Analytical functions: conformal map and applications to hydrodynamics

This homework deals with the application of conformal maps to the study of two-dimensional hydrodynamics. A conformal map is a geometrical transformation which preserves all (oriented) crossing angles between lines. In dimension $d\geq 3$ a conformal map is necessarily composed from the following limited number of transformations: translations, rotations, homothetic transformation and special conformal transformation (which is the composition of a reflection and an inversion in a sphere). However in two dimensions, $d=2$ , the space of conformal mappings is much larger and one can show that, given an open set $\Omega \in {\mathbb {C} }$ , any holomorphic function $f:\Omega \rightarrow {\mathbb {C} }$ such that $\$f'(z)\neq 0\$$ , $\forall z\in \Omega$ defines a conformal map from $\Omega$ to $f(\Omega )$ . The aim of this HW is to exploit this property to study some hydrodynamic flows in two spatial dimensions.