Airy Functions

7/28/2019 Airy Functions

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There are four varieties of Airy functions: , , , and . Of these, and

are by far the most common, with and being encountered much less frequently. Airy

functions commonly appear in physics, especially in optics, quantum mechanics, electromagnetics,

and radiative transfer.

and areentire functions.

A generalization of the Airy function was constructed by Hardy.

The Airy function and functions are plotted above along thereal axis.

The and functions are defined as the twolinearly independent solutions to

(1

)

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

(2

)

where

(3

)
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(4

)

where is a confluent hypergeometric limit function. These functions are implemented in

Mathematicaas AiryAi[z] andAiryBi[z]. Their derivatives are implemented as AiryAiPrime[z]

andAiryBiPrime[z].

For the special case , the functions can be written as

(5

)

(6)

(7

)

where is amodified Bessel function of the first kind and is amodified Bessel function of the

second kind.

Min Max

Re-4 4

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Im-4 4

Plots of in the complex plane are illustrated above.

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Re-3 3

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Im-3 3

Similarly, plots of appear above.

The Airy function is given by the integral

(8

)

and theseries
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(9)

(10)

(Banderieret al. 2000).

For ,

(11)

(12)

where is thegamma function. Similarly,

(13)

(14)

The asymptotic seriesof has a different form in different quadrants of thecomplex plane, a fact

known as thestokes phenomenon.
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Functions related to the Airy functions have been defined as

(15)

(16)

(17)

(18)

(19)

(20)

where is ageneralized hypergeometric function.
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Watson (1966, pp. 188-190) gives a slightly more general definition of the Airy function as the solution

to theAiry differential equation

(21)

which isfiniteat the origin, where denotes thederivative , , and eithersign is

permitted. Call these solutions , then

(22)

(23)

(24)

where is aBessel function of the first kind. Using the identity

(25)

where is amodified Bessel function of the second kind, the second case can be re-expressed

(26)

(27)

In mathematics, the Airy function Ai(x) is a special functionnamed after the British

astronomerGeorge Biddell Airy. The function Ai(x) and the related function Bi(x),

which is also called an Airy function, are solutions to thedifferential equation
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known as the Airy equation or the Stokes equation. This is the simplest second-

orderlinear differential equationwith a turning point (a point where the character of

the solutions changes from oscillatory to exponential).

The Airy function describes the appearance of a star a point source of light as it

appears in a telescope. The ideal point image becomes a series of concentric ripplesbecause of the limited aperture and the wave nature of light (Suiter 1994). It is also

the solution to Schrdinger's equation for a particle confined within a triangular

potential well and for a particle in a one-dimensional constant force field.

Contents

[hide]

1 Definitions

2 Properties

3 Asymptotic formulas 4 Complex arguments

o 4.1 Plots

5 Relation to other special functions

6 History

7 References

8 External links

[edit] Definitions

Plot of Ai(x) in red and Bi(x) in blue.

For real values ofx, the Airy function is defined by the integral
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Although the function is not strictly integrable (the integrand does not decay as tgoes

to +), the improper integral converges because of the positive and negative parts of

the rapid oscillations tend to cancel one another out(this can be checked by

integration by parts).

By differentiating under the integration sign, we find that y = Ai(x) satisfies thedifferential equation

This equation has two linearly independent solutions. The standard choice for the

other solution is the Airy function of the second kind, denoted Bi(x). It is defined as

the solution with the same amplitude of oscillation as Ai(x) as x goes to which

differs in phase by (1/2):

[edit] Properties

The values of Ai(x) and Bi(x) and their derivatives at x = 0 are given by

Here, denotes theGamma function. It follows that theWronskianof Ai(x) and Bi(x)

is 1/.

When x is positive, Ai(x) is positive, convex, and decreasing exponentially to zero,

while Bi(x) is positive, convex, and increasing exponentially. When x is negative,

Ai(x) and Bi(x) oscillate around zero with ever-increasing frequency and ever-

decreasing amplitude. This is supported by the asymptotic formulas below for the

Airy functions.

[edit] Asymptotic formulas

The asymptotic behaviour of the Airy functions as x goes to + is given by
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For the limit in the negative direction we have

Asymptotic expansionsfor these limits are also available. These are listed in

(Abramowitz and Stegun, 1954) and (Olver, 1974).

[edit] Complex arguments

We can extend the definition of the Airy function to the complex plane by

where the integral is over a path Cstarting at the point at infinity with argument -

(1/3) and ending at the point at infinity with argument (1/3). Alternatively, we can

use the differential equation y'' xy = 0 to extend Ai(x) and Bi(x) to entire functions

on the complex plane.

The asymptotic formula for Ai(x) is still valid in the complex plane if the principal

value ofx

2/3

is taken and x is bounded away from the negative real axis. The formulafor Bi(x) is valid provided x is in the sector {xC : |arg x| < (1/3)} for some

positive . Finally, the formulas for Ai(x) and Bi(x) are valid ifx is in the sector

{xC : |arg x| < (2/3)}.

It follows from the asymptotic behaviour of the Airy functions that both Ai(x) and

Bi(x) have an infinity of zeros on the negative real axis. The function Ai(x) has no

other zeros in the complex plane, while the function Bi(x) also has infinitely many

zeros in the sector {zC : (1/3) < |arg z| < (1/2)}.

[edit] Plots
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For negative arguments, the Airy function are related to the Bessel functions:

Here, J1/3 are solutions ofx2y'' + xy' + (x2 1 / 9)y = 0.

The Scorer's functions solve the equation y'' xy = 1 / . They can also be expressed

in terms of the Airy functions:

[edit] History

The Airy function is named after the British astronomerGeorge Biddell Airy, who

encountered it in his study ofoptics(Airy 1838). The notation Ai(x) was introduced

by Harold Jeffreys.
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Airy Functions