Airy Functions and Applications to Physics

A1RY FUNCTION S AND APPLICATIONS TO PHYSICS This page intentionally left blank AIRY FUNCTIONS AND APPLICATIONS TO...

Author: Olivier Vallee | manuel Soares

235 downloads 1866 Views 9MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

A1RY FUNCTION S

AND APPLICATIONS TO PHYSICS

This page intentionally left blank

AIRY FUNCTIONS

AND APPLICATIONS TO PHYSICS

Olivier Vallee U niversite d'orleans, France

Manuel Soares Ministeve de I'Equipement,

DD

E des Yvelines, France

yp World Scientific N E W JERSEY * L O N D O N * SINGAPORE

-

BElJlNG

-

SHANGHAI

HONG KONG * TAIPEI

*

CHENNAI

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9%

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

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

AIRY FUNCTIONS AND APPLICATIONS TO PHYSICS Copyright 0 2004 by Imperial College Press

All rights reserved. This book, or parts thereoJ 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 1-86094-478-7

Printed in Singapore by World Scientific Printers (S) Pte Ltd

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, and even in the necessity, 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 resolved by computers. This trend is now becoming the norm as the importance of computers continues to grow. As a last resort, the special functions employed in physics will have, indeed, to be calculated numerically, even if the analytic formulation of physics is of first importance. The knowledge on Airy functions was periodically the subject of many review articles. Generally these were about their tabulations for the numerical calculation of these functions which is particularly difficult. We shall quote the most known works in this field: the tables of J.C.P. Miller which are from 1946 and the chapter in the Handbook of Mathematical Functions by Abramowitz and Stegun whose first version appeared in 1954. No noteworthy compilation on Airy functions has been published since that time, in particular about the calculus implying these functions. For example, in the last editions of the tables of Gradshteyn and Ryzhik, they are hardly evoked. At the same time, many accumulated results in the scientific literature, remain extremely dispersed and fragmentary, The Airy functions are used in many fields of physics, but the analytical outcomes that have been obtained are not (or weakly) transmitted between the various fields of research which after all remain isolated. Moreover the tables of Abramowitz and Stegun are still the only common reference t o all the authors using these functions. Thus many of the results have been rediscovered, sometimes extremely old findings are the subject of publications and consequently a useless effort for researchers.

V

vi

Airy Functions and Applications to Physics

In this work, we would like to make a rather exhaustive compilation of the current knowledge on the analytical properties of Airy functions. In particular, the calculus implying the Airy functions is developed with care. This is, actually, one of the major objectives of this book. We are however aware of making a great number of repetitions regarding the previous compilations, but, it seemed necessary t o ensure coherence. This book is addressed mainly to physicists (from undergraduate students to researchers). For the mathematical demonstrations, as one will see, we do not have any claim about the rigour.' The aim is the outcome, or the fastest way to reach it. Finally, in the second part of this work, the reader will find some applications t o 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 the works of molecular physics or in physics of surfaces, and vice wersa.

The authors would like to warmly thank Nick Rowswell who considerably improved the content of this book.

0. VallQe &' M. Soares, Fall 2003

'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.

Contents

Preface

V

1. A Historical Introduction : Sir George Biddell Airy

1

2. Definitions and Properties

5

2.1 The Homogeneous Airy Functions . . . . . . . . . . . . . . 2.1.1 The Airy's equation . . . . . . . . . . . . . . . . . . 2.1.2 Elementary properties . . . . . . . . . . . . . . . . . 2.1.2.1 Wronskians of homogeneous Airy functions . 2.1.2.2 Particular values of Airy functions . . . . . 2.1.2.3 Relations between Airy functions . . . . . . 2.1.3 Integral representations . . . . . . . . . . . . . . . . 2.1.4 Ascending and asymptotic series . . . . . . . . . . . 2.1.4.1 Expansion of Ai near the origin . . . . . . . 2.1.4.2 Ascending series of Ai and Bi . . . . . . . . 2.1.4.3 Asymptotic series of Ai and Bi . . . . . . . 2.2 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 . . . . . . . . . 2.2.5.1 Definitions . . . . . . . . . . . . . . . . . . . 2.2.5.2 Differential equations . . . . . . . . . . . . . 2.2.5.3 Asymptotic expansions . . . . . . . . . . . . 2.2.5.4 Functions of positive arguments . . . . . . . vii

5 5 8

8 8 9 9 11 11 12 13 15 15 18 20 20 21 21 22 23 24

...

Vlll


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 a t 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(z)Ai(-z): Airy wavelets . . . . . . 32 3. Primitives and Integrals of Airy Functions 3.1 Primitives Containing One Airy Function . . . . . . . . . . 3.1.1 In terms of Airy functions . . . . . . . . . . . . . . . 3.1.2 Ascending series . . . . . . . . . . . . . . . . . . . . . 3.1.3 Asymptotic series . . . . . . . . . . . . . . . . . . . . 3.1.4 Primitive of Scorer functions . . . . . . . . . . . . . . 3.1.5 Repeated primitives . . . . . . . . . . . . . . . . . . 3.2 Product of Airy Functions . . . . . . . . . . . . . . . . . . . 3.2.1 The method of Albright . . . . . . . . . . . . . . . . 3.2.2 Some primitives . . . . . . . . . . . . . . . . . . . . . 3.3 Other Primitives . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Elementary Integrals . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Particular integrals . . . . . . . . . . . . . . . . . . . 3.5.2 Integrals containing a single Airy function . . . . . . 3.5.2.1 Integrals involving algebraic functions . . . . 3.5.2.2 Integrals involving transcendental functions 3.5.3 Integrals of products of two Airy functions . . . . . . 3.6 Other Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Integrals involving the Volterra p-function . . . . . . 3.6.2 Canonisation of cubic form . . . . . . . . . . . . . . . 3.6.3 Integrals with three Airy functions . . . . . . . . . . 3.6.4 Integrals with four Airy functions . . . . . . . . . . . 3.6.5 Double integrals . . . . . . . . . . . . . . . . . . . . .

37 37 37 38 38 39 40 40 41 43 48 49 50 50 51 51 54 56 60 60 64 65 67 68

Contents

ix

4. Transformations of Airy Functions

71

4.1 Causal Properties of Airy Functions . . . . . . . . . . . . . 4.1.1 Causal relations . . . . . . . . . . . . . . . . . . . . . 4.1.2 Green function of the Airy equation . . . . . . . . . 4.2 The Airy Transform . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Definitions and elementary properties . . . . . . . . . 4.2.2 Some examples . . . . . . . . . . . . . . . . . . . . . 4.2.3 Airy polynomials . . . . . . . . . . . . . . . . . . . . 4.2.4 Summary of Airy transform . . . . . . . . . . . . . . 4.2.5 Airy averaging . . . . . . . . . . . . . . . . . . . . . 4.3 Other Kinds of Transformations . . . . . . . . . . . . . . . . 4.3.1 Laplace transform of Airy functions . . . . . . . . . . 4.3.2 Mellin transform of Airy function . . . . . . . . . . . 4.3.3 Fourier transform of Airy functions . . . . . . . . . . 4.4 Expansion into Fourier-Airy Series . . . . . . . . . . . . . . 5. The Uniform Approximation

91

5.1 Oscillating Integrals . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The method of stationary phase . . . . . . . . . . . . 5.1.2 The uniform approximation of oscillating integral . . 5.1.3 The Airy uniform approximation . . . . . . . . . . . 5.2 Differential Equation of the Second Order . . . . . . . . . . 5.2.1 The JWKB method . . . . . . . . . . . . . . . . . . . 5.2.2 The generalisation of Langer . . . . . . . . . . . . . . 5.3 Inhomogeneous Differential Equations . . . . . . . . . . . .

7. Applications to Classical Physics

91 91 93 94 95 95 97 98 101

6. Generalisation of Airy Functions 6.1 Generalisation of the Airy Integral . . . . . . . . 6.2 Third Order Differential Equations . . . . . . . . 6.2.1 The linear third order differential equation 6.2.2 Asymptotic solutions . . . . . . . . . . . . 6.2.3 The comparison equation . . . . . . . . . 6.3 Differential Equation of the Fourth Order . . . .

71 71 73 74 74 77 82 84 85 85 85 86 87 88

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

101 105 105 106 107 111 115

7.1 Optics and Electromagnetism . . . . . . . . . . . . . . . . . 115 119 7.2 Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The Tricomi equation . . . . . . . . . . . . . . . . . . 119

Airy Functions a n d Applications t o Physics

X

7.2.2 The Orr-Sommerfeld equation . . . . . . . . . . . . . 7.3 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . 7.5 Nonlinear Physics . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Korteweg-de Vries equation . . . . . . . . . . . . . . 7.5.1.1 The linearised Korteweg-de Vries equation . 7.5.1.2 Similarity solutions . . . . . . . . . . . . . . 7.5.2 The second Painlev6 equation . . . . . . . . . . . . . 7.5.2.1 The Painled equations . . . . . . . . . . . . 7.5.2.2 An integral equation . . . . . . . . . . . . . 7.5.2.3 Rational solutions . . . . . . . . . . . . . . . 8. Applications to Quantum Physics 8.1 The Schrodinger Equation . . . . . . . . . . . . . . . . . . . 8.1.1 Particle in a uniform field . . . . . . . . . . . . . . . 8.1.2 The 1x1 potential . . . . . . . . . . . . . . . . . . . . 8.1.3 Uniform approximation of the Schrodinger equation . 8.1.3.1 The JWKB approximation . . . . . . . . . . 8.1.3.2 The Airy uniform approximation . . . . . . 8.1.3.3 Exact vs approximate wave functions . . . . 8.2 Evaluation of the Franck-Condon Factors . . . . . . . . . . 8.2.1 The Franck-Condon principle . . . . . . . . . . . . . 8.2.2 The JWKB approximation . . . . . . . . . . . . . . . 8.2.3 The uniform approximation . . . . . . . . . . . . . . 8.3 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 . . . . . . . 8.4 Airy Transform of the Schrodinger Equation . . . . . . . . . Appendix A

Numerical Computation of the Airy Functions

121 124 127 129 129 129 131 132 132 134 135 137 137 137 140 144 145 146 148 152 153 154 157 162 163 164 166 169 173 177

A.l The Homogeneous Functions . . . . . . . . . . . . . . . . . 177 A.2 The Inhomogeneous Functions . . . . . . . . . . . . . . . . 180 Bibliography

183

Index

193

Chapter 1

A Historical Introduction : Sir George Biddell Airy

George Biddell Airy was born July 27, 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). He was a brilliant student although being a sizar,’ and finally graduated in 1823 as a senior wrangler. Three years later, he was elected to occupy the celebrated Lucasian chair of mathernatics. Nevertheless, his salary as Lucasian professor was too small to marry Richarda, as her father said. 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 first works at this time were about the mass of Jupiter and also about the irregular motions of Earth and Venus. In 1834, Airy started his first mathematical studies on the diffraction phenomenon and optics. Due to the diffraction phenomenon, 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 has nothing to do with the purpose of the present book. In June 1835, Airy became the 7thRoyal Astronomer and director of the Greenwich observatory, succeeding John Pond. Under his administration, modern equipment was installed, leading the observatory to its worldwide fame assisted by the quality of its published data. Airy also introduced the study of sun spots and the magnetism of Earth, he built a new apparatus for the observation of the Moon, and also for cataloging the stars. The question of absolute time was also a broad challenge, Airy defined the “Airy Transit Circle”, that became in 1884: Greenwich Mean Time. But the ‘With the meaning that he paid a reduced fee but worked as a servant to richer students. 1

2

A i r y Functions and Applications t o Physics

renown of Airy is also due to the “Neptune affair”. During the decade 1830-40, astronomers were interested in the perturbations of Uranus that were discovered in 1781. In France, Franqois Arago suggested to Urbain Le Verrier finding a new planet that might cause the perturbations of Uranus. In England, the young John Adams was doing the same calculations with a slight advance, however Airy was doubtful on the issue of such a work. Adams tried twice to meet Airy in 1845 but was unsuccessful: the first time Airy was away, the second time Airy was taking dinner and did not like 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 September 20, 1846. A polemic started then between Airy and Arago, between France and England, and also against Airy himself. The polemic spread out with the name of the planet itself Airy wanting t o name the new planet Oceanus. The name of 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. The experiment stood in the comparison of gravity forces on a single pendulum a t the entrance of a pit and a t its ground. This experiment was carried out near South Shields in a mine of 1250 feet in depth. Taking into account the elliptical form and the rotation of the Earth, Airy found a density of 6.56, which is not so far - considering the epoch - from the usually admitted 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 eighty-nine years old and was unwilling to revise his calculations. Late in 1881, Sir George left his astronomer position at Greenwich for retirement. He died January 2, 1892. The autobiography of Sir George, edited by his son Wilfred, was published in 1896 (“Autobiography of G.B. AIRY”,W. Airy ed., 1896). The name of Airy is associated with many phenomena such as the Airy spiral (optical phenomenon visible in quartz crystals), the Airy spot in diffraction phenomena or the Airy stress function he introduced in his work on elasticity, different as well from the Airy functions that we shall discuss in this 2After he declined the offer on three occasions, arguing the fees.

A Historical Introduction

3

Fig. 1.1 Sir George Biddell Airy (after the Daily Graphic, January 6 , 1892).

book. Among of the most-known books he wrote, we may quote “Mathematical tracts o n physical astronomy” (1826) and ‘LPopularastronomy” (1849). 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 by 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

W ( m )= /cos

7r

[? (w3 - m u ) ] dw,

0

which is now called the Airy function. This is the object of the present book. W is the solution of the differential equation

The numerical calculation of Airy functions is somewhat tricky, even today!

4


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 years after, Stokes (1851, 1858) introduced the asymptotic series of W ( m ) ,of its derivative and of the zeros. Practically no reseach was endeavoured on Airy function until the work of Nicholson (1909), Brillouin (1916) and Kramers (1926) who contributed broadly to a better knowledge of this function. In 1928 Jeffreys introduced the notation used nowadays

which is the solution of the homogeneous differential equation, called the Airy’s equation y” = xy.

Clearly, this equation may be considered as an approximation of the differential equation of the second order y”

+ F(z)y = 0,

where F is any function of x. If F ( x ) is expanded in a neighbourhood of a point x = 20,we have to the first order (F’(x0) # 0) y”

+ [F(xo)+ (X - xo)F’(xo)] y = 0.

Then with a change of variable, we find the Airy’s equation. This method is particularly useful in a neighbourhood of a zero of F ( x ) . The point xo 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 of linear differential equations of the second order [Jeffreys (1942)], such as the stationary Schrodinger equation. Finally we can note that Airy functions are Bessel functions (or linear combinations of these functions) of order 1/3. The relation between both of the Airy’s equation and the Bessel equation is done with the change of variable [ = $x3/2,yielding 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”!

Chapter 2

Definitions and Properties

This chapter is devoted to general definitions and properties of Airy functions 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 The Homogeneous Airy Functions

2.1.1

The Airy’s equation

We consider the following homogeneous second order differential equation called the Airy’s 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

this is equivalent to solve the first order differential equation

v’

+

Z2V

= 0.

We thus obtain the solution to the equation (2.1), except a normalisation constant,

5

6

Airy Functions and Applications t o Physics

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 go into the regions of the complex plane z , where the real part of z3 is positive (shading regions of the complex plane).

F'rom symmetry considerations, it is useful to work with the paths Co, C1 and Cz. Clearly the integration paths C1 and Cz lead t o solutions that tend t o infinity when x goes t o infinity. When we consider the path Co and the associated solution, we can deform this curve until it joins the imaginary axis. Now we define the Airy function Ai by +im

-ica

If 1,j , j2 are the cubic roots of unity (that is to say j = eizTl3)the functions defined by the paths C1 and Cz are respectively the functions Ai(jx) and Ai(j2x). We have between these solutions, two by two linearly independents for they satisfy the same second order differential equation, the relation Ai(x)

+ jAi(jz) + j2Ai(j2x)= 0.

(2.3)

Now, in place of the functions Ai(jz) and Ai(j2x), we define the function S i ( x ) ,linearly independent of Ai(z), which has the interesting property to be real when x is real

B i ( x ) = ij2Ai(j2x)- ijAi (jx) .

(2.4)


7

Similarly t o Ai(z) (cf. formula (2.3)), we have the relation

~ i ( z+)jBi(jz) + j2Bi(j2z)= 0.

(2.5)

On Figs. 2.1 and 2.2, the plots of the functions Ai(z),Bi(z),and of their derivatives Ai’(z) and Bi’(z) are given.

-15

-10

-5

0

5

X

Fig. 2.1 Plot of the Airy function

Ai (full line) and its derivative (dotted line).

2 -

-1 5

-1 0

-5

0

5

X

Fig. 2.2

Plot of the Airy function

Bi (full line) and its derivative (dotted line).

a 2.1.2 2.1.2.1

A i r y Functions and Applications to Physics

Elementary properties Wronskians of homogeneous Airy functions

The Wronskian W { f , g} of two functions f (x) and g(z) is defined by ddx)

w{f,g} = f W , z For the Airy functions Ai and [Abramowitz & Stegun (1965)] 0

2.1.2.2

Bi, we have the following Wronskians

1 W{Ai(x), B ~ ( x )= }n

Particular values of Airy functions

The values a t the origin of homogeneous Airy functions are Bi(0) Ai(0) = -=

,h

Bi'(0)

-Ail(()) = --

,/3

= 0.355 028 053 887 817 239

(2.10)

(i)= 0.258 819 403 792 806 798

(2.11)

32/31? (;) 31/3r

and therefore

-1 Ai(O)Ai'(O)= -

2n ,he

(2.12)

More generally, we have for the higher derivatives [Crandal (1996)l

Ai(")(O)= (-l)ncn sin(n(n + 1)/3),

(2.13)

+ 1)/6)],

(2.14)

and

Bi(n)(0)= ~ [+ sin(n(4n l where the coefficient c, is


9

2.1.2.3 Relations between Airy functions The following relations are deduced from the formulae (2.3), (2.4) and (2.5) [Miller (1946); Abramowitz & Stegun (1965)]

2.1.3

Integral representations

An integral definition of Ai(x) was given by the formula (2.2). This function can also be defined by the following formulae [Abramowitz & Stegun (1965)] 00

11cos

Ai(x) = n

(g +

XZ)

dz

(2.19)

0

(2.21)

-m 00

e-c

Ai(x) = -/e-szz cos 7r

2 (f> dz, x > 0, 5 = -x3l2. 3

(2.23)

0

More generally, we have

(2.24)


10

We give also, the useful formula

Olver (1974) gives the expressions

5 / cosh (g im

Ai(x) = 17r

-

(2.26)

xz) dz,

0

and for x > 0 X1/2

Ai(-x)

= -T c o s 7r

(

[x3l2 z3

+ z 2 - ?)] 3 dz.

(2.27)

-1

Copson (1963) the expression x > 0,

2

E = -x3I2, 3

(2.28)

0

and Reid (1995) the following expression (x > 0)

(2.29)

For the function Bi,we have the integral representation (2.30)

Other formulae [Gordon (1969); Schulten et al. (1979)l having a great interest for the numerical computation of Airy functions are obtained by setting p(x) =

7r1/2

1 211/6 32/3 x2/3 e-"Ai

[(F)

'I3] , z > 0.

(2.31)

11


In fact, the Bessel function Kv(x) verifies the relation [Gradshetyn & Ryahik (1965)l

i

e-"Kv(u) du etKv(t) u + t J;I ="JtcoY(*v)

,~

1

( v 0 by the relations [Miller (1946); Abramowitz & ~ i ( -=~M) ( ~cos ) [qX)1 Bi(-x) = M ( x )sin [19(x)] A ~ ' ( - x= ) N ( x )cos [+(.)I Bi'(-x) = N ( x )sin [+(.)I ,

(2.87) (2.88)

(2.89) (2.90)

and the inverse relations

M ( x ) = [Ai2(-z)+ Bi2(-z)]1/2

(2.91) (2.92)

N ( x ) = [Ail2(-.) + BiI2(-x)]'I2 Bit(-x) +(x)= arctan [Air(-x) ' ~

2.2.5.2

]

(2.93) (2.94)

Differential equations

Moduli and phases are solutions to the following differential equations, for x > 0 [Miller (1946);Abramowitz & Stegun (1965)]

(2.95) (2.96) (2.97) (2.98)

M 0' tan(q5 - 0) = M'

1 sin($ - 0) = nMN

1 MI'+ X M - -= O ~ T ~ M ~ (M2)"'+4x(M2)'+2M2 =0

(2.99) (2.100)

(2.101) (2.102) (2.103)


23

+

This last expression may be alternatively written fY2 f (6, x} = x,where (0, x} is the Schwarzian derivative of 0 with respect t o x, as we shall see in $5.2. 2.2.5.3

Asymptotic expansions

For x >> 1,the asymptotic series for the moduli and phases of Airy functions are [Miller (1946); Abramowitz & Stegun (1965)] (2.104)

1

1.3.5

1.3.5.7.9.11

- 1.3.5.7.9.11.13.15.17 3!963x9

5

-

P ( 2 ) M x1'2

7T

k=O

(-l)"' 12"!

6 k d6k-1

23k

+...)

(i)3k

(2x)-3k

(2.105)

1.3 7 1.3.5.7.9 13 1!96x3 2!962 x6 1.3.5.7.9.11.13.15 19 i3!963 29

lr

--...)

and

q X x) -l4 -r -32x 3 / 2

5 1 I--4(2x)3

1 +--1105 96 (2x)6

--.. I

(2.106)

82825 1 1282031525 1 128 (2x)9 -I- 14336 (2x)12 37T - -23'2 2 $(x) M 4

3

[

7 13 1463 1 1+ - -4(2x) 96 ( 2 ~ ) ~

+--495271 640 where

(a)k

1

px)g

- . .I

(2.107)

206530429 1 , 2048 ( 2 ~ ) ~ ~

is the Pochhammer symbol defined in s2.1.4, formula (2.39).

24


2.2.5.4 Functions of positive arguments When the argument of the Airy functions is positive, they do not oscillate, but increase or decrease exponentially. It is then convenient to “exponentially” normalise these functions [Alexander & Manolopoulos (1987)] (2.108)

(2.109) (2.110)

(2.1 11) with 5 = $x3/2, x > 0. We can then, as above, define the moduli g ( x ) and m(z), and the hyperbolic phases x(x) and ~ ( x of ) Airy functions by ai(x) = Z ( X )cosh [ ~ ( z ) ]

(2.112)

bi(z) = X ( x ) sinh [ ~ ( x ) ] u~‘(x)= W ( 2 )cosh [ ~ ( x ) ] bi’(x) = T ( z )sinh [~(x)] ,

(2.113) (2.114) (2.115)

and the inverse relations

M ( z ) = [bi2(z)- a i 2 ( z ) ]lI2

(2.116)

x(x) = arctanh -

(2.117)

-

I;: :;[ I;:;:[

N(x) = [bP(z) - a P ( z ) ] lI2

(2.118)

~ ( x= ) arctanh - .

(2.119)

We can also calculate the asymptotic expansion of these functions. For x >> 1, we have 25 +-72x312 7

+...I

(2.120)

1

(2.121) (2.122)

1 q(x)=--ln3+ 2

...

(2.123)


2.3 2.3.1

25

The Inhomogeneous Airy Functions Definitions

In this section, we consider the inhomogeneous differential equation of the second order [Scorer (1950); Abramowitz & Stegun (1965)] yrr - xy = &+,

(2.124)

the resolution of which is done by a similar method to the homogeneous one. The solutions are the inhomogeneous functions G i ( x ) and H i ( x ) (also called Scorer functions), according t o the sign of the right member - or respectively. The integral representations of these functions are given by

+

1.1 (: 03

G i ( x )=

sin

7r

+ xt) dt

(2.125)

0

and: M

(2.126) We can alternatively define these functions from the primitives of the homogeneous functions [Scorer (1950); Olver (1974)]

/ /

/ /

+M

Gi(z)= Bi(z)

X

Ai(t)dt + Ai(x) Bi(t)dt 0

X

X

H i ( x ) = Bi(x)

(2.127)

X

Ai(t)dt - Ai(rc)

-m

Bi(t)dt.

(2.128)

-m

These functions are related to the homogeneous Airy function B i ( x ) by the important relation

+

Bi(x) = G i ( x ) H i ( x ) .

(2.129)

We have plotted on Figs. 2.5 and 2.6 the inhomogeneous Airy functions G i ( x ) ,H i ( x ) and their derivatives Gi‘(x) and Hi’(x). The general solutions of Eq. (2.124) are then y ( ~= ) aAi(x)

+ b B i ( ~+) G i ( x ) ,

(2.130)

26

A i r y F u n c t i o n s a n d Applications t o P h y s i c s

1

0

-1 -1 5

-1 0

-5

5

0

X

Fig. 2.5 Plot of the inhomogeneous Airy function Gi (full line) and its derivative Gi’ (dotted line).

-15

-10

-5

0

5

X

Fig. 2.6 Plot of the inhomogeneous Airy function H i (full line) and its derivative Hi’ (dotted line).

or

Y(X) = c A i ( z )

+ dBi(z)+ Hi(%),

(2.131)

according to the equation (2.129); a , b, c and d being integration constants.

27


2.3.2 2.3.2.1

Properties of inhomogeneous Airy functions Values at the origin

The values a t the origin of inhomogeneous Airy functions and of their derivatives are given by [Scorer (1950); Gordon (1970)]

Gi (0) = Hi(0)= Ai(0) -

1

(2.132)

,/3 37/6r(p)' 1 Gi' (0) = Hi' (0) = --Ai' (0) 2

&

L__

2

(3)

351~

(2.133) '

2.3.2.2 Other integral representations Besides the definitions (2.125) and (2.126) of Gi and Hi,these functions may be given in terms of the following integrals, for x > 0 [Gordon (1970)] (2.134)

(2.135)

where K I p ( t ) is the modified Bessel function and 5 = Zx3l2. We also have for Gi(x) the following integral representation [Lee (1980)] cos ($tz

7r

+ $) dt.

(2.136)

0

It should be noted that the function Hi($) is a particular case of the Faxen integral (1921),

/

00

Fi (a,p; y) =

etny-'tP-'dt,

0 5 %(a)< 1, !R(,B)

> 0,

0

that is to say (2.137)


28

Ascending and asymptotic series

2.3.3

2.3.3.1

Ascending series

We can write the integral representation of the function Hi(-z) (formula (2.126)) [Scorer (1950); Lee (1980)], as

/

oc)

1

Hi(x)= 7r

xk k!

-

k=O

tkee-t3/3dt.

0

We recognise, on this relation, the expression of the gamma function r ( z )= e-"uZ-ldu. From which we deduce the ascending series

sow

(2.138) This result may be alternatively written 1

+ C 4 d - z ) + $4,

Hi(-z) = C 3 f ( - z ) with:

c3

(2.139)

= Hi(O), c4 = Hi'(O), and the series

f(.)

=1

6 + 3!1 3 + --z1.4 + 1.4.7 + ... 6! 9! --5

-Z9

2 2.5 g(z) = z (1 +,-&x3 --z 7!

+

h ( - z ) = - z 2 ( -1+ - T 3 2 5!

+--z3.6

2.5.8 +zg + . . . 3.6.9

8!

Note that the series o f f and g are identical t o the series defined in 52.1.4.2. The ascending series of G i ( z )can be deduced from the ones of H i ( z ) and Bi(z) (formula (2.38)) thanks to the relation (2.129). We obtain

1 . 1 Gi(-z) = -Bz(z) - -h(z). 3 7r

(2.140)

For the derivatives, we have

(2.141) The ascending series of Gi'(z) is deduced (like for Gi(z)) thanks t o the ascending series of Si'(-z)and Hi'(z).


29

2.3 -3.2 As yrnptotic series The asymptotic expansions of the functions G i ( x ) and H i ( - x ) are, for

x >> 1 [Scorer (1950); Lee (1980)]: (2.142)

8! 3.629

1 2! 5! = -T (Xl + - +x3 - + - 3x6

Hi(-x)

M

1 T X

c co

n=O

+

(-l)n(3n)! 1 3"n! x3n .

1

2!

8!

5!

7TX

(2.143)

+ ...

From the two preceding formulae and from the relation (2.129), we can obtain the expansion of Gi(-x) and H i ( x ) . So we have

Gi(-x)

= B ~ ( - x) Hi(-x),

and:

Hi(%)= B ~ ( x-) G i ( x ) , the asymptotic expansions of B i ( x ) and B i ( - x ) being given by the formulae (2.46) and (2.50). Olver (1954) gives the asymptotic series under an equivalent form 1

[ c 1

=1+ 2 73-X TX

O0

(3s+2)! s! (3.3)"

s=o

s=o

1

, x-++oo

(2.144)

1

(2.145)

, x -+

-m.

2.3.4 Zeros of the Scorer functions In an interesting paper on the zeros of the Scorer functions, Gil, Segura and Temme (2003) gave several important results on the subject. Here, we limit ourselves to two: The Scorer function Hi has no real zero, but infinite many complex zeros on the half line phz = n / 3 , and at the complex conjugated values. The derivative Gi' has exactly one positive zero at g' = 0.609075417.. ., as it is seen on Fig. (2.5).


30

For the other values and properties of these zeros, we leave the reader to the paper of Gil et al. (2003).

2.4

2.4.1

Squares and Products of Airy Functions Digerential equation and integral representation

The homogeneous differential equation of the third order y"' - 4xy' - 2y = 0

(2.146)

has three linearly independent solutions [Aspnes (1966); Reid (1995)l: Ai2(x),Ai(x)Bi(x)and B i 2 ( z ) whose , Wronskian is 2 w { ~ i ~ ( ~ ) , ~ i ( ~ ) ~ i ( ~ ) ,=~ 7r3 i ~ .( ~ ) } The solution

(2.147)

Ai2(x)may be written as the integral

Changing the variables

1 s = -(v-u); t = v + u , 2

and integrating the variable s , we obtain an integral representation of Ai2(x),

P...('+tx+n)rn . 12 4 d 3

A i 2 ( x )= 27r3/20

(2.148)

Similarly, we have an integral representation of Ai(x)Bi(x) W

A i ( z ) B i ( ~=)2n3/2 / s i n ( c12+ t x + z )4 d t& 0

(2.149)


31

An interesting formula for Ai2(z)is given by

/ (g+ 00

1

Ai2(z)= - Jo 4 7 4

t z ) t dt.

(2.150)

0

We can generalise the preceding result in the case of the product of Airy functions with different arguments [Vallke et al. (1997)]:

Ai (u)Ai(u)

(2.151)

+-)

03

1

-

-. 41h

4t 0

This result allows us t o obtain +m

1 Ai2(z)= 21/37r .

Ai

b2/3(t2 + x)] dt

(2.152)

--M

--M

Ai(x)Ai(x*)= IAi(x)12

(2.154)

/

I).(%

1 Ai k2I3( t 2-+ $00

-

2113~

cosh [2t S ( 2 ) ]dt.

--M

The following relations can also be established [Aspnes (1966): cf. $3.5.21

Ai2(z)= -

(2.155)

0

/

M

1 Ai(x)Bi(x)= 22/3T Ai (2'l3x

-5 t)

(2.156)

0

1 A i ( x ) A i ' ( z )= -

(2.157)

2n

0


32

and [Aspnes (1967): cf. $3.5.21

1 M

Ai2(z) = -Gi (22/3x - t) 2 ~ / 3 ~

dt

(2.158)

0 co

(2.159)

Ai(z)Bi(z) = 0

Finally, Reid (1995) gives the relations Ai2(z) + Eli2(%)= -

dt

(2.160)

0

(2.161)

(2.162)

2.4.2

A remarkable identity

Similar to the relations in the complex plane (cf. Eqs. (2.3), (2.5)), Voros (1999) proved the noteworthy result

where

Do = - 2 (Ai2(z))‘, ~ D1 = -27r (Ai2(j2z))’, D2 = - 2 (Ai2(jz))’. ~ 2.4.3

T h e product A i ( x ) A i ( - x ) : Airy wavelets

As we shall see in $4.2, the Airy function allows the definition of a semigroup of transformation. Unfortunately the weak decreasing of the Airy function for the negative values of the variable (as x-1/4 in average) deters the use of this transformation as a numerical filter.

33


However, as we have seen in the preceding paragraph an integral representation of the product A i ( z ) A i ( - Z )corresponds to the Fourier transform

1

Ai(z)Ai(-z)= -

2413~

Ai [2-4/3u2]eixUdu.

(2.164)

This product and its Fourier transform are well localized with a fast decreasing, for we have

when u -+ fa. Then, this product and its Fourier transform form a couple of square integrable functions. The decreasing of the Fourier transform being faster than a Gaussian, we can consider this product as a continuous basis of wavelets [Holschneider (1995)l. As a matter of fact, when we take twice the derivative of Ai (z)Ai (-z) , we obtain

-m

The three first moments of this function cancel:

s

+-M

Ai’(z)Ai’(-z)zndZ = 0 ,

n = 0, 1,2,

(2.166)

--M

then its Fourier transform behaves parabolically near the origin. On Figs. 2.7 and 2.8, we plotted the wavelet Ai’(z)Ai’(-z) and its Fourier transform (for the positive values of the Fourier variable). The maximum value of the Fourier transform is given by the solution of the equation: Ai(zo) zoAi’(z0) = 0,i.e. zo= 0.88405, and uo = 1.4925 for the Fourier variable. This continuous basis of wavelets is very close to what is called the “Mexican hat”, that is to say the Maar wavelets, but with a specificity of a faster decreasing of the Fourier transform for the Airy wavelets.

+


34

0.05

0

-5

0

5

X

Fig. 2.7 Plot of the wavelets A i ’ ( z ) A i ’ ( - z ) .

2

0

4

X

Fig. 2.8 Fourier transform of the wavelets AZ’(z)AZ’(-z).

Finally, we give the normalisation of the mother wavelet (see

Eq. (2.164))

/ (y ) (5) +m

Ai2

Q -m

Ai2

dz = 1 12n’

(2.167)

whose demonstration will be given in $3.6.4 (see also in Reid (1995)).


35

Exercises 1. Prove, with an appropriate change of function on the Airy differential equation (Eq. (2.1)), that the logarithmic derivative of Ai(x)satisfies the Riccati equation u’ + u2 = x. Find a differential equation of the first order of which the solution is @.Conclude in relation with $2.2.2. 2. Find the solution of the differential equation du

u- = 2xu dx

+1

in terms of Airy functions (choose a convenient boundary condition). Hint: see Davis (1962) in connection with the period of the van der Pol oscillator.

3. Show that if

< = Q x3I2

and

where WA,,(.) and MA,,(.) are the Whittaker functions. Find similar relations for the derivatives of Airy functions. Find relations between Airy functions and the confluent hypergeometric functions.

4. Find a particular solution to the differentia1 equation y” - xy = A‘

4.).

Hint: Calculate the derivatives of Ai(x) up to the third order. 5. From the Wronskian relation

1

W { A ~ ( x )B, ~ ( x )=} -,

i?

prove that the Wronskian of squares and product of Airy functions is 2

2

W {Ai2(~),Ai(~)Bi(~), Bi (z)} = -,

x3

see Eq. (2.147). Hint: The Wronskian of a canonical differential equation of the third order is given in $6.2.1.


Chapter 3

Primitives and Integrals of Airy Functions

3.1

Primitives Containing One Airy Function

I n terms of Airy functions

3.1.1

From the formulae (2.127) and (2.128), we deduce the expressions of the primitives of Ai(z) and B i ( z ) [Abramowitz & Stegun (1965)]

/

1 Ai ( t ) d t = - + 7r [Ail(z) Gi (x)- Ai (x)Gi’(x)] 3

(3.1)

0

- _ _ - 7r [Ai’(x)H i

3

i

(x)- Ai (z) Hi’(z)]

1 3

AZ (-t)dt = -- - 7r [Ai’ (-z) Gi (-x) - Ai (-z) Gi’(-z)]

(3.2)

0

2

=3

/ Bi

+ 7r [Ai’(-z) H i (-z)

- Ai (-z)

Hi’ (-x)]

(t)dt = 7r IBi’ (x)Gi (z) - Bi (z)Gi’(x)]

(3.3)

0

= -7r

[Bi’(z)Hi (z) - Bi (z) Hi’().I

i

Bi (-t)dt = -T [Bi’(-Z) Gi (-Z)- Bi (-Z) Gi’(-x)]

0

=K

[Si‘(-z) Hi (-z) - Bi (-x) Hi’(-z)] . 37

(3.4)


38

3.1.2

Ascending series

The ascending series of the primitives of Airy functions are [Abramowitz & Stegun (1965)]

s

Ai (t)dt = c ~ (x) F - c ~ (x) G

(3.5)

0

X

1

+

Bi (t)dt = V6 [qF(x) c2G ( x ) ]

(3.7)

0 X

/ B i (-t)dt = --A[ c ~ (-x) F

+ c2G (-.)I,

(3.8)

0

the series F ( x ) and G ( x ) being defined by integration term-by-term of the series f and g (cf. 52.1.4)

('> 3 (3k + I)! .$k+l

oc)

F ( s )= C 3 k k=O

k

1 1.4 =z+-x4+-x 4! 7!

1 2!

2

=-x2+-x5+-x

5!

2.5

8!

+-1.4.7 lo! d o +. . .

2.5.8

+- ll! X I 1 + . . .

where the constants c1 and c2 are defined in 52.1.4.2 c1 = Ai(0) and c2 = Ai'(0).

3.1.3

Asymptotic series

For x >> 1 (and E = $ x 3 / ' ) , the first terms of the asymptotic series of the primitives of the homogeneous Airy functions are [Abramowitz & Stegun

Integrals of Airy Functions

39

(1965)] X

1 3 2 f ie-E x3/4

JAi(t)dt

M -

0

9 241 + - . . .) (l - 48s3/2 4608x3

(3.9)

+ . . .)sin (E - ):

(3.10)

41

~

X

2

5

7

1

0

23/2 1 - 460823 8761 ..

-

(I+

M

-(1 + -

-

.)

COS

(t -

31

X

J Bi(t)dt

eE fix314

41 48x312

+

9 241

) + . . .)

(3.11)

4608rC3 . * *

0

i

5

cos

(c - ):

(3.12)

0

+ (1+--1 x3i2

8761 + . . . ) s i n ( < - : ) ] . 4608x3

These series are obtained by integrating term-by-term the series defined in s2.1.4.

3.1.4 Primitive of Scorer functions Gordon (1970) also gives some primitives implying the inhomogeneous function Gi(x). The primitive Gi [ a ( x+ P)] dx seems unable t o be expressed simply in terms of Airy functions. Nevertheless, we can calculate 0

1 . J zGi [a(. + p)]dx = + 2Gz' [a(z+ P)] an-

-P 0

1

+ P)]dx x x = - (- - P) + -Gal on- 2

/

+

Gi [a(. P)] d z

x2Gi [a(.

(3.14) [a(x

a2

1 --Gi [a(. a3

(3.13)

+ P)] + P2

J

+ P)]

Gi [a(.

+ P)] dx


40

(T

~ / ~ ~ G i [ a ( s + p ) ] d xX= - - - p , X

+p2

(3.15)

ax

+ - pX +-p - Gi IC2

+

Gi’[a(% + p)]

p2

a2 2X

a3

[Q(X

+ (2a-3- p3) 3.1.5

+ P)1

J Gi

[Q(X

+ P)] dz.

Repeated primitives

For all the primitives given below, the integration constant has been omitted. If y is any linear combination of Airy function and y’ its derivative, we note y1 its primitive. Then we have ylda: = X O

(3.16)

Y ~ y’

S

1 zylda: = -(z2yl -ZIJ’+Y) 2 1 / z 2 y 1 d x = ?(x3yl - x y ’ + 2 ( z y - y l ) ) .

./

(3.17)

(3.18)

From which we find

SXs” 3.2

1 yidz’dd’ = -(z2yl - xy’ - y), 2

etc.

(3.19)

Product of Airy Functions

It is sometimes possible to easily calculate the primitive of a product of Airy function. For example, we can calculate

I

=

p

Ai2(x)dz,

from an integration by parts

I = [xAi2(z)]:

-2

zAi(z)Ai’(z)dz.


41

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

I = [xAi2(x)]:- 2 X

= [xAi2(x)]:- [Ai‘2(x)]:

Since lim xAi2(x)= 0 and lim Ai’(x) = 0, we finally obtain X+OO

1-00

I

= -xAi2(x)

+ Aif2(x).

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

3.2.1

The method of Albright

We want to calculate integrals of the kinds

J xny2dx, J x n y l y d x and J xny12dx,

(3.20)

where y is a linear combination of the functions Ai(x) and B i ( x ) ( i . e . y ( x )= aAi(x)+psi(.)), the sign prime ( I ) stands for the differentiation with respect t o x. Albright builds the following table, where D stands for the operator d/dx, and where y“ is replaced by x y (according to the Airy equation

(2.1)) Y2

DY DY‘Y

YlY 2

YI2

ZY2

1

1

DXY’Y Dryt2 Dx2YL Dx2y1y Dx2yt2

XYI2

X2Y2

1

1

X2Y1Y

X2Y’2

2 2

DyI2 DXY2

XY’Y

1 1

2 2

1 2 2

1

2

The properties of this table are such, that we are able t o calculate the

42

Airy Functions and Applications

t o Physics

primitives of the kinds (3.20). For example, t o calculate

we just have to subtract lines (4) and (5) of the table. So we obtain

D (xy2 - yI2)

= y2,

and then the result Eq. (3.24)

F

= xy2 - yI2,

except the integration constant. For the following primitive G

=

/

yi2dx,

three lines of the table have to be considered, these can be written into the matrix form

(

1 1 0

D xy“ yly)

=

0 2 0 2)

xzy2

It is then sufficient to inverse the system

(

=

:;2)

f

”)

(1; i2

2 -2 -1

X2Y‘Y

($)

(3.21)

x2y‘y

D

(:$)

(3.22)

x2y2

leading immediately to the result

+

xy” - xzy2) . 3 In the next section we set out some particular cases of primitives obtained by the method of Albright and with the help of the primitive

G = -1 (2y’y

L

=

/

xnyiydx,

calculated from an integration by parts

L

= xny2 -

J

nxn-1 y2 dx - L , (3.23)

43


3.2.2

Some primitives

If y is a linear combination of Ai(x) and Bi(x) and n a positive integer. y2dX = xy2 - yI2

1

xy2dx =

1 3

(yly

(3.24)

- xy/2

+ .zY2)

(3.25) (3.26) (3.27) (3.28)

J x2y'ydx = 61 (x2y2 -

1 yf2dX= - (2y'y 3

- 2y'y

+ 2xy")

+ X Y ' ~- x2y2)

(3.29) (3.30) (3.31)

x 2 y '2 dx = 1 (4x2y'y - 4yt2 7

1 Jxny2dx = -(nx"-'y'y 2n 1

+

- z:7Ly'2

+ ,n+ly2)

+ x3yI2- X4Y2)

(3.32)

- n(n - 1)/xn-2y'ydx

(3.33) (3.34)

or


44

+(n

+ 2) (xny’y

-n

1

zn-’y’ydr)]

If A(x) and B ( z ) are any two linear combinations of Ai(x) and B i ( z ) ,and n is a positive integer A(x)B(z)dz = xA(z)B(z) - A’(x)B’(x)

0

0

0

/ /

(3.36)

1

A’(z)B(x)dx = - [A(z)B(z) + xA’(x)B(z) 2 A’(x)B‘(z)dz

=

- xA(z)B’(x)] (3.37)

1 3

- [A’(x)B(x) + A(x)B’(x)+

(3.38)

xA‘ (z)B’(z) - z2A(z) B(x)] 1 xA(z)B(x)dx = - [A’(x)B(z) 6

+ A(x)B’(z) 2xA’(z)B’( x) + 2x2A(z)B (z)]

0

(3.39)

-

1 4

zA’(x)B(z)dx = - [2A’(z)B’(x)

0

0

/

(3.40)

-z2 A(z)B’ (x)]

15 [ I (zA‘(z)B(z) + zA(x)B’(x) (3.41) -A(x)B(x)) + z2A’(x)B’(x) - z3A(z)B(x)]

xA’(z)B’(x)da: =

znA(z)B(z)dz -

0

+ z2A’(z)B(z)

(3.42)

1

+ 1) [nxn-’ (A’(z)B(x) + A(x)B’(z)) -2znA’( x)B’(X) + 2xn” A(x)B( X) -n(n - 1) xn-’ (A’(z)B(x) + A(z)B’(z)) dx]

2(2n

/xnA(z)B(x)dx =

2

(3.43)

+

[zn-’ (A’(x)B(x) A(z)B’(z)) - 2

/

-(n - 1)

zn-’ (A’(x)B(x)

s

zn-’A’(z)B’(z)dz

1

+ A(z)B’(x))dz

, n >_ 2

45


0

/z"A'(z)B(x)dx

=

x"A(z)B(x) - n

J

xn-'A(x)B(x)dx

(3.44)

(3.45)

-n /xn-' (A'(z)B(x)

+ A(x)B'(x)) dx]}.

The formulae (3.36) to (3.45) are true for any A(x) and B ( x ) . In particular, if A(z) = Ai(2) and B ( x ) = Bi(rc),we can simplify some of these expressions by using the Wronskian relationship (formula (2.6)),

1 AiBi' - Ai'Bi = -. 7r

For example, the formula (3.37) becomes

Some of the preceding primitives can be calculated for a more general form [Gordon (1969, 1970, 1971)l. If A is a linear combination of Ai(x) and S i ( z ) ,we have the following primitives


46

/

x2A [a(z

/

x3A [a(x

+ P)] dz = (y2 - ’A’

+ P)] dx = x2 - Px a2

[a(x

+ P2

+ P)]

A’ [a(x

(3.48)

+ P)]

(3.49)

+ (2aP3 - P3) J A [a(x + P)] d z When A and B are any linear combinations of Ai(x) and Bi(x), we have the primitives


47

and for different arguments

0

/

z A [a(.

+

P1)]

B [a(.

+ 0211 dx

X

+ [ a2(PI - P 2 )

(3.54)

2

+ a5

(PI - P 2 ) 3

1

For completeness, we also give the integral’ (3.55) 3(P1

a3(P2 - P d 2

]

+

+

A[a(z PI)] B[a(z P,)]

with

Finally, we give some primitives involving the primitive y1 of any linear ‘It is not given in the work of Gordon.


48

combination of Airy function y, (3.56)

(3.57) (3.58) where y' is the derivative of y.

3.3

Other Primitives

Albright & Gavathas (1986) give other kinds of primitives of Airy functions. The expression

/

7r Bi2(x) ,dx = 2 Ai2(x) Biz(,)

+

[Ai::Yijfir)j

'

(3.59)

is obtained by differentiation and thanks to the Wronskian of Ai and Bi (formula (2.6)). More generally, we have

./

Ain-1 (x)Bi"-l (x) 7r Bin(z) ,dx=. n Az"(x) Bin(,) [Ain(x) Bi"(x)]

+

+

'

(3.60)

and from a similar method, we obtain dx

(3.61)

Ai(x) Bi(x) dx Bi(x) = 7rlnAi(x)Bi(x) Ai(x)

(3.62)

-7r-

J' J'

Bin(x)

A Z " + ~ ( X= )~~

7r

(3.63)

Bi(x)

(AL(1))

.

(3.64)

Albright & Gavathas (1986) build more general results by considering two functions: namely f and F such that f = F'. We have then

(3.65)

49


and (3.66) These last results are easily verified. The equations (3.59) to (3.64) are nothing but particular cases of this general result. There are also some interesting particular cases, such as

/

dx Bi(x) = lr tan-' Ai(x) ' Ai2(x) Bi2(x)

(3.67)

+

and

1

Bin(x) Ain+2(x)

3.4

Bi(x)

(Az(z))

nfl

lr exp dx = n+l

(3)"" .

(3.68)

Miscellaneous

Isolated values of primitives involving Airy functions, may be given here as well. For instance, we can quote -

Ail2 Ai2 - -.

(3.69)

X

There are also some results related to a solution of the heat equation

u ( x ,t ) = exp(2t3/3 - xt) Ai(t2 - x). This function has particular and interesting properties. In the case where x = 0, we have for example

J exp(2 t3/3) Ai(t2)dt = exp(2 t3/3) [tAi(t2)- Ai'(t2)], +

(3.71)

+ Ai'(t2)]dt = exp(2 t3/3) Ai'(t2).

(3.72)

1 2 t exp(2t3/3) [tAi(t2) Ai'(t2)] dt = exp(2 t3/3) A i ( t 2 ) ,

1

2t2 exp(2 t3/3) [tAi(t2)

(3.70)

We shall return to this solution in 57.4.

50


Elementary Integrals

3.5

3.5.I Particular integrals From the asymptotic expansion of 83.1.3, we may deduce the values of the defined integrals

P

fa

ca

J’

Ai(t) dt = - , 3

1 Ai(t) dt = - , then 3

0

-ca

J’ Ai(t) dt = 1

(3.73)

-ca

and

P

Bi(t) dt = 0,

T B i ( t ) dt

M CQ.

(3.74)

0

-ca

The Airy function Ai is not square integrable on R but

/

Ai2(t)dt =

1 32/3r2(

(3.75)

4)

‘

0

For the cube of the Airy function, we have [Reid (1997a)l

1 1 (t)dt = --J?(-), 4n2 3

(3.76)

-ca

and

J

Ai2(t)Bi(t)dt

=

1 3

-r2(-). 4n2&

-ca

(3.77)

Reid (1997b) gives integrals with fourth power, for example

Ai4(t),dt

=

1 -In 24n2

3.

(3.78)

0

The reader may find other integrals in the papers by Reid (1997a,b).


3.5.2

51

Integrals containing a single Airy function

3.5.2.1 Integrals involving algebraic functions According to the notation of Aspnes (1966), we set

/

00

Ail(x) =

Ai(t)dt,

X

which can be expressed in terms of inhomogeneous functions (formulae (3.1) and (3.73)),

ail(^) = 7r [ A i ( x ) G i ' ( ~-) Ai'(x)Gi(x)]. We have now, for n

>0

1 M

M

tnAi'(t

+ z)dt = --72

0

J /

tn-lAi(t

+ x)dt

tnAi(t

+ x)dt,

(3.79)

0

-

dx

0

and for n > -1

/

00

00

t"Aii(t 0

+ x)dt = n+l

t"+'Ai(t

+ x)dt

(3.80)

0

= T/tnAi(t

+ x)dtdx,

x o

with the particular case

Aii(t

+ z)dt = Ai'(x) + x A i i ( 2 ) .

(3.81)

0

These two kinds of integrals can be treated from the calculation of t"Ai(t z)dt. Now from the equation of Airy (formula (2.1)), we can

soM

+

52


deduce TtnAi(t

+ z)dt

(3.82)

0

d2Ai(t

+ z) dt

-

z

7

tn-lAi(t

+ z)dt

0

0

[& /

00

=

- X]

+ z)dt.

tn-lAi(t

0

In the two last equations, the exponent of the variable t is reduced of one unity until the iteration leads to 0 2 n > -1. It should be noted that for z = 0 the integral (3.82) can be explicitly written [Olver (1974)l

tnAi(t)dt =

I'(n

+ 1)

3(n+3)/3r

(F)

(3.83)

'

0

We can obtain the moments of the Airy function in calculating [Gislason (197311

1

+co

lim E+O

Ai(z)zne-E"2dz, n E N,

-co

that is to say (3k)! 3k k!

/ - A i ( z )z3'dz = -

A i ( z )z3"'dz -co

Ai(z) z3'+'d,

=

(3.84)

= 0.

(3.85)

-co

We can also explicitly write the integrals (3.79) to (3.82) for n = -1/2,


53

with the help of the definition (2.20) of Ai(z),

7

Ai(x + t)-

=-

0

0

We obtain, by comparison to Eq. (2.148) dt

Ai(x + t)-

lh

0

= 22/37rAi2

(3.86)

We retrieve here the formula (2.155). In a similar manner, we shall have dt Ai(x - t ) 0

fi

= 22/37rAi

(&)Bi(&).

(3.87)

Differentiating and integrating the equation (3.82), we obtain

Ai’(x 0

7

+ t)-dt = 27rAi 4

(6) (-) A?

X

(3.88)

dt Ail(x + t)-

0

(3.89)

4

We can also quote the integrals [Aspnes (1967)]

(3.91) that are obtained thanks to the causal relations between the Airy functions (cf. 54.1.1). It should be noted we can encounter the formula (3.86) under %ome of the following integrals may be found thanks to the relationship between the Fredholm equations f(z t ) = g(z) w f(z)= g’(z t)

+ $

-+ JF

+

2.


54

a different form [Berry (1977a)l 00

L

= /=dx Ai(x)

= T A i (u2 - y) du.

-Y

(3.92)

-cc

Thanks to the definition (2.20) of the Airy function Ai, we can express this function as

With the following change of variables

we obtain

-cc

in other words

(22/3) .

L

=22/3n~i2 Y

(3.93)

3.5.2.2 Integrals involving transcendental functions Considering first the integral

/

+W

M =

Ai (x2

+ a) eikxdx,

(3.94)

--oo

that can be written using again (2.20) for the

M

= -!L 27r

//

Ai function

ei[t3/3+t(z2fa)]eikzd~dt.

wz

The calculation of this integral is not so easy if one integrates first on the variable z. However, it becomes simpler if we employ the method described in 53.6.5, that is to say if we canonise the cubics t3/3 +x2t. When we make

55


the following change of variable

we immediately obtain the result (see formulae (2.151) to (2.154))

We can also, thanks to the integral representation (2.20) of Ai(z), obtain the relation [Widder (1979)] e" tAi (t) d t = eU3I3.

(3.96)

-m

We can also quote the following integral T e - t 2 / 4 u A i(t) dt = 2

6 e2u3/3Ai(21') .

(3.97)

In this last expression, we find, thanks to the asymptotic formula (2.44), the relation (3.73) Ai(t) dt = 1, for u -+ 00. Now we give some other integrals involving one Airy function and transcendentals without demonstration

s_',"

Ai(-zt)lntdt=L 9x [ $ ( : ) + + ( i ) - l n ( g ) ] ,

(3.98)

where ~ ( . stands ) for the logarithmic derivative of the gamma function. (3.99)

01me-t3/3A2

(-f)dt = 27r

(3.100)

(3.101)

56


3.5.3 Integrals of products of two A i ~ functions g In a similar manner as in 53.5.2, according to the method of Aspnes, we can establish the relation [Aspnes (1966)l T t n A i 2 (t

+ x) dt

(3.104)

0 -

72[Ad”- 2x1 7tn-’Ai2 2n+1

2dx2

(t + z) dt, n > 0.

0

We extend this relation to the one containing Ai’ for we have Ai(x)Ai’(x) 3&Ai2(x) and Ai’2(x) = !j - x] Ai2(x). In particular for n = -1/2 we find

[-&

=

00

(3.105) M

J

Ai(t

+ z)Ai’(t + z)-dt

4

0

= -2-4/3Ai

(2

2 2/3

1

(3.106)

00

AiI2(t + x)0

dt

4

We can also quote the important integrals

(3.107)


1

57

/ Ai [y] Ai +m

x+a

x + b dx

I 4 -m

(3.108)

and +m

1

laPI

/ [5]

dx

Gi

Ai

(3.109)

-m

=i 1

1

ifp=a

ZQ36-a Ip3 - a 1 3

I‘ I 3 Gi

[

if

&‘;+3]

P > a,

This last is calculated thanks to the integral representation (4.7) of Ai+iGi, p being the Cauchy principal value. 0 In order to calculate the following integral [Biennieck (1977)], which is a generalisation of formula (3.108)

/ [ +m

I,

=

x”Ai

x+a

Ai

[y] P > dz,

CY,

p > 0,

(3.110)

-m

the general method consists in taking for the Airy functions their integral representation (2.20), namely

Then we obtain


58

with +W

-W

and

//

f(z)g(y)h(,)(z

+ y)dzdy =

s

f(z)dn)(-z)dz,

its nth derivative. We finally

where 6 is the Dirac delta function and find the relation

-W

Another method starts from the equation (3.110) and from the Airy equation (2.1) leading to the recurrence relation

In+l = P3-

d2I ,

(3.113)

- bIn.

I0 being given by the relation (3.108), we can then deduce (3.114)

I1 =

ba3 - ap3 p3

- a3

(3.115)

I0 2a3p3

I0

+

I;

(3.116)

We also quote the following integral [Balazs & Zipfel (1973)], which appears, in particular, in the semiclassical calculation of the Wigner function (cf. 58.3) 0

F

=

/

+

Ai(z t)Ai(z- t)ei2Ptdt.

(3.117)


59

To calculate this integral, we use again the integral representation of Ai (2.20) to thus obtain

The integration on the variable t leads to a Dirac function, allowing us to derive the relation i(8p3/3+2pz)

F=

i[2u3/3+2Pu'+(2s+4p')u]du ,

2n -ca

and after the change of variable w3 = 2u3

-ca

The integral

(3.118) --w

is simply calculated by putting u = t - a , providing

-ca

in other terms (see formula (2.25)) (3.119) In our case, we obtain, after simplification (3.120)


60

More generally the same method yields for ,8 > a

(3.121)

and for

P =a (3.122)

[?

(u-b)2

a3X3

x exp{-i

-

4a3X

~

+-X ( a2+ b ) + -sgn(.X)]} 4 7r

Finally, we give the following integral

LmAi

3.6 3.6.1

(T)

Ai(t)&dt

=

-1 ~ i ' ( 2 ~ / ~ , /a:z > ) , 0. 22/3fi

~

(3.123)

Other Integrals Integrals involving the Volterra p-function

The Volterra p-function is relatively intriguing in the bestiary of special functions, for it has received only a few considerations, despite its remarkable properties. In this section, we would like to add some properties linked to Airy functions [VallBe (2002)l. In his work on integral equations with logarithmic kernel, Volterra introduced a function that can be generalised by [Erddyi et al. (198l)I

(3.124) We are going t o calculate the following integral in using the definition


of

61

the

(3.125)

Using the Mellin transform of the Airy function (see Eq. (3.83)):

We then find for the right member of Eg. (3.125)

3

r(p+i) r ( q + i )

*

Making the change t = 3 z , we obtain

from which we deduce the result

(3.126) This can also be written

Therefore we have found that p(x,p,O) is an eigenfunction of the above integral equation for the kernel

(3.128) This kernel is easily recognised to satisfy (as a similarity solution) the evolution equation

(3.129) which is the linearised Korteweg-de Vries equation (I-KdV) [Ablowitz (198l)l.


62

Volterra functions may be the eigenfunctions of many other integral equations involving the Airy function, by using the recurrence relations between Volterra functions

xp(x, P, a ) = p(z,P

+ 1,a + 1)+ ( a+ 1)p ( z , P, a + 1).

(3.130b)

From the Airy equation : A P - u Ai = 0, we can write

We then make an integration by parts:

I"

Ai"(u) p(yu, P, a ) du =

1

(3.132)

03

A i W P(YU, P, (.>I:

-

Y

A+)

P'(Y% P, a )du

The first term of the right member cancel, for u = 0 due to the p function and for u 4 00 thanks to the Airy function. Now we use the relationship (3.130a) and we find r03

Making a second integration by parts:

Thus using the result from Eq. (3.126), we obtain (3.134) There are other ways to prove this result. The first one is to take the derivative of Eq. (3.126)

Srn

Ai(u) p(yu, P, a ) du = dY 0 which is equal to

leading to the same result.

I"

u Az(u) p(yu, ,B, a - 1)du,


63

Another method is to use the recurrence relation between Volterra pfunctions (3.130b). Eq. (3.134) may be written into the other form

Then p ( x , P, -1) is deduced to be the eigenfunction of an integral equation with the kernel

-Ai X (3t)4/3

(A) (3t)1/3

'

(3.136)

which is again a solution t o the Eq. (3.129). By the same way, we can find other interesting results like

and (3.138) where A i ~ ( u = )

sumAi(t) dt.

More generally, we have a formula for the nth derivative of the Airy function

y3 Ai(n)(u)p(yu, P, a )du = 3P(-7)n p(-, 3

a-n

P, -). 3

(3.139)

Each of the equations (3.137)-(3.139), may be put into a form, which satisfies an eigenfunction equation with a kernel that is a similarity solution to the I-KdV equation. Finally, we give a result for Ai2 from its Mellin transform [Reid (1995)l

which reads

This equation may alternatively be written as


64

It is then found that p(x,,f3, i) is an eigenfunction of this equation, with the kernel

I"[

( 1 21t ) V Ai2 (12t)1/3

(3.142)

which again satisfies the 1-KdV equation.

3.6.2

Canonisation of cubic form

In this section, we are going to derive a canonisation method of cubic forms as described by Turnbull [Turnbull (1960)l. With the help of the following linear transformation

x=ax+py Y =yx+sy with

A = 06

-

,f3r # 0.

We are looking for the transformation of the cubic form: F ( z ,y ) = Ax3 + 3 B x 2 y + 3 C x y 2 + Dy3

+

into its canonical form F ( X ,Y ) = X 3 Y 3 .For this purpose, we consider the covariants and the invariant of the cubics: 0

The Hessian, i.e. the determinant of the second derivatives

x = (AC - B ~2) + ( A D - B 0

C ) +~( B~D - c2) y2 ;

The cubic covariant is obtained by taking the Jacobian of the cubics and of the Hessian

+ 2 B 3 ) x3 + 3 (ABD - 2 A C 2 + B2C)x 2 y +3 ( 2 B 2 D - A C D - BC2)xy2 + ( 3 B C D - AD2 - 2 C 3 ) y 3

%? = ( A 2 D - 3 A B C

0

The invariant of the cubics is built by taking the Hessian of the Hessian, which is the discriminant of the cubics F ,

9 = ( A D - BC)2- 4 (AC - B 2 ) ( B D - C2) . There is a noteworthy relation (which is a so called syzygy) between the invariant and the covariants

F 2 9 = g2+ 4H3.


65

For similar quantities of the canonical cubics F ( X ,Y) = X 3

+ Y3, we have

3co = A2XY,

4R = A3 (X3- Y 31 ,

9 =A6.

By comparison between the cubics F and the cubic covariant %, identifying the terms x3 and y3 with the terms X3 and Y 3respectively, we finally obtain

, with the condition

9 = ( A D - BC)2- 4 (AC - B2)( B D - C2) = A6 # 0. It should be noted, in the case where 9 = 0, one cannot use the general canonical form F ( X , Y ) = X 3 Y3.In this case, the canonical form of F ( z ,y) is either X 3 or X2Y.

+

3.6.3

Integrals with three Airy functions

We consider the following integral (where a , c and e are different from zero) [Vallhe (1982)]

/

+m

L=

Ai(az + b)Ai(ca:+ d)Ai(ea: + f)dz.

(3.143)

--oo

Using once again the integral representation (2.20) of the Airy function

An integration on the variable x results in a Dirac delta function allowing us to express w as a function of u and u

L

=

1///ei(~3+~3+w3)/3,i(b~+du+f~)~(a?L + cz, + ew)dudvdw. 47r2 w3


66

We then have an integral of the kind

with

I

A = 1 - a3/e3, B = -a2c/e3,

c = -ac2/e3, D

= 1 - c3/e3,

V = b-af/e, W = d - cf/e.

The calculation of this integral is carried out in 53.6.5, formula (3.148), and the result reads

with '

K

= e6

+ a6 + c6 - 2a3c3 - 2a3e3 - 2c3e3,

p3 = f [D- -& ( A D 2 - 3 B C D + 2C3)], y3 = f

[A- & ( A 2 D - 3 A B C + 2 B 3 ) ) l ,

h3 = f ID +

& (AD2- 3 B C D + 2C3)].

We give, as a particular case, the integral with the product of three Airy functions [VallBe et al. (1997)]

A i ( ~ ) A i (y x)AZ(Z- X)dX

where E is the golden mean

E

=

q.

(3.144)


67

Integrals with four Airy functions

3.6.4

Let us consider the integral [Aspnes (1966)]

/

+oo

I=

(y) Ai2 ( b-x 7 dx )

Ai2 a + x

(3.145)

--m

with a reads

> 0 and p > 0.

With the help of the formula (2.155), this integral

x A i [ ;2213 j-(b-x)+v

1

.

By now using the formula (3.108), we have 03

I= 47r2 ( a 3

+ p 3 ) 113

/$[$, Ai [

+ + + Pv +p 3 y 3

22/3(a b) (a3

(YU

1

Using again the formula (2.155), this expression becomes

We find here the primitive of the Airy function A i (cf. formula (3.105))

dt 0

We finally obtain

J

+oo

(a> (7)

Ai2 a + x

Ai2

dx =

-W

(3.146)


68

3.6.5

Double integrals

Let us consider the following double integral [VallBe (1982)]

G=

//

+

Ai(aa: by + p ) A i ( c z x Ai(ea:

+dy +q)

(3.147)

+ f y + r)Ai(ga: + hy + w)dzdy,

where a , c, e , g are non-zero constants. We make use of the integral representation (2.20) of Ai(z),and we rearrange the terms, producing

From the integral representation of the Dirac delta function b ( z ) = 1 +m 2K eiZwdw,we have

s-,

x 6 (at

+ ct‘ + es + gs’) b (bt + dt’ + f s + hs’)

An integration over s et

s’,

then gives us

with

K = eh - f g # 0, A = ( e h - f g ) 3 (bg - ah)3 + ( af - be)3, D = (eh - f g ) 3 (dg - ~ h+ (cf ) ~- ~ i e ) ~ , B = (bg - ah)’(dg - ch) (af - be)’(cf - de), V = v(bg - ah) w(af - be) + p ( e h - f g ) , W = v(dg - ch) w(cf - d e ) + q ( e h - f g ) .

+ + + +

+


69

The next step is t o canonise the cubic form (cf. §3.6.2), leading to

G = - /K / d3X d Y 47r2A

exp

x3

Y3

(3.149)

w2

with

A6 = 9 = (AD - BC)' - 4 (AC - B2)(BD - C2) Finally the integral (3.147) may be written as

K3

G=-A ai (

VS-Wy A

)Ai(

).

W a - vp ~

(3.150)

Exercises 1. Generalise the table of Albright (see 53.2.1) to calculate the following primitives (n=1, 2)

1

x"y: dx,

1

xnylg dx,

1

xnyly'dx.

2. Calculate the following integrals in terms of Ai and Gi functions

Jdw

+ + c ) dt,

cos ( a t 3 bt

J d m s i n ( a t 3 + b t + c ) dt.

3. Calculate

0

4. Use the relations between Bessel and Airy functions (cf. 52.2.4) to express the following integral in terms of Airy functions

when v =

and v =

2,a > 0.

5. Check the formula (3.144), with the method described in $3.6.3.


Chapter 4

Transformations of Airy Functions

4.1 Causal Properties of Airy Functions 4.1.1

Causal relations

With the help of the integral representation of Ai(z), Gi(z),A i 2 ( z ) and Ai(z)Bi(z)respectively given by the formulae (2.20), (2.125), (2.148) and (2.149), we can write [Scorer (1950); Aspnes (1967)]

These functions being analytic in the complex plane, the real and imaginary parts may be written as a Hilbert transform. We then have

(4.3) -W

and conversely

--m

71


72

where p is the Cauchy principal value. From the relation (4.2), we can also deduce

Ai (x')Bi (x')

dx',

-m

and conversely

-m

With the help of the relation (4.1), we can write m

So that when a lim

a-0

la1

4

[Ai

0, this last relation becomes

($) + iGi (31=

M

~ e i u x d=u S(x)

1 1 + i-p-, 7 r x

0

that is to say

1 lim -AZ a-0 la1

. 3 :

(a ) = S(x),

and

It can be seen that these last two expressions are the beginning of the expansions given by Lee (1980) a3 3

-S(3)(X)

a6

- -S(6)(X) 18

+ ...

(4.10)

and

1

-Gi la1

(i)

(4.11)


4.1.2

73

Green function of the Airy equation

In this section, we are going to build the Green function satisfying the differential equation [Moyer (1973 ); Burnett & Belsley (1983)]

(g

- z) G ( x ,2’)= S(x - d).

(4.12)

The integral expression of this function given by Lukes & Somaratna (1969), is

G(z,x’)= -i

i

U(x,z’,t)dt,

(4.13)

0

with

We can write, thanks to the integral representation (2.20) of the function

Ai(z)

for 0 5 arg(z - z’) 5 n/2,and

for 0 5 arg(z‘ - z ) 5 n/2. Comparing Eqs. (4.13-4.15), we obtain

Ai (z‘ei2x/3)Ai (z) if

z

2 x‘

G ( ~ , = 2niei2=/3x Ai (z’)Ai (zei2x/3)if z 5 2’. Thanks t o the relation


74

we can then write G ( x ,x’) as a function of Ai and Bi Ai(z)Bi(z’)

+ iAi(z)Ai(z’) if z 2 5’

Ai(z’)Bi(x)

+ iAi(z’)Ai(z) if z 5 x’.

G ( x ,x’) = 2 ~ i i2?r/3 e x

4.2

The Airy Transform

4.2.1

Definitions and elementary properties

We define the family of functions (4.16)

On the one hand, one of the most important properties of this family is (formula (4.8)) lim (w,(x)} = 6(x),

(4.17)

0r-O

where 6(z)is the Dirac delta function and a E R. On the other hand, from the formula (3.108), we can set the relation giving the convolution product of two functions w,(x) and w p ( z )

+

with y3 = a3 p3. In other words, the family of functions defined by Eq. (4.16) is stable under the convolution product and then leads to a semigroup of convolution. Note also the functions w, are normalised (formula (3.73)) T A i ( x ) d x = T w , ( x ) d z = 1. --w

(4.19)

-co

Therefore if f is a function of x, and f its Fourier transform, we write p,, the Airy transform of f , as the convolution product


75

which is also written

-W

Consequently, f ++ ( P is ~ a particular class of functional transform that can be reversed by the formula

f(.)

= (Pa

* W-a(X)

= J1a [cpa(4].

(4.22)

In order to sum up the results, Eqs. (4.20) and (4.22) define the Airy transform, where the inverse satisfies A&, = &A_, = 3. As physical applications of this transform, we can quote a work by Hunt [Hunt (1981)] concerning molecular physics, and a work [Bertoncini et al. (1990)l using the Airy transform, as a tool in the calculation of the Green function of non-equilibrium high field quantum transport. As a matter of fact, the Airy transform was mathematically introduced by Widder [Widder (1979)] from another approach, i.e. by considering

{ &I .

the self adjoint Schrodinger operator: 2 The solutions of the eigenvalue equation are Ai(( - x). These functions form a continuous set of eigenfunctions, with the eigenvalues (. Indeed, from the relation (3.108), we have the orthogonality condition

j.

Ai(( - z)Ai((’ - z)dz = b(( - ( I ) .

-m

The definition of the Airy transform then follows

(4.23) -m

and the inverse transform

(4.24) Therefore, in the transformation defined by Eqs. (4.20) and (4.22), (Y appears as a scaling parameter, relatively to the basic transform defined by the above Eqs. (4.23) and (4.24), but in this case with (Y > 0.


76

From the definition of the Airy transform, some elementary properties can be derived:

(i) Translation If pa(.) is the transform of f(z),then p a ( z + + ) is the transform of f (x-t .I. (ii) Scaling If pa(.) is the transform of f ( z ) ,then pak(kz) is the transform of f ( k z ) .

(iii) Derivative If cp,(z) is the transform of f(z), then p’,(z) is the transform of f‘(z). (iv) Iteration If pa(.) is the transform of f(x) by the function w,(z), then the transform of pa(.) by the function w p ( z ) is py(z),where y3 = a3 p3. In other words

+

cpr

=

(f * w,)

* wp = f * ( W a * u p ) = f * WY.

(4.25)

(v) Convolution

If pa(.) is the transform of f(z)by the function w,(z) and

$ p ( z ) is the transform of g(x) by the function up(.), then the convolution product (pa * $p is the transform o f f * g by the function w y ( z ) , where y3 = a3 ,B3

+

(Pa

* $p

= (f * w,)

* (g * w p ) = (f * g) * (Ua * u p ) = (f * 9) * wy.

(4.26)

As the Airy functions { A i ( J- z), J E R} form a continuous basis of orthogonal functions, we have: (vi) The Plancherel-Parceval rule If pa(.) and $ a ( z )are the transforms of the real functions f(z)and g(z) by w,(z), then for all real a (4.27)


77

The proof of all these properties of the Airy transform can be shown rigourously, with the methods described in the paper of Widder. It will be the same for the examples that are presented in the next section. 4.2.2

Some examples

Let us start with the transform of a constant. From Eq. (4.19), we immediately deduce that the transform of f(x) = 1 is pa(.) = 1. Flom these, we can examine the periodic functions by putting

f(x) = eiEx. We then obtain, with the integral representation (2.20) of Ai and the Dirac delta representation

-03

Now from this equation, we can obtain the transform of periodic functions (4.28)

cos (6.)

A

-.scos

kx+ T) ,313

(4.29)

This result may be employed to transform a periodic function of which we have the Fourier expansion

f(x) =

C cnei.rrnx.

The Airy transform of this function is then

cpa(x)=

bn(a)eiknx,

with

b, = c,e

i(7rn~x)~

We now consider the case of a normalised Gaussian function (4.30)

78


Few algebra gives the result of the transform

(4.31)

-4

0

4

-4

0

1

0

4

I

....'.

I -4

Fig. 4.1 Airy transform of a Gaussian: the transform (full lines) given from the formula (4.31)is compared to the normalised Airy function we(.) (formula (4.16)),for a = 0.3, a: = 0.6, a: = 0.9.

Transformations of Airy F u n c t i o n s

79

On Fig. 4.1, the Airy function w,(x) (dotted lines) and the transform of the Gaussian function pa(.) (full lines) are plotted for different values of the parameter a. For the smaller values of a , the transform resembles a Gaussian, but it behaves rapidly, as an Airy function, when a increases, losing its Gaussian character even asymptotically. The last example concerns the step function O(x), its Airy transform is

/

+W

pa(.) = 1 la1

O(y)Ai

(?)

X-Y

-W

/

xla

dy

=

Ai(u)du.

(4.32)

-W

0

-4

4

X

Fig. 4.2 Airy transform of the step function B(z).

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

when

when x + -co. We can also give an important result allowing the transform of many functions to be calculated analytically.

80


Lemma 4.1 If the Airy transform of a function f is p a , then the Airy transform of xf is

Actually the derivation rule implies

but w, is a solution of the Airy equation wg have:

-

Sw, = 0. So we readily

which proves the result. As we have seen, the Airy transform of 1 is 1, the transform of x is x,' and the Airy transform of x2 is x2. This chain of transforms may be continued in order to form a family of polynomials: the Airy polynomials Pin(x).Some properties of these polynomials are given in the next section. Airy polynomials may be used to calculate the Airy transform of functions that can be written as a powerseries M

(4.34) n=O

The result in the transformation is a power-series M

(4.35) n=O

We can also use the preceding lemma 4.1 to calculate the Airy transform of a function f , knowing the transform of x f. We have just to solve an inhomogeneous differential equation

IT0 prove this result, and the one given by Eq. (4.35),we should have to introduce a converging factor.


81

For example, we know that the Airy transform of 1 is 1. We can therefore calculate the transform of f(z) = 1/x. So that = A, [x;] = 1), and: $J

/ (5),I:[

+m

p,(y) = p;1

iAi

dz = ; 7rGi

where is the Cauchy principal value and Gi is the inhomogeneous function (cf. 52.3.1) and a > 0. This last relation is nothing else but formula (4.4). Another result is useful to calculate Airy transforms, it consists of a generalisation of the Plancherel-Parceval rule Lemma 4.2 If the Airy transform of the function f(z)is p a ( z ) , then these two functions have the same autocorrelation function

/ f(Y)f(. +

+a

F(z)=

--w

Y)dY = T P f f ( U ) P &

+ u)du.

(4.36)

-m

The proof stems from the substitution of the value of p a into the right member of Eq. (4.36),

The formula (3.108) then leads to the announced result. If @(0

-W

Y Y2

(5) exp [ i (ky + a 3 k 3 / 3 ) )

a"Pin

&Ai

(z) +W

1- 2 J Ai(u)du Yl" +W

1 - J Ai(u)du Y/"

1 2

e-x2

Ai(x)

%["I

EG' 2? a

[&(Y + &)]

+XP

+ 1)-''3

(a3

+

Ai [ ( a 3

Ai

(: + &) y]

Gi ( x ) Ai2 ( x ) Ai ( x )Bi ( x )

(4a3

+ 1 ) - 1 / 6 A i 2[ ( 4 a 3 + 1)-'/3y]


4.2.5

85

Airy averaging

Some years ago Englert and Schwinger (1984) introduced a method they called the Airy averaging, in the context of an improvement of the ThomasFermi statistical model. The definition of the Airy averaging of a function fb)is

In their paper [Englert and Schwinger (1984)], these authors gave several properties of Airy averaging, such as (z)Ai

== 0,

( z 2 ) A i = 0,

(X3)Ai = 2

( X n f l ) A i = n(n - l)(Zn-')Ai

(zf(z))Ai =

d2

(Gf (z))Ai

Clearly, from the definition and properties, the Airy averaging is a particular case of an inverse Airy transform for we have

In particular, we have ( 2 3 n ) A i = (-l)nPzn(0),and all other properties of Airy averaging can be deduced from those of Airy transform.

4.3 4.3.1

O t h e r Kinds of Transformations Laplace transform of A i r y functions

In 1982 Davison & Glasser calculated the Laplace transform of the functions Ai(-z) and B i ( - z ) , z > 0, for their application in surface physics (Schrodinger equation in a uniform electric field). In 1983, Leach calculated the Laplace transform of A i ( f z ) and B i ( - z ) , 3: > 0, for their application to magnetohydrodynamics, from their integral representation. Exton (1985) and Wille (1986) calculated the Laplace transform of the Meijer function

; : ;1

G = GE;" ( z

;L):

. We shall not detail the definition of this func-

tion [see Gradsteyn & Ryzhik (1965)], but we are concerned by a particular case of G G;$ (z3l a , b ) = 2 ~ ~ ( " + ~ ) / ~(K2, -~b ~, /with ~ ) ( a ,b) = (',O), ($,0) or ($, K , being, the modified Bessel function (cf. 52.2.4). We

i),

Airy Functions and Applications to

86

Physics

therefore obtain the Laplace transforms of Airy functions 00

IAi-

e-pxAi(-x)dx

(4.47)

( p ) = J e-pxBi(-x)dx

(4.48)

(p)=

0

/

00

I A ~(+p ) =

(4.49)

e-PxAi(x)dx

0

with a

=

$. r ( x ) is the gamma function, y ( a , x ) is the incomplete gamma X

X

function y(a,x) = Se-"u"-'du,

and $(a,x)

=

Jeuua-'du.

We can see

0

0

that it is possible to express these transforms in terms of the confluent hypergeometric function F , thanks to the relations Xa

y ( a ,x) = -e-2 CY

4.3.2

xl--a

F (1,1+ a;x) and $(a,x) = -F (1- a,2 - a;x). 1-a

Mellin transform of Airy function

The Mellin transform of a function f(x),with the notation f*(s), is defined by the integral: f*(s)= Tf(x)x'-'dx. 0

We shall not detail the calculus of the following transforms, but rather exhibit the following Table 4.2 of Mellin transforms, mostly determined by Reid (1995).


87

Table 4.2 Mellin transform of Airy functions [Reid (1995)]. We have defined p = ;(I - 2s), y = ;(I s).

+

f

+W

J f(z)z"-'dz

f*(s)=

n

Table 4.3 Fourier transform of Airy functions

4.3.3

Fourier transform of Airy functions

The Fourier transform of a function f(z), with the notation f ( w ) , is defined by the integral

1

fm

f(w)=

--w

f(z)e-iwzdz.


88

In Table 4.3, we have given some Fourier transform of Airy functions.

4.4

Expansion into Fourier-Airy Series

We recall that the zeros of the Airy function Ai, { a n , n = 1 , 2 , . . . } are placed on the real negative axis of the complex plane. (cf. s2.2.1). Let US consider the integral

Inn/ =

i

Ai (x+ an) Ai (X + a,() dx.

(4.50)

0

In the case where n # n’ the formula (3.53) allows us to obtain

I,,,

=

Ai (an)Ai’ ( a n # )- Ai‘ (an)Ai (a,#) = 0, a, - a,,

(4.51)

whereas if n = n’, the formula (3.50) gives: oc,

Ai2 (x + a,) dx = Ail2 (a,).

(4.52)

0

In both cases, we have used the property of the Airy function to decrease exponentially toward infinity (see the asymptotic expansions (2.44) and (2.45)). Therefore the functions {Ai (x a,) /Ail (a,) , n E W} form an orthonormal basis on the interval [0, co[[Titchmarsh (1962)l. Then for any integrable function f (x),piecewise continuous, we can write the expansion

+

(4.53) The coefficients c, of this expansion are determined by the relation

/

1 c, = - f ( x ) A i(x a,) d z Ail (a,) 0

+

(4.54)


89

.-.. .:. .'...

0.12

0.08

0.04

1

2

3

4

5

1

2

3

4

5

0.12

0.08

'

0.04.

Fig. 4.3 Reconstitution of the function: s A i ( z ) (dotted lines) by a Fourier-Airy series (full lines) bound by 2, 4, 16 and 64 terms.

90


As a n example, we can consider the function f(z)= z A i ( z ) . The above integral Eq. (4.54) gives (see formula (3.54)) for the coefficient c, 2

Cn = - [Ai(O)- anAi’(0)].

ax Fig. 4.3 illustrates the reconstitution of f(z)for partial sums with 2, 4, 16 and 64 terms of the series.

Exercises 1. Calculate the Airy transform of eiX”Ai(z)Hint:See formula (3.121) 2. Calculate the Airy transform of the Airy wavelet Ai’(z)Ai’(-z) in the case cy = 1. Is it still a wavelet? Hint: Use the integral (3.144), and then the property of the derivative of Airy transform (see 34.2.1). 3. Show that the heat equation &u = BXxu,is invariant under an Airy transform. Solve the transformed equation with the initial condition iia(y,O) = S(y) i.e. the fundamental solution. Then take the inverse transform of this solution. What is the initial condition in the inverse transform u ( z ,O)? Hint: See 87.4. 4. Calculate the Airy transform of Hermite polynomials. Are they still orthogonal? 5 . Find the coefficients, in a Fourier-Airy expansion, of the function f(z) = z2Ai(z). Plot the reconstitution of f(z)for partial sums with 2, 4, 16 and

64 terms of the series.

Chapter 5

The Uniform Approximation

5.1 5.1.1

Oscillating Integrals The method of stationary phase

In this paragraph, we are going to present the stationary phase method, which was studied during the early XXth by Stokes, Kelvin and Brillouin. Erdklyi (1956) and later Copson (1967) detailed this method. A more recent review paper by Knoll & Schaeffer (1977), can also be quoted. Let us consider an integral of the form

I

= ]g(.z)eiAf(’)dz, a

where X is an arbitrary large parameter. If the function occurring in this integral is analytic, in a given domain of the complex plane, the integration contour can be deformed. The aim is to obtain an approximation of I in the limit X -+ 00, it is then advantageous to keep the contour in regions where the integrand is as small as possible. If a topography of the complex plane with IeiXf(’) as an altitude is introduced, it is then easier to speak, by analogy, of valley and top. Thus the most favourable contours will be those that remain as far as possible in the valleys, except for the transitions from one valley to another, ie. at points like zi such that: f’(-ti) = 0. This is the reason why this method is also called the steepest descent method. The points zi are called the stationary points, they are the points where the integrand is maximum giving the most important contribution to the value of the integral I . Let us consider a function f with only one stationary point 20 such that: f’(z0) = 0. We can expand g ( z ) to the zero order (other terms are neglected) and f (2) to

I

91


92

the second order d z ) = g(zo> f ( r )M f(z0)

+q q y r o ) .

The integral I may be written b

I

M

/g(Zo)e Rf(zO)ei~(z-;0)2 f”(ZO)dz. a

The integration can now be extended from -co to 00, because we take into account only the neighbourhood of the stationary point CW

--oo

When doing the change of variable

we obtain

with

s_’,” e-”’du

= fi.This yields finally to the expression

which can be generalised in the case where f ( z ) admits several stationary points zi

This last expression can also be written

The Uniform Approximation

5.1.2

93

The uniform approximation of oscillating integral

The limit when X

-+

00

of an integral of the kind [Knoll & Schaeffer (1977)]

I

=

s

g(z)e j X f (")dz

(5.1)

is given in the frame of the stationary phase approximation as

where the stationary points zi are defined by f'(za) = 0. But this approximation is no more valid if two stationary points z1 and z2 are coalescing, i.e. if (ZI) - f (z2)I is of the order 1/X. The uniform approximation gives, however uniformly valid solutions, even in a neighbourhood of coalescing stationary points. We make the change of variable

If

f ( Y ) = f (Z(Y)>

7

such that the integrand is transformed into a simpler form allowing an analytic evaluation of the integral. The stationary points zi become

and the amplitude

S(Y) = 9 (Y(Z))

d4Y) d

y

e

The integral Eq. (5.1) is now written in terms of the new variables

I

=

/

j(y)eiXJ(Y)dy,

which is tantamount to the integral (5.1), for it does not employ any a p proximation. The change of variables is chosen in such a manner that

So we recover far from the stationary points the approximation (5.2) given by the stationary phase method.


94

Now, we are going to study a particular case of the uniform treatment of the integral (5.1): the uniform approximation by Airy function.

5.1.3

The Airy uniform approximation

Numerous variations on this method can be found in scientific literature, for instance Child (1974) or Knoll & Schaeffer (1977). Let us choose for .f a cubic form

with the stationary points we obtain

y1,2=

*1

(6.8) (6.9) (6.10)

lNote that the functions Ci,Si and Ei (so defined) have nothing to do with the functions: cosine integral, sine integral and exponential integral.

103

Generalasation of Airy Functions

and CZ,(--LY)

=

2n sin

(a)p-l/n(2anl2)+ J1/,(2N"/')]

(6.11)

(2an/'> - J I / n (2a"/">]

(6.12)

&,(-a) =

2n sin (5)

Ei,(-a) =

n sin (I)

na1/2

[J-I/,,

(6.13)

For example, in the case n = 4, we obtain

where P (z, y ) is the integral of Pearcey [Pearcey (1946); Connor & Farrelly ( 1980)], given by

If n is odd, we obtain similarly, for a > 0, CZ(,CY)

=

(6.14)

2n sin (&)

-

Ci,(-a) =

2a1/2cos 72-

7ra1/2

(2)

2n sin (&)

(6.15)

K1/n ( 2 ~ " ' ~ )

[J_,/, (2anl2)

+ J1/, (2an/2)]

.

(6.16)

The function Cin(a)( n odd) verifies then, the differential equation

(6.17) In particular for n = 3, we find the Airy equation (2.1). The expressions for Si and Ei are more complicated. Sin(-a) can be written under the form ( a > 0)


104

amn ( m + - & ) r( m +

Sin(@)= -

;

;+ &)

(6.18)

(6.19)

Sin(a)( n odd) satisfies the differential equation -d2Sin

da2

n2an-2

n-3

Sin = - n a T

(6.20)

In particular for n = 3, we find again the Airy inhomogeneous differential equation (2.124). The function Ein(a) (n odd) may be written as the series

Ein(a)( n odd) satisfies the differential equation

(6.22)

Generalisation of Airy Functions

105

6.2 Third Order Differential Equations 6.2.1

The linear third order differential equation

In this section, some generalities about the linear third order differential equation are given (6.23) that can be written in the canonical form (Ince (1956)] .’I(+

f (.)z‘

+ g(5)z = 0 ,

(6.24)

with the help of the following change of function y ( z ) = ~ ( 5exp )

(-:

/z

P(t)dt) .

If L is the operator associated to the above equation L [z]= zlll+ f (.)Zl+ then the adjoint operator

g(x)z,

(6.25)

reads (6.26)

Now we have the following theorem Theorem 6.1 If zi (i = 1,2,3) are the three linearly independent solutions of the equation L[z] = 0, then the adjoint equation L[z] = 0 admits as solutions the minors of the Wronskian of the equation L[z] = 0: where … is the completely antisymmetric tensor. The Wronskian of the equation L[z] = 0 is defined by the determinant z1 .2

w(zI,z2,%3)= z{ .I1

z3

24 Z!J

= Ct.

.ll

1

2

3

The proof of this theorem is given by a simple substitution. More likely the result may be generalised to the canonical differential equation of any order. In particular, it is clearly found in the case of the second order differential equation (which is self-adjoint) where the monirs are written this time


106

6.2.2

Asymptotic solutions

Let us consider the differential equation [Langer (1955a,b)]

z"'

+ X 2 ( f ( x ) z '+ g(+)

= 0,

(6.27)

where X is a large parameter. We are looking for the solutions t o this equation when X + 00, and when f (x)has a transition point 2 0 , i.e. when f ( z o ) = 0. Moreover, we suppose that: f'(xo) # 0 and g(x0) # 0. We start with a change of the dependent and of the independent variables, as was done for the second order differential equation (cf. 55.2)

{u

= u(x)

2 =

+/$(u).

The equation (6.27) so becomes

where 4' is the derivative of 4 with respect to the variable u and u' the derivative of u with respect to the variable x. The quantity S(u) is proportional to the Schwarzian derivative {u,x}

Now we complete the choice of the change of variable, setting u = f ( x ) / d 2 , namely (6.29) L

xo

The equation (6.28) thus becomes

4"'

+

[PU

+ S ( U ) ] 4' + [ h z h ( ~+)

4 = 0,

(6.30)

with: h(u)= g ( X ( U ) ) / U ' ~ .However, we have not used the asymptotic limit X + 00, which allows the problem to be simplified. In particular, we may compare Eq. (6.30) to the following reference (comparison) equation, where p = constant,

a"' + X2U@'

+ P p a = 0.

(6.31)

Genenzlisation of Airy Functions

107

For this purpose, we carry out the asymptotic expansion $(u)= m(u)@(u)+ O($), which is introduced into Eq. (6.31), leading (up to terms (3($)) to the identification

um’(u)= ( p - h(u))a(.).

(6.32)

After an integration, we find (6.33) The convergence of the integral in the relation (6.33) is ensured by setting h(0) = p, allowing the asymptotic solution of Eq. (6.27) to be written as

where G(u) is the solution of Eq. (6.31) and u is given by the expression (6.29). Moreover, we have obtained a uniform solution at the transition point 20. Especially, in a neighbourhood of 20, f(z) “N f’ ( 2 0 )(x - Q), and according to the relation Eq. (6.29)

Hence u‘(x0) = f’(xo)+. Consequently, as h(u) = g (x(u))/d3, we obtain h(0) = g (zo)/f‘ (20)= PThese results lead naturally to the study of the comparison equation. 6.2.3

The comparison equation

The aim of this section is to give some analytic solutions to the equation [Langer (1955a,b)]

y”’- xy’- py = 0.

(6.35)

A first remark about this equation is that a scaling of the x variable leads to the comparison equation (6.31) in putting x = -X2/3Z.


108

A solution of this equation can be obtained with the Laplace method [Ince (1956)]. In writing

with the condition e-z3/3

(6.36)

we obtain the integral representation (6.37) where the integration paths Ci are given on the figure below.

We consider now some particular cases of the comparison equation (6.35) 0

The case where p = 0 leads t o the Airy equation (2.1): (y‘)‘’ - xy‘ = 0, producing the general solution (cf. 92.1) yo = a

0

1

Ai(x)dx

+b

1

Bi(z)dx

+ c,

(6.38)

where a, b, c are integration constants; The case where p = 1 corresponds to the inhomogeneous Airy equation (2.124) (cf. 92.3) y” - xy = C / T , giving the solution2 y1 = aAi(x)

+ bBi(2)+ CHZ(Z).

(6.39)

It should be noted that if y is a solution of y”’ - zy’ - py = 0, then y’ = z is a solution of z”’ - IZ’ - ( p 1)”= 0.

+


0

109

The case where p = 1 / 2 involves the square of Airy functions (cf. 52.4) y 1 p = aAi2 (2-2/3x)

+ bBi2 P b 2 I 3 x )

(6.40)

t c ~( 2i- 2 / s X ) ~i ( 2 - 2 1 3 ~ ) .

0

Note that this case is self-adjoint. The case where p = -1/2 involves also the square of Airy functions y - 1 1 ~= a [Ad‘2( T 2 1 3 x ) - 2 - 2 / 3 x A i 2

(6.41)

+b [Bi’2( 2 - 2 / 3 x ) - 2 - 2 / 3 x B i 2 ( 2 - 2 / 3 x ) ] - 2 - 2 P X ~ i2 - W X

(

+

>~i (2-2PX 11 .

Now we consider the more general case, where p is any relative integer. We set A(z) = a A i ( x ) bBi(x),A’(x) its derivative and A l ( x ) its primit i ~ e It . ~can be seen (according to the above results) that y o ( x ) = A1(x). So if we set zn = y-, for the parameter p = -n, n E N,we obtain the recurrence relation: zk = nz,-1. Thus we can verify that the first solutions of Eq. (6.35) are

+

~1 = X A l ( 2 ) - A’(x) 22 = x2A1(x)- A(x) - ~ A ’ ( x ) 23

(6.42)

= (x3- 2 ) A 1 ( x ) - xA(x)- x2A‘(x).

We can then generate the solutions of Eq. (6.35), thanks to the recurrence relation

allowing these solutions to be written explicitly as Zn(X)

= Pin(Z)Al(x)- QZ,-~(X)A(X) - Ri,-l(x)A’(z),

where the polynomials Pi,,

(6.44)

Qi, and Ri, obey the recurrence relations

Pi, = xPi,-l Qi, = xQi,-l-

- ( n - l ) ( n - 2)Pin-3

+

n(n l)Qin--d Ri, = ~Rin-1- n(n - 1)Rin-3.

(6.45) (6.46) (6.47)

The values of these polynomials for n = 0, 1 , 2 , are respectively 1 , x , x 2 . 3This primitive includes the additive integration constant c 4The properties of the polynomial Pi, were already given in 34.2.3.

110

A i r y Functions and Applacations to Physics

It is interesting to note that if the integration constants a and b cancel, the solutions are only given by the polynomial z, = Pi,(x), which is nothing else but the Airy polynomial (cf. $4.2.3). These solutions also allow us t o obtain the ones corresponding t o the adjoint equation, i.e. for p = n+ 1, n E N, y”‘ - xy’ - ( n

+ 1)y = 0.

(6.48)

This equation has the following noteworthy recurrence properties

According to the theorem given in 56.2.1, when we have the solutions for p = -n, we consequently obtain the solutions of Eq. (6.48). For example, in the case where p = 0, the solution Yo(x) = a

/

Ai(x)dz

+b

/

Bi(z)dx + C ,

allows us t o find the solution in the case where p = 1

with a = ca‘-c’a,

P = c’b-cb’, y = ab’-a’b,

/

thanks t o the relation (2.128)

X

H i ( x ) = Bi(x)

--M

/ X

Ai(t)dt - Ai(x)

Bi(t)dt.

--M

Another example is found from the first recurrence relation (6.49a), it allows the nth order derivative of an Airy function to be calculated. As a matter of fact, the first derivatives are Ail, Ai” = zAi, Ai”’ = Ai + xAi’, and for n 2 3 one has PinQin-s) Ai(x)

(6.50)

With the same method, we find the derivatives of the Scorer function H i ( x )

Generalisation of

Airy Functions

111

with the integral representation (for n 2 3) M

(6.51)

6.3

Differential Equation of the Fourth Order

Langer has used the comparison equation (6.35), as a reference equation, for the study of the asymptotic solutions to the Orr-Sommerfeld equation (a fourth order differential equation), which describes the hydrodynamic instabilities of a Poiseuille flow. In this case the parameter A2 corresponds to the Reynolds number of the fluid. A somewhat more elaborated asymptotic method [Drazin & Reid (198l)l involves, as a comparison equation, a fourth order differential with two parameters y"" - xy" - ay' - py = 0.

(6.52)

In the case where p = 0, we find again Eq. (6.35). In the general case, a method of Laplace yields to solutions of (6.52) with the following integral representations Yk(a,P;x) =

J e-Z3/3+XZ-p/ZZCY-2dz,

(6.53)

ck

where the contours Ck are conveniently chosen (cf. 56.2.3). These solutions, which constitute generalisations of Airy functions, were studied in detail by Rabenstein (1958). We refer the reader to this article for further information. However, we notice that the adjoint equation of (6.52) may be written zl/ll

d2 dx2

+ az'

- pz = 0,

(6.54)

+ ( a - 2)z' - p z = 0.

(6.55)

- -(xz)

or zl'l' - xzll


112

We see consequently the noteworthy result, for a! = 1, Eq. (6.52) is selfx) adjoint. This result (6.55) allows us to generate the solutions of y(2-a, 0; as soon as we know the solutions y(a, /3; x) with Q > 0, as previously for Eq. (6.35). The equation (6.35), as well as Eq. (6.52), deserve a remark concerning their Airy transform (see 54.2). In fact, this transform allows these equations to be rewritten in a particularly simpler form. We start with the third order equation (6.35). Taking into account the properties of the Airy transform (§4.2), we have

z[xf] = xp + p”.

(6.56)

Hence, in this transform, Eq. (6.35) becomes a first order differential equation

d v + p p = 0. dx

(6.57)

2-

A solution of equation (6.35) is then +W

y

dx’

= A [p] = J’ Ai(x -XI)-.

(6.58)

X‘P

-W

We can easily recognise this is the solution Pi,(x) of Eq. (6.35) in the case p = -72, namely the Airy polynomials. This method also works for the equation (6.52). The formula (6.56) gives

xcp“+ ap‘

+ pp = 0,

which is a Bessel equation of which solutions are for /3

(6.59)

>0 (6.60)

where is a Bessel function which is chosen to ensure the convergence (and the continuity at x = 0) of the integral in the inverse Airy transform, so a solution of (6.52) reads

/

f m

y=

Ai(x - x’)2’(1--a)’2Zl-a

(i6) dx‘.

(6.61)

-W

For instance, in the selfadjoint case a! = 1, one has Jo for x’ > 0 and XI < 0.

10

for


113

In order t o close this section on the generalisation of Airy functions, we quote the functions defined by the integrals [see Drazin & Reid (1981)]

(6.62) They have noteworthy recurrence properties that were used recently in search of analytic solutions t o non-linear equations [Laurenzi (1993)].

Exercises 1. Find three linearly independent solutions to the equations yJJ’- zy’

3 = 0. 2

f -y

Hints: See footnote p. 108 and examples in 36.2.3, Eq. (6.41). 2. Find a third order differential equation for the following functions W

Ai’(z

y(z) = 0

s

+ t)-dt

4’

m

Z(X)

=

dt Ail(z+t)-

&.

0

3. Find a differential equation of the fourth order of which a solution is

/ ($ + W

y(z) =

cos

zt - p2 t

1) 5,

+-

0

Then, using a scaling, compare the result with Eq. (6.55). Hint: Express y(z) in term of Airy functions.


Chapter 7

Applications to Classical Physics

7.1

Optics and Electromagnetism

Airy functions were introduced by G.B. Airy in 1838 in his article about the calculation of the light intensity in the neighbourhood of a caustic. We shall establish here the expression of this intensity (not in the way followed by Airy), but by the more ‘Lmodern”approach of Landau & Lifchitz (1964). Let us consider a monochromatic source and an aperture in an opaque screen. According to the laws of geometrical optics, beyond this screen, space is shared in two zones: the dark zone presenting a clear border with the enlightened zone. However the phenomenon of diffraction, as intense as the wavelength of the source, is large compared to the dimension of the opening, complicates the distribution of the light intensity in the neighbourhood of this border. According to the Huygens principle, we consider that each element of the surface d S of the opening is the source of a spherical wave. That is to say, u being the amplitude of the field on dS and lc the wave number of the source of light, the electromagnetic field in a point P located at a distance R from the opening, is proportional to the sum of these spherical waves up

c(

ppn,

(7.1)

where dS, is the projection of d S on the normal plane, corresponding to the direction of the ray resulting from the source, and arriving on the surface of the opening (cf. Fig. 7.1). If moreover we consider that u is constant on the surface of the opening, the field in P is up

0:

1 115

116


Fig. 7.1 Diffraction by an opening in an opaque screen.

Let us consider now the particular case where the point P is located in the neighbourhood of a caustic, which is the line separating the shaded zone from the lit zone, in the ideal case of geometrical optics. Let us note that the caustic is also defined, in geometrical optics, as being the surface where the light intensity becomes infinite. We consider, at first, only one section V? in the plane (x,y) of the surface defined previously. The caustic V?’ is then the envelope of %?. We look to establish the light intensity in the neighbourhood of the point 0, which is the contact point between the ray M’O and the caustic. D indicates the length of the segment [M’O].P is the point located in the neighbourhood of 0 where we shall calculate the intensity of the electromagnetic field: this point is located by its ordinate y (cf. figure 7.2). We note C the curvature center of the caustic and p the radius of curvature. The variable R in the expression (7.2) is then the distance from a point M of the wave surface V? at the point P. The geometrical properties of the envelope make it possible to establish the equality of the angles ( M ,H , MI) and (O,C,O’), which is noted a. If we suppose that we are located at a point P far from the wave surface, i.e. a is small enough to be expanded up to the third order: sina M B - B3/6, and that the points 0 and 0’ are

Applications t o Classical Physics

117

sufficiently close to consider that the radius of curvature p is constant on the arc 00', we can establish the relations

R = M P M M O - ysina M O z M O ' + psina MO' M M'O' - ap. Thus we get

R M D - Y ~ - -P c,Y3 6

(7.3)

and the formula (7.2)is reduced to up

0;

1

,ikR

-7i-da.

M /

Fig. 7.2 Calculation of the light intensity at the point P in the neighbourhood of the caustic V .

The factor 1/R varies slowly according to a, so we will neglect this variation in front of the exponential M $ (this is the principle of the stationary phase method, cf. 55.1). We obtain finally the expression of the amplitude at the point P

k

Let us note that we find here the expression given by Hochstadt (1973), which carries out the same type of calculation by considering plane waves


118

reflected by a concave surface. By carrying out the change of variable

t=

(F)

a, this expression becomes

We recognise the integral definition (2.20) of the Airy function Ai. The amplitude can thus be written

By reintroducing a proportionality factor, the light intensity in the vicinity of a caustic becomes I p = A Ai2

[($)

‘I3 y]

(7.4)

It is noticeable, in particular, that the intensity is not maximum on the caustic, but in the lit zone, at a position determined by the first maximum of

(

the function Ai: %)‘I3 y 2~ -1.02. For “large” values of y, the expression (7.4) becomes (cf. s2.1.4.3) ( 2 k 2 / p ) 1/2 y 3/2

1

.

(7.5)

In the shaded zone, the intensity decreases exponentially. For the negative values of y, the formula (7.4) gives the asymptotic expression

In the lit zone, the intensity oscillates quickly according to y. Its average value is

i.e. we find here the intensity given by the geometrical optics.

Applications t o Classical Physics

7.2

119

Fluid Mechanics

7.2.1

The Tricomi equation

The stationary, two-dimensional flow (in a (x,y) plane) of a compressible gas obeys the Tchaplyguine equation [Landau & Lifchitz (1971)]

d2@

d82

+

6%

v2

1-V2/C2d02

d@ +v--0

dv

l

(7.7)

where v is the velocity of gas, 0 the angle between v‘ and the x axis, c the speed of sound in gas. @ is a function of speed defined by

vd,.

The relation (7.8) defines, in where d, is the velocity potential v’ = fact, a Legendre transformation which brings back the non-linear equation of the motion to Eq. (7.7), which is linear. The price to be paid for this linearisation, is that the boundary conditions become non-linear [Landau & Lifchitz (1971)]. At the transonic limit, the velocity of gas approaching the speed of sound, the third term of the equation (7.7) becomes negligible in front of the second term. Moreover, without going into the details of this system, we can introduce a new variable q, known as “variable of Tchaplyguine” [Hayasi (1971)], which depends on the velocity v (as well as the critical speed of the sound and other parameters), such that the equation (7.7) becomes (7.9) This partial derivative equation is called the Tricomi equation, or EulerTricomi. Gramtcheff pursues a mathematical study of this equation and its relationship to the Airy equation [Gramtcheff (19Sl)l. Let us note first of all, that for 77 > 0, Eq. (7.9) is a hyperbolic equation, and for q < 0, an elliptic one. Then, let us consider the Fourier transform & ( q , w ) of the function @(Vl@)

6(q,w ) =

1

ePiwe@(q, 0)dQ.

120


By applying this transformation to Eq. (7.9), we obtain

By the change of variable

E = qw2/3, we find then the Airy equation (7.10)

The solution of this equation being of the type 6 = Ai(-E) = Ai(-77~'/~), the solution of the Tricomi equation, is determined by the inverse Fourier transform @(q,0) = / e i w B A i ( - q ~ 2 / 3 ) f ( ~ ) d w ,

where f(w) is the Fourier transform of the initial angular profile.' In fact, the general integral of the Tricomi equation can be written (7.11)

where g is an arbitrary function such that g' takes the same values at the ends of the integration path C. Assuming the Fourier transform of g, the relation (7.11) becomes

c -00 By gathering the terms in z , this last relation can also be written Cm

(7.12) -m

C

Under the condition of choosing conveniently the integration path C, we find here the integral expression of the Airy function Ai. lNotice that the separation of the variables method leads to the same kind of solutions.


7.2.2

121

The O w - S o m m e r f e l d equation

Plane flow of an incompressible viscous fluid The Navier equation for an incompressible viscous fluid comes from the general equations of the motion of a fluid

where the density p = po is a constant. We will also suppose that the external force comes from a potential. By introducing the vorticity

7

1W'=-VAU', 2 and by noticing that

iii.Gii= 2;

A

u'+

1+ -v (u'I2, 2

we can transform the Navier equation (7.2.2) into the vorticity equation (7.13) In the particular case of the plane flow w' = w 6 , the velocity field can be obtained from the current function +(z, y, t ) by the relations

For the vorticity component, we obtain (7.14) 1

= -2 =

0211, (-a24 ax2 + @)

1 2

--A+.

Taking the curl of the first member of Eq. (7.13) yields to (7.15)

A i r y F u n c t i o n s a n d Applications to P h y s i c s

122

Consequently, taking into account Eqs. (7.14) and (7.15), the vorticity equation is reduced to the quasilinear, fourth order equation for the current function (7.16) In this equation, we scale the variables, so that the Reynolds number Re appears instead of the viscosity. The equation (7.16), in the extreme cases Re = 0 and Re = 03, leads to remarkable equations of the fluid. For instance, the case Re = 0 produces the biharmonic equation

A21C,= 0 , which presents the disadvantage (in the nonstationary case) to lead to an ill-posed problem. As for the cases Re = 00, which gives the Euler equation in the stationary case

d$aA+ ay ax

d$aA$ =0, ax ay

yielding to the resolution of

A$ = F(+), where F is an arbitrary function of the current $. The case Re >> 1 leads to the Prandtl equation for the boundary layer, but also to the study of the stability of a plane flow and to the Orr-Sommerfeld equation.

Stability of an almost parallel flow In this paragraph, we concern oursleves with the perturbation of the velocity field along the x-axis, parallel to the flow. We suppose thus that we have upstream

i

u, = V(y) = u, = 0.

%# 0

The velocity field V(y) should not be unspecified for being strictly parallel. In particular, the pressure scalar field should depend only on z, and the pressure gradient should be constant, because of the equation of motion,

2

v- d2U dy2

d~ - constant. pdx


123

Thus U(y) should be a quadratic function of y. Then two particular cases emerge 0 0

the Couette flow where V(y) = y; the Poiseuille flow where U(y) = 1 - y2,

that we wrote by standardising the variables conveniently. Hence, we seek the perturbed solution $l(tl 2 , y ) by writing that

is the solution of Eq. (7.16), where rl is a “small” parameter. We find thus 41, by keeping only the dominant terms according to q, satisfying the linear equation (7.17) Because we are seeking a perturbation solution of the speed distribution U(y), it is natural to look for a solution $1 in the form of a wave that propagates along to the x axis with a speed c

By introducing the relation (7.18) into the perturbed equation, we obtain (7.19)

which is nothing else but the Orr-Sommerfeld equation. It is necessary to add to this equation the boundary conditions 4(y) = #(y) = 0 when y = yl or y = y2. Thus to obtain a non-trivial solution to this problem, it is necessary t o find a relation between the various parameters S ( a , c; Re) = 0,

this relation defining the eigenvalues of the problem. The reader will find a detailed analysis of the Orr-Sommerfeld equation in the work by Drazin & Reid (1981). Among the cases where the Airy functions are involved as solutions of the Orr-Sommerfeld equation (7.19), we shall see only two of them.

124


At first, let us consider the case of a Couette flow. We have V(y) = y, and thus V1‘(y) = 0. The equation is simplified by writing (7.20)

(“-cY2)4=$. dz2

So we obtain &2

(2 - 2) .1c, dz2

Zlc) = 0,

(7.21)

where we note z = y - c and E = (ic~Re)-”~.The boundary conditions are 4’ = = 0 at z = f l - c. The solution of Eq. (7.21) is expressed in terms of Airy functions and the solution of Eq. (7.20) can then be obtained by the use of the method of variation of constants [Drazin & Reid (198l)l. For the second case, we consider the case of large Reynolds numbers, or more precisely, we look for asymptotic solutions according to the parameter

2

&3 =

1 iaRe

-O ca if r 5 0,

is similar to the previous one. Nevertheless, in this potential well, the states are bounded and the energy spectrum is discrete: the energy En corresponds to the level n. The Schrodinger equation of the nth state of the particle wave function is

Performing the change of variable

we get the Airy equation (cf. (8.2))

The solution is (excluding the Bi term)

The energy levels are determined by the condition $ J ~ ( O )= 0

Then, the En are determined by the zeros an of the Airy function (cf.

140


32.2.1)l

i. e.

EO N 2.33811 ( h 2 / 2 m )1/3 El

N

4.08795 ( h 2 / 2 m )1/3

E2

N

5.52056 ( h 2 / 2 m )‘I3

E3 21 6.78671 (h2/2m)1/3

The normalisation factor N is determined by the orthonormalisation condition of the wave functions

i” 0

with the wave function

$n(r)$k(r)dr = 6 (En - Ern), $n

which is written

Then we get (cf. 34.4) N = (@)‘I6 finally given by

&.

The solution of Eq. (8.4) is

We can see on Fig. 8.1 the potential V ( r )= T and the first energy levels, with m = 1/2 and ti = 1. 8.1.2

The 1x1 potential

Let us consider the one-dimensional system with a particle in the potential V(z) = 1z1, z E 1-m,+m[. The bound states are determined by solving ‘We note: E, 0: -a,+l, and not: E, 0: -a,, because the numbering of the levels begins from zero for the ground state, meanwhile a , is the nth zero of Ai.

Applications t o Q u a n t u m Physics

141

8

6

4

2

2

4

6

8

10

Fig. 8.1 The linear potential: in this figure are the first energy levels and the corresponding wave functions.

the Schrodinger equation for the wave function + n ( x ) of the particle in the state n

While proceeding in the same way as above for the particle in the potential V(T= ) T , we have, for 2 > 0, the wave function

where the normalisation constant N and the energy levels En remain to be determined. (1) (z) The energy levels are defined by matching two wave functions $Jn and q ! ~ ~ ? ( at x ) z = 0. At this step, the respective wave functions and their derivatives must be equal (except for the sign)

It is then necessary to distinguish two cases, according t o the parity of the quantum number n


142

n is even: the wave function has an even number of nodes, the axis x = 0 is a symmetry axis, and consequently the derivative of $ ~ ~ ( xis) zero for x = O (i.e. &(x) presents a local extremum in x = 0, cf. Fig. 8.2). We thus have (with the atomic units 2m = h = 1 to simplify) Gn(1) (x)= NAZ (-x - En), x < 0, and $n(2) (x)= N A i (x - En), x > 0. The relation (8.9) enables us t o obtain b

Ai’ [-En] = 0 , i. e.

where a; indicates the nth zero of the Ai’ function (cf $2.2.1). b n is odd: the wave function has an odd number of nodes, which are located on the x = 0 axis. There is no longer an axial symmetry, but a central symmetry a t the origin (cf. Fig. 8.2). The wave functions are $n(1) (x) = - N A z ( - x - En),z < 0, et (2) $n (x)= N A i (x - En), x > 0. The relation (8.8) gives us

Ai [-en]

= 0,

i. e. E n = -an+lr

where a, is the nth zero of the Ai function, that we already met in the case of the potential V ( T )= T . The coefficient of normalisation N is determined by the condition

jY

$ n ( ~ ) $ k ( ~= ) d6~( E n - Em),

--oo

which is reduced to

N 2 T A i (x - En)Ai (x - Em)dx 0

Then we obtain (cf $4.4 and formula (3.50))

N=

1 aAi(aL)

= 6 (En- Em).

Applications t o Q u a n t u m P h y s i c s

143

if n is even, and

if n is odd. Finally, we reach the solution of the Schrodinger Eq. (8.6) (by restoring the constant ( % r ~ / f i ~ ) ~ / ~ ) b

n even:

(8.10)

that is to say

(g)

1/3

En = - u ; + ~ b

.

n odd:

(8.11) that is to say

X

-6

-4

-2

2

4

6

Fig. 8.2 The 1x1 potential. On this figure, the first energy levels and the corresponding wave functions are represented.

144


We show on Fig. 8.2 the potential V(x) = 1x1 and the first energy levels, with m = 1 / 2 and h = 1. 8.1.3

Uniform approximation of the Schrodinger equation

The one-dimensional Schrodinger equation (or the radial equation in the three-dimensional case) can be written [Berry & Mount (1972); Eu (1984)] Y/’

(r) + P2 T Y = 0,

?-

(8.12)

E [O, +m[,

where p(r) is the momentum of the particle. We find here the equation (5.5) where q1(x) = p 2 ( r ) , qZ(x) = 0 and X = l / h . By the same way that for the $5.2, we carry out the changes of variable and function according to x = x(r) y = zR(z). The equation (8.12) then becomes

““1

-- z’

(8.13)

dx2 d2r d r

+ 1 d2R [Rdxz

) +-

d 2 r d x -1na dx2dr ( i x

P2( r ) h2

($)2]z=0’

where z’ = dz/dx and z‘’ = d2z/dx2. We choose the change of function so that the term z’ disappears. We then obtain -1/2

R=(2)

.

Thus we are able to transform Eq. (8.13) into an equation depending only on x and p where the Schwarzian derivative appears {x, r}: {x,r} = - - where x‘, x“, thus obtain

2‘”

are the successive derivatives of x as a function of r . We

(8.14) In the semiclassical limit (the Planck constant h leads formally towards 0), the term containing the Schwarzian derivative can be neglected in front of

145

Applications t o Q u a n t u m Physics

the other term. The generalised JWKB approximation consists in preserving only the first term in z , with a suitable choice of change of variable (8.15) This choice will be driven by considerations under the topological nature (in particular the turning points) of the momentum p ( r ) . The general method consists in choosing (8.16) so that we know the solutions of Eq. (8.15). Then the integration of Eq. (8.16) determines the change of variable r

2

/ 0

lo), we can carry out the Gauss quadrature with less integration points and use the asymptotic series (cf. $2.1.4.3), without loss of precision. We can easily test the validity of these various methods, in particular a t the points of connection, with the help of the Wronskian of the Airy functions (formula (2.6)) 1

Ai(z)Bi’(z)- A i ’ ( z ) B i ( z )= li-

There are other methods to numerically calculate the Airy functions. Among them, Kriiger (1981), proposed the following method for the calcu-


179

lation of Ai(z). Let us consider the definition of the Airy function too

--oo

By carrying out the change of variable t obtain

s

t + icu, in the complex plane, we

--t

+co+ia

~ i (=~-ea3/3-ax 1)

2n

e-atz+i[t3/3+(r-(Y2)t]dt

-co+io!

We can then use the Poisson summation formula to get

and

+(t>= ; 1ea3/3--ax--at2

[3 + t t3

COS

(2 - a21)]

,

00

r=l

R ( p ) = R(-p) = eaPAi(x + p )

+ e-aPAi(z - p ) .

The expression (A.5) is an exact trapezoidal summation formula. Under some conditions, detailed by Kriiger (1981), we can neglect the term A(p), and limit the summation to a range N , so that we obtain for cu = 1

k=O

where

$(t) = -e- x-tz+1/3 cos 7F

[7 + t(" t3

- l)]

and N = 24,these two formulae give the value of A i ( x ) for -10 < 2 < 0 with a relative error lower than 3.10-8, and for 2 > 0, an error of about 10-l2.

If we choose h =


180

A.2

The Inhomogeneous Functions

As previously, the functions G i ( z ) and H i ( z ) can be calculated using the sums, for z > 0 (cf. formulae (2.136) and (2.126))

.

n

i=l

Gi(-z) and Hi(%), result from the two preceding equations and of the formulae (A.3) and (A.4) thanks t o the relation between Bi,Gi and Hi (formula (2.129)) Bi(z) = Gi(z)+ H i ( z ) . The neighbourhood of the origin will be calculated by using the ascending series defined in the 52.3.3: formulae (2.139) for Hi,(2.38) (series of Si)and the relation (2.129) for Gi(x). In Table A.3, we give the partition of R for the calculation of the inhomogeneous Airy functions, and in Table A.4 the weight factors of integration wi and the corresponding abscissae zi,calculated thanks to the algorithm of Gordon (1968).

Exercises Check your favourite computer library for the computation of Airy functions with the following tests:

1. Check the Wronskian relationships W { A i ( z ) ,B i ( z ) }= and W { A i 2 ( z ) , A i ( z ) B i ( z ) , B i 2 (=~ 7 )2} .

2. Compare the logarithmic derivative of Airy functions with a numerical solution to the Riccati equation u’

+ u2 = x .

3. In the complex plane: check the relation Ai(x) +jAi(jx) +j2Ai(j2x)= 0, and the corresponding one for

Bi. Check the relation (2.163) of 52.4.2.


181

Table A.3 Partition of R for the calculation of the Airy inhomogeneous functions. Gi(x)

Hi(x)

A.7 and 2.129

integration A.7

integration A.7 and 2.129

integration A.7

series 2.139 and 2.129

series 2.139

X

< -6 < x < -4

x

-6

- 4 < ~ < 6 6<x

Airy Functions and Applications to Physics

Elliptic functions and applications

Elliptic functions and applications

Special functions and their applications

Special functions: an introduction to classical functions of mathematical physics

Anthocyanins: Biosynthesis, Functions, and Applications

Bessel functions and their applications

Anthocyanins: Biosynthesis, Functions, and Applications

Bernstein functions: Theory and applications

Convex functions and their applications

Laser physics and applications

Superconductivity: Physics and Applications

Cstar-Algebras and Applications to Physics

Geometric Algebra and Applications to Physics

Geometric Algebra and Applications to Physics

Bessel Functions and Their Applications (Analytical Methods and Special Functions)

Applications of Fourier Transforms to Generalized Functions

Applications of Fourier transforms to generalized functions

Green's Functions with Applications

Green's functions with applications

Spectral Functions in Mathematics and Physics

Special Functions of Mathematical Physics and Chemistry

Zeta Functions, Topology and Quantum Physics

Spectral functions in mathematics and physics

Special Functions of Mathematical Physics: A UNIFIED INTRODUCTION with applications

Special Functions of Mathematical Physics and Chemistry

Up The Airy Mountain

Spectral functions in mathematics and physics

Spectral functions in mathematics and physics

Green's Functions in Quantum Physics

Vector-Valued Functions and their Applications (Mathematics and its Applications)

Airy Functions and Applications to Physics

Elliptic functions and applications

Elliptic functions and applications

Special functions and their applications

Special functions: an introduction to classical functions of mathematical physics

Anthocyanins: Biosynthesis, Functions, and Applications

Bessel functions and their applications

Anthocyanins: Biosynthesis, Functions, and Applications

Bernstein functions: Theory and applications

Convex functions and their applications

Laser physics and applications

Superconductivity: Physics and Applications

Cstar-Algebras and Applications to Physics

Geometric Algebra and Applications to Physics

Geometric Algebra and Applications to Physics

Bessel Functions and Their Applications (Analytical Methods and Special Functions)

Applications of Fourier Transforms to Generalized Functions

Applications of Fourier transforms to generalized functions

Green's Functions with Applications

Green's functions with applications

Spectral Functions in Mathematics and Physics

Special Functions of Mathematical Physics and Chemistry

Zeta Functions, Topology and Quantum Physics

Spectral functions in mathematics and physics

Special Functions of Mathematical Physics: A UNIFIED INTRODUCTION with applications

Special Functions of Mathematical Physics and Chemistry

Up The Airy Mountain

Spectral functions in mathematics and physics

Spectral functions in mathematics and physics

Green's Functions in Quantum Physics

Vector-Valued Functions and their Applications (Mathematics and its Applications)

Recommend Documents