This book gives a survey of diverse properties and applications of Airy functions. Although G.-B. Airy (1801–1892) was astronomer, the main applications of Airy functions are not in astronomy but, among others, in Airy’s other favourite subject optics, and in electromagnetism. In 1838 Airy introduced the function \[ W(m)=\int_{0}^{\infty} \cos[\frac{\pi}{2}(x^{3}-mx)] \,dx\tag{uu} \] to calculate the light intensity in the neighbourhood of a caustic. By an excessively laborious quadrature process, Airy published tables for the computation of (uu). After Stokes’ introduction of the asymptotic series in the 1850s, J. W. Nicholson [“On the relation of { Airy}’s integral to the {Bessel} functions”, Phil. Mag. (6) 18, 6–17 (1909; JFM 40.0515.02)] discovered the exact connection with Bessel functions of order \(\pm\frac{1}{3}\).
In modern notation, \(Ai(z)\) and \(Bi(z)\) are two independent solutions to \(y''-zy=0\). \[ \lim_{y\to+\infty}Ai(z)=0, \quad \lim_{y\to+\infty}Bi(z)=\infty. \] N. Bleistein and R. A. Handelsman [Asymptotic expansions of integrals (Second edition. Dover) (1986), p. 268] found the asymptotic expansion \(Ai(z)\sim \frac{z^{-\tfrac{1}{4}}} {2\sqrt{\pi}} \exp[-\frac{2z^{\tfrac{3}{2}}}{3}]\), \(z\to+\infty\).
Returning to the book under review, every chapter contains exercises. Chapter 1 gives a historical introduction. Chapter 2 is about definitions and various properties like zeros and asymptotic expansions. Chapter 3 contains primitive functions. Chapter 4 treats Airy transforms and various transforms of Airy functions. Chapter 5 gives a brief introduction to oscillating integrals and its application to second order differential equations. Chapter 6 gives some generalizations of Airy functions. In chapter 7 applications to classical physics are given. Chapter 8 treats applications to quantum physics. There are Mathematica add-on packages for computations of the zeros of these functions [S. Wolfram, The Mathematica book, Cambridge: Cambridge University Press (1999; Zbl 0924.65002)].