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