Watson's (1966, pp. 188-190) definition of an Airy function is the solution to the Airy Differential Equation
(1) |
(2) |
(3) | |||
(4) |
(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) |
(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) |
See also Airy-Fock Functions
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 Eric W. Weisstein