Airy functions

Particular solutions of the Airy equation.

The first Airy function (or simply the Airy function) is defined by

For complex values of

where is a contour in the complex -plane. The second Airy function is defined by

The functions and are real for real .

A second collection of Airy functions was introduced by V.A. Fock [V.A. Fok]:

in this case is called the Airy–Fok function (Airy–Fock function). The following identities hold:

(1)

Any two of and are linearly independent.

The most important Airy function is (or ). Its asymptotic behaviour on the real axis is given by

so decreases rapidly for and oscillates strongly for . The functions and increase exponentially as . For complex the Airy functions have the following asymptotic expansions as :

(2)

where

The asymptotic expansion of is of the form (2), but it is valid in the sector

Here is arbitrary, the branches of and are positive on the semi-axis , and the asymptotic expansions are uniform with respect to and can be differentiated term by term any number of times. In the remaining sector the asymptotic expansion of is expressed in terms of those of and by means of (1); hence, the asymptotic expansion of has a different form in different sectors of the complex -plane. This fact was first established by G.G. Stokes [2] and is called the Stokes phenomenon.

The Airy functions occur in the study of integrals of rapidly-oscillating functions, of the form

for . Here and are smooth functions, is real and is a real parameter. If for small values of the phase has two close non-degenerate stationary points and that coincide for , for example, if

then for small values of , as , the contribution to the asymptotics of the integral coming from a neighbourhood of the point can be expressed in terms of the Airy function and its derivative (see [6]). Integrals of this kind occur in the study of short-wave fields near a simple focus (see [7] and [8]); the Airy functions arose in connection with the study of this problem [1].

Consider the second-order differential equation

(3)

where is a smooth real-valued function on the interval and is a large parameter. The zeros of are called turning points (or transfer points) of the equation (3). Let

(such a point is called simple),

Set

Equation (3) has linearly independent solutions and such that, as ,

uniformly with respect to .

This result has been generalized in various directions: asymptotic series have been obtained for the solutions, the case has been studied (for example, if can be expanded in an asymptotic series as ), and the asymptotic behaviour of the solutions near multiple turning points has been investigated. Other generalizations concern the equation

(4)

where the function is analytic in a domain of the complex -plane. Let be the maximal connected component of the level line

emanating from a turning point and containing no other turning points; then is called a Stokes line. If (that is, (4) is the Airy equation), then the Stokes lines are the rays and . Analogously, if is a simple turning point of (4), then there are three Stokes lines and emanating from it and the angle between adjacent lines at is equal to . Let be a neighbourhood of from which a neighbourhood of the Stokes line , , has been removed. For a suitable numbering of the , equation (4) has three solutions , , such that, as ,

for .

The Airy functions also occur in the study of asymptotic solutions of ordinary differential equations and systems of higher order near simple turning points.

References

[1]	G.B. Airy, Trans. Cambridge Philos. Soc. , 6 (1838) pp. 379–402
[2]	G.G. Stokes, Trans. Cambridge Philos. Soc. , 10 (1857) pp. 105–128
[3]	V.A. Fok, "Tables of the Airy functions" , Moscow (1946) (In Russian)
[4]	A. Segun, M. Abramowitz, "Handbook of mathematical functions" , Appl. Math. Ser. , 55 , Nat. Bur. Standards (1970)
[5]	V.M. Babich, V.S. Buldyrev, "Asymptotic methods in the diffraction of short waves" , Moscow (1972) (In Russian) (Translation forthcoming: Springer)
[6]	M.V. Fedoryuk, "The saddle-point method" , Moscow (1977) (In Russian)
[7]	E.M. Lifshits, "The classical theory of fields" , Addison-Wesley (1951) (Translated from Russian)
[8]	V.P. Maslov, M.V. Fedoryuk, "Quasi-classical approximation for the equations of quantum mechanics" , Reidel (1981) (Translated from Russian)
[9]	A.A. Dorodnitsyn, "Asymptotic laws of distribution of the characteristic values for certain types of second-order differential equations" Uspekhi Mat. Nauk , 6 : 7 (1952) pp. 3–96 (In Russian)
[10]	W. Wasov, "Asymptotic expansions for ordinary differential equations" , Interscience (1965)
[11]	M.V. Fedoryuk, "Asymptotic methods for linear ordinary differential equations" , Moscow (1983) (In Russian)

Comments

The Airy function can be expressed in terms of modified Bessel functions of the third kind:

The function satisfies the differential equation , cf. [a2].

References

[a1]	F.W.J. Olver, "Asymptotics and special functions" , Acad. Press (1974)
[a2]	N.N. Lebedev, "Special functions and their applications" , Dover, reprint (1972) (Translated from Russian)