Airy functions

Particular solutions of the Airy equation.

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

$$\mathop{\rm Ai} (x) = \frac{1} \pi \int\limits _ { 0 } ^ \infty \cos \left ( \frac{t ^ {3} }{3} + x t \right ) \ d t .$$

For complex values of $z$

$$\mathop{\rm Ai} (z) = \frac{1}{2 \pi } \int\limits _ \gamma \mathop{\rm exp} \left ( zt - \frac{t ^ {3} }{3} \right ) \ d t ,$$

where $\gamma = ( \infty e ^ {- 2 \pi i / 3 } , 0 ] \cup [ 0 , + \infty )$ is a contour in the complex $t$- plane. The second Airy function is defined by

$$\mathop{\rm Bi} (z) = i \omega ^ {2} \mathop{\rm Ai} ( \omega ^ {2} z ) - i \omega \mathop{\rm Ai} ( \omega z ) ,\ \omega = e ^ {2 \pi i / 3 } .$$

The functions $\mathop{\rm Ai} (x)$ and $\mathop{\rm Bi} (x)$ are real for real $x$.

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

$$v (z) = \frac{\sqrt \pi }{2} \mathop{\rm Ai} (z) ,$$

$$w _ {1} (z) = 2 e ^ {i \pi / 6 } v ( \omega z ) ,$$

$$w _ {2} (z) = 2 e ^ {- i \pi / 6 } v ( \omega ^ {-1} z ) ;$$

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

$$\tag{1 } v (z) = \frac{w _ {1} (z) - w _ {2} (z) }{2i} ,\ \ \overline{ {w _ {1} (z) }}\; = w _ {2} ( \overline{z}\; ) .$$

Any two of $v (z) , w _ {1} (z)$ and $w _ {2} (z)$ are linearly independent.

The most important Airy function is $v (z)$( or $\mathop{\rm Ai} (z)$). Its asymptotic behaviour on the real axis is given by

$$v (x) = \frac{1}{2} \frac{x ^ {-1/4} }{\sqrt \pi } \ \mathop{\rm exp} \left ( - \frac{2}{3} x ^ {3/2} \right ) [ 1 + O ( x ^ {-3/2} ) ] , \ x \rightarrow + \infty ,$$

$$v (x) = \frac{| x | ^ {-1/4} }{\sqrt \pi } \left [ \sin \left ( \frac{2}{3} | x | ^ {3/2} + \frac \pi {4} \right ) + O ( | x | ^ {-3/2} ) \right ] ,$$

$$\ x \rightarrow - \infty ,$$

so $v (x)$ decreases rapidly for $x > 0 , x \gg 1$ and oscillates strongly for $x < 0 , | x | \gg 1$. The functions $w _ {1} (x)$ and $w _ {2} (x)$ increase exponentially as $x \rightarrow + \infty$. For complex $z$ the Airy functions have the following asymptotic expansions as $| z | \rightarrow \infty$:

$$\tag{2 } v (z) \sim \frac{1}{2 \sqrt \pi } z ^ {-1/4} \mathop{\rm exp} \left ( - \frac{2}{3} z ^ {3/2} \right ) \times$$

$$\times \sum _ { n=0 } ^ \infty (-1) ^ {n} a _ {n} z ^ {-3n/2} \ \textrm{ for } | \mathop{\rm arg} z | \leq \pi - \epsilon ,$$

$$w _ {1} (z) \sim \frac{1}{\sqrt \pi } z ^ {-1/4} \mathop{\rm exp} \left ( \frac{2}{3} z ^ {3/2} \right ) \times$$

$$\times \sum _ { n=0 } ^ \infty a _ {n} z ^ {-3n/2} \ \textrm{ for } \ | \mathop{\rm arg} z - \frac \pi {3} | \leq \pi - \epsilon ,$$

where

$$a _ {n} = \ \frac{\Gamma \left ( 3 n + \frac{1}{2} \right ) 9 ^ {-n} }{( 2 n ) ! } .$$

The asymptotic expansion of $w _ {2} (z)$ is of the form (2), but it is valid in the sector

$$\left | \mathop{\rm arg} \left ( z + \frac \pi {3} \right ) \ \right | \leq \pi - \epsilon .$$

Here $\epsilon \in ( 0 , \pi )$ is arbitrary, the branches of $\sqrt z$ and $z ^ {1/4}$ are positive on the semi-axis $( 0 , \infty )$, and the asymptotic expansions are uniform with respect to $\mathop{\rm arg} z$ and can be differentiated term by term any number of times. In the remaining sector $| \mathop{\rm arg} -z | < \epsilon$ the asymptotic expansion of $v (z)$ is expressed in terms of those of $w _ {1} (z)$ and $w _ {2} (z)$ by means of (1); hence, the asymptotic expansion of $v (z)$ has a different form in different sectors of the complex $z$- 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

$$I ( \lambda , \alpha ) = \int\limits _ { a } ^ { b } \textrm{ e } ^ {i \lambda S ( x , \alpha ) } f ( x , \alpha ) d x ,$$

for $\lambda > 0 , \lambda \rightarrow \infty$. Here $f$ and $S$ are smooth functions, $S$ is real and $\alpha$ is a real parameter. If for small values of $\alpha \geq 0$ the phase $S$ has two close non-degenerate stationary points $x _ {1} ( \alpha )$ and $x _ {2} ( \alpha )$ that coincide for $\alpha = 0$, for example, if

$$S ( x , \alpha ) = \alpha x - x ^ {3} + O ( x ^ {4} ) \ \textrm{ as } x \rightarrow 0 ,$$

then for small values of $\alpha \geq 0$, as $\lambda \rightarrow + \infty$, the contribution to the asymptotics of the integral coming from a neighbourhood of the point $x = 0$ can be expressed in terms of the Airy function $v$ 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

$$\tag{3 } y ^ {\prime\prime} + \lambda ^ {2} q (x) y = 0 ,$$

where $q (x)$ is a smooth real-valued function on the interval $I = [ a , b ]$ and $\lambda > 0$ is a large parameter. The zeros of $q (x)$ are called turning points (or transfer points) of the equation (3). Let

$$a < x _ {0} < b ,\ \ q ( x _ {0} ) = 0 ,\ \ q ^ \prime ( x _ {0} ) \neq 0$$

(such a point is called simple),

$$q (x) \neq 0 \ \textrm{ for } \ x \in I , x \neq x _ {0} ,\ \ q ^ \prime ( x _ {0} ) > 0 .$$

Set

$$\xi (x) = \left ( \frac{2}{3} \int\limits _ {x _ {0} } ^ { x } \sqrt {q (t) } d t \right ) ^ {2/3} ,\ \ \mathop{\rm sign} \xi (x) = \mathop{\rm sign} ( x - x _ {0} ) ,$$

$$Y _ {0} (x) = ( \xi ^ \prime (x) ) ^ {-1/2} \mathop{\rm Ai} ( - \lambda ^ {2/3} \xi (x) ) ,$$

$$Y _ {1} (x) = ( \xi ^ \prime (x) ) ^ {-1/2} \mathop{\rm Bi} ( - \lambda ^ {2/3} \xi (x) ) .$$

Equation (3) has linearly independent solutions $y _ {0} (x)$ and $y _ {1} (x)$ such that, as $\lambda \rightarrow + \infty$,

$$y _ {j} (x) = Y _ {j} (x) \left [ 1 + O \left ( \frac{1} \lambda \right ) \right ] ,\ \ a \leq x \leq x _ {0} ,\ j = 0 , 1 ,$$

$$y _ {0} (x) = Y _ {0} (x) \left [ 1 + O \left ( \frac{1} \lambda \right ) \right ] + Y _ {1} (x) O \left ( \frac{1} \lambda \right ) ,$$

$$y _ {1} (x) = Y _ {1} (x) \left [ 1 + O \left ( \frac{1} \lambda \right ) \right ] + Y _ {0} (x) O \left ( \frac{1} \lambda \right ) ,$$

$$x _ {0} \leq x \leq b ,$$

uniformly with respect to $x$.

This result has been generalized in various directions: asymptotic series have been obtained for the solutions, the case $q = q ( x , \lambda )$ has been studied (for example, if $q ( x , \lambda )$ can be expanded in an asymptotic series $q \sim \sum _ {n=0} ^ \infty \lambda ^ {-n} q _ {n} (x)$ as $\lambda \rightarrow + \infty$), and the asymptotic behaviour of the solutions near multiple turning points has been investigated. Other generalizations concern the equation

$$\tag{4 } w ^ {\prime\prime} + \lambda ^ {2} q (x) w = 0 ,$$

where the function $q (z)$ is analytic in a domain $D$ of the complex $z$- plane. Let $l$ be the maximal connected component of the level line

$$\mathop{\rm Re} \int\limits _ {z _ {0} } ^ { z } \sqrt {q (t) } \ d t = 0 ,$$

emanating from a turning point $z _ {0}$ and containing no other turning points; then $l$ is called a Stokes line. If $q = - z$( that is, (4) is the Airy equation), then the Stokes lines are the rays $( - \infty , 0 )$ and $( 0 , e ^ {\pm i \pi / 3 } )$. Analogously, if $z _ {0}$ is a simple turning point of (4), then there are three Stokes lines $l _ {1} , l _ {2}$ and $l _ {3}$ emanating from it and the angle between adjacent lines at $z _ {0}$ is equal to $2 \pi / 3$. Let $S _ {j}$ be a neighbourhood of $z _ {0}$ from which a neighbourhood of the Stokes line $l _ {j}$, $j = 1 , 2 , 3$, has been removed. For a suitable numbering of the $S _ {j}$, equation (4) has three solutions $\widetilde{w} _ {j} (z)$, $j = 1 , 2 , 3$, such that, as $\lambda \rightarrow + \infty$,

$$\widetilde{w} _ {j} (z) \sim \frac{1}{\sqrt {\xi (z) } } v ( - \lambda ^ {2/3} \omega ^ {j} \xi (z) ) ,\ \ \omega = e ^ {2 \pi i / 3 } ,$$

for $z \in S _ {j}$.

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:

$$\mathop{\rm Ai} (x) = \ \frac{1}{\pi \sqrt 3 } \sqrt x K _ {1/3} \left ( \frac{2}{3} x ^ {2/3} \right ) .$$

The function $\mathop{\rm Ai} (z)$ satisfies the differential equation $w ^ {\prime\prime} (z) = z w (z)$, cf. [a2].

References

[a1]	F.W.J. Olver, "Asymptotics and special functions" , Acad. Press (1974) Zbl 0303.41035
[a2]	N.N. Lebedev, "Special functions and their applications" , Dover, reprint (1972) (Translated from Russian) Zbl 0271.33001