Chapter I. Some Basic Properties of Linear Differential Equations in the Complex Domain |
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1. |
Preparatory Remarks |
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2. |
The Basic Existence Theorem and its Consequences |
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3. |
Circuit Relations About Singular Points |
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Chapter II. Regular Singular Points |
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4. |
Method of Solution |
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5. |
Solutions at a Regular Singular Point |
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Chapter III. Asymptotic Power Series |
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6. |
Introductory Remarks on Irregular Singular Points |
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7. |
Definition of Asymptotic Power Series |
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8. |
Elementary Properties of Asymptotic Series |
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9. |
The Existence of Asymptotic Series |
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Chapter IV. Irregular Singular Points |
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10. |
Introduction |
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11. |
Formal Simplification |
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12. |
Analytic Symplification and Asymptotic Solution |
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13. |
Miscellaneous Remarks |
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14. |
Proof of the Main Asymptotic Existence Theorem when all Eigenvalues are Distinct |
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15. |
The Stokes Phenomenon |
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Chapter V. Generalizations by Means of Jordan's Canonical Form |
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16. |
Jordan's Canonical Form |
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17. |
Solutions at Regular Singular Points: General Case |
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18. |
Proof of Theorem 12.1: General Case |
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19. |
Asymptotic Solution at an Irregular Singularity: General Case |
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Chapter VI. Some Special Asymptotic Methods |
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20. |
Introduction |
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21. |
Calculating Asymptotic Expansions from Convergent Power Series |
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22. |
Solution by Laplace Contour Integrals |
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23. |
The Saddlepoint Method |
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Chapter VII. Asymptotic Expansions with Respect to a Parameter |
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24. |
Introduction |
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25. |
Formal Theory |
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26. |
Analytic Symplification |
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27. |
Proof of Theorem 26.1 |
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28. |
Shearing Transformations |
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Chapter VIII. Turning Point Problems |
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29. |
Problems Reducible to Airy's Equation: Formal Theory |
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30. |
Problems Reducible to Airy's Equation: Analytic Theory |
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31. |
Short Report on Other Turning Point Problems |
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Chapter IX. Nonlinear Equations |
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32. |
Introduction |
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33. |
Solution by Asymptotic Power Series |
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34. |
Transformation into a Linear Differential Equation |
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35. |
Solution by Exponential Series |
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36. |
Nonlinear Equations with a Parameter |
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Chapter X. Singular Perturbations |
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37. |
Boundary Value Problems for Linear Equations |
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38. |
Boundary Value Problems for Linear Equations: The Method of Višik and Lyusternik |
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37. |
Initial Value Problems for Nonlinear Equations: Qualitative Theory |
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40. |
Series Expansions for the Initial Value Problem |
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41. |
Nonlinear Two-Point Boundary Value Problems |
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42. |
Decomposition of General Linear Systems of Singular Perturbation Type |
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43. |
Periodic Solutions of Singular Perturbation Problems: General Remarks |
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44. |
Periodic Solutions of Singular Perturbation Problems: Linear Theory |
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45. |
Series Expansions for Periodic Solutions of Singular Perturbation Problems |
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Chapter XI. Integration of Differential Equations by Factorial Series |
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46. |
Factorial Series and Laplace Integrals |
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47. |
Solution of Differential Equations of Rank One by Factorial Series |
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48. |
Remarks on the Solution of Differential Equations of Higher Rank by Factorial Series |
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Appendix: A Brief Summary of Some Recent Research |
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Bibliography |
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Subject Index |
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