Airy Functions and Applications to Physics Book

Preface

1 A Historical Introduction: Sir George Biddell Airy 1

2 Definitions and Properties 5

2.1 Homogeneous Airy functions 5

2.1.1 The Airy equation 5

2.1.2 Elementary properties 7

2.1.3 Integral representations 9

2.1.4 Ascending and asymptotic series 11

2.2 Properties of Airy functions 16

2.2.1 Zeros of Airy functions 16

2.2.2 The spectral zeta function 17

2.2.3 Inequalities 20

2.2.4 Connection with Bessel functions 20

2.2.5 Modulus and phase of Airy functions 22

2.3 Inhomogeneous Airy functions 25

2.3.1 Definitions 25

2.3.2 Properties of inhomogeneous Airy functions 27

2.3.3 Ascending series and asymptotic expansion 28

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 expansions 38

3.1.4 Primitives 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 42

3.3 Other primitives 47

3.4 Miscellaneous 49

3.5 Elementary integrals 50

3.5.1 Particular integrals 50

3.5.2 Integrals containing a single Airy function 50

3.5.3 Integrals of products of two Airy functions 55

3.6 Other integrals 59

3.6.1 Integrals involving the Volterra μ-function 59

3.6.2 Canonisation of cubic forms 62

3.6.3 Integrals with three Airy functions 63

3.6.4 Integrals with four Airy functions 65

3.6.5 Double integrals 66

4 Transformations of Airy Functions 69

4.1 Causal properties of Airy functions 69

4.1.1 Causal relations 69

4.1.2 Green's function of the Airy equation 70

4.1.3 Fractional derivatives of Airy functions 72

4.2 The Airy transform 73

4.2.1 Definitions and elementary properties 73

4.2.2 Some examples 76

4.2.3 Airy polynomials 81

4.2.4 A particular case: correlation Airy transform 83

4.3 Other kinds of transformations 94

4.3.1 Laplace transform of Airy functions 94

4.3.2 Mellin transform of Airy functions 95

4.3.3 Fourier transform of Airy functions 96

4.3.4 Hankel transform and the Airy kernel 97

4.4 Expansion into Fourier---Airy series 98

5 The Uniform Approximation 101

5.1 Oscillating integrals 101

5.1.1 The method of stationary phase 101

5.1.2 The uniform approximation of oscillating integrals 103

5.1.3 The Airy uniform approximation 104

5.2 Differential equations of the second order 104

5.2.1 The JWKB method 104

5.2.2 The Langer generalisation 106

5.3 Inhomogeneous differential equations 108

6 Generalisation of Airy Functions 111

6.1 Generalisation of the Airy integral 111

6.1.1 The generalisation of Watson 111

6.1.2 Oscillating integrals and catastrophes 114

6.2 Third order differential equations 118

6.2.1 The linear third order differential equation 118

6.2.2 Asymptotic solutions 119

6.2.3 The comparison equation 120

6.3 A differential equation of the fourth order 124

7 Applications to Classical Physics 127

7.1 Optics and electromagnetism 127

7.2 Fluid mechanics 130

7.2.1 The Tricomi equation 130

7.2.2 The Orr-Sommerfeld equation 132

7.3 Elasticity 135

7.4 The heat equation 137

7.5 Nonlinear physics 139

7.5.1 Korteweg-de Vries equation 139

7.5.2 The second Painleve equation 143

8 Applications to Quantum Physics 147

8.1 The Schrodinger equation 147

8.1.1 Particle in a uniform field 147

8.1.2 The ?x? potential 151

8.1.3 Uniform approximation of the Schrodinger equation 154

8.2 Evaluation of the Frank-Condon factors 162

8.2.1 The Franck-Condon principle 163

8.2.2 The JWKB approximation 163

8.2.3 The uniform approximation 166

8.3 The semiclassical Wigner distribution 170

8.3.1 The Weyl-Wigner formalism 172

8.3.2 The one-dimensional Wigner distribution 173

8.3.3 The two-dimensional Wigner distribution 175

8.3.4 Configuration of the Wigner distribution 178

8.4 Airy transform of the Schrodinger equation 181

Appendix A Numerical Computation of the Airy Functions 185

A.1 Homogeneous functions 185

A.2 Inhomogeneous functions 187

Bibliography 191

Index 201