Asymptotic methods in analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam [u.a.]
North-Holland Publ. Co. [u.a.]
1970
|
| Ausgabe: | 3. ed. |
| Schriftenreihe: | Bibliotheca mathematica
4 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | 200 S. |
Internformat
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| 245 | 1 | 0 | |a Asymptotic methods in analysis |c by N. G. de Bruijn |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b North-Holland Publ. Co. [u.a.] |c 1970 | |
| 300 | |a 200 S. | ||
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| 490 | 1 | |a Bibliotheca mathematica |v 4 | |
| 650 | 7 | |a Analysis |2 swd | |
| 650 | 7 | |a Asymptotische Methode |2 swd | |
| 650 | 4 | |a Mathematical analysis | |
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Datensatz im Suchindex
| _version_ | 1804136096640008192 |
|---|---|
| adam_text | CONTENTS
Preface v
Ch. 1. INTRODUCTION 1
1.1. What is asymptotics ? 1
1.2. TheO symbol 3
1.3. The o symbol 10
1.4. Asymptotic equivalence 10
1.5. Asymptotic series 11
1.6. Elementary operations on asymptotic scries . . . 14
1.7. Asymptotics and Numerical Analysis 18
1.8. Exercises 19
Ch. 2. IMPLICIT FUNCTIONS 21
2.1. Introduction 21
2.2. The Lagrange inversion formula 22
2.3. Applications 23
2.4. A more difficult case 25
2.5. Iteration methods 28
2.6. Roots of equations 30
2.7. Asymptotic iteration 31
2.8. Exercises 33
Ch. 3. SUMMATION 34
3.1. Introduction 34
3.2. Case a 34
3.3. Case b 36
3.4. Casec 37
3.5. Case d 38
X CONTENTS
3.6. The Euler Maclaurin sum formula 40
3.7. Example 42
3.8. A remark 42
3.9. Another example 43
3.10. The Stirling formula for the F iunction in the
complex plane 46
3.11. Alternating sums 49
3.12. Application of the Poisson sum formula .... 52
3.13. Summation by parts 56
3.14. Exercises 58
Ch. 4. THE LAPLACE METHOD FOR INTEGRALS . . 60
4.1. Introduction 60
4.2. A general case 63
4.3. Maximum at the boundary 65
4.4. Asymptotic expansions 66
4.5. Asymptotic behaviour of the / function .... 69
4.6. Multiple integrals 71
4.7. An application 72
4.8. Exercises 75
Ch. 5. THE SADDLE POINT METHOD 77
5.1. The method 77
5.2. Geometrical interpretation 79
5.3. Peakless landscapes 82
5.4. Steepest descent 83
5.5. Steepest descent at end point 86
5.6. The second stage 86
5.7. A general simple case 87
5.8. Path of constant altitude 89
5.9. Closed path 90
5.10. Range of a saddle point 91
5.11. Examples 93
5.12. Small perturbations 96
5.13. Exercises 101
CONTENTS XI
Ch. 6. APPLICATIONS OF THE SADDLE POINT
METHOD , 102
6.1. The number of class partitions of a finite set . . . 102
6.2. Asymptotic behaviour of dn 104
6.3. Alternative method 108
6.4. The sum S(s, n) 109
6.5. Asymptotic behaviour of P 112
6.6. Asymptotic behaviour of Q 115
6.7. Conclusions about S(s, n) 118
6.8. A modified Gamma Function 119
6.9. The entire function G0(s) 123
6.10. Conclusions about G(s) 131
6.11. Exercises 133
Ch. 7. INDIRECT ASYMPTOTICS 134
7.1. Direct and indirect asymptotics 134
7.2. Tauberian theorems 137
7.3. Differentiation of an asymptotic formula .... 139
7.4. A similar problem 141
7.5. Karamata s method 143
7.6. Exercises 147
Ch. 8. ITERATED FUNCTIONS 148
8.1. Introduction 148
8.2. Iterates of a function 148
8.3. Rapid convergence 151
8.4. Slow convergence 153
8.5. Preparation 154
8.6. Iteration of the sine function 157
8.7. An alternative method 160
8.8. Final discussion about the iterated sine 164
8.9. An inequality concerning infinite series 166
8.10. The iteration problem 169
8.11. Exercises 175
XII CONTENTS
Ch. 9. DIFFERENTIAL EQUATIONS 176
9.1. Introduction 176
9.2. A Riccati equation 177
9.3. An unstable case 184
9.4. Application to a linear second order equation . . 186
9.5. Oscillatory cases 189
9.6. More general oscillatory cases 195
9.7. Exercises 198
INDEX 199
|
| adam_txt |
CONTENTS
Preface v
Ch. 1. INTRODUCTION 1
1.1. What is asymptotics ? 1
1.2. TheO symbol 3
1.3. The o symbol 10
1.4. Asymptotic equivalence 10
1.5. Asymptotic series 11
1.6. Elementary operations on asymptotic scries . . . 14
1.7. Asymptotics and Numerical Analysis 18
1.8. Exercises 19
Ch. 2. IMPLICIT FUNCTIONS 21
2.1. Introduction 21
2.2. The Lagrange inversion formula 22
2.3. Applications 23
2.4. A more difficult case 25
2.5. Iteration methods 28
2.6. Roots of equations 30
2.7. Asymptotic iteration 31
2.8. Exercises 33
Ch. 3. SUMMATION 34
3.1. Introduction 34
3.2. Case a 34
3.3. Case b 36
3.4. Casec 37
3.5. Case d 38
X CONTENTS
3.6. The Euler Maclaurin sum formula 40
3.7. Example 42
3.8. A remark 42
3.9. Another example 43
3.10. The Stirling formula for the F iunction in the
complex plane 46
3.11. Alternating sums 49
3.12. Application of the Poisson sum formula . 52
3.13. Summation by parts 56
3.14. Exercises 58
Ch. 4. THE LAPLACE METHOD FOR INTEGRALS . . 60
4.1. Introduction 60
4.2. A general case 63
4.3. Maximum at the boundary 65
4.4. Asymptotic expansions 66
4.5. Asymptotic behaviour of the /" function . 69
4.6. Multiple integrals 71
4.7. An application 72
4.8. Exercises 75
Ch. 5. THE SADDLE POINT METHOD 77
5.1. The method 77
5.2. Geometrical interpretation 79
5.3. Peakless landscapes 82
5.4. Steepest descent 83
5.5. Steepest descent at end point 86
5.6. The second stage 86
5.7. A general simple case 87
5.8. Path of constant altitude 89
5.9. Closed path 90
5.10. Range of a saddle point 91
5.11. Examples 93
5.12. Small perturbations 96
5.13. Exercises 101
CONTENTS XI
Ch. 6. APPLICATIONS OF THE SADDLE POINT
METHOD , 102
6.1. The number of class partitions of a finite set . . . 102
6.2. Asymptotic behaviour of dn 104
6.3. Alternative method 108
6.4. The sum S(s, n) 109
6.5. Asymptotic behaviour of P 112
6.6. Asymptotic behaviour of Q 115
6.7. Conclusions about S(s, n) 118
6.8. A modified Gamma Function 119
6.9. The entire function G0(s) 123
6.10. Conclusions about G(s) 131
6.11. Exercises 133
Ch. 7. INDIRECT ASYMPTOTICS 134
7.1. Direct and indirect asymptotics 134
7.2. Tauberian theorems 137
7.3. Differentiation of an asymptotic formula . 139
7.4. A similar problem 141
7.5. Karamata's method 143
7.6. Exercises 147
Ch. 8. ITERATED FUNCTIONS 148
8.1. Introduction 148
8.2. Iterates of a function 148
8.3. Rapid convergence 151
8.4. Slow convergence 153
8.5. Preparation 154
8.6. Iteration of the sine function 157
8.7. An alternative method 160
8.8. Final discussion about the iterated sine 164
8.9. An inequality concerning infinite series 166
8.10. The iteration problem 169
8.11. Exercises 175
XII CONTENTS
Ch. 9. DIFFERENTIAL EQUATIONS 176
9.1. Introduction 176
9.2. A Riccati equation 177
9.3. An unstable case 184
9.4. Application to a linear second order equation . . 186
9.5. Oscillatory cases 189
9.6. More general oscillatory cases 195
9.7. Exercises 198
INDEX 199 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:16:21Z |
| indexdate | 2024-07-09T20:50:58Z |
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| language | English |
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| series | Bibliotheca mathematica |
| series2 | Bibliotheca mathematica |
| spelling | Bruijn, Nicolaas Govert de 1918-2012 Verfasser (DE-588)12271024X aut Asymptotic methods in analysis by N. G. de Bruijn 3. ed. Amsterdam [u.a.] North-Holland Publ. Co. [u.a.] 1970 200 S. txt rdacontent n rdamedia nc rdacarrier Bibliotheca mathematica 4 Analysis swd Asymptotische Methode swd Mathematical analysis Numerical analysis Asymptotische Methode (DE-588)4287476-2 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s Asymptotische Methode (DE-588)4287476-2 s DE-604 Bibliotheca mathematica 4 (DE-604)BV001887077 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015338154&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bruijn, Nicolaas Govert de 1918-2012 Asymptotic methods in analysis Bibliotheca mathematica Analysis swd Asymptotische Methode swd Mathematical analysis Numerical analysis Asymptotische Methode (DE-588)4287476-2 gnd Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4287476-2 (DE-588)4001865-9 |
| title | Asymptotic methods in analysis |
| title_auth | Asymptotic methods in analysis |
| title_exact_search | Asymptotic methods in analysis |
| title_exact_search_txtP | Asymptotic methods in analysis |
| title_full | Asymptotic methods in analysis by N. G. de Bruijn |
| title_fullStr | Asymptotic methods in analysis by N. G. de Bruijn |
| title_full_unstemmed | Asymptotic methods in analysis by N. G. de Bruijn |
| title_short | Asymptotic methods in analysis |
| title_sort | asymptotic methods in analysis |
| topic | Analysis swd Asymptotische Methode swd Mathematical analysis Numerical analysis Asymptotische Methode (DE-588)4287476-2 gnd Analysis (DE-588)4001865-9 gnd |
| topic_facet | Analysis Asymptotische Methode Mathematical analysis Numerical analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015338154&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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