Solution
= Solution
For $z=e^{i\theta}$, $w=i\tan(\theta/2)$, which is purely imaginary; $z=-1$ supplies the point at infinity. Conversely, the <inverse map> is $z=(1+w)/(1-w)$. Requiring $|w|=1$ is equivalent to $|z-1|=|z+1|$, hence $\boxed{\operatorname{Re}z=0}$, including the appropriate infinite point in the extended-plane formulation. The <unit circle> and <imaginary axis> are exchanged, but the <real axis> is not.