A Course in Complex Analysis and Riemann Surfaces

Schlag presents a textbook for a one-quarter first-year graduate course in analysis for mathematics majors that covers real analysis and measure theory, and functional analysis and applications. It also discusses elementary aspects of complex analysis such as the Cauchy integral theorem, the residue theorem, Laurent series, the Riemann mapping theorem, and more advanced material selected from Riemann surface theory. He does not try to complete with specialized textbooks or classic treatises on these topics, he says, but offers a fairly detailed yet fast-paced introduction to those parts of the theory of one complex variable that seem most useful in other areas of mathematics such as geometric group theory, dynamics, algebraic geometry, number theory, and functional analysis. Annotation ©2014 Ringgold, Inc., Portland, OR (protoview.com)

Table of Contents

Chapter 1 . From i to z: the basics of complex analysis

Chapter 2. From z to the Riemann mapping theorem

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