A Course in Complex Analysis

A comprehensive graduate-level textbook that takes a fresh approach to complex analysis A Course in Complex Analysis explores a central branch of mathematical analysis, with broad applications in mathematics and other fields such as physics and engineering. Ideally designed for a year-long graduate course on complex analysis and based on nearly twenty years of classroom lectures, this modern and comprehensive textbook is equally suited for independent study or as a reference for more experienced scholars. Saeed Zakeri guides the reader through a journey that highlights the topological and geometric themes of complex analysis and provides a solid foundation for more advanced studies, particularly in Riemann surfaces, conformal geometry, and dynamics. He presents all the main topics of classical theory in great depth and blends them seamlessly with many elegant developments that are not commonly found in textbooks at this level. They include the dynamics of Möbius transformations, Schlicht functions and distortion theorems, boundary behavior of conformal and harmonic maps, analytic arcs and the general reflection principle, Hausdorff dimension and holomorphic removability, a multifaceted approach to the theorems of Picard and Montel, Zalcman’s rescaling theorem, conformal metrics and Ahlfors’s generalization of the Schwarz lemma, holomorphic branched coverings, geometry of the modular group, and the uniformization theorem for spherical domains. Written with exceptional clarity and insightful style, A Course in Complex Analysis is accessible to beginning graduate students and advanced undergraduates with some background knowledge of analysis and topology. Zakeri includes more than 350 problems, with problem sets at the end of each chapter, along with numerous carefully selected examples. This well-organized and richly illustrated book is peppered throughout with marginal notes of historical and expository value. Presenting a wealth of material in a single volume, A Course in Complex Analysis will be a valuable resource for students and working mathematicians. Cover Contents Preface Chapter 1. Rudiments of complex analysis 1.1 What is a holomorphic function? 1.2 Complex analytic functions 1.3 Complex integration 1.4 Cauchy’s theory in a disk 1.5 Mapping properties of holomorphic functions Problems Chapter 2. Topological aspects of Cauchy’s theory 2.1 Homotopy of curves 2.2 Covering properties of the exponential map 2.3 The winding number 2.4 Cycles and homology 2.5 The homology version of Cauchy’s theorem Problems Chapter 3. Meromorphic functions 3.1 Isolated singularities 3.2 The Riemann sphere 3.3 Laurent series 3.4 Residues 3.5 The argument principle Problems Chapter 4. Möbius maps and the Schwarz lemma 4.1 The Möbius group 4.2 Three automorphism groups 4.3 Dynamics of Möbius maps 4.4 Conformal metrics 4.5 The hyperbolic metric Problems Chapter 5. Convergence and normality 5.1 Compact convergence 5.2 Convergence in the space of holomorphic functions 5.3 Normal families of meromorphic functions Problems Chapter 6. Conformal maps 6.1 The Riemann mapping theorem 6.2 Schlicht functions 6.3 Boundary behavior of Riemann maps Problems Chapter 7. Harmonic functions 7.1 Elementary properties of harmonic functions 7.2 Poisson’s formula in a disk 7.3 Some applications of Poisson’s formula 7.4 Boundary behavior of harmonic functions 7.5 Harmonic measure on the circle Problems Chapter 8. Zeros of holomorphic functions 8.1 Infinite products 8.2 Weierstrass’s theory of elementary factors 8.3 Jensen’s formula and its applications 8.4 Entire functions of finite order Problems Chapter 9. Interpolation and approximation theorems 9.1 Mittag-Leffler’s theorem 9.2 Elliptic functions 9.3 Rational approximation 9.4 Finitely connected domains Problems Chapter 10. The holomorphic extension problem 10.1 Regular and singular points 10.2 Analytic continuation 10.3 Analytic arcs and reflections 10.4 Two removability results Problems Chapter 11. Ranges of holomorphic functions 11.1 Bloch’s theorem 11.2 Picard’s theorems 11.3 A rescaling approach to Picard and Montel 11.4 Ahlfors’s generalization of the Schwarz-Pick lemma Problems Chapter 12. Holomorphic (branched) covering maps 12.1 Covering spaces 12.2 Holomorphic coverings and inverse branches 12.3 Proper maps and branched coverings 12.4 The Riemann-Hurwitz formula Problems Chapter 13. Uniformization of spherical domains 13.1 The modular group and thrice punctured spheres 13.2 The uniformization theorem 13.3 Hyperbolic domains 13.4 Conformal geometry of topological annuli Problems Bibliography Image credits Index « This textbook is intended for a year-long graduate course on complex analysis, a branch of mathematical analysis that has broad applications, particularly in physics, engineering, and applied mathematics. Based on nearly twenty years of classroom lectures, the book is accessible enough for independent study, while the rigorous approach will appeal to more experienced readers and scholars, propelling further research in this field. While other graduate-level complex analysis textbooks do exist, Zakeri takes a distinctive approach by highlighting the geometric properties and topological underpinnings of this area. Zakeri includes more than three hundred and fifty problems, with problem sets at the end of each chapter, along with additional solved examples. Background knowledge of undergraduate analysis and topology is needed, but the thoughtful examples are accessible to beginning graduate students and advanced undergraduates. At the same time, the book has sufficient depth for advanced readers to enhance their own research. The textbook is well-written, clearly illustrated, and peppered with historical information, making it approachable without sacrificing rigor. It is poised to be a valuable textbook for graduate students, filling a needed gap by way of its level and unique approach. »--Résumé de l'éditeur "This textbook is intended for a year-long graduate course on complex analysis, a branch of mathematical analysis that has broad applications, particularly in physics, engineering, and applied mathematics. Based on nearly twenty years of classroom lectures, the book is accessible enough for independent study, while the rigorous approach will appeal to more experienced readers and scholars, propelling further research in this field. While other graduate-level complex analysis textbooks do exist, Zakeri takes a distinctive approach by highlighting the geometric properties and topological underpinnings of this area. Zakeri includes more than three hundred and fifty problems, with problem sets at the end of each chapter, along with additional solved examples. Background knowledge of undergraduate analysis and topology is needed, but the thoughtful examples are accessible to beginning graduate students and advanced undergraduates. At the same time, the book has sufficient depth for advanced readers to enhance their own research. The textbook is well-written, clearly illustrated, and peppered with historical information, making it approachable without sacrificing rigor. It is poised to be a valuable textbook for graduate students, filling a needed gap by way of its level and unique approach"--Provided by publisher