In this thesis we will observe the Riemann mapping theorem in an alternative way through the theory of discrete analytic functions. The fact that conformal mapping sends infinitesimal circles to circles will be used to construct biholomorphism between non-empty simply connected open subset of the complex plane and the open unit disk. We will describe a method called circle packing, which will help us to define a sequence of discrete mappings which can be continuously extended to a triangulation of both domains. Finally, we will prove that this sequence converges to a conformal mapping, which conicides with the one from the Riemann mapping theorem.
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