| 释义 |
Airy FunctionsWatson's (1966, pp. 188-190) definition of an Airy function is the solution to the Airy Differential Equation
 | (1) |
 denotes the Derivative  ,  , and either Sign is permitted. Call these solutions  , then
 | (2) |
where  is a Bessel Function of the First Kind and  is a Modified Bessel Function of the FirstKind. Using the identity
 | (5) |
 is a Modified Bessel Function of the Second Kind, the second case can be re-expressed
A more commonly used definition of Airy functions is given by Abramowitz and Stegun (1972, pp. 446-447) and illustratedabove. This definition identifies the  and  functions as the two Linearly Independentsolutions to (1) with  and a Minus Sign,
 | (9) |
 | (10) |
In the above plot,  is the solid curve and  is dashed. For zero argument,
 | (13) |
 is the Gamma Function. This means that Watson's expression becomes
 | (14) |
The Asymptotic Series of has a different form in different Quadrants of theComplex Plane, a fact known as the Stokes Phenomenon. Functions related to the Airy functions have beendefined as
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. |
