## Airy Functions

Watson's (1966, pp. 188-190) definition of an Airy function is the solution to the Airy Differential Equation

 (1)

which is Finite at the Origin, where denotes the Derivative , , and either Sign is permitted. Call these solutions , then
 (2)

 (3) (4)

where is a Bessel Function of the First Kind and is a Modified Bessel Function of the First Kind. Using the identity
 (5)

where is a Modified Bessel Function of the Second Kind, the second case can be re-expressed
 (6) (7) (8)

A more commonly used definition of Airy functions is given by Abramowitz and Stegun (1972, pp. 446-447) and illustrated above. This definition identifies the and functions as the two Linearly Independent solutions to (1) with and a Minus Sign,

 (9)

The solutions are then written
 (10)

where
 (11) (12)

In the above plot, is the solid curve and is dashed. For zero argument,
 (13)

where is the Gamma Function. This means that Watson's expression becomes
 (14)

A generalization has been constructed by Hardy.

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)

References

Abramowitz, M. and Stegun, C. 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.

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.

Spanier, J. and Oldham, K. B. The Airy Functions Ai() and Bi().'' 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.