A Course in Complex Analysis and Riemann Surfaces

[BOOK DESCRIPTION]

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 fairly 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]

Preface                                            vii
Acknowledgments                                    xv
    Chapter 1 From i to z: the basics of           1  (40)
    complex analysis
      §1.1 The field of complex numbers       1  (3)
      §1.2 Holomorphic, analytic, and         4  (5)
      confermal
      §1.3 The Riemann sphere                 9  (2)
      §1.4 Mobius transformations             11 (4)
      §1.5 The hyperbolic plane and the       15 (3)
      Poincare disk
      §1.6 Complex integration, Cauchy        18 (5)
      theorems
      §1.7 Applications of Cauchy's           23 (10)
      theorems
      §1.8 Harmonic functions                 33 (3)
      §1.9 Problems                           36 (5)
    Chapter 2 From z to the Riemann mapping        41 (44)
    theorem: some finer points of basic complex
    analysis
      §2.1 The winding number                 41 (4)
      §2.2 The global form of Cauchy's        45 (2)
      theorem
      §2.3 Isolated singularities and         47 (9)
      residues
      §2.4 Analytic continuation              56 (4)
      §2.5 Convergence and normal families    60 (3)
      §2.6 The Mittag-Leffler and             63 (6)
      Weierstrass theorems
      §2.7 The Riemann mapping theorem        69 (5)
      §2.8 Runge's theorem and simple         74 (5)
      connectivity
      §2.9 Problems                           79 (6)
    Chapter 3 Harmonic functions                   85 (44)
      §3.1 The Poisson kernel                 85 (6)
      §3.2 The Poisson kernel from the        91 (4)
      probabilistic point of view
      §3.3 Hardy classes of harmonic          95 (5)
      functions
      §3.4 Almost everywhere convergence      100(5)
      to the boundary data
      §3.5 Hardy spaces of analytic           105(4)
      functions
      §3.6 Riesz theorems                     109(2)
      §3.7 Entire functions of finite order   111(6)
      §3.8 A gallery of conformal plots       117(5)
      §3.9 Problems                           122(7)
    Chapter 4 Riemann surfaces: definitions,       129(50)
    examples, basic properties
      §4.1 The basic definitions              129(2)
      §4.2 Examples and constructions of      131(12)
      Riemann surfaces
      §4.3 Functions on Riemann surfaces      143(3)
      §4.4 Degree and genus                   146(2)
      §4.5 Riemann surfaces as quotients      148(3)
      §4.6 Elliptic functions                 151(9)
      §4.7 Covering the plane with two or     160(4)
      more points removed
      §4.8 Groups of Mobius transforms        164(10)
      §4.9 Problems                           174(5)
    Chapter 5 Analytic continuation, covering      179(46)
    surfaces, and algebraic functions
      §5.1 Analytic continuation              179(6)
      §5.2 The unramified Riemann surface     185(4)
      of an analytic germ
      §5.3 The ramified Riemann surface of    189(3)
      an analytic germ
      §5.4 Algebraic germs and functions      192(7)
      §5.5 Algebraic equations generated      199(7)
      by compact surfaces
      §5.6 Some compact surfaces and their    206(5)
      associated polynomials
      §5.7 ODEs with meromorphic              211(10)
      coefficients
      §5.8 Problems                           221(4)
    Chapter 6 Differential forms on Riemann        225(44)
    surfaces
      §6.1 Holomorphic and meromorphic        225(2)
      differentials
      §6.2 Integrating differentials and      227(3)
      residues
      §6.3 The Hodge-* operator and           230(6)
      harmonic differentials
      §6.4 Statement and examples of the      236(8)
      Hodge decomposition
      §6.5 Weyl's lemma and the Hodge         244(6)
      decomposition
      §6.6 Existence of nonconstant           250(8)
      meromorphic functions
      §6.7 Examples of meromorphic            258(8)
      functions and differentials
      §6.8 Problems                           266(3)
    Chapter 7 The Theorems of Riemann-Roch,        269(36)
    Abel, and Jacobi
      §7.1 Homology bases and holomorphic     269(4)
      differentials
      §7.2 Periods and bilinear relations     273(7)
      §7.3 Divisors                           280(5)
      §7.4 The Riemann-Roch theorem           285(4)
      §7.5 Applications and general           289(3)
      divisors
      §7.6 Applications to algebraic curves   292(3)
      §7.7 The theorems of Abel and Jacobi    295(8)
      §7.8 Problems                           303(2)
    Chapter 8 Uniformization                       305(48)
      §8.1 Green functions and Riemann        306(4)
      mapping
      §8.2 Perron families                    310(4)
      §8.3 Solution of Dirichlet's problem    314(3)
      §8.4 Green's functions on Riemann       317(9)
      surfaces
      §8.5 Uniformization for                 326(9)
      simply-connected surfaces
      §8.6 Uniformization of                  335(3)
      non-simply-connected surfaces
      §8.7 Fuchsian groups                    338(11)
      §8.8 Problems                           349(4)
  Appendix A Review of some basic background       353(18)
  material
      §A.1 Geometry and topology              353(10)
      §A.2 Algebra                            363(2)
      §A.3 Analysis                           365(6)
Bibliography                                       371(6)
Index                                              377