Airy Functions

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 are entire functions.

A generalization of the Airy function was constructed by Hardy.

The Airy function and functions are plotted above along the real axis.

The and functions are defined as the two linearly independent solutions to

(1)

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

(2)

where

			(3)
			(4)

where 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 , the functions can be written as

			(5)
			(6)
			(7)

where is a modified Bessel function of the first kind and is a modified Bessel function of the second kind.

Min Max Re Im

Plots of in the complex plane are illustrated above.

Min Max Re Im

Similarly, plots of appear above.

The Airy function is given by the integral

(8)

and the series

			(9)
			(10)

(Banderier et al. 2000).

For ,

			(11)
			(12)

where is the gamma function. Similarly,

			(13)
			(14)

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

Functions related to the Airy functions have been defined as

			(15)
			(16)
			(17)
			(18)
			(19)
			(20)

where 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

(21)

which is finite at the origin, where denotes the derivative , , and either sign is permitted. Call these solutions , then

(22)

			(23)
			(24)

where is a Bessel function of the first kind. Using the identity

(25)

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

			(26)
			(27)
			(28)

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