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

(1) |

(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) |

(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) |

(10) |

(11) | |||

(12) |

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

(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) |

**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.

© 1996-9

1999-05-25