| Preface |
| PART ONE Review of Complex Analysis |
| Introductory Survey |
| Chapter 1. Analytic Behavior |
| Differentiation and Integration |
| 1-1. Analyticity |
| 1-2. Integration on curves and chains |
| 1-3. Cauchy integral theorem |
| Topological Considerations |
| 1-4. Jordan curve theorem |
| 1-5. Other manifolds |
| 1-6. Homologous chains |
| Chapter 2. Riemann Sphere |
| Treatment of Infinity |
| 2-1. Ideal point |
| 2-2. Stereographic projection |
| 2-3. Rational functions |
| 2-4. Unique specification theorems |
| Transformation of the Sphere |
| 2-5. Invariant properties |
| 2-6. Möbius geometry |
| 2-7. Fixed-point classification |
| Chapter 3. Geometric Constructions |
| Analytic Continuation |
| 3-1. Multivalued functions |
| 3-2. Implicit functions |
| 3-3. Cyclic neighborhoods |
| Conformal Mapping |
| 3-4. Local and global results |
| 3-5. Special elementary mappings |
| PART TWO Riemann Manifolds |
| Definition of Riemann Manifold through Generalization |
| Chapter 4. Elliptic Functions |
| Abel's Double-period Structure |
| 4-1. Trigonometric uniformization |
| 4-2. Periods of elliptic integrals |
| 4-3. Physical and topological models |
| Weierstrass' Direct Construction |
| 4-4. Elliptic functions |
| 4-5. Weierstrass' Ã function |
| 4-6. The elliptic modular function |
| Euler's Addition Theorem |
| 4-7. Evolution of addition process |
| 4-8. Representation theorems |
| Chapter 5. Manifolds over the z Sphere |
| Formal Definitions |
| 5-1. Neighborhood Structure |
| 5-2. Functions and differentials |
| Triangulated Manifolds |
| 5-3. Triangulation structure |
| 5-4. Algebraic Riemann manifolds |
| Chapter 6. Abstract Manifolds |
| 6-1. Punction field on M |
| 6-2. Compact manifolds are algebraic |
| 6-3. Modular functions |
| PART THREE Derivation of Existence Theorems |
| Return to Real Variables |
| Chapter 7. Topological Considerations |
| The Two Canonical Models |
| 7-1. Orientability |
| 7-2. Canonical subdivisions |
| 7-3. The Euler-Poincaré theorem |
| 7-4. Proof of models |
| Homology and Abelian Differentials |
| 7-5. Boundaries and cy |
| 7-6. Complex existence theorem |
| Chapter 8. Harmonic Differentials |
| Real Differentials |
| 8-1. Cohomology |
| 8-2. Stokes' theorem |
| 8-3. Conjugate forms |
| Dirichlet Problems |
| 8-4. The two existence theorems |
| 8-5. The two uniqueness proofs |
| Chapter 9. Physical Intuition |
| 9-1. Electrostatics and hydrodynamics |
| 9-2. Special solutions |
| 9-3. Canonical mappings |
| PART FOUR Real Existence Proofs |
| Evolution of Some Intuitive Theorems |
| Chapter 10. Conformal Mapping |
| 10-1. Poisson's integral |
| 10-2. Riemann' s theorem for the disk |
| Chapter 11. Boundary Behavior |
| 11-1. Continuity |
| 11-2. Analyticity |
| 11-3. Schottky double |
| Chapter 12. Alternating Procedures |
| 12-1. Ordinary Dirichlet problem |
| 12-2. Nonsingular noncompact problem |
| 12-3. Planting of singularities |
| PART FIVE Algebraic Applications |
| Resurgence of Finite Structures |
| Chapter 13. Riemann's Existence Theorem |
| 13-1. Normal integrals |
| 13-2. Construction of the function field |
| Chapter 14. Advanced Results |
| 14-1. Riemann-Roch theorem |
| 14-2. Abel's theorem |
| Appendix A. Minimal Principles |
| Appendix B. Infinite Manifolds |
| Table 1: Summary of Existence and Uniqueness Proofs |
| Bibliography and Special Source Material |
| Index |
