 Solution

At large $|\zeta |$, $z=\zeta +O({\zeta }^{-1})$, so the map tends to the identity. Its exterior inverse has the expansion $\zeta =z-a^2/z+O(z^{-3})$. Therefore the physical complex potential of a two-dimensional potential flow is $w(z)=W(\zeta (z))$, withIts leading term is unchanged, so the far-field velocity still has magnitude $U$ and inclination $\alpha$.

The chain rule gives the complex velocity:The parameter $\zeta$ denotes the unique exterior preimage of $z$. If an expression solely in $z$ is desired, takewhere the exterior holomorphic square root is chosen so that $\sqrt{z^2-4a^2}\sim z$ at infinity. Its branch points lie inside the ellipse. This is uniform potential flow around an ellipse.