Airy Functions and Applications to Physics

Cover......Page 1
Title......Page 4
Preface......Page 6
Contents......Page 8
1. A Historical Introduction: Sir George Biddell Airy......Page 12
2.1.1. The Airy equation......Page 16
2.1.2.1. Wronskians of homogeneous Airy functions......Page 18
2.1.2.3. Relations between Airy functions......Page 19
2.1.3. Integral representations......Page 20
2.1.4.2. Ascending series of Ai and Bi......Page 22
2.1.4.4. The Stokes phenomenon......Page 23
2.2.1. Zeros of Airy functions......Page 27
2.2.2. The spectral zeta function......Page 28
2.2.4. Connection with Bessel functions......Page 31
2.2.5.1. Definitions......Page 33
2.2.5.3. Asymptotic expansions......Page 34
2.2.5.4. Functions of positive arguments......Page 35
2.3.1. Definitions......Page 36
2.3.2.2. Other integral representations......Page 38
2.3.3.1. Ascending series......Page 39
2.3.4. Zeros of the Scorer functions......Page 40
2.4.1. Differential equation and integral representation......Page 41
2.4.3. The product Ai(x)Ai(-x): Airy wavelets......Page 43
Exercises......Page 45
3.1.1. In terms of Airy functions......Page 48
3.1.3. Asymptotic expansions......Page 49
3.1.4. Primitives of Scorer functions......Page 50
3.2. Product of Airy functions......Page 51
3.2.1. The method of Albright......Page 52
3.2.2. Some primitives......Page 53
3.3. Other primitives......Page 58
3.4. Miscellaneous......Page 60
3.5.2.1. Integrals involving algebraic functions......Page 61
3.5.2.2. Integrals involving transcendental functions......Page 65
3.5.3. Integrals of products of two Airy functions......Page 66
3.6.1. Integrals involving the Volterra μ-function......Page 70
3.6.2. Canonisation of cubic forms......Page 73
3.6.3. Integrals with three Airy functions......Page 74
3.6.4. Integrals with four Airy functions......Page 76
3.6.5. Double integrals......Page 77
Exercises......Page 78
4.1.1. Causal relations......Page 80
4.1.2. Green\'s function of the Airy equation......Page 81
4.1.3. Fractional derivatives of Airy functions......Page 83
4.2.1. Definitions and elementary properties......Page 84
4.2.2. Some examples......Page 87
4.2.3. Airy polynomials......Page 92
4.2.4. A particular case: correlation Airy transform......Page 94
4.2.4.1. Properties of the correlation Airy transform......Page 95
4.2.4.2. Some examples......Page 97
4.2.4.3. Self-reciprocal transforms......Page 101
4.2.4.4. Application to the Airy kernel......Page 102
4.2.4.5. The CAT of Hermite functions......Page 103
4.3.1. Laplace transform of Airy functions......Page 105
4.3.2. Mellin transform of Airy functions......Page 106
4.3.3. Fourier transform of Airy functions......Page 107
4.3.4. Hankel transform and the Airy kernel......Page 108
4.4. Expansion into Fourier-Airy series......Page 109
Exercises......Page 111
5.1.1. The method of stationary phase......Page 112
5.1.2. The uniform approximation of oscillating integrals......Page 114
5.2.1. The JWKB method......Page 115
5.2.2. The Langer generalisation......Page 117
5.3. Inhomogeneous differential equations......Page 119
Exercises......Page 121
6.1.1. The generalisation of Watson......Page 122
6.1.2. Oscillating integrals and catastrophes......Page 125
6.2.1. The linear third order differential equation......Page 129
6.2.2. Asymptotic solutions......Page 130
6.2.3. The comparison equation......Page 131
6.3. A differential equation of the fourth order......Page 135
Exercises......Page 136
7.1. Optics and electromagnetism......Page 138
7.2.1. The Tri.comi equation......Page 141
7.2.2.1. Plane flow of an incompressible viscous fluid......Page 143
7.2.2.2. Stability of an almost parallel flow......Page 145
7 .3. Elasticity......Page 146
7 .4. The heat equation......Page 148
7.5.1. Korteweg-de Vries equation......Page 150
7.5.1.1. The linearised Korteweg-de Vries equation......Page 151
7.5.1.2. Similarity solutions......Page 152
7.5.2.1. The Painleve equations......Page 154
7.5.2.2. An integral equation......Page 155
7.5.2.3. Rational solutions......Page 156
Exercises......Page 157
8.1.1.1. The stationary case......Page 158
8.1.1.2. The time dependent case......Page 161
8.1.2. The lxl potential......Page 162
8.1.3. Uniform approximation of the Schrodinger equation......Page 165
8.1.3.1. The JWKB approximation......Page 166
8.1.3.2. The Airy uniform approximation......Page 167
8.1.3.3. Exact vs approximate wave functions......Page 168
8.2. Evaluation of the Franck-Condon factors......Page 173
8.2.2. The JWKB approximation of Franck-Condon factors......Page 174
8.2.3. The uniform approximation of Franck-Condon factors......Page 177
8.3. The semiclassical Wigner distribution......Page 181
8.3.1. The Weyl-Wigner formalism......Page 183
8.3.2. The one-dimensional Wigner distribution......Page 184
8.3.3. The two-dimensional Wigner distribution......Page 186
8.3.4. Configuration of the Wigner distribution in the phasespace......Page 189
8.4. Airy transform of the Schrodinger equation......Page 192
Exercises......Page 194
A.1. Homogeneous functions......Page 196
A.2. Inhomogeneous functions......Page 198
Exercises......Page 199
Bibliography......Page 202
Index......Page 212