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 |