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The geometric representation of complex numbers
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algebraic analytic continuations analytic function angle applied arbitrary boundary bounded Cauchy's theorem choose circle closed curve coefficients compact set complement complex number condition conformal mapping conjugate consider constant contained corresponding cycle defined definition denote derivative differential equation direct analytic continuations end points entire function equal exists extended plane finite number follows formula function elements function f(z half plane harmonic function hence homotop imaginary inequality infinite initial branch integral interval lemma linear transformation mapping maximum principle multiple neighborhood notation obtain open sets pole polynomial proof prove radius of convergence rational function real axis real numbers rectangle region 12 removable singularity residue Riemann surface roots satisfies sequence simple simply connected simply connected region single-valued singularity solution subharmonic subset sufficient Suppose symmetric tion topological uniform convergence uniformly vanishes variable whole plane z-plane