Titlebook: Geometric Function Theory; Explorations in Comp Steven G. Krantz Textbook 2006 Birkh?user Boston 2006 Complex analysis.Green‘s function.Poi

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Boundary Regularity of Conformal Mapself) back to the unit disk, or vice versa. But many of the more delicate questions require something more. If we wish to study behavior of functions at the boundary, or growth or regularity conditions, then we must know something about the boundary behavior of the conformal mapping.

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The Boundary Behavior of Holomorphic Functionsebesgue’s first publications on measure theory, Fatou proved a seminal result about the almost-everywhere boundary limits of bounded, holomorphic functions on the disk. Interestingly, be was able to render the problem as one about convergence of Fourier series, and he solved it in that language.

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The Cauchy-Riemann Equationsplex derivative, give an important connection between the real and complex parts of a holomorphic function. Certainly conformality, harmonicity, and many other fundamental ideas are effectively explored by way of the Cauchy—Riemann equations.

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Automorphism Groups of Domains in the Planevalent and powerful in modern approaches to the subject. Certainly Alexandre Grothendieck and Saunders Mac Lane carried this idea to new heights in their modern formulations of algebraic geometry and algebraic topology.

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,Seeing through Pamela’s Clothes,tic continuation, division problems, approximation theorems (Runge, Mergelyan), and the Cauchy—Riemann equations are just some of the devices that we have for taking a local construction and making it global.