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Ordinary Differential Equations in the Complex Domain

Einar Hille
Dover Publications
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Chapter 1.
  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.
    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 Equations
    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 Boutroux
    12.4 The Emden and the Thomas-Fermi Equations
    12.5 Quadratic Systems
    12.6 Other Autonomous Polynomial Systems