<s>[INST] Inverse of the Joukowski map $\phi(z) = z + \frac{1}{z}$ We know the Joukowski map $$\phi(z) = z + \frac{1}{z}$$ which maps the upper semidisc of radius $1$ in the lower half plane, and the lower semidisc of radius $1$ in the upper half plane.
What is the inverse of this function ? We obtain $z ^{2}-zy + 1 = 0$ and this equation has $2$ solutions, which is the right one ?
 [/INST] 
We obtain $z^2−zy+1=0$ and this equation has $2$ solutions, which is the right one?

The solutions are given by
$$z = \frac{y - \sqrt{y^2-4}}{2},$$
where the different solutions correspond to the different choices of the square root.
We want a holomorphic inverse, so we need a holomorphic branch of $\sqrt{y^2-4}$ on the image of $\phi$, and the choice of the square root at one point determines whether we land inside the unit circle or outside. Since $\phi(\frac12) = \frac52$, we want
$$1 = \frac{5}{2} - \sqrt{\left(\frac{5}{2}\right)^2-4} = \frac{5}{2} - \sqrt{\frac{25-16}{4}},$$
so we need the branch of $\sqrt{y^2-4}$ on $\mathbb{C}\setminus [-2,2]$ that takes the value $\frac{3}{2}$ in $y = \frac{5}{2}$.
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