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 |