Chapter I. Some Basic Properties of Linear Differential Equations in the Complex Domain 

1. 
Preparatory Remarks 

2. 
The Basic Existence Theorem and its Consequences 

3. 
Circuit Relations About Singular Points 

Chapter II. Regular Singular Points 

4. 
Method of Solution 

5. 
Solutions at a Regular Singular Point 

Chapter III. Asymptotic Power Series 

6. 
Introductory Remarks on Irregular Singular Points 

7. 
Definition of Asymptotic Power Series 

8. 
Elementary Properties of Asymptotic Series 

9. 
The Existence of Asymptotic Series 

Chapter IV. Irregular Singular Points 

10. 
Introduction 

11. 
Formal Simplification 

12. 
Analytic Symplification and Asymptotic Solution 

13. 
Miscellaneous Remarks 

14. 
Proof of the Main Asymptotic Existence Theorem when all Eigenvalues are Distinct 

15. 
The Stokes Phenomenon 

Chapter V. Generalizations by Means of Jordan's Canonical Form 

16. 
Jordan's Canonical Form 

17. 
Solutions at Regular Singular Points: General Case 

18. 
Proof of Theorem 12.1: General Case 

19. 
Asymptotic Solution at an Irregular Singularity: General Case 

Chapter VI. Some Special Asymptotic Methods 

20. 
Introduction 

21. 
Calculating Asymptotic Expansions from Convergent Power Series 

22. 
Solution by Laplace Contour Integrals 

23. 
The Saddlepoint Method 

Chapter VII. Asymptotic Expansions with Respect to a Parameter 

24. 
Introduction 

25. 
Formal Theory 

26. 
Analytic Symplification 

27. 
Proof of Theorem 26.1 

28. 
Shearing Transformations 

Chapter VIII. Turning Point Problems 

29. 
Problems Reducible to Airy's Equation: Formal Theory 

30. 
Problems Reducible to Airy's Equation: Analytic Theory 

31. 
Short Report on Other Turning Point Problems 

Chapter IX. Nonlinear Equations 

32. 
Introduction 

33. 
Solution by Asymptotic Power Series 

34. 
Transformation into a Linear Differential Equation 

35. 
Solution by Exponential Series 

36. 
Nonlinear Equations with a Parameter 

Chapter X. Singular Perturbations 

37. 
Boundary Value Problems for Linear Equations 

38. 
Boundary Value Problems for Linear Equations: The Method of Višik and Lyusternik 

37. 
Initial Value Problems for Nonlinear Equations: Qualitative Theory 

40. 
Series Expansions for the Initial Value Problem 

41. 
Nonlinear TwoPoint Boundary Value Problems 

42. 
Decomposition of General Linear Systems of Singular Perturbation Type 

43. 
Periodic Solutions of Singular Perturbation Problems: General Remarks 

44. 
Periodic Solutions of Singular Perturbation Problems: Linear Theory 

45. 
Series Expansions for Periodic Solutions of Singular Perturbation Problems 

Chapter XI. Integration of Differential Equations by Factorial Series 

46. 
Factorial Series and Laplace Integrals 

47. 
Solution of Differential Equations of Rank One by Factorial Series 

48. 
Remarks on the Solution of Differential Equations of Higher Rank by Factorial Series 

Appendix: A Brief Summary of Some Recent Research 

Bibliography 

Subject Index 


