Difference between revisions of "Airy equation"
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− | + | {{MSC|33C10}} | |
+ | {{TEX|done}} | ||
− | + | The Airy equation is the second-order linear ordinary differential equation | |
− | + | \[ | |
− | It occurred first in G.B. Airy's research in optics | + | y'' - xy = 0. |
− | + | \] | |
− | + | 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 $\pm 1/3$: | |
− | + | \[ | |
− | Since the Airy equation plays an important role in various problems of physics, mechanics and asymptotic analysis, its solutions are regarded as forming a distinct class of special functions (see [[Airy functions|Airy functions]]). | + | y(x) = |
− | + | c_1 \sqrt{x} J_{1/3}\left(\frac{2}{3}\mathrm{i}x^{3/2}\right) + | |
− | + | c_2 \sqrt{x} J_{-1/3}\left(\frac{2}{3}\mathrm{i}x^{3/2}\right). | |
− | + | \] | |
− | + | 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: | have the following fundamental properties: | ||
− | 1) Every solution is an entire function of | + | 1) Every solution is an entire function of $z$ and can be expanded in a power series |
− | + | \[ | |
− | + | w(z) = | |
− | + | w(0) | |
− | + | \left( | |
− | + | 1 + \frac{z^3}{2.3} + \frac{z^6}{(2.3).(5.6)} + \cdots | |
− | which converges for all | + | \right) |
− | + | + | |
− | + | w'(0) | |
+ | \left( | ||
+ | z + \frac{z^4}{3.4} + \frac{z^7}{(3.4).(6.7)} + \cdots | ||
+ | \right), | ||
+ | \] | ||
+ | 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==== | + | ====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) | ||
+ | |- | ||
+ | |} |
Latest revision as of 18:03, 26 July 2012
2010 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 \[ w(z) = w(0) \left( 1 + \frac{z^3}{2.3} + \frac{z^6}{(2.3).(5.6)} + \cdots \right) + w'(0) \left( z + \frac{z^4}{3.4} + \frac{z^7}{(3.4).(6.7)} + \cdots \right), \] 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) |
Airy equation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Airy_equation&oldid=27208