Branched circle packings, discrete complex polynomials, and the approximation of analytic functions

The aim of this dissertation was to investigate circle packings more general than those which had been studied in the past. In particular, new techniques were developed to explore these broader concepts. We found necessary and sufficient conditions for the existence of finite and infinite branched circle packings; we proved the uniqueness of hexagonal circle packings of finite valence; we also obtained a finite-valence generalization of the Ring Lemma. As a result, the theory of circle packings was extended to incorporate branched circle packings. Moreover, the extended theory was shown to be, in many aspects, a discrete parallel of the theory of analytic functions. Especially, via branched circle packings, we created objects termed discrete Blaschke products and discrete complex polynomials which are discrete analogues of their classical counterparts. Then we proved that any classical complex polynomial (resp. Blaschke product) can be approximated uniformly on compacta by discrete complex polynomials (resp. Blaschke products).