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基本説明
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.
Full Description
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.
Contents
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Acknowledgments
Chapter 1. Fundamental concepts
Chapter 2. Complex line integrals
Chapter 3. Applications of the Cauchy integral
Chapter 4. Meromorphic functions and residues
Chapter 5. The zeros of a holomorphic function
Chapter 6. Holomorphic functions as geometric mappings
Chapter 7. Harmonic functions
Chapter 8. Infinite series and products
Chapter 9. Applications of infinite sums and products
Chapter 10. Analytic continuation
Chapter 11. Topology
Chapter 12. Rational approximation theory
Chapter 13. Special classes of holomorphic functions
Chapter 14. Hilbert spaces of holomorphic functions, the Bergman kernel, and biholomorphic mappings
Chapter 15. Special functions
Chapter 16. The prime number theorem
Appendix A: Real analysis
Appendix B: The statement and proof of Goursat's theorem
References
Index
