Airy Functions and Applications to Physics [2nd ed.] 10184816548X, 139781848165489

Table of contents :
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

Citation preview

Airy Functions and Applications to Physics 2nd Edition

P709 tp.indd 1

10/11/11 10:29 AM

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2nd Edition

Airy Functions and Applications to Physics Olivier Vallée

Université d’Orléans, France

Manuel Soares DDEA des Yvelines, France

ICP

P709 tp.indd 2

Imperial College Press

10/11/11 10:29 AM

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

AIRY FUNCTIONS AND APPLICATIONS TO PHYSICS (2nd Edition) Copyright © 2010 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-1-84816-548-9 ISBN-10 1-84816-548-X

Printed in Singapore.

YeeSern - Airy functions and Applications.pmd 1

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Preface

The use of special functions, and in particular of Airy functions, is rather common in physics. The reason may be found in the need to express a physical phenomenon in terms of an effective and comprehensive analytical form for the whole scientific community. However, for almost the last twenty years, many physical problems have been solved by computers, a trend which is now becoming the norm as the importance of computers continues to grow. Indeed as a last resort, the special functions employed in physics will have to be calculated numerically, even if the analytic formulation of physics is of prime importance. Airy functions have periodically been the subject of many review articles, most of which focus on tabulations for the numerical calculation of these functions, as this is particularly difficult. The most well-known publications in this field are the tables by J. C. P. Miller, dating from 1946, and the chapter in the Handbook of Mathematical Functions by Abramowitz and Stegun, first published in 1954. Since that time, no noteworthy compilation of Airy functions has been published1 in particular about the calculus involved in these functions. For example, in the latest editions of the tables by Gradshteyn and Ryzhik, they are hardly mentioned. Similarly, many results that have accrued over time in the scientific literature, remain extremely scattered and fragmentary. Although Airy functions are used in many fields of physics, the analytical outcomes that have been obtained are not (or only poorly) transmitted among the various fields of research, which basically remain isolated. Moreover the tables by Abramowitz and Stegun are still the only common 1 Note however that recently an equivalent compilation has become available on the web: the NIST Digital Library of Mathematical Functions – Chap. 9 Airy & Related Functions by F. W. J. Olver. Internet address: http://dlmf.nist.gov/

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reference for all the authors using these functions. Thus many of the results have been rediscovered, and sometimes extremely old findings are the subject of publications, which represent a wasted effort for researchers. The aim of this work is to make a rather exhaustive compilation of current knowledge on the analytical properties of Airy functions. In particular, the calculus involved in the Airy functions is carefully developed. This is, in fact, one of the major objectives of this book. While we are aware of repeating previous compilations to a large extent, this seemed necessary to ensure coherence. This book is addressed mainly to physicists (from advanced undergraduate students to researchers). We make no claim about the rigour of the mathematical demonstrations, as the reader will see.2 The main aim is the result, or the fastest way to reach it. Finally, in the second part of this work, the reader will find some applications to various fields of physics. These examples are not exhaustive; they are only given to succinctly illustrate the use of Airy functions in classical or in quantum physics. For instance, we point out to the physicist in fluid mechanics that he can find what he is looking for in works on molecular physics or surface physics, and vice versa. O. Vall´ee & M. Soares, Fall 2003 Preface to the second edition. In this second edition we have corrected various misprints of the first edition. Moreover we have added, here and there, new material such as catastrophe theory for the generalisation of the Airy function, additional results concerning the Airy transform and applications for instance to the Airy kernel, etc. It is our hope that this new edition will keep the book up to date in this still useful field. We would like to thank Pr. Vladimir Varlamov and Pr. Sir Michael Berry for pointing out several omissions and flaws in the first edition. O. Vall´ee & M. Soares, January 2010

2 As a matter of fact, the Airy function can be considered as a distribution (generalised function) whose Fourier transform is an imaginary exponential. Also most of the integrals evoked in this work should be evaluated with the help of a convergence factor.

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Contents

Preface

v

1.

A Historical Introduction: Sir George Biddell Airy

1

2.

Definitions and Properties

5

2.1

2.2

2.3

2.4

3.

Homogeneous Airy functions . . . . . . . . . . . . . . . 2.1.1 The Airy equation . . . . . . . . . . . . . . . . . 2.1.2 Elementary properties . . . . . . . . . . . . . . . 2.1.3 Integral representations . . . . . . . . . . . . . . 2.1.4 Ascending and asymptotic series . . . . . . . . . Properties of Airy functions . . . . . . . . . . . . . . . . 2.2.1 Zeros of Airy functions . . . . . . . . . . . . . . 2.2.2 The spectral zeta function . . . . . . . . . . . . 2.2.3 Inequalities . . . . . . . . . . . . . . . . . . . . . 2.2.4 Connection with Bessel functions . . . . . . . . 2.2.5 Modulus and phase of Airy functions . . . . . . Inhomogeneous Airy functions . . . . . . . . . . . . . . 2.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . 2.3.2 Properties of inhomogeneous Airy functions . . 2.3.3 Ascending series and asymptotic expansion . . . 2.3.4 Zeros of the Scorer functions . . . . . . . . . . . Squares and products of Airy functions . . . . . . . . . 2.4.1 Differential equation and integral representation 2.4.2 A remarkable identity . . . . . . . . . . . . . . . 2.4.3 The product Ai(x)Ai(−x): Airy wavelets . . . .

Primitives and Integrals of Airy Functions vii

. . . . . . . . . . . . . . . . . . . .

5 5 7 9 11 16 16 17 20 20 22 25 25 27 28 29 30 30 32 32 37

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viii

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Airy Functions and Applications to Physics

3.1

3.2

3.3 3.4 3.5

3.6

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Primitives containing one Airy function . . . . . . 3.1.1 In terms of Airy functions . . . . . . . . . 3.1.2 Ascending series . . . . . . . . . . . . . . . 3.1.3 Asymptotic expansions . . . . . . . . . . . 3.1.4 Primitives of Scorer functions . . . . . . . 3.1.5 Repeated primitives . . . . . . . . . . . . . Product of Airy functions . . . . . . . . . . . . . . 3.2.1 The method of Albright . . . . . . . . . . . 3.2.2 Some primitives . . . . . . . . . . . . . . . Other primitives . . . . . . . . . . . . . . . . . . . Miscellaneous . . . . . . . . . . . . . . . . . . . . . Elementary integrals . . . . . . . . . . . . . . . . . 3.5.1 Particular integrals . . . . . . . . . . . . . 3.5.2 Integrals containing a single Airy function 3.5.3 Integrals of products of two Airy functions Other integrals . . . . . . . . . . . . . . . . . . . . 3.6.1 Integrals involving the Volterra µ-function 3.6.2 Canonisation of cubic forms . . . . . . . . 3.6.3 Integrals with three Airy functions . . . . . 3.6.4 Integrals with four Airy functions . . . . . 3.6.5 Double integrals . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

Transformations of Airy Functions 4.1

4.2

4.3

4.4

Causal properties of Airy functions . . . . . . . . . . 4.1.1 Causal relations . . . . . . . . . . . . . . . . 4.1.2 Green’s function of the Airy equation . . . . 4.1.3 Fractional derivatives of Airy functions . . . The Airy transform . . . . . . . . . . . . . . . . . . . 4.2.1 Definitions and elementary properties . . . . 4.2.2 Some examples . . . . . . . . . . . . . . . . . 4.2.3 Airy polynomials . . . . . . . . . . . . . . . 4.2.4 A particular case: correlation Airy transform Other kinds of transformations . . . . . . . . . . . . 4.3.1 Laplace transform of Airy functions . . . . . 4.3.2 Mellin transform of Airy functions . . . . . . 4.3.3 Fourier transform of Airy functions . . . . . 4.3.4 Hankel transform and the Airy kernel . . . . Expansion into Fourier–Airy series . . . . . . . . . .

37 37 38 38 39 40 40 41 42 47 49 50 50 50 55 59 59 62 63 65 66 69

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

69 69 70 72 73 73 76 81 83 94 94 95 96 97 98

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Contents

5.

The Uniform Approximation

101

5.1

101 101 103 104 104 104 106 108

5.2

5.3 6.

6.2

6.3

111

Generalisation of the Airy integral . . . . . . . . . 6.1.1 The generalisation of Watson . . . . . . . . 6.1.2 Oscillating integrals and catastrophes . . . Third order differential equations . . . . . . . . . . 6.2.1 The linear third order differential equation 6.2.2 Asymptotic solutions . . . . . . . . . . . . 6.2.3 The comparison equation . . . . . . . . . . A differential equation of the fourth order . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

Applications to Classical Physics 7.1 7.2

7.3 7.4 7.5

8.

Oscillating integrals . . . . . . . . . . . . . . . . . . . . . 5.1.1 The method of stationary phase . . . . . . . . . . 5.1.2 The uniform approximation of oscillating integrals 5.1.3 The Airy uniform approximation . . . . . . . . . . Differential equations of the second order . . . . . . . . . 5.2.1 The JWKB method . . . . . . . . . . . . . . . . . 5.2.2 The Langer generalisation . . . . . . . . . . . . . Inhomogeneous differential equations . . . . . . . . . . . .

Generalisation of Airy Functions 6.1

7.

ix

Optics and electromagnetism . . . . . Fluid mechanics . . . . . . . . . . . . 7.2.1 The Tricomi equation . . . . . 7.2.2 The Orr–Sommerfeld equation Elasticity . . . . . . . . . . . . . . . . The heat equation . . . . . . . . . . . Nonlinear physics . . . . . . . . . . . . 7.5.1 Korteweg–de Vries equation . 7.5.2 The second Painlev´e equation

127 . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

Applications to Quantum Physics 8.1

8.2

111 111 114 118 118 119 120 124

The Schr¨ odinger equation . . . . . . . . . . . . . . 8.1.1 Particle in a uniform field . . . . . . . . . . 8.1.2 The |x| potential . . . . . . . . . . . . . . . 8.1.3 Uniform approximation of the Schr¨odinger equation . . . . . . . . . . . . . . . . . . . Evaluation of the Franck–Condon factors . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

127 130 130 132 135 137 139 139 143 147

. . . . 147 . . . . 147 . . . . 151 . . . . 154 . . . . 162

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Airy Functions and Applications to Physics

8.3

8.4

8.2.1 The Franck–Condon principle . . . . . . 8.2.2 The JWKB approximation . . . . . . . . 8.2.3 The uniform approximation . . . . . . . The semiclassical Wigner distribution . . . . . . 8.3.1 The Weyl–Wigner formalism . . . . . . . 8.3.2 The one-dimensional Wigner distribution 8.3.3 The two-dimensional Wigner distribution 8.3.4 Configuration of the Wigner distribution Airy transform of the Schr¨odinger equation . . .

Appendix A A.1 A.2

Numerical Computation of the Airy Functions

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

163 163 166 170 172 173 175 178 181 185

Homogeneous functions . . . . . . . . . . . . . . . . . . . 185 Inhomogeneous functions . . . . . . . . . . . . . . . . . . 187

Bibliography

191

Index

201

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Chapter 1

A Historical Introduction: Sir George Biddell Airy

George Biddell Airy was born 27 July 1801 at Alnwick in Northumberland (North of England). His family was rather modest, but thanks to the generosity of his uncle Arthur Biddell, he went to study at Trinity College, University of Cambridge. Although a sizar,1 he was a brilliant student and finally graduated in 1823 as a senior wrangler. Three years later he was appointed to the celebrated Lucasian chair of mathematics. However, his salary as Lucasian professor was too small to marry Richarda, the young lady’s father objected, so he applied for a new position. In 1828, Airy obtained the Plumian chair, becoming professor of Astronomy and director of the new observatory at Cambridge. His early work at this time concerned the mass of Jupiter and the irregular motions of the Earth and Venus. In 1834, Airy started his first mathematical studies on the diffraction phenomenon and optics. Due to diffraction, the image of a point through a telescope is actually a spot surrounded by rings of smaller intensity. This spot is now called the Airy spot; the associated Airy function, however, has nothing to do with the purpose of this book. In June 1835, Airy became the 7th Royal Astronomer and director of the Greenwich observatory, succeeding John Pond. Under his administration, modern equipment was installed, leading the observatory to worldwide fame assisted by the quality of its published data. Airy also introduced the study of sun spots and of the Earth’s magnetism, and built new apparatus for the observation of the Moon, and for cataloguing the stars. The question of absolute time was also a major challenge: Airy defined the Airy Transit Circle, which in 1884 became the Greenwich Mean Time. However the renown of Airy is also due to the Neptune affair. During the decade 1830– 1 Meaning

that he paid a reduced fee in exchange for working as a servant to richer

students. 1

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40, astronomers were interested in the perturbations of Uranus, which had been discovered in 1781. In France, Fran¸cois Arago suggested to Urbain Le Verrier that he should seek for a new planet that might have caused the perturbations of Uranus. In England, the young John Adams was doing the same calculations with a slight advance. Airy however was dubious about the outcome of his work. Adams tried twice to meet Airy in 1845 but was unsuccessful: the first time Airy was away, the second time Airy was having dinner and did not wish to be disturbed. Finally, Airy entrusted the astronomer James Challis with the observation of the new planet from the calculations of Adams. Unfortunately, Challis failed in his task. At the same time, Le Verrier asked the German astronomer Johann Galle in Berlin to locate the planet from his data: the new planet was discovered on 20 September 1846. A controversy then started between Airy and Arago, between France and England, and also against Airy himself. The dispute became more acrimonious concerning the name of the planet itself, Airy wanting to name the new planet Oceanus. The name Neptune was finally given. The story goes that, in the end, Adams and Le Verrier became good friends. In 1854 Airy attempted to determine the mean density of the Earth by comparing the gravity forces on a single pendulum at the top and the bottom of a pit. The experiment was carried out near South Shields in a mine 1250 feet in depth. Taking into account the elliptical form and the rotation of the Earth, Airy deduced a density of 6.56, which is not so far — considering the epoch — from the currently accepted density 5.42. Airy was knighted in 1872, and so became Sir George Biddell.2 At this time, Airy started a lunar theory. The results were published in 1886, but in 1890 he found an error in his calculations. The author was then 89 years old and was unwilling to revise his calculations. Late in 1881, Sir George retired from his position as astronomer at Greenwich. He died January 2, 1892. The autobiography of Sir George, edited by his son Wilfred, was published in 1896 [W. Airy (1896)]. The name of Airy is associated with many phenomena such as the Airy spiral (an optical phenomenon visible in quartz crystals), the Airy spot in diffraction phenomena or the Airy stress function which he introduced in his work on elasticity, which is different again from the Airy functions that we shall discuss in this book. Among the most wellknown books he wrote, we may mention Mathematical tracts on physical astronomy (1826) and Popular astronomy (1849) [W. Airy (1896)]. 2 After

having declined the offer on three occasions, objecting to the fees.

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A Historical Introduction

Fig. 1.1

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3

Sir George Biddell Airy (after the Daily Graphic, January 6, 1892).

Airy was particularly involved in optics: for instance, he made special glasses to correct his own astigmatism. For the same reason, he was also interested in the calculation of light intensity in the neighbourhood of a caustic [Airy (1838), (1849)]. For this purpose, he introduced the function defined by the integral Z∞ W (m) =

cos

hπ 2

ω 3 − mω


i

dω,

0

which is now called the Airy function. This is the object of the present book. W is the solution to the differential equation π2 mW. 12 The numerical calculation of Airy functions is somewhat tricky, even today! However in 1838, Airy gave a table of the values of W for m varying from −4.0 to +4.0. Thence in 1849, he published a second table for m varying from −5.6 to +5.6, for which he employed the ascending series. The problem is that this series is slowly convergent as m increases. A few W 00 = −

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years later, Stokes (1851, 1858) introduced the asymptotic series of W (m), of its derivative and of the zeros. Practically no research was undertaken on the Airy function until the work by Nicholson (1909), Brillouin (1916) and Kramers (1926) who contributed significantly to our knowledge of this function. In 1928 Jeffreys introduced the notation used nowadays  3  Z∞ 1 t Ai(x) = cos + xt dt, π 3 0

which is the solution of the homogeneous differential equation, called the Airy equation y 00 = xy. Clearly, this equation may be considered as an approximation of the differential equation of the second order y 00 + F (x)y = 0, where F is any function of x. If F (x) is expanded in the neighbourhood of a point x = x0 , we have to the first order (F 0 (x0 ) 6= 0) y 00 + [F (x0 ) + (x − x0 )F 0 (x0 )] y = 0. Then with a change of variable, we find the Airy equation. This method is particularly useful in the neighbourhood of a zero of F (x). The point x0 defined by the relation F (x0 ) = 0 is called a transition point by mathematicians and a turning point by physicists. Turning points are involved in the asymptotic solutions to linear differential equations of the second order [Jeffreys (1942)], such as the stationary Schr¨odinger equation. Finally we can note that Airy functions are Bessel functions (or linear combinations of these functions) of order 1/3. The relation between the Airy equation and the Bessel equation is performed with the change of variable ξ = 32 x3/2 , leading Jeffreys (1942) to say: “Bessel functions of order 1/3 seem to have no application except to provide an inconvenient way of expressing this function!”

AiryFunctions

Chapter 2

Definitions and Properties

This chapter is devoted to general definitions and properties of Airy func­ tions as they can be, at least partially, found in the chapter concerning these functions in the Handbook of Mathematical Functions by Abramowitz & Stegun (1965).

2.1

2.1.1

Homogeneous Airy functions

The Airy equation

We consider the following homogeneous second order differential equation, called the Airy equation

y"-xy = 0.

(2.1)

This differential equation may be solved by the method of Laplace, i.e. in seeking a solution as an integral y = j* exzv(z)dz, c

which is equivalent to solving the first order differential equation v' + z2v = 0.

We thus obtain the solution to Eq. (2.1), except for a normalisation con­ stant,

y = y exz~z3/3dz. c

The integration path C is chosen such that the function v(z) must vanish at the boundaries. This is the reason why the extremities of the path must 5

6

Airy Functions and Applications to Physics

Fig. 2.1

Integration paths associated to the solution of the Airy equation (2.1).

tend to infinity in the regions of the complex plane z, where the real part of z3 is positive (shaded regions of the complex plane). From symmetry considerations, it is useful to work with the paths Co, C+ and C_. Clearly the integration paths C+ and C_ lead to solutions that tend to infinity when x goes to infinity. When we consider the path Cq and the associated solution, we can deform this curve until it joins the imaginary axis. We now define the Airy function Ai by +ioo J4j(x) = J_ [ exz~z3/3dz. (2.2) 27T1 J — ioo If 1, j, j2 are the cubic roots of unity (that is to say j = e127r/3) the functions defined by the paths C+ and C_ are respectively the functions AiQ#) and Ai(j2x). Combining these solutions, two by two linearly independents for they satisfy the same second order differential equation, we have the relation + jAi (jx) + j2Ai(j2x) = 0.

(2.3)

Now, in place of the functions AiQ#) and Ai(j2x), we define the function Bi(x), linearly independent of Ai(x), which has the interesting property of being real when x is real Bi(x) = ij2Xi(j2a;) - ij4i (jx).

(2.4)

Similarly to Ai(x) (cf. formula (2.3)), we have the relation Bi(x) + jBi(ja:) + j2Bi(j2x) = 0.

(2.5)

On Figs. (2.2) and (2.3), the plots of the functions Ai(x), Bi(x), and of their derivatives Ai!(x) and Bi'(x) are given.

Definitions and Properties

-15

Fig. 2.2

-10

-5 x

7

0

5

Plot of the Airy function Ai (solid line) and its derivative (dotted line).

Fig. 2.3 Plot of the Airy function Bi (solid line) and its derivative (dotted line).

2.1.2 2.1.2.1

Elementary properties Wronskians of homogeneous Airy functions

The Wronskian W {f,g} of two functions f(x) and g(x) is defined by W{f,g} = f(x)^--^-g(x).

Airy Functions and Applications to Physics

8

For the Airy functions Ai and Bi, we have the following Wronskians [Abramowitz & Stegun (1965)] W {Ai(rr), Bi(x)} = — 7T (

2.1.2.2

/

(2-6) p 17T/ 6

\ 1

wl«(0) and C2 = — 4i'(0), and the series

1 3 3! *

fc=0 /v.3fc+l

fc=0

I

(3fc + l)!

1.4 6

1.4.7 9

2 4 2.5 7 2.5.8 in I!" + rx + -ioT*

12

Airy Functions and Applications to Physics

where the Pochhammer symbol (a)n is defined by

(a)0 = 1, (a)n =

i ya)

= a(a + l)(a + 2)... (a + n - 1).

(2.40)

The ascending series of the derivatives Ai'(x) and Bi'(x) are obtained from the differentiation of the series f(x) and g(x) term-by-term. We obtain therefore Aif(x) = c^x) - c2gf(x)

(2.41)

Bi'(x) = a/3 [cif(x) + c2g’{x)],

(2.42)

and the series

f

,

x2 lx5 lx8 “ T+ + 2.3.5.6~8" + ’ ’'

, 9(^ =

2.1.4.3

+

1 X8

l3y

1 XQ 1 xg +1.3.4.6T +1.3.4.6.7.9 9"+ ■ ■ ■ ■

Asymptotic expansion of Ai and Bi

The asymptotic expansions of Ai and Bi are calculated with the steepest descent method [Olver (1954); Chester et al. (1957)]. We will calculate the asymptotic expansion of Ai(x) for x > 0. Definition (2.2) of Ai allows us to write

Co

where Co is the contour defined in §2.1.1. Setting

t = y/x + ill, — OO < U < OO, we obtain

OO 7re^>k(£) = / e-u ^cos

du

o _ 2

e~’

/

V8

\

Definitions and Properties

13

with £ = |rr3/2. The cosine function may be replaced by its expansion: +oo 1 r 2 / vQ i;12 \ J e v (j - ^x3/2 + ~ • • J dv. —oo Integrating term-by-term . ar1/2 / 3.5 5.7.9.11 \ ~ 2a;1/4 ( “ l!144a;3/2 + 2!1442a;3 “ ’ ’' J ’ we obtain the formula given below (2.45). The other expansions are evalu­ ated similarly. For x 1 and s > 1, one defines r (3s + 1/2) (2s + l)(2s + 3)... (6s - 1) Uo =-------------------------- =---------------------------------------------(2.43) 54ss!r (s + 1/2) 216ss! 6s + 1 (2.44) Vs = -------------Us, 6s-1 and the other expansion, according to the notation of Olver (1954) _ V1 Us -

3,5 1 5-7-9-n 1 + 1!216 ~z + 2!2162 z2 , 7.9.11.13.15.17 1 + 3!2163 z3 + ’ ’'

3.7 1 5.7.9.13 1 11216 z “ 2!2162 z2

7.9.11.13.15.19 1 3!2163 z3 OO 5.7.9.11 1 f(z) = £(-i)s^ = i2!2162 z2 3=0 Z , 9.11.13.15.17.19.21.23 1 + 4!2164 z4 OO 3=0

OO

3.5 1 11216 z

Z

v2s =

7?(z) = £(-1)s z2s ~

7.9.11.13.15.17 1 3!2163 5.7.9.13 1 + 2!2162 z2

s=0

9.11.13.15.17.19.21.25 1 4!2164 z4 + ’ ’ ’

OO 3=0

Z

3.7 1 11216 z , 7.9.11.13.15.19 1 + 3!2163 z3

14

Airy Functions and Applications to Physics

Hence we obtain the asymptotic expansions of the Airy functions and of their derivatives (with £ = |rr3/2)

Ai(x) ~ 27T1/2;El/4e Ai'(x) ~

Bi(x) < Bi'(x)

(2-45)

x1/4*,

(2-46) (2-47)

TT1/2^/4*3

x1/4 ,

Ai(—x) " 7T1/2a;l/4 [Sm 0 _ z)

(2.48)

+cos (^'Z)' (?)]

(2.49)

^1/4 r • if Ai'(—x) ‘~ 7T1/2 [Sln V ’-l)7?(0-cos(e-^ )s(0]

Bi'(-x) '

2.1.4.4

2:1/4 r • (c [sm

-)s(o+coS (e--) *(£)]•

___

4)^«) + ™(«

1

Bi(—x) ~ w[“(!

1

1

(2.50)

(2-51) (2.52)

The Stokes phenomenon

The Stokes phenomenon is linked to the asymptotic series of Airy functions (and other similar functions) which have different form in different sec­ tors of the complex plane. In this subsection, we will briefly describe this “phenomenon” that Stokes discovered in his investigations on the asymp­ totic expansion of Airy functions [Stokes (1851), (1858)]. Here we follow the analysis by Copson (1963), (1967) of the Stokes phenomenon for the asymptotic expansion of Airy functions. Equation (2.45) may also be written as 1 „-jz3/2 V' r(3n + (-!)” 27TZ1/4 1 n—Q 32nx(2n)! 7 as |z| —> oo in the sector |phz| < 7r. In order to extend the range of values of phz, we rearrange Eq. (2.3)

Ai(z) = -jAz(jz) - j2Az(j2z). Taking j = e217r/3 and j2 = e417r/3, with — |tt < phz < — |tt, we have —% < ph(jz) < |tt and —|tt < ph(j2z) < 7r then for Az(jz) and Ai(j2z),

Definitions and Properties

15

we can use the asymptotic expansion given above (Eq. (2.53)). Therefore, we may write Ai(^)-F(z)-iG(z),

where

vr j. Other partic­ ular cases of this zeta function may be obtained by taking the successive derivatives of Eq. (2.71). For instance, we have

00

i

i

(2.75)

The same can be done for the derivative of the Airy function (Az"(0) = 0) oo Ai'(z) = Ai'(Q) l+ (2.76) KI/ n—1

P

for which we define the zeta function1 (s > |) OO 1 (2'77>

n=l1 nl Some particular cases are given in Table (2.1) [Crandal (1996); Voros (1999)]. The values for n = 1 are obtained by analytic continuations. In this table, we can see the noteworthy result: the sum of the inverse cubes of the zeros of Ai'(z) is unity. Flajolet & Louchard (2001) studied the area Airy distribution which occurs in a number of combinatorial structures, such as path length in trees, area below random walks, displacement in parking sequence,.... They found interesting results for the spectral function Z~(s) which is 1The notation: Z±(s) comes from Voros (1999).

Airy Functions and Applications to Physics

20

Table 2.1 Some particular values of the Airy zeta functions Z±(s). n 1 2 4 3 Z~(n)

— K,

Z+{n)

0

K2

1 K,

1 - K3 2 K

— Av4

1 K2

1

closely related to this area distribution. In particular, they extended the definition Eq. (2.74) by analytic continuation, and using the Mellin trans­ form M

7T Ai'(z) + k (s) = Z-(l —s) sin tvs 1 Ai(z)

where M represents a Mellin transform, they showed that Z"(-l) = Z-(—2) = Z-(—4) = ... = 0.

Other results may be found in the paper by Flajolet & Louchard (2001).

2.2.3

Inequalities

Some of the previous results may be used to prove inequalities for the Airy function Az, which is concave in the domain (—ai,oo) [Salmassi (1999)] x Ai2(x) < Ai 2(x).

(2.78)

For two different arguments

Ai(x)Ai(y) < Ai2

,

(2-79)

and for the scaling

Ai(x)a AitO)1-0 < Ai(ax);

0 < a < 1.

(2.80)

Similar inequalities may be found for the function Bi.

2.2.4

Connection with Bessel functions

As mentioned in the introduction, Airy functions Ai and Bi may be alter­ natively written in terms of Bessel functions I and J of order 1/3, and of

Definitions and Properties

21

order 2/3 for their derivatives, £ = |&3/2 [Jeffreys (1942); Olver (1974)] [/_1/3 (0 - 11/3 (£)]

(2.81)

[j_1/3 (0 + J1/3 (0]

(2.82)

Ai'(x)= -|[l_2/3(e)-/2/3(0]

(2.83)

Ai(x) =

= Ai(-x) =

= -l/U2/3(£) '* v o

Ai'(-x) = ~ [J_2/3 (£) - J2/3 (£)]

(2.84)

=

17-1/3 (0 +11/3

(2’85)

Bi{~x) =

t* 7-1/3 (e) " JV3 (€)]

(2.86)

Bi'(x) =

[l_2/3 (0 +12/3 (0]

(2.87)

Bi'(-x) =

[j_2/3 (£) + J2/3 (£)]

(2.88)

Bi(x) =

22

Airy Functions and Applications to Physics

The modified Bessel function Kfiz) is defined by

2

sm(7rz/)

and the Hankel functions H^^z) and H„2\z), by

H^\z) = Ju(z) + iYy(z),

H^2>(z) = Jy(z)-iYy(z),

with the Weber function y / \ _

2.2.5 2.2.5.1

~ J-ifiz)

Modulus and phase of Airy functions Definitions

We define the modulus M(x) and the phase (fix) of the functions A fix) and Bfix), and the modulus N(x) and the phase (/fix) of the functions Aifix) and Bifix), for any x > 0 by the relations [Miller (1946); Abramowitz & Stegun, (1965)]

Afi—x) = M(x) cos [0(a;)]

(2.89)

Bfi—x) = M(x) sin [0(x)]

(2.90)

Aifi—x) = N(x) cos [(#)]

(2-91)

Bifi—x) = N(x) sin [(x)],

(2.92)

and the inverse relations

M(x) = [Az2(—#) + Bi2

e{x) = arc tan

(- + y)

(3.17)

•

y x2y1dx=^(x3y1-xyl + ‘l(xy-yl)).

(3.18)

From which we find X rx' 1 / yidx'dx" = ~(x2yi-xy'-y),

/

3.2

etc.

(3.19)

Product of Airy functions

It is sometimes possible to easily calculate the primitive of a product of Airy functions. For example, we can calculate oo I = y* Ai2(x)dx, X

from an integration by parts I=

oo — 2 j xAi(x)Aif(x)dx. X

Thanks to the Airy equation (2.1), we can write oo I=

—2

Ai\x)Ai" (x)dx

Since lim xAi2(x) = 0 and lim Ai'(x) = 0, we finally obtain x—>oo x—>oo I = —x Ai2 (x) + Ai,2(x).

However, this kind of calculation is not always so straightforward. We shall therefore detail the method of Albright (1977) in the next section. This method allows us to calculate the primitives of linear combinations of the homogeneous Airy functions Ai and Bi.

Integrals of Airy Functions

3.2.1

41

The method of Albright

We wish to calculate integrals of the kinds (3.20)

where y is a linear combination of the functions Ai(x) and Bi(x) (i.e. y(x) = aAi(x) + /3Bz(a;)), the sign “ prime ” (') stands for differentia­ tion with respect to x. Albright builds the following table, where D stands for the operator d/d#, and where y" is replaced by xy (according to the Airy equation (2.1)) y2

Dy2 ^>y'y Dy'2 Dxy2 Vxy'y Dxy'2

y'y

y'2

xy

1

1

9

t xy y

xy

f“2

x2 y2

x2 y/ y

x 2 y /2

2

2

1

2 1

1

1

1

2

Dx2y2 Dx2y'y Dx2y'2

2

2

2

1

2

The properties of this table are such that we are able to calculate the primitives of the kinds in (3.20). For example, to calculate

F = j y2dx,

we just have to subtract lines (4) and (3) of the table. So we obtain

D (xy2 - y'2) = y2, and then the result, Eq. (3.24)

F = xy2 — y'2, except for the integration constant. For the following primitive G = j y,2dx, three lines of the table have to be considered; these can be written into the matrix form

/ y'y \ /i 1 0 \ / y'2 ' D I xy'2 I = I 1 0 2 I I xy2 \x2y2) \0 2 2 / \x2y'y

(3-21)

42

Airy Functions and Applications to Physics

It is then sufficient to inverse the system / y'2 \ . (~± —2 2 \ / y'y 1 /2 xy2 I = h i 2 2 —2 j D xy" \ \x2y'y) \ 2 —2 -1 / \£ 2 y 2 leading immediately to the result G = | (2y'y + xy'2 - x2y2).

(3.22)

In the next section we set out some particular cases of primitives ob­ tained by the method of Albright and with the help of the primitive

L= calculated from an integration by parts

— L,

L=x

L = | (xny2 — n f xn 1y2dx) .

3.2.2

(3.23)

Some primitives

If y is a linear combination of Ai(x) and Bi(x) and n a positive integer: • j y2dx = xy2 --y'2 • y zy2da; = j (y'y - xy'2 + x2y2) • y £2y2da; = | n 1 2 | 2/2.32 2 1 xy y- -y 1 -x y + x y * • y y'y&z = 1 2 = 2V • y xy'ydx

= -y'2 2y

• y x2y'ydx = j (x2y2 - 2y'y + 2xy'2) • y y'2da: = | (2y'y + xy'2 - x2y2) • y z?/2dz = | 3 [xy'y - |y2) + x2y'2 - x3y2

* / x2y,2 0 oo oo / tnAi'(t + x)dt = — n y*

0

Ai(t + x)dt

(3.82)

0

=

oo [ tnAi(t + x)dt, dx J o

and for n > — 1 oo oo [ tnAiAt + x)dt = —[ tn+1Ai(t + x)dt J n+1 J 0 0 oo oo = y y tnAi(t + x)dtdx,

(3.83)

x o

with the particular case oo y Aii(t d-x)dt = Aif(x) d-x Aii(x).

(3.84)

o These two kinds of integrals can be treated from the calculation of P’W + x)dt. Now, from the Airy equation (formula (2.1)), we can deduce oo y tnAi(t + x)dt (3.85)

0 oo =y o

=

oo - Xy tn~rAi(t + x)dt

o OO — x [ tn~1Ai(td-x)dt. dxz J J o

In the last two equations, the exponent of the variable t is reduced by a unity until the iteration leads to 0 > n > — 1.

52

Airy Functions and Applications to Physics

It should be noted that, for x = 0, the integral (3.85) can be explicitly written [Olver (1974)] oo (3-86) V 3 7

0

We can obtain the moments of the Airy function by calculating [Gislason (1973)]

+oo lim / Ai(x)xne~ex d#, n G N, —oo that is to say

-|-oo j Ai(x)x3kdx = ^£

+ oo +oo / Ai(x) £3fc+1d£ = / Ai(x) x3k+2dx = 0.

—oo

(3.87)

(3.88)

—oo

We can also explicitly write the integrals (3.82) to (3.85), for n = —1/2, with the help of the definition (2.21) of Ai(x)2

We obtain, by comparison with Eq. (2.150):

[ Ai(x + f)—= = 22/37rAz2 (7^77) . J y/t \2Z/AJ 0

(3.89)

Here we retrieve the formula (2.157). In a similar manner, we shall have / ■4i(I -f)^ =

(2^)Bi (2^)'

(3'90)

0 2 Some of the following integrals may be found, thanks to the relationship between the Fredholm equations Jo°° f(x + t) -^= = g(x) f(x) = f^° gf(x + t) -^=.

Integrals of Airy Functions

53

Differentiating and integrating Eq. (3.85), we obtain oo

(3.91)

0 oo y Aii(x +1)-^=

(3.92)

o

We can also mention the integrals [Aspnes (1967)] that are obtained thanks to the causal relations between the Airy functions (cf. §4.1.1) +oo / + t)^ = 22/37rAi (^) Bi (3.93) 0 +oo y Gi(x ~t)^ = -22/3ttA'P (^3) •

(3.94)

0

It should be noted that we can encounter formula (3.89) in a different form [Berry (1977a)] 00 +00 l=y =/Ai (u2 ~du(3-95) -y

-00

Thanks to definition (2.21) of the Airy function Ai, we can express this function as

ei[”3/3+(«2-yHd uAVt Making the following change of variables

u = 2-2/3 (Y - X) V = 2-2/3 (Y + X), we obtain

+00 L = _[ ei[x3/3+Y3/3_2-/3y(X+Y)]dxdy 2?t21/3 J in other words i=22/3^2(^)-

(3.96)

54

3.5.2.2

Airy Functions and Applications to Physics

Integrals involving transcendental functions

Considering first the integral +oo M= j Ai (x2 + a) e ikxdx,

(3.97)

—co that can be written, again using (2.21) for the Ai function,

M=f- yyei[‘3/3+< * 2 +a)]eifcxd^dt. R2 The calculation of this integral is not so easy if one integrates first on the variable x. However, it becomes simpler if we employ the method described in §3.6.5, that is to say if we canonise the cubics t3/3-\-x2t. When we make the following change of variable X = 2-1/3(t-a;) Y = 2-1/3(f + x), we immediately obtain the result (see formulae (2.153) to (2.156))

M = 22/3irAi [2"2/3(a - fc)] Ai [2"2/3(a + &)] .

(3.98)

We can also, thanks to the integral representation (2.21) of Ai(x), obtain the relation [Widder (1979)] +oo e“tJ4i(f)dt = e“3/3. (3.99)

Further, we can mention the following integral +oo e-*2/4"^ (t) dt = 2y/iru e2u3/3Ai (u2).

(3.100)

In this last expression, we find, thanks to the asymptotic formula (2.45), the relation (3.76) Ai(t) dt = 1, for u —> oo. We can now give some other integrals involving one Airy function and transcendentals without demonstration 2\ /1 \ / r3 \' 2 + V, _in , (3.101) Ai(-xt) Intdt = — where ^(-) stands for the logarithmic derivative /•OO • / e_2t3/3x3 ?li(t2) dt = JO T3 1 / /-------- \ 1/3 —. - (1 + ^/l — ) —x 6 Jl - x6 L * V /

of the gamma function.

(3.102)

/ /-------(1 + \/l — a:6 V

Integrals of Airy Functions 2tt

dt = —Ai(x)Ai(—x).

(3.103)

= -J^^Ai^x2).

(3.104)

• r°e-t3/12Ai'(xt)

Jo

yt

Jo •

V 3

(x Ai(x2) — Ai'(x2)).

e f Z12 Az(a;t) -^= = —\r^re2x

[

•

e-t Z12 Ai (xt) t Vtdt = — 2yJ~^-e2x (x Ai(x2) + Ai'(a;2)) .

°

3.5.3

(3.105)

* 3

tyt

/

55

(3.106)

Integrals of products of two Airy functions

In a similar manner to that in §3.5.2, following the method of Aspnes, we can establish the relation [Aspnes (1966)] tnAi2 (t + x) dt

n 2n + 1

(3.107)

Id2 2 d#2

oo / tn~1Ai2 (t + x) d£, n > 0.

o

We can extend this relation to one containing Ai', since we have Ai(x)Ai'(x) = ^-^Ai2(x) and Ai'2(x) = | Ai2(x).

In particular, for n = —1/2, we find oo / Ai2(t + = ^Aii (22/3x)

0 oo / Ai(t + x)Ai'(t + a;)~| = —2-4/3 Ai (22^3x^

(3.109)

o oo /^'2(t + a;)^|

(3.110)

0 = •

(3.108)

+ 22/3^! (22/3^)} .

We can also mention the important integrals

Airy Functions and Applications to Physics

56

(3.111)

' 8(b - a)

if /3 = a

__ 1___ Ai [__ —__ if /3 > o, k |/33—on3!1/3___ |_ (/33—a3)1/3

and

(3.112)

if /3 = a

% b—a _____ i

a

r____ _______ i

|^3_q3|1/3____ [(/^-q3)1/3 J

if /3 > a,

Eq. (3.112) is calculated from the integral representation (4.7) of Ai + iGi, p representing the Cauchy principal value. • In order to calculate the following integral [Biennieck (1977)], which is a generalisation of formula (3.111):

the general method consists of taking the integral representation of the Airy functions (2.21), namely +oo X + a = M f ei[(Qt)3/3+(x+a)t] (3.114) a 27T J —oo Then we obtain In = jj dtdt'

R2

-|-oo ei[(at)3/3+(^')3/3+at+6t'] j

—oo

with

xnei^t+t')dx

ix(t+t')dX)

Integrals of Airy Functions

57

and

where 6 is the Dirac delta function and 3^ its nth derivative. We finally find the relation +oo (3.115)

Another method starts from Eq. (3.113) and from the Airy equation (2.1), leading to the recurrence relation d2 T In+l=!33—^-bIn do2 with Iq being given by the relation (3.111), we can then deduce

Io =

b—a _(/33 - a3)1/3 _

(/33 _ a3)l/S

ba3 — affP Il = /33 — a3 01

h =

ba3 — a/33 /33 — a3

(3.116)

(3.117)

(3.118) 2 Io

2a3/?3 , f33 — a3 0

(3.119)

• Note also the following integral [Balazs & Zipfel (1973)], which ap­ pears, in particular, in the semiclassical calculation of the Wigner function (cf. §8.3) +oo F = y Ai(x + t')Ai(x - t')e'2ptdt. (3.120)

—oo To calculate this integral, we again use the integral representation of Ai (2.21) to obtain +oo F= J_ / ei(u3/3+v3/3+ux+vx)ei(2p+u-v)tdu(ivdL 4tt2 J — OO Integration on the variable t leads to a Dirac function, making it possible to derive the relation i(8p3/3+2px) +?° __________ / ei[2u/3+2P^+(2^+4p2)n]jw 2% J — oo

Airy Functions and Applications to Physics

58

and, after the change of variable w3 = 2u3, +oo i(8p3/3+2Px) F =----------------- 2-1/3 y ei[cu3/3+21/3po>2+22/3(a;+2p2)a>]du; 2%

The integral +oo G = y ei(t3/3+at2+6t)dt,

(3.121)

—oo is simply calculated by putting u = t + a, providing +oo G = eia(2a2/3-6) y ei(“3/3+(6-“2)“)du, — OO

in other words (see formula (2.26)) G = 27reia(2a2/3-,))J4i (b - a2).

(3.122)

In our case, after simplification, we obtain F = 2“1/3Ai [22/3 (x + p2)] .

Conversely, we have +oo y Ai(a + x2)e'bx da; = 2-1/3Ai

Ai

(3.123)

•

(3.124)

But also a non-trivial result obtained by Berry et al. (1979), and indepen­ dently by Trinkaus & Drepper (1977). +oo y Ai(a - x2ybx da; = 2-1/3 a

1 = ------------- 777 exp (id3—a3)1/3

x Ai

1 / G03 - a3)1/3 \

a

X2a3(33 \ (33 — a3 )

Integrals of Airy Functions

59

and for /3 = a

(3.127)

2^|A|1/2 N3/2

X exp

f

. Fo3A3

(a — 6)2

A(a + 6)

tt

+ —sgn(aA)

Finally, we give the following integral /

3.6

Ai (?)

= 22/3^ x4i'(22/3v^), x > 0.

(3.128)

Other integrals

Integrals involving the Volterra ^-function

3.6.1

The Volterra yn-function is relatively intriguing in the bestiary of special functions, for it has received little attention, despite its remarkable prop­ erties. In this section, we would like to add some properties linked to Airy functions [Vallee (2002)]. In his work on integral equations with logarithmic kernel, Volterra in­ troduced a function that can be generalised by Erdelyi et al. (1981) rG0 + 1) r(f + a + 1) di-

^u, (3, a) =

(3-129)

We are going to calculate the following integral by using the definition of the fi-function. 00

Ai(u) fi(yu, /3, a) du r°° = / dt

Jo

t?

f°° (yu)a+t / Ai(u) —----- -du. r(/5 +1) Jo r(t + a +1) J

Using the Mellin transform of the Airy function (see Eq. (3.86)) [°° . n , r(n + l) Jo Aluu du~ 3("+3)/3r(^)-

We then find, for the right member of Eq. (3.130),

00 dt t'3 (73/3) 00 thanks to the Airy function. Now using the first of the relationships (3.135), we find pOO

pOO

/

u Ai(u) p^yu, /3, a) du = — 7 /

Jo

Jo

A/(u)//(7U,/3, o — 1) du.

(3.138)

61

Integrals of Airy Functions

Making a second integration by parts /»OO /•OO / uAi(u) Ft'yu, /3, a) du = y2 / Ai(u) Jo Jo Thus using the result from Eq. (3.131), we obtain

(3, a — 2) du.

/* uAi(u) ft^u, (3, a) du = 3^72/z(^-,/3, Q ^). (3.139) Jo 3 3 There are other ways to prove this result. The first one is to take the derivative of Eq. (3.131) d J*00 /»oo — / Ai(u) /jJ^u, /3, a) du = / u Ai(u) ft, a ~ 1) du, d? Jo Jo which is equal to d7 [

3

3 J

3

3

leading to the same result. Another method is to use the recurrence relation between Volterra jjlfunctions (3.135). Eq. (3.139) may be written into the other form

—/■ (3t)4/3 Jo

.(3^)1/3_

/jjx, 0, a) dx = 3^72 /jjt, 0, —v—)• o

(3.140)

Then fjjx, 0, —1) is deduced to be the eigenfunction of an integral equation with the kernel (3t)4/3AZ ((3t)1/3) ’ (3'141)

which is again a solution to Eq. (3.134). In the same way, we can find other interesting results such as

[ Jo

Ai\u)/j,(^u,^,a)du=(3.142) 6 6

and [ Jo

Aii(u)/z(7u,/3,a)du = -3^-M(V^’^V“)’ 7 3 3

(3.143)

where Ai^u) = Ai(t) dt. More generally, we have a formula for the nth derivative of the Airy function jT Ai^(u) ^U,/3,a)du = ^(-yr

(3.144)

Each of the equations (3.142) to (3.144), may be put into a form which satisfies an eigenfunction equation with a kernel that is a similarity solution

Airy Functions and Applications to Physics

62

to the 1-KdV equation. Finally, we give a result for Ai2 from its Mellin transform [Reid (1995)]

r Jo

l

X nA _ 2r(n+l) )U U ^F12(2«+7)/6r(2^tZ)’

which reads r00

/

q/3 = 5f2 +4J^3. For similar quantities of the canonical cubics F(X, T) = X3 + Y3, we have ' Jf = A2XY, < & = A3 (X3 - Y3), . ^ = A6.

By comparing the cubics F and the cubic covariant Y>, identifying the terms x3 and y3 with the terms X3 and V3 respectively, we finally obtain ' a3 = | [A + ^(?12£> - 3ABC + 2B3)] ,

73 = I

- 3ABC + 2B3)] ,

/33 = | [£> - ^(AD2 - 3BCD + 2C3)] ,

k 63 = | [£> + -f^(AD2 - 3BCD + 2C3)] ,

with the condition Q) = (AD - BC)2 - 4 (AC - B2) (BD - C2) = A6

0.

It should be noted, in the case where = 0, that one cannot use the general canonical form F(X, Y) = X3 + F3. In this case, the canonical form of F(x,y) is either X3 or X2Y.

3.6.3

Integrals with three Airy functions

Consider the following integral (where a, c and e are all non-zero) [Vallee (1982)] +oo L = / Ai(ax + b)Ai(cx + d)Ai(ex + f)dx. (3.148)

Airy Functions and Applications to Physics

64

Once again using the integral representation (2.21) of the Airy function

£=

[ ei(w3+v3+w3)/3ei(ax+0u

1 R4

x e^cx^v^ex^wdudvdwdx. An integration on the variable x results in a Dirac delta function allowing one to express w as a function of u and v z,=_L yyy ei(u3+«3+w3)/3ei(6u+dv+/w)5(au+cv+eW)dMdvdw. R3 We then have an integral of the kind

fei(Ax3/3+Bx2y+Cxy2+Dy3/3+Vx+W2/) 4?r2 |e|

R2

with A = 1 - u3/e3, B = —n2c/e3, C = —nc2/e3, D = 1 — c3/e3, V = b-af/e, W = d — cf/e.

We consider here that the quantity K = e6 + a6 + c6 - 2n3c3 - 2u3e3 - 2c3e3 > 0.

The calculation of this integral is then carried out in §3.6.5, formula (3.154), and the result reads 1 . / (be - af)S - (de - cf)y\ ((de- cf)a - (be - af)/3 K1/6 \ K1^ 7 \ K1/^ with ' a3 = | [A + ^(A2D - 3ABC + 2B3)] ,

jd3 = | [D - ~^(AD2 - 3BCD + 2C3)] ,

73 = 1 [4 _ -^(A2D - 3ABC + 2B3)] , 63 = 1 [D + -A=(AD2 - 3BCD + 2C3)] .

If K < 0 the integral will involve the irregular companion Bi of the Airy function Ai. As an example, let us consider the integral /»+oo M= / Ai(x)Ai(x + a)Ai(x + b)dx

Integrals of Airy Functions

65

The calculation of this integral proceeds as follows. We first use the integral representation of the product of two Airy functions (cf. formula (2.153)) ... .... , , Ai(x)Ai(x + a) =

1

f+ca . .(x + t2 +a/2\ iatji j Ai -----J e dt.

So we have 1 f+ca e iat dt J-^ f+°° Ai 4 .(x t2+a/2\ “ 21/3^ ( +2-2/3 ) A^. X + ...

leading, thanks to Eq. (3.111), to

We then use the non-trivial result, formula (3.125), to obtain

with the notation c = (|)1/3(6 — a/2) and d=(|)1/6a, and 91, the real part. A little algebra yields M = 3-1/69t pi p-1/3 (6 + ja)] Bi [3-1/3(6 + j2a)] } ,

(3.149)

where j and j2 are the cubic roots of unity. We give another particular case of an integral with the product of three Airy functions [Vallee et al. (1997)]

+oo

Z

Ai(x)Ai(y — x)Ai(z — £)d£

(3.150)

-oo = 5-1/6Az ^5-1/3 (y/e — sz)j Ai [5-1/3 (z/e — ey)^

where e is the golden mean e = v/^~1.

3.6.4

Integrals with four Airy functions

Let us consider the integral [Aspnes (1966)]

-(-oo

(3.151)

—oo

66

Airy Functions and Applications to Physics

with a > 0 and (d > 0. With the help of the formula (2.157), this integral reads r22/3 ----- (a + x) + u r22/3 x Ai —^—(6 — x) + v

By now using the formula (3.111), we have

22/3(a + 6) + au + flv a/3 f du f dv 4?r2 (ce3 + /33)1/3 J V^J (a3+/33)1/3 v 7 o o With the help of formula (2.157), this expression becomes OO p (a+b\„+™ ■ 2ir J y/w |_(a3+ /33)1/3 I

We find here the primitive of the Airy function Ai (cf. formula (3.108)) OO Ai-^x) = 2 / Ai2 (t + 2-2/3^

o We finally obtain q2/3

(Q + 0 (q3 + /33)1/3_ ’ (3.152)

3.6.5

Double integrals

Let us consider the following double integral [Vallee (1982)] G = yy * Ai(ax A by A p)Ai(cx + dy + q)

(3.153)

R2

x Ai(ex + fy + r)Ai(gx + hy + w)dxdy, where a, c, e, g are non-zero constants. Making use of the integral repre­ sentation (2.21) of Ai(x) and rearranging the terms, produces G= mi dtdt'dsds' ei[(t3+t'3+s3+s'3)/3+Pt+9t'+rs+wS'] R4 x yy ei[x(6zt+ct/+es+ps/)+i/(6t+dt/+/s+/is/)] £xdy.

R2

67

Integrals of Airy Functions

From the integral representation of the Dirac delta function 8(z) = x r+oo ei2wj we have Z7T J —OQ 7

dtdt'dsds' e^t3+t'3+s3+s'3)/3+pt+gt'+rs+ws'] R4 x 8 (at + ct' + es + gsf) 8 (bt + dtf + fs + hs').

An integration over s and s', then gives us

Q _ K Jdxdy e[(Ax3/3+Bx2y+Cxy2+Dy3/3+Vx+wy)

(3.154)

R2 with

' K = Ch~ f9^0, A = (eh- fg)3 + (bg - ah)3 + (af - be)3, D = (eh — fg)3 + (dg - ch)3 + (cf - de)3, B = (bg — ah)2(dg — ch) + (af — be)2(cf — de), V = v(bg — ah) + w(af — be) + p(eh — fg), W = v(dg — ch) + w(cf — de) + q(eh — fg).

The next step is to canonise the cubic form (cf. §3.6.2), leading to k3 r r dXdrex>>r[iGr / x3 + T y3 G=4^//

(3155)

R2 +

Vd-VC; Wa- V0\J\ X ~x + X—~ Y , A A J

with A6 =

= (AD - BC)2 - 4 (AC - B2) (BD - C2) > 0.

Finally, the integral (3.153) may be written as K3 A. (VS-4 ,(Wa-V/3\ G = ~rAi ------------- Ai ------ -—- . A \ A J \ A J

(3.156) 7

Exercises (1)

Generalise the table of Albright (see §3.2.1) to calculate the following primitives (n=l, 2) xny2dx,

xnytydx,

xnyiy' dx.

68

Airy Functions and Applications to Physics

(2)

Calculate the following integrals in terms of Ai and Gi functions /•OO /»OO / cos (a t3 + b t + c) dt, / sin (a t3 + b t + c) dt. Jo Jo

(3)

Calculate dt Vt

(4)

Use the relations between Bessel and Airy functions (cf. express the following integral in terms of Airy functions OO / [Ij/Cr) + Ky(x) d# = aea ^(a),

§2.2.4) to

o when v = | and z/ = |, a > 0.5 (5) Check the formula (3.150), with the method described in §3.6.3.

Chapter 4

Transformations of Airy Functions

4.1

4.1.1

Causal properties of Airy functions Causal relations

Using the integral representation of Ai(x), Gi(x), Ai2(x) and Ai(x)Bi(x) given by the formulae (2.21), (2.127), (2.150) and (2.151) respectively, we can write [Scorer (1950); Aspnes (1967)] oo

Ai(x) + iGi(x) = 1 y e 0, this last relation becomes oo If. = / emd *u

lim -1- | |q| I

7T

11 = 5(x) + i-p-,

J

7T

X

0

that is to say r 1 -At alim -— |a|

U)=5(a:)’

(4-8)

and lim ± Gi ((x\ x' G(x,x') = 27riei27r/3 x

k Ai (x') Ai (a;e127r/3) if x < xf.

Thanks to the relation (see formula (2.15))

ei27r/3^ (zei27r/3) =

[Ai(z) - iBi(z)],

G(x,x') can be written as a function of Ai and Bi

G(x,x) =

f Ai(x)Bi(x') + iAi(x)Ai(xf)

if x > x1

k Ai^x^Bi^x) + \A.i(xr)Ai(x)

if x < xf.

—7T X

Airy Functions and Applications to Physics

72

4.1.3

Fractional derivatives of Airy functions

In a series of papers, Varlamov [Varlamov (2007), (2008-a,b,c); Temme & Varlamov (2009)] was particularly interested in the fractional derivatives of Airy functions and their applications. In order to give a rapid overview of this subject, we first introduce a def­ inition of fractional derivatives. For real a > — 1, the fractional derivative of order a of a function f is defined by the relation

D“/W = D“{/(x)} = -

dC,

(4.16)

subject to the convergence of the integral and where f is the Fourier trans­ form of f. We give now several results established by Varlamov. He first introduces some functions which are linear combinations of products of Airy functions w+(x) = Ai(x)Bi(x) + Ai2(x),

(4.17)

w_(x) = Ai(x)Bi(x) — Ai2(x).

(4.18)

and

He then proves the following relations

D-1/2G«(a;) = kw_ (^3) ,

(4.19)

D-1/242(;r) = kw+ (^3) ,

(4.20)

where k = 21/6a/tt. Moreover

DV W) =

(5^) .

(4.21)

(^) ■

(4.22)

More generally, we have for a > —1/2

DQ[24i2(a;)] = k^-^Ai&^x) - D“-1/2G?(22/3a:)],

(4.23)

DQ[Xi(a;)Bi(a;)] = ka\Da~1/2Ai(22/3x) + Da-1/2Gz(22/3,7))],

(4.24)

and

22(«-l)/3

/Cq — /o • From these results some very nice relationships may be found. Let

AViiere

di = ^Oa(Ai{x)Bi{x)), K>ct

d2 = ^Da(Ai2(x)), l^ot

Transformations of Airy Functions

a = (D *

1/2^) (22/3a;),

73

(22/3x) .

b=

(4.25)

Then we have the following relationships [Varlamov (2008-b)]

+ d>2 = 2(a2 + 62),

d2 —

= 4a6

and

did? = a2 — b2.

(4.26)

The results given in §(3.3) are generalised by using Eq. (3.70). Varlamov (2008-b) obtains the non-trivial result

/,Da”1/2Ai(o:).Da+1/2A?(3:) + Da-1/2ffi(^).Da+1/2ffi(a:) [Da~1/2Ai(x) + D^~1/2Gi(x)]2

J

fO-^Ai^ + D^Gi^y

= 2/3

/Da [Ai(y')Bi(y')]\

\

J

D“[4z2(y)]

)'

where y = 2-2/3a; and Ff = /, from which many particular cases may be found. Several other results concerning the fractional derivatives of Airy functions can be found in the work by Varlamov.

4.2

4.2.1

The Airy transform Definitions and elementary properties

We define the family of functions (—) ,

^a(^) =

|a|

ael.

(4.27)

\af

One of the most important properties of this family is (see formula (4.8)) lim {cja(a?)} = £(&),

Ou—>0

(4.28)

where 5{x) is the Dirac delta function and a E R. Moreover, from for­ mula (3.111), we can set the relation giving the convolution product of two functions cuQ(#) and a^(x) * ^(^) = ^7(^)5

(4.29)

with 73 = o3 + /?3. In other words, the family of functions defined by Eq. (4.27) is stable under the convolution product and then leads to a semigroup of convolution. Note also that the functions are normalised (formula (3.76)) (jja{x)dx = 1.

(4.30)

74

Airy Functions and Applications to Physics

Therefore if f is a function of x, and f its Fourier transform, we write (/>□,, the Airy transform of /, as the convolution product Vaix) = f * u a(x)= Aa [/(x)],

(4.31)

which is also written +oo

¥>a(x) =

[ ei(a3e/3+ix>lf^)d^ lit J

(4-32)

Consequently, f i-> (pa is a particular class of functional transform that can be reversed by the formula

f(x) =

W-a(x) = Aa [Q(x) =

(4-43)

0

-4

4

X

Fig. 4.2

Airy transform of the step function 0(x).

The transformed function oscillates for x < 0 and goes exponentially to 1 for x > 0 (Fig. (4.2)), as can be seen from the following asymptotic expansions

«1-

—3/4 _2/^\3/2 e 3ka)

when x —> Too, and

when x —> — oo. We can also give an important result allowing the transform of many functions to be calculated analytically:

Proposition 4.7. If the Airy transform of a function f is (pa, then the Airy transform of xf is Aa

[x/(a:)] = x 0

M

f(x + k)

/(fcx)

) * /'( x/(x)

yv