Namespaces
Variants
Actions

Airy functions

From Encyclopedia of Mathematics
Revision as of 16:09, 1 April 2020 by Ulf Rehmann (talk | contribs) (tex encoded by computer)
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


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)
[a2] N.N. Lebedev, "Special functions and their applications" , Dover, reprint (1972) (Translated from Russian)
How to Cite This Entry:
Airy functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Airy_functions&oldid=45054
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article