= Uniform potential flow around an ellipse
{title2=$q=U\frac{e^{-i\alpha}\zeta^2-e^{i\alpha}}{\zeta^2-a^2}$}
For $0<a<1$, the <Joukowski map> $z=\zeta+a^2/\zeta$ maps the exterior <unit circle> to an <ellipse> of horizontal and vertical semiaxes $1+a^2$ and $1-a^2$. The <Milne-Thomson circle theorem> gives $W(\zeta)=U(e^{-i\alpha}\zeta+e^{i\alpha}/\zeta)$ for a <uniform flow> inclined by $\alpha$. The <chain rule> produces the displayed <complex velocity> in the physical plane. Since $z\sim\zeta$, the far-field flow retains its speed and direction. Its two <stagnation points> are
$$
z=\pm[(1+a^2)\cos\alpha+i(1-a^2)\sin\alpha],
$$
obtained from $\zeta=\pm e^{i\alpha}$. The map derivative is nonzero on and outside the <unit circle>, so these zeros are preserved without introducing a boundary singularity.
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