| Chapter 1. |
|   | Introduction |
| l. Algebraic and Geometric Structures |
|   | 1.1. Vector Spaces |
|   | 1.2. Metric Spaces |
|   | 1.3. Mappings |
|   | 1.4. Linear Transformations on C into Itself ; Matrices |
|   | 1.5. Fixed Point Theorems |
|   | 1.6. Functional Inequalities |
| II. Analytical Structures |
|   | 1.7. Holomorphic Functions |
|   | 1.8. Power Series |
|   | 1.9. Cauchy Integrals |
|   | 1.10. Estimates of Growth |
|   | 1.11. Analytic Continuation; Permanency of Functional Equations |
| Chapter 2. |
|   | Existence and Uniqueness Theorems |
|   | 2.1. Equations and Solutions |
|   | 2.2. The Fixed Point Method |
|   | 2.3. The Method of Successive Approximations |
|   | 2.4. Majorants and Majorant Methods |
|   | 2.5. The Cauchy Majorant |
|   | 2.6. The Lindelöf Majorant |
|   | 2.7. The Use of Dominants and Minorants |
|   | 2.8. Variation of Parameters |
| Chapter 3. |
|   | Singularities |
|   | 3.1. Fixed and Movable Singularities |
|   | 3.2. Analytic Continuation; Movable Singularities |
|   | 3.3. Painlevé's Determinateness Theorem; Singularities |
|   | 3.4. Indeterminate Forms |
| Chapter 4. |
|   | Riccati's Equation |
|   | 4.1. Classical Theory |
|   | 4.2. Dependence on Internal Parameters; Cross Ratios |
|   | 4.3. Some Geometric Applications |
|   | 4.4. "Abstract of the Nevanlinna Theory, I " |
|   | 4.5. "Abstract of the Nevanlinna Theory, II " |
|   | 4.6. The Malmquist Theorem and Some Generalizations |
| Chapter 5. |
|   | Linear Differential Equations: First and Second Order |
|   | 5.1. General Theory: First Order Case |
|   | 5.2. General Theory: Second Order Case |
|   | 5.3. Regular-Singular Points |
|   | 5.4. Estimates of Growth |
|   | 5.5. Asymptotics on the Real Line |
|   | 5.6. Asymptotics in the Plane |
|   | 5.7. Analytic Continuation; Group of Monodromy |
| Chapter 6. |
|   | Special Second Order Linear Dulerential Equat |
|   | 6.1. The Hypergeometric Equation |
|   | 6.2. Legendre's Equation |
|   | 6.3. Bessel's Equation |
|   | 6.4. Laplace's Equation |
|   | 6.5. The Laplacian; the Hermite-Weber Equation; Functions of the Parabolic Cylinder |
|   | 6.6. The Equation of Mathieu; Functions of the Elliptic Cylinder |
|   | 6.7. Some Other Equations |
| Chapter 7. |
|   | Representation Theorems |
|   | 7.1. Psi Series |
|   | 7.2. Integral Representations |
|   | 7.3. The Euler Transform |
|   | 7.4. Hypergeometric Euler Transforms |
|   | 7.5. The Laplace Transform |
|   | 7.6. Mellin and Mellin-Barnes Transforms |
| Chapter 8. |
|   | Complex Oscillation Theory |
|   | 8.1. Stunnian Methods; Green's Transform |
|   | 8.2. Zero-free Regions and Lines of Influence |
|   | 8.3. Other Comparison Theorems |
|   | 8.4. Applications to Special Equations |
| Chapter 9. |
|   | Linear nth Order and Matrix Differential Equations |
|   | 9.1. Existence and Independence of Solutions |
|   | 9.2. Analyticity of Matrix Solutions in a Star |
|   | 9.3. Analytic Continuation and the Group of Monodromy |
|   | 9.4. Approach to a Singularity |
|   | 9.5. Regular-Singular Points |
|   | 9.6. The Fuchsian Class; the Riemann Problem |
|   | 9.7. Irregular-Singular Points |
| Chapter 10. |
|   | The Schwarzian |
|   | 10.1. The Schwarzian Derivative |
|   | 10.2. Applications to Conformal Mapping |
|   | 10.3. Algebraic Solutions of Hypergeometric Equations |
|   | 10.4. Univalence and the Schwarzian |
|   | 10.5. Uniformization by Modular Functions |
| Chapter 11. |
|   | First Order Nonlinear Differential Equations |
|   | 11.1. Some Briot-Bouquet Equations |
|   | 11.2. Growth Properties |
|   | 11.3. Binomial Briot-Bouquet Equations of Elliptic Function Theory |
|   | Appendix. Elliptic Functions |
| Chapter 12. |
|   | Second Order Nonlinear Differential Equations and Some Autonomous Systems |
|   | 12.1 Generalities; Briot-Bouquet Equations |
|   | 12.2 The Painlevé Transcendents |
|   | 12.3 The Asymptotics of Bout |
|   | 12.4 The Emden and the Thomas-Fermi Equations |
|   | 12.5 Quadratic Systems |
|   | 12.6 Other Autonomous Polynomial Systems |
| Bibliography |
| Index |
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