View source for Airy equation

{{MSC|33C10}}
{{TEX|done}}

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 {{Cite|Ai}}. Its general solution can be expressed in terms of [[Bessel functions|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|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
\[
w(z) = 
w(0)
\left( 
1 + \frac{z^3}{2 \cdot 3} + \frac{z^6}{(2\cdot 3) \cdot(5 \cdot 6)} + \cdots
\right)
+
w'(0)
\left( 
z + \frac{z^4}{3.4} + \frac{z^7}{(3 \cdot 4) \cdot(6 \cdot7)} + \cdots
\right),
\]
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==== 
{|
|-
|valign="top"|{{Ref|AbSt}}||valign="top"| M. Abramowitz (ed.)  I.A. Stegun (ed.), ''Handbook of mathematical functions'', ''Appl. Math. Series'', '''55''', Nat. Bureau of Standards,, U.S. Department Commerce (1964)
|-
|valign="top"|{{Ref|Ai}}||valign="top"| G.B. Airy, "On the intensity of light in the neighbourhood of a caustic" ''Trans. Cambridge Philos. Soc.'', '''6''' (1838) pp. 379–402
|-
|valign="top"|{{Ref|BaBu}}||valign="top"| 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)
|-
|}