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 for Ordinary Differential Equations in the Complex Domain

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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