Introduction to asymptotics and special functions:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Academic Press
1974
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 297 Seiten graph. Darst. |
| ISBN: | 0125258569 |
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MARC
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| 245 | 1 | 0 | |a Introduction to asymptotics and special functions |c F. W. J. Olver, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland and National Bureau of Standards, Washington, D.C. |
| 264 | 1 | |a New York, NY [u.a.] |b Academic Press |c 1974 | |
| 300 | |a XII, 297 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Développements asymptotiques | |
| 650 | 4 | |a Fonctions spéciales | |
| 650 | 4 | |a Équations différentielles - Solutions numériques | |
| 650 | 4 | |a Asymptotic expansions | |
| 650 | 4 | |a Calculus | |
| 650 | 4 | |a Functions, Special | |
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Record in the Search Index
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| adam_text | CONTENTS
Preface ix
Preface to Asymptotics and Special Functions xi
1 Introduction to Asymptotic Analysis
1 Origin of Asymptotic Expansions 1
2 The Symbols ~, o, and O 4
3 The Symbols ~, o, and O (continued) 6
4 Integration and Differentiation of Asymptotic and Order
Relations 8
5 Asymptotic Solution of Transcendental Equations:
Real Variables 11
6 Asymptotic Solution of Transcendental Equations:
Complex Variables 14
7 Definition and Fundamental Properties of Asymptotic
Expansions 16
8 Operations with Asymptotic Expansions 19
9 Functions Having Prescribed Asymptotic Expansions 22
10 Generalizations of Poincare s Definition 24
11 Error Analysis; Variational Operator 27
Historical Notes and Additional References 29
2 Introduction to Special Functions
1 The Gamma Function 31
2 The Psi Function 39
3 Exponential, Logarithmic, Sine, and Cosine Integrals 40
4 Error Functions, Dawson s Integral, and Fresnel Integrals 43
5 Incomplete Gamma Functions 45
6 Orthogonal Polynomials 46
7 The Classical Orthogonal Polynomials 48
8 The Airy Integral 53
V
vj Contents
9 The Bessel Function Jv(z) 55
10 The Modified Bessel Function/v(z) 60
11 The Zeta Function 61
Historical Notes and Additional References 64
3 Integrals of a Real Variable
1 Integration by Parts 66
2 Laplace Integrals 67
3 Watson s Lemma 71
4 The Riemann Lebesgue Lemma 73
5 Fourier Integrals 75
6 Examples; Cases of Failure 76
7 Laplace s Method 80
8 Asymptotic Expansions by Laplace s Method; Gamma
Function of Large Argument 85
9 Error Bounds for Watson s Lemma and Laplace s Method 89
10 Examples 92
11 The Method of Stationary Phase 96
12 Preliminary Lemmas 98
13 Asymptotic Nature of the Stationary Phase Approximation 100
14 Asymptotic Expansions by the Method of Stationary Phase 104
Historical Notes and Additional References 104
4 Contour Integrals
1 Laplace Integrals with a Complex Parameter 106
2 Incomplete Gamma Functions of Complex Argument 109
3 Watson s Lemma 112
4 Airy Integral of Complex Argument; Compound
Asymptotic Expansions 116
5 Ratio of Two Gamma Functions; Watson s Lemma for Loop
Integrals 118
6 Laplace s Method for Contour Integrals 121
7 Saddle Points 125
8 Examples 127
9 Bessel Functions of Large Argument and Order 130
10 Error Bounds for Laplace s Method; the Method of Steepest
Descents 135
Historical Notes and Additional References 137
Contents vii
5 Differential Equations with Regular Singularities; Hyper geometric
and Legendre Functions
1 Existence Theorems for Linear Differential Equations:
Real Variables 139
2 Equations Containing a Real or Complex Parameter 143
3 Existence Theorems for Linear Differential Equations:
Complex Variables 145
4 Classification of Singularities; Nature of the Solutions
in the Neighborhood of a Regular Singularity 148
5 Second Solution When the Exponents Differ by an Integer
or Zero 150
6 Large Values of the Independent Variable 153
7 Numerically Satisfactory Solutions 154
8 The Hypergeometric Equation 156
9 The Hypergeometric Function 159
10 Other Solutions of the Hypergeometric Equation 163
11 Generalized Hypergeometric Functions 168
12 The Associated Legendre Equation 169
13 Legendre Functions of General Degree and Order 174
14 Legendre Functions of Integer Degree and Order 180
15 Ferrers Functions 185
Historical Notes and Additional References 189
6 The Liouville Green Approximation
1 The Liouville Transformation 190
2 Error Bounds: Real Variables 193
3 Asymptotic Properties with Respect to the Independent
Variable 197
4 Convergence of ¦f(F) at a Singularity 200
5 Asymptotic Properties with Respect to Parameters 203
6 Example: Parabolic Cylinder Functions of Large Order 206
7 A Special Extension 208
8 Zeros 211
9 Eigenvalue Problems 214
10 Theorems on Singular Integral Equations 217
11 Error Bounds: Complex Variables 220
12 Asymptotic Properties for Complex Variables 223
13 Choice of Progressive Paths 224
Historical Notes and Additional References 228
viii Contents
7 Differential Equations with Irregular Singularities; Bessel
and Confluent Hypergeometric Functions
1 Formal Series Solutions 229
2 Asymptotic Nature of the Formal Series 232
3 Equations Containing a Parameter 236
4 Hankel Functions; Stokes Phenomenon 237
5 The Function Yv(z) 241
6 Zeros of yv(z) 244
7 Zeros of Yv(z) and Other Cylinder Functions 248
8 Modified Bessel Functions 250
9 Confluent Hypergeometric Equation 254
10 Asymptotic Solutions of the Confluent Hypergeometric
Equation 256
11 Whittaker Functions 260
12 Error Bounds for the Asymptotic Solutions in the General
Case 262
13 Error Bounds for Hankel s Expansions 266
14 Inhomogeneous Equations 270
15 Struve s Equation 274
Historical Notes and Additional References 277
Answers to Exercises 279A
References 281A
Index of Symbols 289A
General Index 291A
|
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| discipline | Mathematik |
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| id | DE-604.BV003714237 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:04:16Z |
| institution | BVB |
| isbn | 0125258569 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002364226 |
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| owner_facet | DE-91G DE-BY-TUM DE-824 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-29T DE-83 DE-188 |
| physical | XII, 297 Seiten graph. Darst. |
| psigel | TUB-www |
| publishDate | 1974 |
| publishDateSearch | 1974 |
| publishDateSort | 1974 |
| publisher | Academic Press |
| record_format | marc |
| spelling | Olver, Frank W. J. 1924-2013 Verfasser (DE-588)139038736 aut Introduction to asymptotics and special functions F. W. J. Olver, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland and National Bureau of Standards, Washington, D.C. New York, NY [u.a.] Academic Press 1974 XII, 297 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Développements asymptotiques Fonctions spéciales Équations différentielles - Solutions numériques Asymptotic expansions Calculus Functions, Special Asymptotik (DE-588)4126634-1 gnd rswk-swf Spezielle Funktion (DE-588)4182213-4 gnd rswk-swf Asymptotik (DE-588)4126634-1 s Spezielle Funktion (DE-588)4182213-4 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002364226&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Olver, Frank W. J. 1924-2013 Introduction to asymptotics and special functions Développements asymptotiques Fonctions spéciales Équations différentielles - Solutions numériques Asymptotic expansions Calculus Functions, Special Asymptotik (DE-588)4126634-1 gnd Spezielle Funktion (DE-588)4182213-4 gnd |
| subject_GND | (DE-588)4126634-1 (DE-588)4182213-4 |
| title | Introduction to asymptotics and special functions |
| title_auth | Introduction to asymptotics and special functions |
| title_exact_search | Introduction to asymptotics and special functions |
| title_full | Introduction to asymptotics and special functions F. W. J. Olver, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland and National Bureau of Standards, Washington, D.C. |
| title_fullStr | Introduction to asymptotics and special functions F. W. J. Olver, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland and National Bureau of Standards, Washington, D.C. |
| title_full_unstemmed | Introduction to asymptotics and special functions F. W. J. Olver, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland and National Bureau of Standards, Washington, D.C. |
| title_short | Introduction to asymptotics and special functions |
| title_sort | introduction to asymptotics and special functions |
| topic | Développements asymptotiques Fonctions spéciales Équations différentielles - Solutions numériques Asymptotic expansions Calculus Functions, Special Asymptotik (DE-588)4126634-1 gnd Spezielle Funktion (DE-588)4182213-4 gnd |
| topic_facet | Développements asymptotiques Fonctions spéciales Équations différentielles - Solutions numériques Asymptotic expansions Calculus Functions, Special Asymptotik Spezielle Funktion |
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