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accumulation point analytic continuation analytic function angle arbitrarily boundary values bounded Cauchy Integral Cauchy's Integral Theorem circle of convergence coefficients complex function complex numbers complex value consider const constant continuous function converges absolutely converges uniformly corresponding cross ratio defined definition denote derivative differentiable entire function essential singularity extend finite number flow follows formula function element function-element functional equation geometrical given harmonic function Hence infinite infinity interior inverse Lemma Let f(z limit linear transformation mapping meromorphic function napping neighborhood obtain one-to-one origin polygon polynomial positive power series proof prove rational function real and imaginary real axis real functions real numbers regular function representation residue Riemann surface sequence series converges simple closed curve simple poles simply connected simply-connected subdivision suppose tends to zero theorem unit circle upper half-plane vanishes z-plane