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A Course in Complex Analysis and Riemann Surfaces

Wilhelm Schlag University of Chicago, Chicago, IL
Available Formats:
Hardcover ISBN: 978-0-8218-9847-5
Product Code: GSM/154
384 pp
List Price: $84.00 MAA Member Price:$75.60
AMS Member Price: $67.20 Sale Price:$54.60
Electronic ISBN: 978-1-4704-1716-1
Product Code: GSM/154.E
384 pp
List Price: $79.00 MAA Member Price:$71.10
AMS Member Price: $63.20 Sale Price:$51.35
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $126.00 MAA Member Price:$113.40
AMS Member Price: $100.80 Sale Price:$81.90
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A Course in Complex Analysis and Riemann Surfaces
Wilhelm Schlag University of Chicago, Chicago, IL
Available Formats:
 Hardcover ISBN: 978-0-8218-9847-5 Product Code: GSM/154 384 pp
 List Price: $84.00 MAA Member Price:$75.60 AMS Member Price: $67.20 Sale Price:$54.60
 Electronic ISBN: 978-1-4704-1716-1 Product Code: GSM/154.E 384 pp
 List Price: $79.00 MAA Member Price:$71.10 AMS Member Price: $63.20 Sale Price:$51.35
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $126.00 MAA Member Price:$113.40
AMS Member Price: $100.80 Sale Price:$81.90
• Book Details

Volume: 1542014
MSC: Primary 30;

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.

Graduate students interested in complex analysis, conformal geometry, Riemann surfaces, uniformization, harmonic functions, differential forms on Riemann surfaces, and the Riemann-Roch theorem.

• Chapters
• Chapter 1. From $i$ to $z$: The basics of complex analysis
• Chapter 2. From $z$ to the Riemann mapping theorem: Some finer points of basic complex analysis
• Chapter 3. Harmonic functions
• Chapter 4. Riemann surfaces: Definitions, examples, basic properties
• Chapter 5. Analytic continuation, covering surfaces, and algebraic functions
• Chapter 6. Differential forms on Riemann surfaces
• Chapter 7. The theorems of Riemann-Roch, Abel, and Jacobi
• Chapter 8. Uniformization
• Appendix A. Review of some basic background material

• Reviews

• [T]his is an extremely valuable textbook for graduate classical complex analysis in one variable both for lecturers and students not following the increasing standardization trends of student's curricula...The presentation style is excellent, a very well contemplated pleasant reading throughout, rich in interesting outlooks. I recommend this work to all the mathematical libraries at universities as an extremely helpful material in teaching or studying complex analysis.

László L. Stachó, ACTA Sci. Math.
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• Get Permissions
Volume: 1542014
MSC: Primary 30;

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.

Graduate students interested in complex analysis, conformal geometry, Riemann surfaces, uniformization, harmonic functions, differential forms on Riemann surfaces, and the Riemann-Roch theorem.

• Chapters
• Chapter 1. From $i$ to $z$: The basics of complex analysis
• Chapter 2. From $z$ to the Riemann mapping theorem: Some finer points of basic complex analysis
• Chapter 3. Harmonic functions
• Chapter 4. Riemann surfaces: Definitions, examples, basic properties
• Chapter 5. Analytic continuation, covering surfaces, and algebraic functions
• Chapter 6. Differential forms on Riemann surfaces
• Chapter 7. The theorems of Riemann-Roch, Abel, and Jacobi
• Chapter 8. Uniformization
• Appendix A. Review of some basic background material
• [T]his is an extremely valuable textbook for graduate classical complex analysis in one variable both for lecturers and students not following the increasing standardization trends of student's curricula...The presentation style is excellent, a very well contemplated pleasant reading throughout, rich in interesting outlooks. I recommend this work to all the mathematical libraries at universities as an extremely helpful material in teaching or studying complex analysis.

László L. Stachó, ACTA Sci. Math.
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