Map \(\DD^c \intersect \HH\) to \(\HH\), sending
\(-1\to -1\)
\(i\to 0\)
\(1\to 1\)
Solution
The Joukowski map \[ J(z)={1\over 2}\qty{z + \inverseof{z}} \] does this. For \(z=re^{i\theta}\) with \(r>1\) and \(0<\theta<\pi\), \[ \Im J(z)={1\over2}\qty{r-\inverseof{r}}\sin\theta>0, \] and \(J\) is injective on \(\theset{\abs z>1}\) with image \(\CC\setminus[-1,1]\), so \(J\) maps \(\DD^c\intersect\HH\) bijectively onto \(\HH\). Moreover \(J(-1)=-1\), \(J(i)=0\) and \(J(1)=1\).