Asymptotic analysis of differential equations:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
London
Imperial College Press
2005
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 286 S. graph. Darst. |
| ISBN: | 1860945872 1860946127 |
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| adam_text | ASYMPTOTIC ANALYSIS OF DIFFERENTIAL EQUATIONS ROSCOE B WHITE PRINCETON
UNIVERSITY, USA IMPERIAL COLLEGE PRESS CONTENTS PREFACE V ACKNOWLEDGMENT
VII LIST OF FIGURES XV 1. DOMINANT BALANCE 1 1.1 INTRODUCTION 2 1.2
SOLUTIONS USING KRUSKAL-NEWTON GRAPHS 3 1.2.1 THIRD ORDER 3 1.2.2 NON
POLYNOMIAL FORM 5 1.2.3 HIGHER ORDER 7 1.2.4 HIDDEN POINTS 8 1.3
PROBLEMS 9 2. EXACT SOLUTIONS 11 2.1 INTRODUCTION 11 2.2 CONSTANT
COEFFICIENTS 13 2.3 INHOMOGENEOUS LINEAR EQUATIONS 14 2.4 THE FREDHOLM
ALTERNATIVE 16 2.5 THE DIFFUSION EQUATION 18 2.6 EXACT SOLUTIONS TO
NONLINEAR EQUATIONS 19 2.7 PHASE PLANE ANALYSIS T 19 2.8 PROBLEMS 22 3.
COMPLEX VARIABLES 25 3.1 ANALYTICITY 25 3.2 CAUCHY INTEGRAL THEOREM 26
3.3 SERIES REPRESENTATION 28 3.4 THE RESIDUE THEOREM 29 X ASYMPTOTIC
ANALYSIS OF DIFFERENTIAL EQUATIONS 3.5 ANALYTIC CONTINUATION 31 3.6
INVERSE FUNCTIONS 32 3.7 PROBLEMS 34 4. LOCAL APPROXIMATE SOLUTIONS 37
4.1 INTRODUCTION 37 4.2 CLASSIFICATION 38 4.2.1 ORDINARY POINT 39 4.2.2
REGULAR SINGULAR POINT 39 4.2.3 IRREGULAR SINGULAR POINT 42 4.3
ASYMPTOTIC SERIES 45 4.3.1 PROPERTIES 46 4.3.2 TRUNCATION: A SERIES
ABOUT X * 0 48 4.3.3 TRUNCATION: A SERIES ABOUT X = OO 50 4.3.4
TRUNCATION: A SERIES ABOUT X = 0 51 4.3.5 ASYMPTOTIC OSCILLATION 55 4.4
CONSTRUCTION OF ASYMPTOTIC SERIES 55 4.4.1 THE ERROR FUNCTION 56 4.5
ORIGIN OF THE DIVERGENCE 58 4.6 IMPROVING SERIES CONVERGENCE 60 4.6.1
SHANKS TRANSFORMATION 60 4.6.2 EULER SUMMATION 61 4.6.3 BOREL SUMMATION
62 4.7 PROBLEMS 62 5. PHASE INTEGRAL METHODS 67 5.1 INTRODUCTION 67 5.2
CONNECTION FORMULAE: ISOLATED ZERO 71 5.3 DERIVATION OF STOKES CONSTANTS
73 5.4 RULES FOR CONTINUATION 76 5.5 CAUSALITY 77 5.6 BOUND STATES AND
INSTABILITIES 78 5.7 SCATTERING 81 5.8 EIGENVALUE PROBLEMS 85 5.9 THE
BUDDEN PROBLEM 86 5.10 THE ERROR FUNCTION 88 5.11 PROBLEMS 89 6.
PERTURBATION THEORY 91 6.1 INTRODUCTION 92 6.2 EIGENVALUES OF A
HERMITIAN MATRIX , . . . 93 6.3 BROKEN SYMMETRY DUE TO TUNNELING 97
CONTENTS XI 6.4 PROBLEMS 100 7. ASYMPTOTIC EVALUATION OF INTEGRALS 103
7.1 INTRODUCTION 103 7.2 END POINT 105 7.3 SADDLE POINT 108 7.4 PROBLEMS
112 8. THE EULER GAMMA FUNCTION 115 8.1 INTRODUCTION 116 8.2 THE
STIRLING APPROXIMATION 117 8.3 THE EULER-MASCHERONI CONSTANT 119 8.4
SINE PRODUCT IDENTITY 120 8.5 CONTINUATION OF T(Z) 122 8.6 ASYMPTOTIC
T(Z) 122 8.7 EULER PRODUCT FOR T 125 8.8 INTEGRAL REPRESENTATION FOR
L/T(Z) 126 8.9 . T(NX) 128 8.10 THE EULER BETA FUNCTION 128 8.11
PROBLEMS 129 9. INTEGRAL SOLUTIONS 131 9.1 CONSTRUCTING INTEGRAL
SOLUTIONS 131 9.1.1 INTEGRATION BY PARTS 132 9.1.2 FINDING A DISCRETE
DIFFERENCE EQUATION 134 9.1.3 CONSTRUCTION FROM A SERIES 135 9.2 CAUSAL
SOLUTIONS 136 9.3 COMMENTS 138 9.4 PROBLEMS 138 10. EXPANSION IN BASIS
FUNCTIONS 141 10.1 LEGENDRE FUNCTIONS 142 10.1.1 LOCAL ANALYSIS 142
10.1.2 EULER INTEGRAL REPRESENTATION 143 10.1.3 RECURRENCE RELATIONS 145
10.1.4 LAPLACE INTEGRAL REPRESENTATION 147 10.1.5 GENERATING FUNCTION .
. .* 149 10.1.6 LEGENDRE POLYNOMIALS 150 10.2 ORTHOGONAL POLYNOMIALS 151
10.3 WAVELETS 154 10.3.1 INTRODUCTION 154 10.3.2 SCALING FUNCTION 154
10.3.3 WAVELET BASIS CONSTRUCTION 158 XII ASYMPTOTIC ANALYSIS OF
DIFFERENTIAL EQUATIONS 10.3.4 DETERMINING THE EXPANSION COEFFICIENTS 160
10.3.5 EXAMPLES 162 10.3.6 TIME-FREQUENCY ANALYSIS 166 10.4 PROBLEMS 168
11. AIRY 171 11.1 WKB ANALYSIS 172 11.2 FOURIER LAPLACE INTEGRAL
REPRESENTATION 174 11.3 ASYMPTOTIC LIMITS 176 11.3.1 LARGE NEGATIVE Z
176 11.3.2 LARGE POSITIVE Z 178 11.3.3 SMALL Z 180 11.4 MELLIN
INTEGRAL REPRESENTATION 181 11.5 MATCHING LOCAL SOLUTIONS 183 11.6
THEWRONSKIAN 185 11.7 PROBLEMS 185 12. BESSEL 187 12.1 LOCAL ANALYSIS
187 12.1.1 LOCAL AT ZERO 187 12.1.2 ANALYTIC CONTINUATION IN V 189
12.1.3 LOCAL AT INFINITY 189 12.2 WKB ANALYSIS 190 12.3 INTEGRAL
REPRESENTATIONS 191 12.3.1 FOURIER LAPLACE REPRESENTATION 191 12.3.2 THE
HANKEL INTEGRALS 192 12.3.3 ASYMPTOTIC LIMITS , 193 12.3.4 RELATION TO
BESSEL AND NEUMANN FUNCTIONS 195 12.3.5 THE WRONSKIAN 197 12.3.6
SOMMERFELD INTEGRAL REPRESENTATION 198 12.3.7 MELLIN INTEGRAL
REPRESENTATION 201 12.4 GENERATING FUNCTION 202 12.5 MATCHING LOCAL
SOLUTIONS 202 12.6 IMAGINARY ARGUMENT 203 12.7 PROBLEMS 204 13.
WEBER-HERMITE 207 13.1 LOCAL ANALYSIS AT INFINITY 208 13.2 LOCAL
ANALYSIS AT ZERO 209 13.3 WKB ANALYSIS 210 13.4 EULER INTEGRAL
REPRESENTATION 212 13.5 FOURIER-LAPLACE INTEGRAL REPRESENTATION I 213
CONTENTS XIII 13.6 ORTHOGONALITY 217 13.7 FOURIER-LAPLACE INTEGRAL
REPRESENTATION II 218 13.8 THE WRONSKIAN 221 13.9 MELLIN INTEGRAL
REPRESENTATION 222 13.10 PROBLEMS 223 14. WHITTAKER AND WATSON 225 14.1
LOCAL ANALYSIS AT INFINITY 226 14.2 LOCAL ANALYSIS AT ZERO 227 14.3
EULER INTEGRAL REPRESENTATION 227 14.4 RELATION BETWEEN M X ^{Z) AND
W ^(Z) 230 14.5 FOURIER LAPLACE INTEGRAL REPRESENTATION 231 14.6 MELLIN
INTEGRAL REPRESENTATION 232 14.7 SPECIAL CASES 235 14.7.1 THE ERROR
FUNCTION 235 14.7.2 THE LOGARITHMIC INTEGRAL FUNCTION 235 14.8 PROBLEMS
235 15. INHOMOGENEOUS DIFFERENTIAL EQUATIONS 237 15.1 THE DRIVEN
OSCILLATOR 238 15.2 THE STRUVE EQUATION 238 15.2.1 LOCAL ANALYSIS AT
ZERO 239 15.2.2 LOCAL ANALYSIS AT INFINITY 239 15.2.3 INTEGRAL
REPRESENTATION 240 15.3 RESISTIVE RECONNECTION 240 15.4 RESISTIVE
INTERNAL KINK 241 15.5 A CAUSAL INHOMOGENEOUS PROBLEM 242 15.6 PROBLEMS
247 16. THE RIEMANN ZETA FUNCTION 249 16.1 INTRODUCTION 249 16.2 CO) AND
C(L - S) 251 16.3 THE EULER PRODUCT FOR ((S) 25 3 16.4 DISTRIBUTION OF
PRIME NUMBERS 254 16.5 PUBLIC KEY CODES 258 16.6 STIRLING REVISITED 259
16.7 PROBLEMS 262 17. BOUNDARY LAYER PROBLEMS 263 17.1 INTRODUCTION 263
17.2 LAYER LOCATION 264 17.3 LAYER AT LEFT BOUNDARY 267 17.4 LAYER IN
DOMAIN CENTER 269 XIV ASYMPTOTIC ANALYSIS OF DIFFERENTIAL EQUATIONS 17.5
LAYER AT RIGHT BOUNDARY 271 17.6 NESTED BOUNDARY LAYERS 272 17.7
PROBLEMS 275 APPENDIX LAGRANGE S THEOREM 277 BIBLIOGRAPHY 279 INDEX 283
|
| adam_txt |
ASYMPTOTIC ANALYSIS OF DIFFERENTIAL EQUATIONS ROSCOE B WHITE PRINCETON
UNIVERSITY, USA IMPERIAL COLLEGE PRESS CONTENTS PREFACE V ACKNOWLEDGMENT
VII LIST OF FIGURES XV 1. DOMINANT BALANCE 1 1.1 INTRODUCTION 2 1.2
SOLUTIONS USING KRUSKAL-NEWTON GRAPHS 3 1.2.1 THIRD ORDER 3 1.2.2 NON
POLYNOMIAL FORM 5 1.2.3 HIGHER ORDER 7 1.2.4 HIDDEN POINTS 8 1.3
PROBLEMS 9 2. EXACT SOLUTIONS 11 2.1 INTRODUCTION 11 2.2 CONSTANT
COEFFICIENTS 13 2.3 INHOMOGENEOUS LINEAR EQUATIONS 14 2.4 THE FREDHOLM
ALTERNATIVE 16 2.5 THE DIFFUSION EQUATION 18 2.6 EXACT SOLUTIONS TO
NONLINEAR EQUATIONS 19 2.7 PHASE PLANE ANALYSIS T 19 2.8 PROBLEMS 22 3.
COMPLEX VARIABLES 25 3.1 ANALYTICITY 25 3.2 CAUCHY INTEGRAL THEOREM 26
3.3 SERIES REPRESENTATION 28 3.4 THE RESIDUE THEOREM 29 X ASYMPTOTIC
ANALYSIS OF DIFFERENTIAL EQUATIONS 3.5 ANALYTIC CONTINUATION 31 3.6
INVERSE FUNCTIONS 32 3.7 PROBLEMS 34 4. LOCAL APPROXIMATE SOLUTIONS 37
4.1 INTRODUCTION 37 4.2 CLASSIFICATION 38 4.2.1 ORDINARY POINT 39 4.2.2
REGULAR SINGULAR POINT 39 4.2.3 IRREGULAR SINGULAR POINT 42 4.3
ASYMPTOTIC SERIES 45 4.3.1 PROPERTIES 46 4.3.2 TRUNCATION: A SERIES
ABOUT X * 0 48 4.3.3 TRUNCATION: A SERIES ABOUT X = OO 50 4.3.4
TRUNCATION: A SERIES ABOUT X = 0 51 4.3.5 ASYMPTOTIC OSCILLATION 55 4.4
CONSTRUCTION OF ASYMPTOTIC SERIES 55 4.4.1 THE ERROR FUNCTION 56 4.5
ORIGIN OF THE DIVERGENCE 58 4.6 IMPROVING SERIES CONVERGENCE 60 4.6.1
SHANKS TRANSFORMATION 60 4.6.2 EULER SUMMATION 61 4.6.3 BOREL SUMMATION
62 4.7 PROBLEMS 62 5. PHASE INTEGRAL METHODS 67 5.1 INTRODUCTION 67 5.2
CONNECTION FORMULAE: ISOLATED ZERO 71 5.3 DERIVATION OF STOKES CONSTANTS
73 5.4 RULES FOR CONTINUATION 76 5.5 CAUSALITY 77 5.6 BOUND STATES AND
INSTABILITIES 78 5.7 SCATTERING 81 5.8 EIGENVALUE PROBLEMS 85 5.9 THE
BUDDEN PROBLEM 86 5.10 THE ERROR FUNCTION 88 5.11 PROBLEMS 89 6.
PERTURBATION THEORY 91 6.1 INTRODUCTION 92 6.2 EIGENVALUES OF A
HERMITIAN MATRIX , . . . 93 6.3 BROKEN SYMMETRY DUE TO TUNNELING 97
CONTENTS XI 6.4 PROBLEMS 100 7. ASYMPTOTIC EVALUATION OF INTEGRALS 103
7.1 INTRODUCTION 103 7.2 END POINT 105 7.3 SADDLE POINT 108 7.4 PROBLEMS
112 8. THE EULER GAMMA FUNCTION 115 8.1 INTRODUCTION 116 8.2 THE
STIRLING APPROXIMATION 117 8.3 THE EULER-MASCHERONI CONSTANT 119 8.4
SINE PRODUCT IDENTITY 120 8.5 CONTINUATION OF T(Z) 122 8.6 ASYMPTOTIC
T(Z) 122 8.7 EULER PRODUCT FOR T 125 8.8 INTEGRAL REPRESENTATION FOR
L/T(Z) 126 8.9 . T(NX) 128 8.10 THE EULER BETA FUNCTION 128 8.11
PROBLEMS 129 9. INTEGRAL SOLUTIONS 131 9.1 CONSTRUCTING INTEGRAL
SOLUTIONS 131 9.1.1 INTEGRATION BY PARTS 132 9.1.2 FINDING A DISCRETE
DIFFERENCE EQUATION 134 9.1.3 CONSTRUCTION FROM A SERIES 135 9.2 CAUSAL
SOLUTIONS 136 9.3 COMMENTS 138 9.4 PROBLEMS 138 10. EXPANSION IN BASIS
FUNCTIONS 141 10.1 LEGENDRE FUNCTIONS 142 10.1.1 LOCAL ANALYSIS 142
10.1.2 EULER INTEGRAL REPRESENTATION 143 10.1.3 RECURRENCE RELATIONS 145
10.1.4 LAPLACE INTEGRAL REPRESENTATION 147 10.1.5 GENERATING FUNCTION .
. .* 149 10.1.6 LEGENDRE POLYNOMIALS 150 10.2 ORTHOGONAL POLYNOMIALS 151
10.3 WAVELETS 154 10.3.1 INTRODUCTION 154 10.3.2 SCALING FUNCTION 154
10.3.3 WAVELET BASIS CONSTRUCTION 158 XII ASYMPTOTIC ANALYSIS OF
DIFFERENTIAL EQUATIONS 10.3.4 DETERMINING THE EXPANSION COEFFICIENTS 160
10.3.5 EXAMPLES 162 10.3.6 TIME-FREQUENCY ANALYSIS 166 10.4 PROBLEMS 168
11. AIRY 171 11.1 WKB ANALYSIS 172 11.2 FOURIER LAPLACE INTEGRAL
REPRESENTATION 174 11.3 ASYMPTOTIC LIMITS 176 11.3.1 LARGE NEGATIVE Z
176 11.3.2 LARGE POSITIVE Z 178 11.3.3 SMALL \Z\ 180 11.4 MELLIN
INTEGRAL REPRESENTATION 181 11.5 MATCHING LOCAL SOLUTIONS 183 11.6
THEWRONSKIAN 185 11.7 PROBLEMS 185 12. BESSEL 187 12.1 LOCAL ANALYSIS
187 12.1.1 LOCAL AT ZERO 187 12.1.2 ANALYTIC CONTINUATION IN V 189
12.1.3 LOCAL AT INFINITY 189 12.2 WKB ANALYSIS 190 12.3 INTEGRAL
REPRESENTATIONS 191 12.3.1 FOURIER LAPLACE REPRESENTATION 191 12.3.2 THE
HANKEL INTEGRALS 192 12.3.3 ASYMPTOTIC LIMITS , 193 12.3.4 RELATION TO
BESSEL AND NEUMANN FUNCTIONS 195 12.3.5 THE WRONSKIAN 197 12.3.6
SOMMERFELD INTEGRAL REPRESENTATION 198 12.3.7 MELLIN INTEGRAL
REPRESENTATION 201 12.4 GENERATING FUNCTION 202 12.5 MATCHING LOCAL
SOLUTIONS 202 12.6 IMAGINARY ARGUMENT 203 12.7 PROBLEMS 204 13.
WEBER-HERMITE 207 13.1 LOCAL ANALYSIS AT INFINITY 208 13.2 LOCAL
ANALYSIS AT ZERO 209 13.3 WKB ANALYSIS 210 13.4 EULER INTEGRAL
REPRESENTATION 212 13.5 FOURIER-LAPLACE INTEGRAL REPRESENTATION I 213
CONTENTS XIII 13.6 ORTHOGONALITY 217 13.7 FOURIER-LAPLACE INTEGRAL
REPRESENTATION II 218 13.8 THE WRONSKIAN 221 13.9 MELLIN INTEGRAL
REPRESENTATION 222 13.10 PROBLEMS 223 14. WHITTAKER AND WATSON 225 14.1
LOCAL ANALYSIS AT INFINITY 226 14.2 LOCAL ANALYSIS AT ZERO 227 14.3
EULER INTEGRAL REPRESENTATION 227 14.4 RELATION BETWEEN M X ^{Z) AND
W\^(Z) 230 14.5 FOURIER LAPLACE INTEGRAL REPRESENTATION 231 14.6 MELLIN
INTEGRAL REPRESENTATION 232 14.7 SPECIAL CASES 235 14.7.1 THE ERROR
FUNCTION 235 14.7.2 THE LOGARITHMIC INTEGRAL FUNCTION 235 14.8 PROBLEMS
235 15. INHOMOGENEOUS DIFFERENTIAL EQUATIONS 237 15.1 THE DRIVEN
OSCILLATOR 238 15.2 THE STRUVE EQUATION 238 15.2.1 LOCAL ANALYSIS AT
ZERO 239 15.2.2 LOCAL ANALYSIS AT INFINITY 239 15.2.3 INTEGRAL
REPRESENTATION 240 15.3 RESISTIVE RECONNECTION 240 15.4 RESISTIVE
INTERNAL KINK 241 15.5 A CAUSAL INHOMOGENEOUS PROBLEM 242 15.6 PROBLEMS
247 16. THE RIEMANN ZETA FUNCTION 249 16.1 INTRODUCTION 249 16.2 CO) AND
C(L - S) 251 16.3 THE EULER PRODUCT FOR ((S) 25 3 16.4 DISTRIBUTION OF
PRIME NUMBERS 254 16.5 PUBLIC KEY CODES 258 16.6 STIRLING REVISITED 259
16.7 PROBLEMS 262 17. BOUNDARY LAYER PROBLEMS 263 17.1 INTRODUCTION 263
17.2 LAYER LOCATION 264 17.3 LAYER AT LEFT BOUNDARY 267 17.4 LAYER IN
DOMAIN CENTER 269 XIV ASYMPTOTIC ANALYSIS OF DIFFERENTIAL EQUATIONS 17.5
LAYER AT RIGHT BOUNDARY 271 17.6 NESTED BOUNDARY LAYERS 272 17.7
PROBLEMS 275 APPENDIX LAGRANGE'S THEOREM 277 BIBLIOGRAPHY 279 INDEX 283 |
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| dewey-full | 515.35 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
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| id | DE-604.BV021325754 |
| illustrated | Illustrated |
| index_date | 2024-07-02T14:00:17Z |
| indexdate | 2024-07-09T20:35:43Z |
| institution | BVB |
| isbn | 1860945872 1860946127 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014646099 |
| oclc_num | 66901304 |
| open_access_boolean | |
| owner | DE-824 DE-355 DE-BY-UBR DE-739 DE-11 DE-188 |
| owner_facet | DE-824 DE-355 DE-BY-UBR DE-739 DE-11 DE-188 |
| physical | XVIII, 286 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Imperial College Press |
| record_format | marc |
| spelling | White, Roscoe B. Verfasser aut Asymptotic analysis of differential equations Roscoe B. White London Imperial College Press 2005 XVIII, 286 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Differential equations Asymptotic theory Asymptotische Methode (DE-588)4287476-2 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Analysis (DE-588)4001865-9 s Asymptotische Methode (DE-588)4287476-2 s Differentialgleichung (DE-588)4012249-9 s DE-604 b DE-604 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014646099&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | White, Roscoe B. Asymptotic analysis of differential equations Differential equations Asymptotic theory Asymptotische Methode (DE-588)4287476-2 gnd Analysis (DE-588)4001865-9 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4287476-2 (DE-588)4001865-9 (DE-588)4012249-9 |
| title | Asymptotic analysis of differential equations |
| title_auth | Asymptotic analysis of differential equations |
| title_exact_search | Asymptotic analysis of differential equations |
| title_exact_search_txtP | Asymptotic analysis of differential equations |
| title_full | Asymptotic analysis of differential equations Roscoe B. White |
| title_fullStr | Asymptotic analysis of differential equations Roscoe B. White |
| title_full_unstemmed | Asymptotic analysis of differential equations Roscoe B. White |
| title_short | Asymptotic analysis of differential equations |
| title_sort | asymptotic analysis of differential equations |
| topic | Differential equations Asymptotic theory Asymptotische Methode (DE-588)4287476-2 gnd Analysis (DE-588)4001865-9 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Differential equations Asymptotic theory Asymptotische Methode Analysis Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014646099&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT whiteroscoeb asymptoticanalysisofdifferentialequations |