Airy function

In the physical sciences, the Airy function Ai(x) is a special function named after the British astronomer George Biddell Airy. The function Ai(x) and the related function Bi(x), which is also called an Airy function, are solutions to the differential equation

known as the Airy equation or the Stokes equation. This is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential).

The Airy function is the solution to Schrödinger's equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniform semiclasssical approximations near a turning point in the WKB method, when the potential may be locally approximated by a linear function of position. The triangular potential well solution is directly relevant for the understanding of many semiconductor devices.

The Airy function also underlies the form of the intensity near a directional caustic, such as the rainbow. Historically, this was the problem that led Airy to develop this function.

Definitions

Plot of Ai(x) in red and Bi(x) in blue.

For real values of x, the Airy function is defined by the improper integral

which converges because the positive and negative parts of the rapid oscillations tend to cancel one another out (as can be checked by integration by parts).

By differentiating under the integration sign, we find that y = Ai(x) satisfies the differential equation

This equation has two linearly independent solutions. The standard choice for the other solution is the Airy function of the second kind, denoted Bi(x). It is defined as the solution with the same amplitude of oscillation as Ai(x) as x goes to −∞ which differs in phase by π/2:

Properties

The values of Ai(x) and Bi(x) and their derivatives at x = 0 are given by

Here, Γ denotes the Gamma function. It follows that the Wronskian of Ai(x) and Bi(x) is 1/π.

When x is positive, Ai(x) is positive, convex, and decreasing exponentially to zero, while Bi(x) is positive, convex, and increasing exponentially. When x is negative, Ai(x) and Bi(x) oscillate around zero with ever-increasing frequency and ever-decreasing amplitude. This is supported by the asymptotic formulae below for the Airy functions.

Asymptotic formulae

The asymptotic behaviour of the Airy functions as x goes to +∞ is given by^[1]

For the limit in the negative direction we have^[2]

Asymptotic expansions for these limits are also available. These are listed in (Abramowitz and Stegun, 1954) and (Olver, 1974).

Complex arguments

We can extend the definition of the Airy function to the complex plane by

where the integral is over a path C starting at the point at infinity with argument -(1/3)π and ending at the point at infinity with argument (1/3)π. Alternatively, we can use the differential equation y'' − xy = 0 to extend Ai(x) and Bi(x) to entire functions on the complex plane.

The asymptotic formula for Ai(x) is still valid in the complex plane if the principal value of x^2/3 is taken and x is bounded away from the negative real axis. The formula for Bi(x) is valid provided x is in the sector {x∈C : |arg x| < (1/3)π−δ} for some positive δ. Finally, the formulae for Ai(−x) and Bi(−x) are valid if x is in the sector {x∈C : |arg x| < (2/3)π−δ}.

It follows from the asymptotic behaviour of the Airy functions that both Ai(x) and Bi(x) have an infinity of zeros on the negative real axis. The function Ai(x) has no other zeros in the complex plane, while the function Bi(x) also has infinitely many zeros in the sector {z∈C : (1/3)π < |arg z| < (1/2)π}.

Plots

Relation to other special functions

For positive arguments, the Airy functions are related to the modified Bessel functions:

Here, I_±1/3 and K_1/3 are solutions of x²y'' + xy' − (x² + 1 / 9)y = 0.

For negative arguments, the Airy function are related to the Bessel functions:

Here, J_±1/3 are solutions of x²y'' + xy' + (x² − 1 / 9)y = 0.

The Scorer's functions solve the equation y'' − xy = 1 / π. They can also be expressed in terms of the Airy functions:

Fourier transform

Using the definition of the Airy function Ai(x), it is straightforward to show its Fourier transform is given by

History

The Airy function is named after the British astronomer George Biddell Airy, who encountered it in his study of optics (Airy 1838). The notation Ai(x) was introduced by Harold Jeffreys.

Notes

1. ^ Abramowitz & Stegun (1970), Eqns 10.4.59 and 10.4.63
2. ^ Abramowitz & Stegun (1970), Eqns 10.4.60 and 10.4.64

References

• Abramowitz, Milton; Stegun, Irene A. (1970), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, Ninth printing .
• Airy (1838). On the intensity of light in the neighbourhood of a caustic. Transactions of the Cambridge Philosophical Society, 6, 379–402.
• Olver (1974). Asymptotics and Special Functions, Chapter 11. Academic Press, New York.
• Olivier Vallée and Manuel Soares (2004), "Airy functions and applications to physics", Imperial College Press, London.
• Harold Richard Suiter (1994). Star Testing Astronomical Telescopes: A Manual for Optical Evaluation and Adjustment. Richmond, VA: Willmann-Bell. ISBN 978-0-943396-44-6.  (with many example images)

External links

• Weisstein, Eric W., "Airy Functions" from MathWorld.
• Bessel, Airy, Struve Functions at the Wolfram Functions Site.
• Olver, F. W. J. (2008), "Airy and related functions", in Boisvert, Ronald F.; Clark, Charles W.; Lozier, Daniel M. et al., Digital Library of Mathematical Functions, N.I.S.T.