Ordinary Differential Equations in the Complex Domain

by Hille, Einar

• ISBN13: 9780486696201
• ISBN10: 0486696200
• Format: Paperback
• Copyright: 1997-04-09
• Publisher: Dover Publications

Summary

Graduate-level text offers full and extensive treatments of existence theorems, representation of solutions by series, representation by integrals, theory of majorants, dominants and minorants, questions of growth, much more. Relevant review material on complex analysis supplied. Includes 675 exercises at chapter ends. Bibliography.

Table of Contents

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 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
Bibliography
Index