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Difference between revisions of "Airy equation"

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The second-order linear ordinary differential equation
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{{MSC|33C10}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a0112001.png" /></td> </tr></table>
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The Airy equation is the second-order linear ordinary differential equation
 
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\[
It occurred first in G.B. Airy's research in optics [[#References|[1]]]. Its general solution can be expressed in terms of [[Bessel functions|Bessel functions]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a0112002.png" />:
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y'' - xy = 0.
 
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\]
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a0112003.png" /></td> </tr></table>
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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 $\pm 1/3$:
 
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\[
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]]).
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y(x) =  
 
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c_1 \sqrt{x} J_{1/3}\left(\frac{2}{3}\mathrm{i}x^{3/2}\right) +
The solutions of the Airy equation in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a0112004.png" />,
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c_2 \sqrt{x} J_{-1/3}\left(\frac{2}{3}\mathrm{i}x^{3/2}\right).
 
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\]
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a0112005.png" /></td> </tr></table>
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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]]).
  
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The solutions of the Airy equation in the complex plane $z$,
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\[
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w'' - zw = 0,
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\]
 
have the following fundamental properties:
 
have the following fundamental properties:
  
1) Every solution is an entire function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a0112006.png" /> and can be expanded in a power series
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1) Every solution is an entire function of $z$ and can be expanded in a power series
 
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\[
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a0112007.png" /></td> </tr></table>
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w(z) =  
 
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w(0)
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a0112008.png" /></td> </tr></table>
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\left(
 
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1 + \frac{z^3}{2.3} + \frac{z^6}{(2.3).(5.6)} + \cdots
which converges for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a0112009.png" />.
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\right)
 
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2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a01120010.png" /> is a solution of the Airy equation, then so are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a01120011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a01120012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a01120013.png" />, and any two of these solutions are linearly independent. The following identity holds:
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w'(0)
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\left(
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z + \frac{z^4}{3.4} + \frac{z^7}{(3.4).(6.7)} + \cdots
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\right),
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\]
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which converges for all $z$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011200/a01120014.png" /></td> </tr></table>
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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:
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\[
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w(z) + w(\omega z) + w(\omega^2 z) \equiv 0.
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\]
  
====References====
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====References====  
<table><TR><TD valign="top">[1]</TD> <TD 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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"V.M. Babich,   V.S. Buldyrev,   "Asymptotic methods in the diffraction of short waves" , Moscow (1972) (In Russian) (Translation forthcoming: Springer)</TD></TR><TR><TD valign="top">[3]</TD> <TD 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)</TD></TR></table>
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{|
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|-
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|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)
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|-
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|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
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|-
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|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)
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|-
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|}

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)
How to Cite This Entry:
Airy equation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Airy_equation&oldid=27208
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article