The second-order linear ordinary differential equation
Since the Airy equation plays an important role in various problems of physics, mechanics and asymptotic analysis, its solutions are regarded as forming a distinct class of special functions (see Airy functions).
The solutions of the Airy equation in the complex plane ,
have the following fundamental properties:
1) Every solution is an entire function of and can be expanded in a power series
which converges for all .
2) If is a solution of the Airy equation, then so are and , where , and any two of these solutions are linearly independent. The following identity holds:
|||G.B. Airy, "On the intensity of light in the neighbourhood of a caustic" Trans. Cambridge Philos. Soc. , 6 (1838) pp. 379–402|
|||V.M. Babich, V.S. Buldyrev, "Asymptotic methods in the diffraction of short waves" , Moscow (1972) (In Russian) (Translation forthcoming: Springer)|
|||M. Abramowitz (ed.) I.A. Stegun (ed.) , Handbook of mathematical functions , Appl. Math. Series , 55 , Nat. Bureau of Standards,, U.S. Department Commerce (1964)|
Airy equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Airy_equation&oldid=15069