As I read in The Road to Reality by Roger Penrose, the Joukowsky transform $$w(z) = \frac12\left( z + \frac1z \right)$$ after Nikolai Zhukovsky (transcribed in several versions from Никола́й Его́рович Жуко́вский) can be used to calculate the flow of a non-viscious, incompressible and irrotational flow around an airfoil.
This can be done since the solution of a potential flow around a cylinder is known in full analyticity and the given transform conformally maps a circle on an airfoil-like geometry. I don't understand this argumentation, so:
How is the Joukowsky Transform used to calculate the Flow of an Airfoil?
An example of such a transformation is given in the mentioned Wikipedia article:
Thank you in advance.