-Sherlock Holmes, The Valley of Fear Sir Arthur Conan Doyle The main purpose of our book is to present and explain mathematical methods for obtaining approximate analytical solutions to differential and difference equations that cannot be solved exactly. Preface xiii PART I FUNDAMENTALS Ordinary Differential Equations 3(33) Ordinary Differential Equations 3(2) Initial-Value and Boundary-Value Problems 5(2) Theory of Homogeneous Linear Equations 7(4) Solutions of Homogeneous Linear Equations 11(3) Inhomogeneous Linear Equations 14(6) First-Order Nonlinear Differential Equations 20(4) Higher-Order Nonlinear Differential Equations 24(3) Eigenvalue Problems 27(2) Differential Equations in the Complex Plane 29(7) Problems for Chapter 1 30(6) Difference Equations 36(25) The Calculus of Differences 36(1) Elementary Difference Equations 37(3) Homogeneous Linear Difference Equations 40(9) Inhomogeneous Linear Difference Equations 49(4) Nonlinear Difference Equations 53(8) Problems for Chapter 2 53(8) PART II LOCAL ANALYSIS Approximate Solution of Linear Differential Equations 61(85) Classification of Singular Points of Homogeneous Linear Equations 62(4) Local Behavior Near Ordinary Points of Homogeneous Linear Equations 66(2) Local Series Expansions About Regular Singular Points of Homogeneous Linear Equations 68(8) Local Behavior at Irregular Singular Points of Homogeneous Linear Equations 76(12) Irregular Singular Point at Infinity 88(15) Local Analysis of Inhomogeneous Linear Equations 103(4) Asymptotic Relations 107(11) Asymptotic Series 118(28) Problems for Chapter 3 136(10) Approximate Solution of Nonlinear Differential Equations 146(59) Spontaneous Singularities 146(2) Approximate Solutions of First-Order Nonlinear Differential Equations 148(4) Approximate Solutions to Higher-Order Nonlinear Differential Equations 152(19) Nonlinear Autonomous Systems 171(14) Higher-Order Nonlinear Autonomous Systems 185(20) Problems for Chapter 4 196(9) Approximate Solution of Difference Equations 205(42) Introductory Comments 205(1) Ordinary and Regular Singular Points of Linear Difference Equations 206(8) Local Behavior Near an Irregular Singular Point at Infinity: Determination of Controlling Factors 214(4) Asymptotic Behavior of n! as n ? ? : The Stirling Series 218(9) Local Behavior Near an Irregular Singular Point at Infinity: Full Asymptotic Series 227(6) Local Behavior of Nonlinear Difference Equations 233(14) Problems for Chapter 5 240(7) Asymptotic Expansion of Integrals 247(72) Introduction 247(2) Elementary Examples 249(3) Integration by Parts 252(9) Laplace's Method and Watson's Lemma 261(15) Method of Stationary Phase 276(4) Method of Steepest Descents 280(22) Asymptotic Evaluation of Sums 302(17) Problems for Chapter 6 306(13) PART III PERTURBATION METHODS Perturbation Series 319(49) Perturbation Theory 319(5) Regular and Singular Perturbation Theory 324(6) Perturbation Methods for Linear Eigenvalue Problems 330(5) Asymptotic Matching 335(15) Mathematical Structure of Perturbative Eigenvalue Problems 350(18) Problems for Chapter 7 361(7) Summation of Series 368(49) Improvement of Convergence 368(11) Summation of Divergent Series 379(4) Pade Summation 383(12) Continued Fractions and Pade Approximants 395(5) Convergence of Pade Approximants 400(5) Pade Sequences for Stieltjes Functions 405(12) Problems for Chapter 8 410(7) PART IV GLOBAL ANALYSIS Boundary Layer Theory 417(67) Introduction to Boundary-Layer Theory 419(7) Mathematical Structure of Boundary Layers: Inner, Outer, and Intermediate Limits 426(5) Higher-Order Boundary Layer Theory 431(4) Distinguished Limits and Boundary Layers of Thickness ? &epsis; 435(11) Miscellaneous Examples of Linear Boundary-Layer Problems 446(9) Internal Boundary Layers 455(8) Nonlinear Boundary-Layer Problems 463(21) Problems for Chapter 9 479(5) WKB Theory 484(60) The Exponential Approximation for Dissipative and Dispersive Phenomena 484(9) Conditions for Validity of the WKB Approximation 493(4) Patched Asymptotic Approximations: WKB Solution of Inhomogeneous Linear Equations 497(7) Matched Asymptotic Approximations: Solution of the One-Turning-Point Problem 504(15) Two-Turning-Point Problems: Eigenvalue Condition 519(5) Tunneling 524(10) Brief Discussion of Higher-Order WKB Approximations 534(10) Problems for Chapter 10 539(5) Multiple-Scale Analysis 544(25) Resonance and Secular Behavior 544(5) Multiple-Scale Analysis 549(2) Examples of Multiple-Scale Analysis 551(9) The Mathieu Equation and Stability 560(9) Problems for Chapter 11 566(3) Appendix---Useful Formulas 569(8) References 577(4) Index 581