Function Theory of One Complex Variable

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American Mathematical Soc., 2006 - 504 páginas
Complex analysis is one of the most central subjects in mathematics. It is compelling and rich in its own right, but it is also remarkably useful in a wide variety of other mathematical subjects, both pure and applied. This book is different from others in that it treats complex variables as a direct development from multivariable real calculus. As each new idea is introduced, it is related to the corresponding idea from real analysis and calculus. The text is rich with examples and exercises that illustrate this point. The authors have systematically separated the analysis from the topology, as can be seen in their proof of the Cauchy theorem. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem, and the Bergman kernel. The authors also treat $H^p$ spaces and Painleve's theorem on smoothness to the boundary for conformal maps. This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors.
 

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Contenido

Complex Line Integrals
29
Applications of the Cauchy Integral
69
Meromorphic Functions and Residues
105
The Zeros of a Holomorphic Function
157
Holomorphic Functions as Geometric Mappings
179
Harmonic Functions
207
Infinite Series and Products
255
Applications of Infinite Sums and Products
279
Rational Approximation Theory
363
Special Classes of Holomorphic Functions
385
Exercises
412
Special Functions
449
The Prime Number Theorem
471
Real Analysis
487
The Statement and Proof of Goursats Theorem
493
Index
501

Analytic Continuation
299
Topology
335

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Página 1 - M can be written in one and only one way in the form M = DP, where D is nonsingular and diagonal, and P is a permutation matrix.

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