
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. 
RegularSingular 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 HermiteWeber 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 MellinBarnes Transforms 

Chapter 8. 


Complex Oscillation Theory 


8.1. 
Stunnian Methods; Green's Transform 


8.2. 
Zerofree 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. 
RegularSingular Points 


9.6. 
The Fuchsian Class; the Riemann Problem 


9.7. 
IrregularSingular 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 BriotBouquet Equations 


11.2. 
Growth Properties 


11.3. 
Binomial BriotBouquet Equations of Elliptic Function Theory 


Appendix. Elliptic Functions 

Chapter 12. 


Second Order Nonlinear Differential Equations and Some Autonomous Systems 


12.1 
Generalities; BriotBouquet Equations 


12.2 
The Painlevé Transcendents 


12.3 
The Asymptotics of Boutroux 


12.4 
The Emden and the ThomasFermi Equations 


12.5 
Quadratic Systems 


12.6 
Other Autonomous Polynomial Systems 

Bibliography 

Index 
