Function theory of one complex variable
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 $Hp$ 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
Print Book, English, [2006]
American Mathematical Society, Providence (Rhode Island), [2006]
XIX, 504 p. : il. ; 26 cm.
9780821839621, 0821839624
642570561
Preface to the Third EditionPreface to the Second EditionPreface to the First EditionAcknowledgmentsChapter 1. Fundamental conceptsChapter 2. Complex line integralsChapter 3. Applications of the Cauchy integralChapter 4. Meromorphic functions and residues Chapter 5. The zeros of a holomorphic functionChapter 6. Holomorphic functions as geometric mappings Chapter 7. Harmonic functions Chapter 8. Infinite series and productsChapter 9. Applications of infinite sums and productsChapter 10. Analytic continuationChapter 11. TopologyChapter 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 functionsChapter 16. The prime number theoremAppendix A: Real analysisAppendix B: The statement and proof of Goursat's theoremReferencesIndex
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