**Graduate Studies in Mathematics**

Volume: 154;
2014;
384 pp;
Hardcover

MSC: Primary 30;

Print ISBN: 978-0-8218-9847-5

Product Code: GSM/154

List Price: $79.00

Individual Member Price: $63.20

**Electronic ISBN: 978-1-4704-1716-1
Product Code: GSM/154.E**

List Price: $79.00

Individual Member Price: $63.20

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#### Supplemental Materials

# A Course in Complex Analysis and Riemann Surfaces

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*Wilhelm Schlag*

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.

#### Table of Contents

# Table of Contents

## A Course in Complex Analysis and Riemann Surfaces

- Cover Cover11 free
- Title page i2 free
- Contents iii4 free
- Preface vii8 free
- Chapter 1. From 𝑖 to 𝑧: The basics of complex analysis 118 free
- Chapter 2. From 𝑧 to the Riemann mapping theorem: Some finer points of basic complex analysis 4158
- Chapter 3. Harmonic functions 85102
- Chapter 4. Riemann surfaces: Definitions, examples, basic properties 129146
- Chapter 5. Analytic continuation, covering surfaces, and algebraic functions 179196
- Chapter 6. Differential forms on Riemann surfaces 225242
- Chapter 7. The theorems of Riemann-Roch, Abel, and Jacobi 269286
- Chapter 8. Uniformization 305322
- Appendix A. Review of some basic background material 353370
- Bibliography 371388
- Index 377394 free
- Back Cover Back Cover1402

#### Readership

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