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absolute value an(z branch of log Cauchy-Riemann equations Cauchy's theorem closed curve coefficients complex function theory complex plane complex-valued function connected component constant continuous function converges locally uniformly converges uniformly counterclockwise orientation differentiable domain G example finitely Fix a point follows function f function in G geometric half-plane harmonic conjugate harmonic function Hence holomorphic function holomorphic in G holomorphic map inequality integrand inverse isolated singularity Laurent series Let f Let G limsup linear fractional transformation logarithm metric n-th root nonconstant nonzero complex number north pole obtain open disk open set open set containing open subset open unit disk parameterization piecewise C1 point zq polynomial positive integer positive number power series proof punctured disk radius of convergence rational functions reader real axis real numbers removable singularity Riemann mapping sequence simply connected simply connected domain subset G triangle unit circle vanishes winding number zeros zo)n