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The Geometric Representation of Complex Numbers
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algebraic analytic continuations analytic function angle arbitrary assume boundary bounded Cauchy's theorem Chap choose closed curve coefficients compact set complex numbers conclude condition conformal mapping conjugate consider constant corresponding defined definition denote derivative differential equation direct analytic continuations end points entire function equal exists finite number follows formula function elements function f(z geometric global analytic function harmonic function hence homotopic imaginary inequality infinite initial branch integral inverse lemma linear transformation meromorphic function metric space multiple neighborhood notation obtain open sets poles polygon polynomial positive power series proof prove radius of convergence rational function real axis rectangle region 12 residue Riemann surface roots satisfies sequence simple simply connected simply connected region single-valued singularity solution subharmonic subset Suppose tion topological uniform convergence uniformly upper half plane vanishes variable whole plane z-plane