Construction of the Riemann Map

We do not give the proof of famous Riemann mappings theorem, for every simply connected region in $${\mathbb{C}}$$ but we give a series of increasingly better approximations of the Riemann map in the Riemann mapping theorem. The idea is to approximate a domain by a polygon using a Whitney decomposition, then use the Schwarz–Christoffel formula to compute the map onto the polygonal approximation, and finally use the Carathéodory convergence theorem to obtain the Riemann map in the limit. The suggested method will have easy and effective applications in practice.