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Airy Functions and Applications to Physics

Olivier Vallée and Manuel Soares
Imperial College Press
Publication Date: 
Number of Pages: 
[Reviewed by
Warren Johnson
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One has only to glance through this book to realize that a short treatise developing the basic properties of Airy functions, with a few interesting applications, would be most welcome. If one looks at the book a little longer, one begins to hope that someone else will give it a try. This is an attractive little volume, containing many pretty results which are for the most part not well known, but it has two major flaws.

The preface says that "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." It is not a lack of rigor that bothers me so much, but a lack of demonstration. (I will comment on the prose style later.) The Airy functions Ai(x) and Bi(x) are linearly independent solutions of the differential equation d2y/dx2 = xy. For real x, Ai(x) can be written as 1/π times the integral from zero to infinity of the cosine of xt+(t3/3), with respect to t. It is convenient to define two more functions Gi(x) and Hi(x), the inhomogeneous Airy or Scorer functions. Gi(x) is the same as Ai(x) with sine in place of cosine, while Hi(x) has an exponential function instead with t3/3 replaced by its negative; then Bi(x)=Gi(x)+Hi(x). That these functions satisfy the differential equation is not obvious, and is not made much more obvious by the book under review, though at least in this instance the authors try. Many other fundamental properties are merely quoted, often with a reference to Abramowitz and Stegun, which does not prove them either. Thus the book is an excellent place to learn what is true of Airy functions (I have checked many of the formulas and almost always found them to be accurate), but not why.

For example, the Wronskian of Ai(x) and Bi(x) is not derived. It is easy to see from the differential equation that it must be a constant. One can evaluate the constant by letting x become infinite, if one knows some asymptotics of Airy functions (here the authors give a derivation for Ai(x), but one would do better to look at F. Olver's classic Asymptotics and Special Functions), or by setting x = 0, if one does not mind evaluating some integrals. When I took a course on asymptotics in graduate school we had this Wronskian as a homework problem, and I unintentionally annoyed the professor by doing the latter instead of the former. It comes out to 1/π.

Still more troubling is the book's approach to the power series expansions of Ai(x) and Bi(x). If a(n) is the coefficient of x^n/n! in a power series solution of the differential equation, then it is easy to see that a(n+2)=n a(n-1) for positive n and that a(2)=0. Thus the power series for Ai(x) and Bi(x) are completely determined by the four integrals (or actually six, because Bi(x)=Gi(x)+Hi(x)) that one would compute when evaluating the Wronskian at x = 0. The authors do not derive any of this, not the integrals nor even the recurrence. They also fail to observe that formulas (2.13) and (2.14), which give the nth derivatives of Ai(x) and Bi(x) at zero, should be saying the same thing as formulas (2.37) and (2.38), which give what the authors, following Abramowitz and Stegun, call the "Ascending series" of Ai(x) and Bi(x); in other words, the power series referred to above. They miscopy (2.37) and (2.38) from Abramowitz and Stegun, forgetting that Ai'(0) is negative. They also take (2.36), which they call the "Expansion of Ai(x) near the origin", from Copson's book on asymptotics, not only failing to remark that it must be the same thing as (2.37), but apparently meaning to imply that they are different. They look different, because (2.36) has a unified formula for the coefficients (superficially slightly different than (2.13), but easily seen to be equivalent) whereas (2.37) sieves them mod 3, but it is not difficult to verify that they are the same, or would be the same after fixing the slight error in (2.37). In a book allegedly intended to be used by undergraduate physics students, this borders on criminal negligence.

I'm pretty sure that the book was originally written in French and then translated, presumably by the authors since no translator is mentioned (although they do thank one Nick Rowswell at the end of the preface without saying exactly why); we even find "et" in place of "and" on page 68. The best that can be said of the book's English is that it is better than my French. Occasionally it is rather amusing. The preface begins "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." Far from being embarrassed by these sentences, Imperial College Press repeated them on the back cover, only without the second "of" in the first sentence. Sometimes it is not even clear just what the authors were trying to say, e.g. "In England, the young John [Couch] Adams was doing the same calculations with a slight advance, however Airy was doubtful on the issue of such a work" on page 2. But as bad as the prose is, it is not the real problem.

In spite of the book's flaws, I did enjoy reading it for the many beautiful formulas it contains. Certain integrals involving Airy functions are surprisingly easy because of the simplicity of the differential equation (the Wronskian is sometimes also helpful), and there is an excellent list of these on pages 43 - 45 that was great fun to go through. Anyone who has occasion to use Airy functions should be aware of this book, but also aware of its weaknesses.

Warren Johnson ( is visiting assistant professor of mathematics at Connecticut College. One of his favorite areas of mathematics is special functions.

* A Historical Introduction: Sir George Biddell Airy
* Definitions and Properties
* Primitives and Integrals of Airy Functions
* Transformations of Airy Functions
* The Uniform Approximation
* Generalisation of Airy Functions
* Applications to Classical Physics
* Applications to Quantum Physics