Graduate-level text offers full treatments of existence theorems, representation of solutions by series, theory of majorants, dominants and minorants, questions of growth, much more.
Country of Publication:
Series: Dover Books on Mathematics
09 April 1997
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 Lindelof 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. Painleve'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 Painleve 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