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Limits in the Complex Plane
Integration in the Complex Plane
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absolutely convergent analytic continuation analytic function analytic in G angle arbitrary branch Cauchy-Riemann equations Cauchy's integral theorem Chap closed Jordan curve complex numbers complex plane complex potential conformal mapping constant contained in G contour corresponding defined deleted neighborhood denoted derivative differentiable domain G essential singular point Example f f(z)dz finite number follows from Theorem formula fractional linear transformation function f(z given harmonic functions hence Hint imaginary infinite infinity inverse Laurent expansion Let f(z let z0 limit point line segment neighborhood of z0 obviously piecewise smooth piecewise smooth closed piecewise smooth curve point of G point z0 polygonal curve power series Prob Proof Prove radius of convergence real axis real number Riemann sequence shown in Figure single-valued smooth closed Jordan Suppose f(z term uniformly convergent unique unit disk univalent upper half-plane values variable vector w-plane z0 e G zero