Airy equation

2020 Mathematics Subject Classification: Primary: 33C10 [MSN][ZBL]

The Airy equation is the second-order linear ordinary differential equation \[ y'' - xy = 0. \] It occurred first in G.B. Airy's research in optics [Ai]. Its general solution can be expressed in terms of Bessel functions of order $\pm 1/3$: \[ y(x) = c_1 \sqrt{x} J_{1/3}\left(\frac{2}{3}\mathrm{i}x^{3/2}\right) + c_2 \sqrt{x} J_{-1/3}\left(\frac{2}{3}\mathrm{i}x^{3/2}\right). \] 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 $z$, \[ w'' - zw = 0, \] have the following fundamental properties:

1) Every solution is an entire function of $z$ and can be expanded in a power series \[ w(z) = w(0) \left( 1 + \frac{z^3}{2 \cdot 3} + \frac{z^6}{(2\cdot 3) \cdot(5 \cdot 6)} + \cdots \right) + w'(0) \left( z + \frac{z^4}{3 \cdot 4} + \frac{z^7}{(3 \cdot 4) \cdot(6 \cdot7)} + \cdots \right), \] which converges for all $z$.

2) If $w(z) \not\equiv 0$ is a solution of the Airy equation, then so are $w(\omega z)$ and $w(\omega^2 z)$, where $w=\mathrm{e}^{2\pi\mathrm{i}/3}$, and any two of these solutions are linearly independent. The following identity holds: \[ w(z) + w(\omega z) + w(\omega^2 z) \equiv 0. \]

References

[AbSt]	M. Abramowitz (ed.) I.A. Stegun (ed.), Handbook of mathematical functions, Appl. Math. Series, 55, Nat. Bureau of Standards,, U.S. Department Commerce (1964)
[Ai]	G.B. Airy, "On the intensity of light in the neighbourhood of a caustic" Trans. Cambridge Philos. Soc., 6 (1838) pp. 379–402
[BaBu]	V.M. Babich, V.S. Buldyrev, "Asymptotic methods in the diffraction of short waves", Moscow (1972) (In Russian) (Translation: "Short-Wavelength Diffraction Theory. Asymptotic Methods", Springer, 1991)