Function Theory of One Complex VariableAmerican 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 andexercises 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 $Hp$ 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. |
Contenido
Complex Line Integrals | 29 |
Applications of the Cauchy Integral 6989 | 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 |
| 501 | |
Términos y frases comunes
analytic continuation Assume boundary bounded Calculate Cauchy integral formula Cauchy-Riemann equations Chapter circle closed curve compact sets compact subsets complex analysis complex numbers conformal mapping conformally equivalent connected open set constant continuous function converges uniformly Corollary defined definition derivative differentiable domain entire function essential singularity example Exercise fact Figure finite fixed follows function element function f ƒ is holomorphic harmonic function hence Hint holomorphic function holomorphically simply connected homotopic inequality Laurent series Lemma Let f Let ƒ Let UCC meromorphic function neighborhood nonzero one-to-one open set open set UCC plane pole polynomial proof of Theorem Proposition removable singularity result satisfies sequence series converges simply connected subharmonic Suppose that ƒ topological U₁ uniformly on compact unit disc vanishes zeros of f ας ди მა მე მთ მყ
Pasajes populares
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