g(z)=\operatorname{Im}z(1-|z|^{-2})>0$ there. For any $\operatorname{Im}\zeta>0$, the roots of $z^2-\zeta z+1=0$ have product one and cannot lie on the unit <circle>; exactly one has modulus greater than one, and that root has positive <imaginary part>. This proves that the map is a <bijection>.
Its <inverse function> is
$$
g^{-1}(\zeta)=\frac{\zeta+\sqrt{\zeta-2}\sqrt{\zeta+2}}2,
$$
with each factor taken as the <principal square root>. This product is the <holomorphic square root> of $\zeta^2-4$ that behaves as $\zeta$ at infinity in the <upper half-plane>. The upper unit <semicircle> maps to $[-2,2]$; the exterior real rays map to the remaining real rays. The <derivative> $g'=1-z^{-2}$ vanishes only at the boundary endpoints $z=\pm1$.
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