= 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 $w(z)=W(\zeta(z))$, with
$$
w(z)=Ue^{-i\alpha}z+\frac{U(e^{i\alpha}-a^2e^{-i\alpha})}{z}+O(z^{-3}).
$$
Its leading term is unchanged, so the far-field <velocity> still has magnitude $U$ and inclination $\alpha$.
The <chain rule> gives the <complex velocity>:
$$
\boxed{u-iv=\frac{dW/d\zeta}{dz/d\zeta}
=U\frac{e^{-i\alpha}-e^{i\alpha}/\zeta^2}{1-a^2/\zeta^2}
=U\frac{e^{-i\alpha}\zeta^2-e^{i\alpha}}{\zeta^2-a^2}.}
$$
The parameter $\zeta$ denotes the unique exterior preimage of $z$. If an expression solely in $z$ is desired, take
$$
\zeta(z)=\frac{z+\sqrt{z^2-4a^2}}2,
$$
where 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>.
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