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Asymptotic Expansions for Ordinary Differential Equations

Wolfgang Wasow
Dover Publications
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Dover Phoneix Editions
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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 Two-Point 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
  Subject Index