Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient (eBook)

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1975 | 1. Auflage
289 Seiten
Elsevier Science (Verlag)
978-0-08-087129-5 (ISBN)

Infos

Global theory of a second order linear ordinary differential equation with a polynomial coefficient

Inhalt

Front Cover 1
Global Linear Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient 4
Copyright Page 5
CONTENTS 14
Introduction 6
Chapter 1. Asymptotic expansions 18
1. Definition of asymptotic expansions 18
2. Uniqueness of asymptotic expansions 19
3. Properties of asymptotic expansions 20
4. Summation of formal power series 27
5. Asymptotic expansions of functions depending on parameters 29
Chapter 2. Defition of subdominant solutions 32
6. An existence theorem 32
7. Subdominant solutions 34
8. Case m = 1 and case m = 2 39
9. Proof of Theorem 6.1 43
10. Proof of Lemma 9.1 49
Chapter 3. A relation between Ym and Ym+1 59
11. Remarks on perturbations of asymptotic solutions 59
12. Main problem 62
13. Asymptotic solutions a t x = +8 65
14. Case when P(x.. ) =.x2 +x 70
15. A generalization of Zernike's method 73
16. A subdominant solution of equation (15.2) 76
17. Proof of Theorem 12.1 78
Chapter 4. Asymptotic behavior of Ym(x. a) as am tends to infinity 82
18. Main problem 82
19. Estimates in the sector |arg .| = p - po 83
20. Estimates in the sector |arg .-p| =po 93
Chapter 5. Stokes multipliers 99
21. Preliminary results 99
22. Case m = l and case m = 2 104
23. Properties ata = 0 107
24. Asymptotic behavior of Stokes multipliers as am tends to infinity 117
25. Case when P(x, . ) = x + . 118
26. General case 129
27. A functional equation 139
Chapter 6. A boundary value problem in the complex plane 145
28. Boundary conditions 145
29. Distribution of large eigenvalues 146
30. A unique solution of a non-homogeneous boundary value problem 150
31. Estimate of R(.)g : Part I 157
32. Estimate of R(.)g : Part I1 161
33. Principal parts of R(.)g atits poles 164
34. A completeness property of eigenfunctions 166
Chapter 7. Distribution of zeros 169
35. A relation between zeros and eigenvalues 169
36. Main theorem 171
37. Proof of Theorem 36.1: Part I 173
38. Proof of ‘Theorem 36.1: Part II 178
Chapter 8. A general boundary value problem and associated Riemann surfaces 183
39. A general boundary value problem 183
40. Associated Riemann surfaces 189
41. Construction of a covering surface 193
42. Remarks on non-uniqueness and singularities of solutions 198
43. A remark on the general case 202
44. An application of a line complex 203
45. Proof of Theorem 40.1 208
46. Proof of Theorem 39.1 213
47. Local uniqueness 214
48. A sketch of the proof of Theorem 47.1 215
Chapter 9. Subdominant solutions admitting a prescribed Stokes phenomenon 218
49. Main problems 218
50. Case m = 1 and case m = 2 220
51. Problem 49.1 at a = 0 221
52. Problem 49.2 at a = 0 224
53. Problem 49.1: the general case 228
54. Problem 49.2: the general case 232
55. Subdominant solutions admitting a prescribed Stokes phenomenon 237
56. A general problem of G.D. Birkhoff 242
Chapter 10. Subdominant solutions of the differential equation 250
57. Subdominant solutions 250
58. Formal solutions of the associated Riccati equation 253
59. Asymptotic solutions in a canonical domain as . tends to infinity 259
60. Asymptotic solutions in a canonical domain as x tends to infinity 266
61. Examples of Stokes curves and canonical domains 270
62. Asymptotic representations of subdominant solutions as . tends to infinity 278
63. Simple transition point problems in unbounded domains 283
64. A method due to T.M. Cherry 286
65. Asymptotic solutions in the domain Do 289
66. Remarks on the case when . contains only one of the three S0 , S1 and S2 in the neighborhood of the simple 0 ' sectors 6 transition point Xo 296
67. Examples 298
References 302
Index 306

Produkt-Details

• Erscheint lt. Verlag: 1.1.1975
• Mitarbeit: Herausgeber (Serie): Yasutaka Sibuya
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-087129-1 / 0080871291
• ISBN-13: 978-0-08-087129-5 / 9780080871295