Airy functions and applications to physics:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey [u.a.]
Imperial College Press [u.a.]
2004
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X, 194 S. Ill., graph. Darst. |
| ISBN: | 1860944787 9781860944789 |
Internformat
MARC
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| 020 | |a 9781860944789 |9 978-1-86094-478-9 | ||
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| 035 | |a (DE-599)BVBBV019865328 | ||
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| 100 | 1 | |a Vallée, Olivier |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Airy functions and applications to physics |c Olivier Valleé ; Manuel Soares |
| 264 | 1 | |a New Jersey [u.a.] |b Imperial College Press [u.a.] |c 2004 | |
| 300 | |a X, 194 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Mathematische Physik |0 (DE-588)4037952-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Airy-Funktion |0 (DE-588)4225959-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Airy-Funktion |0 (DE-588)4225959-9 |D s |
| 689 | 0 | 1 | |a Mathematische Physik |0 (DE-588)4037952-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Soares, Manuel |e Verfasser |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013189753&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-013189753 | ||
Datensatz im Suchindex
| _version_ | 1804133387075584000 |
|---|---|
| adam_text | Contents
Preface
v
1.
A Historical Introduction
:
Sir George Biddell Airy
1
2.
Definitions and Properties
5
2.1
The Homogeneous Airy Functions
.............. 5
2.1.1
The Airy s equation
.................. 5
2.1.2
Elementary properties
................. 8
2.1.2.1
Wronskians of homogeneous Airy functions
. 8
2.1.2.2
Particular values of Airy functions
..... 8
2.1.2.3
Relations between Airy functions
...... 9
2.1.3
Integral representations
................ 9
2.1.4
Ascending and asymptotic series
........... 11
2.1.4.1
Expansion of
Ai
near the origin
....... 11
2.1.4.2
Ascending series of
Ai
and Bi
........ 12
2.1.4.3
Asymptotic series of
Ai
and Bi
....... 13
2.2
Properties of Airy Functions
.................. 15
2.2.1
Zeros of Airy functions
................. 15
2.2.2
The spectral
zeta
function
............... 18
2.2.3
Inequalities
....................... 20
2.2.4
Connection with Bessel functions
........... 20
2.2.5
Modulus and phase of Airy functions
......... 21
2.2.5.1
Definitions
................... 21
2.2.5.2
Differential equations
............. 22
2.2.5.3
Asymptotic expansions
............ 23
2.2.5.4
Functions of positive arguments
....... 24
viii
Airy Functions and Applications to Physics
2.3
The Inhomogeneous Airy Functions
.............. 25
2.3.1
Definitions
........................ 25
2.3.2
Properties of inhomogeneous Airy functions
..... 27
2.3.2.1
Values at the origin
.............. 27
2.3.2.2
Other integral representations
........ 27
2.3.3
Ascending and asymptotic series
........... 28
2.3.3.1
Ascending series
................ 28
2.3.3.2
Asymptotic series
............... 29
2.3.4
Zeros of the Scorer functions
.............. 29
2.4
Squares and Products of Airy Functions
........... 30
2.4.1
Differential equation and integral representation
... 30
2.4.2
A remarkable identity
................. 32
2.4.3
The product Ai(x)Ai(—x): Airy wavelets
...... 32
3.
Primitives and Integrals of Airy Functions
37
3.1
Primitives Containing One Airy Function
.......... 37
3.1.1
In terms of Airy functions
............... 37
3.1.2
Ascending series
..................... 38
3.1.3
Asymptotic series
.................... 38
3.1.4
Primitive of Scorer functions
.............. 39
3.1.5
Repeated primitives
.................. 40
3.2
Product of Airy Functions
................... 40
3.2.1
The method of Albright
................ 41
3.2.2
Some primitives
..................... 43
3.3
Other Primitives
........................ 48
3.4
Miscellaneous
.......................... 49
3.5
Elementary Integrals
...................... 50
3.5.1
Particular integrals
................... 50
3.5.2
Integrals containing a single Airy function
...... 51
3.5.2.1
Integrals involving algebraic functions
.... 51
3.5.2.2
Integrals involving transcendental functions
54
3.5.3
Integrals of products of two Airy functions
...... 56
3.6
Other Integrals
......................... 60
3.6.1
Integrals involving the Volterra
μ
-function
......
60
3.6.2
Canonisation of cubic form
............... 64
3.6.3
Integrals with three Airy functions
.......... 65
3.6.4
Integrals with four Airy functions
........... 67
3.6.5
Double integrals
..................... 68
Contents ix
4. Transformations
of Airy Functions
71
4.1
Causal Properties of Airy Functions
............. 71
4.1.1
Causal relations
..................... 71
4.1.2
Green function of the Airy equation
......... 73
4.2
The Airy Transform
...................... 74
4.2.1
Definitions and elementary properties
......... 74
4.2.2
Some examples
..................... 77
4.2.3
Airy polynomials
.................... 82
4.2.4
Summary of Airy transform
.............. 84
4.2.5
Airy averaging
..................... 85
4.3
Other Kinds of Transformations
................ 85
4.3.1
Laplace transform of Airy functions
.......... 85
4.3.2
Mellin transform of Airy function
........... 86
4.3.3
Fourier transform of Airy functions
.......... 87
4.4
Expansion into Fourier-Airy Series
.............. 88
5.
The Uniform Approximation
91
5.1
Oscillating Integrals
...................... 91
5.1.1
The method of stationary phase
............ 91
5.1.2
The uniform approximation of oscillating integral
. . 93
5.1.3
The Airy uniform approximation
........... 94
5.2
Differential Equation of the Second Order
.......... 95
5.2.1
The JWKB method
................... 95
5.2.2
The generalisation of
Langer
.............. 97
5.3
Inhomogeneous Differential Equations
............ 98
6.
Generalisation of Airy Functions
101
6.1
Generalisation of the Airy Integral
.............. 101
6.2
Third Order Differential Equations
.............. 105
6.2.1
The linear third order differential equation
...... 105
6.2.2
Asymptotic solutions
.................. 106
6.2.3
The comparison equation
............... 107
6.3
Differential Equation of the Fourth Order
..........
Ill
7.
Applications to Classical Physics
115
7.1
Optics and
Electromagnetism
................. 115
7.2
Fluid Mechanics
........................ 119
7.2.1
The
Tricomi
equation
.................. 119
X Airy Functions and Applications to Physics
7.2.2
The Orr-Sommerfeld equation
............. 121
7.3
Elasticity
............................ 124
7.4
The Heat Equation
....................... 127
7.5
Nonlinear Physics
........................ 129
7.5.1
Korteweg-de
Vries
equation
.............. 129
7.5.1.1
The linearised Korteweg-de
Vries
equation
. 129
7.5.1.2
Similarity solutions
.............. 131
7.5.2
The second
Painlevé
equation
............. 132
7.5.2.1
The
Painlevé
equations
............ 132
7.5.2.2
An integral equation
............. 134
7.5.2.3
Rational solutions
............... 135
8.
Applications to Quantum Physics
137
8.1
The
Schrödinger
Equation
................... 137
8.1.1
Particle in a uniform field
............... 137
8.1.2
The |x| potential
.................... 140
8.1.3
Uniform approximation of the
Schrödinger
equation
. 144
8.1.3.1
The JWKB approximation
.......... 145
8.1.3.2
The Airy uniform approximation
...... 146
8.1.3.3
Exact vs approximate wave functions
.... 148
8.2
Evaluation of the Franck-Condon Factors
.......... 152
8.2.1
The Franck-Condon principle
............. 153
8.2.2
The JWKB approximation
............... 154
8.2.3
The uniform approximation
.............. 157
8.3
The Semiclassical Wigner Distribution
............ 162
8.3.1
The Weyl-Wigner formalism
.............. 163
8.3.2
The one-dimensional Wigner distribution
....... 164
8.3.3
The two-dimensional Wigner distribution
...... 166
8.3.4
Configuration of the Wigner distribution
....... 169
8.4
Airy Transform of the
Schrödinger
Equation
......... 173
Appendix A Numerical Computation of the Airy Functions
177
A.I The Homogeneous Functions
................. 177
A.
2
The Inhomogeneous Functions
................ 180
Bibliography
183
Index
193
|
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| classification_rvk | SK 950 |
| classification_tum | PHY 013f |
| ctrlnum | (OCoLC)634737160 (DE-599)BVBBV019865328 |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T20:07:54Z |
| institution | BVB |
| isbn | 1860944787 9781860944789 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013189753 |
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| physical | X, 194 S. Ill., graph. Darst. |
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| spelling | Vallée, Olivier Verfasser aut Airy functions and applications to physics Olivier Valleé ; Manuel Soares New Jersey [u.a.] Imperial College Press [u.a.] 2004 X, 194 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Airy-Funktion (DE-588)4225959-9 gnd rswk-swf Airy-Funktion (DE-588)4225959-9 s Mathematische Physik (DE-588)4037952-8 s DE-604 Soares, Manuel Verfasser aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013189753&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vallée, Olivier Soares, Manuel Airy functions and applications to physics Mathematische Physik (DE-588)4037952-8 gnd Airy-Funktion (DE-588)4225959-9 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4225959-9 |
| title | Airy functions and applications to physics |
| title_auth | Airy functions and applications to physics |
| title_exact_search | Airy functions and applications to physics |
| title_full | Airy functions and applications to physics Olivier Valleé ; Manuel Soares |
| title_fullStr | Airy functions and applications to physics Olivier Valleé ; Manuel Soares |
| title_full_unstemmed | Airy functions and applications to physics Olivier Valleé ; Manuel Soares |
| title_short | Airy functions and applications to physics |
| title_sort | airy functions and applications to physics |
| topic | Mathematische Physik (DE-588)4037952-8 gnd Airy-Funktion (DE-588)4225959-9 gnd |
| topic_facet | Mathematische Physik Airy-Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013189753&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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