A Course in Complex Analysis and Riemann Surfaces
Wilhelm Schlag
180 x 240 mm
Year of Publishing
Territorial Rights
Universities Press

Complex analysis is a cornerstone of mathematics, making it an essential element of any area of study in graduate mathematics. Schlag’s treatment of the subject emphasizes the intuitive geometric underpinnings of elementary complex analysis that naturally lead to the theory of Riemann surfaces.

The book begins with an exposition of the basic theory of holomorphic functions of one complex variable. The first two chapters constitute a fairly rapid, but comprehensive course in complex analysis. The third chapter is devoted to the study of harmonic functions on the disk and the half-plane, with an emphasis on the Dirichlet problem. Starting with the fourth chapter, the theory of Riemann surfaces is developed in some detail and with complete rigor. From the beginning, the geometric aspects are emphasized and classical topics such as elliptic functions and elliptic integrals are presented as illustrations of the abstract theory. The special role of compact Riemann surfaces is explained, and their connection with algebraic equations is established. The book concludes with three chapters devoted to three major results: the Hodge decomposition theorem, the Riemann-Roch theorem, and the uniformization theorem. These chapters present the core technical apparatus of Riemann surface theory at this level.

This text is intended as a detailed, yet fast-paced intermediate introduction to those parts of the theory of one complex variable that seem most useful in other areas of mathematics, including geometric group theory, dynamics, algebraic geometry, number theory, and functional analysis. More than seventy figures serve to illustrate concepts and ideas, and the many problems at the end of each chapter give the reader ample opportunity for practice and independent study.

Wilhelm Schlag, University of Chicago, Chicago, IL
• Contents     4
• Preface     8
• Chapter 1. From i to z : The basics of complex analysis     18
• Chapter 2. From z to the Riemann mapping theorem: Some finer points of basic complex analysis      58
• Chapter 3. Harmonic functions       102
• Chapter 4. Riemann surfaces: Definitions, examples, basic properties       146
• Chapter 5. Analytic continuation, covering surfaces, and algebraic functions      196
• Chapter 6. Differential forms on Riemann surfaces       242
• Chapter 7. The theorems of Riemann-Roch, Abel, and Jacobi      286
• Chapter 8. Uniformization      322
• Appendix A. Review of some basic background material      370
• Bibliography      388
• Index      394
• Back Cover      402
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