Joukowski map of the upper exterior of a unit disk ($g(z)=z+z^{-1}$)

The Joukowski map restricts to a conformal map from $\{\mathrm{Im}z>0,|z|>1\}$ onto the upper half-plane. Indeed, $\mathrm{Im}g(z)=\mathrm{Im}z(1-|z{|}^{-2})>0$ there. For any $\mathrm{Im}\zeta >0$, 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 iswith 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 $g^{\prime }=1-z^{-2}$ vanishes only at the boundary endpoints $z=\pm 1$.