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Airy Functions


There are four varieties of Airy functions: Ai(z), Bi(z), Gi(z), and Hi(z). Of these, Ai(z) and Bi(z) are by far the most common, with Gi(z) and Hi(z) being encountered much less frequently. Airy functions commonly appear in physics, especially in optics, quantum mechanics, electromagnetics, and radiative transfer.

Ai(z) and Bi(z) are entire functions.

A generalization of the Airy function was constructed by Hardy.

AiryAiBi

The Airy function Ai(x) and Bi(x) functions are plotted above along the real axis.

The Ai(z) and Bi(z) functions are defined as the two linearly independent solutions to

 y^('')-yz=0.
(1)

(Abramowitz and Stegun 1972, pp. 446-447; illustrated above), written in the form

 y(z)=AAi(z)+BBi(z),
(2)

where

Ai(z)=1/(3^(2/3)Gamma(2/3))_0F_1(;2/3;1/9z^3)-z/(3^(1/3)Gamma(1/3))_0F_1(;4/3;1/9z^3)
(3)
Bi(z)=1/(3^(1/6)Gamma(2/3))_0F_1(;2/3;1/9z^3)+(3^(1/6)z)/(Gamma(1/3))_0F_1(;4/3;1/9z^3),
(4)

where _0F_1(;a;z) is a confluent hypergeometric limit function. These functions are implemented in the Wolfram Language as AiryAi[z] and AiryBi[z]. Their derivatives are implemented as AiryAiPrime[z] and AiryBiPrime[z].

For the special case x>0, the functions can be written as

Ai(x)=1/3sqrt(x)[I_(-1/3)(2/3x^(3/2))-I_(1/3)(2/3x^(3/2))]
(5)
=1/pisqrt(x/3)K_(1/3)(2/3x^(3/2))
(6)
Bi(x)=sqrt(x/3)[I_(-1/3)(2/3x^(3/2))+I_(1/3)(2/3x^(3/2))],
(7)

where I(x) is a modified Bessel function of the first kind and K(x) is a modified Bessel function of the second kind.

AiryAiReImAbs
Min Max
Re
Im Powered by webMathematica

Plots of Ai(z) in the complex plane are illustrated above.

AiryBiReImAbs
Min Max
Re
Im Powered by webMathematica

Similarly, plots of Bi(z) appear above.

The Airy Ai(z) function is given by the integral

 Ai(z)=1/(2pi)int_(-infty)^inftye^(i(zt+t^3/3))dt
(8)

and the series

Ai(z)=1/(3^(2/3)pi)sum_(n=0)^(infty)(Gamma(1/3(n+1)))/(n!)(3^(1/3)z)^nsin[(2(n+1)pi)/3]
(9)
Bi(z)=1/(3^(1/6)pi)sum_(n=0)^(infty)(Gamma(1/3(n+1)))/(n!)(3^(1/3)z)^n|sin[(2(n+1)pi)/3]|
(10)

(Banderier et al. 2000).

For z=0,

Ai(0)=1/(3^(2/3)Gamma(2/3))
(11)
Bi(0)=1/(3^(1/6)Gamma(2/3)),
(12)

where Gamma(z) is the gamma function. Similarly,

Ai^'(0)=-1/(3^(1/3)Gamma(1/3))
(13)
Bi^'(0)=(3^(1/6))/(Gamma(1/3)).
(14)

The asymptotic series of Ai(z) has a different form in different quadrants of the complex plane, a fact known as the stokes phenomenon.

AiryGiReImAiryGiContoursAiryHiReImAiryHiContours

Functions related to the Airy functions have been defined as

Gi(z)=1/piint_0^inftysin(1/3t^3+zt)dt
(15)
=1/3Bi(z)+int_0^z[Ai(z)Bi(t)-Ai(t)Bi(z)]dt
(16)
=1/3Bi(z)-(z^2_1F_2(1;4/3,5/3;1/9z^3))/(2pi)
(17)
Hi(z)=1/piint_0^inftyexp(-1/3t^3+zt)dt
(18)
=2/3Bi(z)+int_0^z[Ai(t)Bi(z)-Ai(z)Bi(t)]dt
(19)
=2/3Bi(z)+(_1F_2(1;4/3,5/3;1/9z^3)z^2)/(2pi),
(20)

where _pF_q is a generalized hypergeometric function.

Watson (1966, pp. 188-190) gives a slightly more general definition of the Airy function as the solution to the Airy differential equation

 Phi^('')+/-k^2Phiz=0
(21)

which is finite at the origin, where Phi^' denotes the derivative dPhi/dz, k^2=1/3, and either sign is permitted. Call these solutions (1/pi)Phi(+/-k^2,z), then

 1/piPhi(+/-1/3;z)=int_0^inftycos(t^3+/-zt)dt
(22)
Phi(1/3;z)=1/3pisqrt(z/3)[J_(-1/3)((2z^(3/2))/(3^(3/2)))+J_(1/3)((2z^(3/2))/(3^(3/2)))]
(23)
Phi(-1/3;z)=1/3pisqrt(z/3)[I_(-1/3)((2z^(3/2))/(3^(3/2)))-I_(1/3)((2z^(3/2))/(3^(3/2)))],
(24)

where J(z) is a Bessel function of the first kind. Using the identity

 K_n(z)=pi/2(I_(-n)(z)-I_n(z))/(sin(npi)),
(25)

where K(z) is a modified Bessel function of the second kind, the second case can be re-expressed

Phi(-1/3;z)=1/3pisqrt(z/3)2/pisin(1/3pi)K_(1/3)((2z^(3/2))/(3^(3/2)))
(26)
=pi/3sqrt(z/3)2/pi(sqrt(3))/2K_(1/3)((2z^(3/2))/(3^(3/2)))
(27)
=1/3sqrt(z)K_(1/3)((2z^(3/2))/(3^(3/2))).
(28)

See also

Airy-Fock Functions, Airy Function Zeros, Airy Zeta Function, Bessel Function of the First Kind, Map-Airy Distribution, Modified Bessel Function of the First Kind, Modified Bessel Function of the Second Kind

Related Wolfram sites

http://functions.wolfram.com/Bessel-TypeFunctions/AiryAi/, http://functions.wolfram.com/Bessel-TypeFunctions/AiryAiPrime/, http://functions.wolfram.com/Bessel-TypeFunctions/AiryBi/, http://functions.wolfram.com/Bessel-TypeFunctions/AiryBiPrime/

Explore with Wolfram|Alpha

References

Abramowitz, M. and Stegun, I. A. (Eds.). "Airy Functions." §10.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 446-452, 1972.Banderier, C.; Flajolet, P.; Schaeffer, G.; and Soria, M. "Planar Maps and Airy Phenomena." In Automata, Languages and Programming. Proceedings of the 27th International Colloquium (ICALP 2000) held at the University of Geneva, Geneva, July 9-15, 2000 (Ed. U. Montanari, J. D. P. Rolim, and E. Welzl). Berlin: Springer, pp. 388-402, 2000.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions." §6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 234-245, 1992.Sloane, N. J. A. Sequences A096714 and A096715 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Airy Functions Ai(x) and Bi(x)." Ch. 56 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 555-562, 1987.Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.

Referenced on Wolfram|Alpha

Airy Functions

Cite this as:

Weisstein, Eric W. "Airy Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AiryFunctions.html

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