Solution
= Solution
For $\zeta=Re^{i\theta}$,
$x=\tfrac c2(R+R^{-1})\cos\theta$ and $y=\tfrac c2(R-R^{-1})\sin\theta$. Thus $a=c(R+R^{-1})/2$, $b=c(R-R^{-1})/2$. Nested level curves and the behaviour at infinity show that exteriors correspond.
= Solution
For $\zeta=Re^{i\theta}$,
$x=\tfrac c2(R+R^{-1})\cos\theta$ and $y=\tfrac c2(R-R^{-1})\sin\theta$. Thus $a=c(R+R^{-1})/2$, $b=c(R-R^{-1})/2$. Nested level curves and the behaviour at infinity show that exteriors correspond.