Preface Acknowledgments Chapter 1.From i to z: the basics of complex analysis §1.1.The field of complex numbers §1.2.Holomorphic, analytic, and conformal §1.3.The Riemann sphere §1.4.M?bius transformations §1.5.The hyperbolic plane and the Poincaré disk §1.6.Complex integration, Cauchy theorems §1.7.Applications of Cauchy's theorems §1.8.Harmonic functions §1.9.Problems Chapter 2.From z to the Riemann mapping theorem: some finer points of basic complex analysis §2.1.The winding number §2.2.The global form of Cauchy's theorem §2.3.Isolated singularities and residues §2.4.Analytic continuation §2.5.Convergence and normal families §2.6.The Mittag-Leffler and Weierstrass theorems §2.7.The Riemann mapping theorem §2.8.Runge's theorem and simple connectivity §2.9.Problems Chapter 3.Harmonic functions §3.1.The Poisson kernel §3.2.The Poisson kernel from the probabilistic point of view §3.3.Hardy classes of harmonic functions §3.4.Almost everywhere convergence to the boundary data §3.5.Hardy spaces of analytic functions §3.6.Riesz theorems §3.7.Entire functions of finite order §3.8.A gallery of conformal plots §3.9.Problems Chapter 4.Riemann surfaces: definitions, examples, basic properties §4.1.The basic definitions §4.2.Examples and constructions of Riemann surfaces §4.3.Functions on Riemann surfaces §4.4.Degree and genus §4.5.Riemann surfaces as quotients §4.6.Elliptic functions §4.7.Covering the plane with two or more points removed §4.8.Groups of M?bius transforms §4.9.Problems Chapter 5.Analytic continuation, covering surfaces, and algebraic functions §5.1.Analytic continuation §5.2.The unramified Riemann surface of an analytic germ §5.3.The ramified Riemann surface of an analytic germ §5.4.Algebraic germs and functions §5.5.Algebraic equations generated by compact surfaces §5.6.Some compact surfaces and their associated polynomials §5.7.ODEs with meromorphic coefficients §5.8.Problems Chapter 6.Differential forms on Riemann surfaces §6.1.Holomorphic and meromorphic differentials §6.2.Integrating differentials and residues §6.3.The Hodge-* operator and harmonic differentials §6.4.Statement and examples of the Hodge decomposition §6.5.Weyl's lemma and the Hodge decomposition §6.6.Existence of nonconstant meromorphic functions §6.7.Examples of meromorphic functions and differentials §6.8.Problems Chapter 7.The Theorems of Riemann-Roch, Abel, and Jacobi §7.1.Homology bases and holomorphic differentials §7.2.Periods and bilinear relations §7.3.Divisors §7.4.The Riemann-Roch theorem §7.5.Applications and general divisors §7.6.Applications to algebraic curves §7.7.The theorems of Abel and Jacobi §7.8.Problems Chapter 8.Uniformization §8.1.Green functions and Riemann mapping §8.2.Perron families §8.3.Solution of Dirichlet's problem §8.4.Green's functions on Riemann surfaces §8.5.Uniformization for simply-connected surfaces §8.6.Uniformization of non-simply-connected surfaces §8.7.Fuchsian groups §8.8.Problems Appendix A.Review of some basic background material §A.1.Geometry and topology §A.2.Algebra §A.3.Analysis Bibliography Index
