Hardcover ISBN:  9780821898475 
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:  9781470417161 
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 

Book DetailsGraduate Studies in MathematicsVolume: 154; 2014MSC: 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 halfplane, 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 RiemannRoch 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 fastpaced 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.ReadershipGraduate students interested in complex analysis, conformal geometry, Riemann surfaces, uniformization, harmonic functions, differential forms on Riemann surfaces, and the RiemannRoch theorem.

Table of Contents

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 RiemannRoch, Abel, and Jacobi

Chapter 8. Uniformization

Appendix A. Review of some basic background material


Additional 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.


Request Exam/Desk Copy

Request Review Copy

Get Permissions
 Book Details
 Table of Contents
 Additional Material
 Reviews

 Request Review Copy
 Request Exam/Desk Copy
 Get Permissions
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 halfplane, 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 RiemannRoch 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 fastpaced 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 RiemannRoch 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 RiemannRoch, 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.