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Origin and Applications of the Theory 1 The development of the idea of analytic functions
The usefulness of the theory
Founders of the theory
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a+ib absolute value algebraic amplitude analytic continuation analytic functions analytic throughout angle approach zero becomes infinite boundary branch Burkhardt-Rasor Cauchy-Riemann equations Cauchy's integral formula Cauchy's integral theorem Chapter circle of convergence closed curve co-ordinates coefficients complex numbers complex variable conformal mapping continuous function corresponding curve integral curve segment defined definite integral definition differential equations double integral equal essential singular point example expression foregoing formula func function analytic geometric given Goursat-Hedrick Hence indefinite integral infinite number infinite series interior Laurent's series limit linear transformation logarithmic lytic many-valued functions mathematical monodromic group negative obtain origin Osgood partial derivatives path Pierpont polynomial power series quotient real functions real number real variable Riemann surface root series converges similarly simply connected simply connected region single-valued and analytic solution Taylor's series theory of functions throughout a neighborhood throughout a region tion tive uniformly convergent unit circle vanish x+iy z-plane