Find a conformal map \(\DD^c\cap\HH\to\HH\), where \(\DD^c=\{z:|z|>1\}\).
Solution
The Joukowski map \[ J(z)=z+{1\over z} \] works. Write \(z=re^{i\theta}\) with \(r>1\) and \(0<\theta<\pi\). Then \[ \Im J(z)=\qty{r-r^{-1}}\sin\theta>0, \] so \(J\) maps the domain into \(\HH\).
It is injective there. Indeed, \[ J(z_1)=J(z_2) \iff (z_1-z_2)\qty{1-{1\over z_1z_2}}=0. \] Since \(|z_1z_2|>1\), the second factor cannot vanish; hence \(z_1=z_2\).
It is also onto. Given \(w\in\HH\), the equation \(J(z)=w\) is \[ z^2-wz+1=0. \] Its two roots have product \(1\). Neither root lies on \(|z|=1\), because \(J(e^{i\theta})=2\cos\theta\in\RR\). Hence exactly one root has modulus greater than \(1\). For that root, the identity \[ \Im w=\qty{|z|-|z|^{-1}}\sin(\arg z)>0 \] forces \(0<\arg z<\pi\), so the root lies in \(\DD^c\cap\HH\).
Finally, \[ J'(z)=1-{1\over z^2}, \] which vanishes only at \(z=\pm1\), outside the domain. Thus \(J\) is a biholomorphic, hence conformal, map from \(\DD^c\cap\HH\) onto \(\HH\).