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| {{about|the Airy special function|the Airy stress function employed in solid mechanics|Stress functions}}
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| In the physical sciences, the '''Airy function''' '''Ai(''x'')''' is a [[special function]] named after the British astronomer [[George Biddell Airy]] (1801–92). The function Ai(''x'') and the related function '''Bi(''x'')''', which is also called the '''Airy function''', but sometimes referred to as the Bairy function, are solutions to the [[differential equation]]
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| :<math>\frac{d^2y}{dx^2} - xy = 0 , \,\!</math>
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| known as the '''Airy equation''' or the '''Stokes equation'''. This is the simplest second-order [[linear differential equation]] with a turning point (a point where the character of the solutions changes from oscillatory to exponential). | |
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| The Airy function is the solution to [[Schrödinger's equation]] for a particle confined within a triangular [[potential well]] and for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniform semiclassical approximations near a turning point in the [[WKB method]], when the potential may be locally approximated by a linear function of position. The triangular potential well solution is directly relevant for the understanding of many semiconductor devices.
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| The Airy function also underlies the form of the intensity near an optical directional [[caustic (optics)|caustic]], such as that of the [[rainbow]]. Historically, this was the mathematical problem that led Airy to develop this special function. The Airy function is also important in [[microscopy]] and [[astronomy]]; it describes the pattern, due to [[diffraction]] and [[interference]], produced by a [[point source]] of light (one which is smaller than the resolution limit of a [[microscope]] or [[telescope]]).
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| ==Definitions==
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| [[Image:Airy Functions.svg|right|thumb|400px|Plot of Ai(''x'') in red and Bi(''x'') in blue]]
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| For real values of ''x'', the Airy function of the first kind can be defined by the [[improper Riemann integral]]:
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| :<math>\mathrm{Ai}(x) = \frac{1}{\pi}\int_0^\infty\cos\left(\tfrac{t^3}{3} + xt\right)\, dt\equiv \frac{1}{\pi}\lim_{b\to\infty} \int_0^b \cos\left(\tfrac{t^3}{3} + xt\right)\, dt,</math>
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| which converges because the positive and negative parts of the [[Riemann-Lebesgue lemma|rapid oscillations tend to cancel one another out]] (as can be checked by [[integration by parts]]).
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| ''y'' = Ai(''x'') satisfies the Airy equation
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| :<math>y'' - xy = 0.</math>
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| This equation has two [[linear independence|linearly independent]] solutions.
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| Up to scalar multiplication, Ai(''x'') is the solution subject to the condition ''y'' → 0 as ''x'' → ∞.
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| The standard choice for the other solution is the Airy function of the second kind, denoted Bi(''x''). It is defined as the solution with the same amplitude of oscillation as Ai(''x'') as ''x'' → −∞ which differs in phase by π/2:
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| :<math>\mathrm{Bi}(x) = \frac{1}{\pi} \int_0^\infty \left[\exp\left(-\tfrac{t^3}{3} + xt\right) + \sin\left(\tfrac{t^3}{3} + xt\right)\,\right]dt.</math>
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| ==Properties==
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| The values of Ai(''x'') and Bi(''x'') and their derivatives at ''x'' = 0 are given by
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| :<math>\begin{align}
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| \mathrm{Ai}(0) &{}= \frac{1}{3^{\frac{2}{3}}\Gamma(\tfrac23)}, & \quad \mathrm{Ai}'(0) &{}= -\frac{1}{3^{\frac{1}{3}}\Gamma(\tfrac13)}, \\
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| \mathrm{Bi}(0) &{}= \frac{1}{3^{\frac{1}{6}}\Gamma(\tfrac23)}, & \quad \mathrm{Bi}'(0) &{}= \frac{3^{\frac{1}{6}}}{\Gamma(\tfrac13)}.
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| \end{align}</math>
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| Here, Γ denotes the [[Gamma function]]. It follows that the [[Wronskian]] of Ai(''x'') and Bi(''x'') is 1/π.
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| When ''x'' is positive, Ai(''x'') is positive, [[convex function|convex]], and decreasing exponentially to zero, while Bi(''x'') is positive, convex, and increasing exponentially. When ''x'' is negative, Ai(''x'') and Bi(''x'') oscillate around zero with ever-increasing frequency and ever-decreasing amplitude. This is supported by the asymptotic formulae below for the Airy functions.
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| The Airy functions are orthogonal<ref>David E. Aspnes, Physical Review, '''147''', 554 (1966)</ref> in the sense that
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| :<math> \int_{-\infty}^\infty \mathrm{Ai}(t+x) \mathrm{Ai}(t+y) dt = \delta(x-y)</math>
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| again using an improper Riemann integral.
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| ==Asymptotic formulae==
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| As explained below, the Airy functions can be extended to the complex plane, giving [[entire function]]s. The asymptotic behaviour of the Airy functions as ''|z|'' goes to infinity at a constant value of arg(''z'') depends on arg(''z''). For |arg(''z'')| < π we have the following [[asymptotic formula]] for Ai(''z''):<ref>{{harvtxt|Abramowitz|Stegun|1970|p=[http://people.math.sfu.ca/~cbm/aands/page_448.htm 448]}}, Eqns 10.4.59 and 10.4.63</ref>
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| :<math>\mathrm{Ai}(z)\sim \frac{e^{-\frac{2}{3}z^{\frac{3}{2}}}}{2\sqrt\pi\,z^{\frac{1}{4}}}</math>
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| and a similar one for Bi(''z''), but only applicable when |arg(''z'')| < π/3:
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| :<math>\mathrm{Bi}(z)\sim \frac{e^{\frac{2}{3}z^{\frac{3}{2}}}}{\sqrt\pi\,z^{\frac{1}{4}}}.</math>
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| A more accurate formula for Ai(''z'') and a formula for Bi(''z'') when π/3 < |arg(''z'')| < π or, equivalently, for Ai(−''z'') and Bi(−''z'') when |arg(''z'')| < 2π/3 but not zero, are:<ref>{{harvtxt|Abramowitz|Stegun|1970|p=[http://people.math.sfu.ca/~cbm/aands/page_448.htm 448]}}, Eqns 10.4.60 and 10.4.64</ref>
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| :<math>\begin{align}
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| \mathrm{Ai}(-z) &{}\sim \frac{\sin \left(\frac23z^{\frac{3}{2}}+\frac{\pi}{4} \right)}{\sqrt\pi\,z^{\frac{1}{4}}} \\[6pt]
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| \mathrm{Bi}(-z) &{}\sim \frac{\cos \left(\frac23z^{\frac{3}{2}}+\frac{\pi}{4} \right)}{\sqrt\pi\,z^{\frac{1}{4}}}.
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| \end{align}</math>
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| When |arg(''z'')| = 0 these are good approximations but are not asymptotic because the ratio between Ai(−''z'') or Bi(−''z'') and the above approximation goes to infinity whenever the sine or cosine goes to zero.
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| [[Asymptotic analysis|Asymptotic expansions]] for these limits are also available. These are listed in (Abramowitz and Stegun, 1954) and (Olver, 1974).
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| ==Complex arguments==
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| We can extend the definition of the Airy function to the complex plane by
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| :<math>\mathrm{Ai}(z) = \frac{1}{2\pi i} \int_{C} \exp\left(\tfrac{t^3}{3} - zt\right)\, dt,</math>
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| where the integral is over a path ''C'' starting at the point at infinity with argument −π/2 and ending at the point at infinity with argument π/2. Alternatively, we can use the differential equation ''y''′′ − ''xy'' = 0 to extend Ai(''x'') and Bi(''x'') to [[entire function]]s on the complex plane.
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| The asymptotic formula for Ai(''x'') is still valid in the complex plane if the principal value of ''x''<sup>2/3</sup> is taken and ''x'' is bounded away from the negative real axis. The formula for Bi(''x'') is valid provided ''x'' is in the sector {''x'' ∈ '''C''' : |arg(''x'')| < (π/3)−δ} for some positive δ. Finally, the formulae for Ai(−''x'') and Bi(−''x'') are valid if ''x'' is in the sector {''x'' ∈ '''C''' : |arg(''x'')| < (2π/3)−δ}.
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| It follows from the asymptotic behaviour of the Airy functions that both Ai(''x'') and Bi(''x'') have an infinity of zeros on the negative real axis. The function Ai(''x'') has no other zeros in the complex plane, while the function Bi(''x'') also has infinitely many zeros in the sector {''z'' ∈ '''C''' : π/3 < |arg(''z'')| < π/2}.
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| ===Plots===
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| {| style="text-align:center" align=center
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| ! <math>\Re \left[ \mathrm{Ai} ( x + iy) \right] </math>
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| ! <math>\Im \left[ \mathrm{Ai} ( x + iy) \right] </math>
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| ! <math>| \mathrm{Ai} ( x + iy) | \, </math>
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| ! <math>\mathrm{arg} \left[ \mathrm{Ai} ( x + iy) \right] \, </math>
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| |-
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| |[[Image:AiryAi Real Surface.png|200px]]
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| |[[Image:AiryAi Imag Surface.png|200px]]
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| |[[Image:AiryAi Abs Surface.png|200px]]
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| |[[Image:AiryAi Arg Surface.png|200px]]
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| |-
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| |[[Image:AiryAi Real Contour.svg|200px]]
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| |[[Image:AiryAi Imag Contour.svg|200px]]
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| |[[Image:AiryAi Abs Contour.svg|200px]]
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| |[[Image:AiryAi Arg Contour.svg|200px]]
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| |}
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| {| style="text-align:center" align=center
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| ! <math>\Re \left[ \mathrm{Bi} ( x + iy) \right] </math>
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| ! <math>\Im \left[ \mathrm{Bi} ( x + iy) \right] </math>
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| ! <math>| \mathrm{Bi} ( x + iy) | \, </math>
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| ! <math>\mathrm{arg} \left[ \mathrm{Bi} ( x + iy) \right] \, </math>
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| |-
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| |[[Image:AiryBi Real Surface.png|200px]]
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| |[[Image:AiryBi Imag Surface.png|200px]]
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| |[[Image:AiryBi Abs Surface.png|200px]]
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| |[[Image:AiryBi Arg Surface.png|200px]]
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| |-
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| |[[Image:AiryBi Real Contour.svg|200px]]
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| |[[Image:AiryBi Imag Contour.svg|200px]]
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| |[[Image:AiryBi Abs Contour.svg|200px]]
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| |[[Image:AiryBi Arg Contour.svg|200px]]
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| |}
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| ==Relation to other special functions==
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| For positive arguments, the Airy functions are related to the [[Bessel function#Modified Bessel functions|modified Bessel functions]]:
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| :<math>\begin{align}
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| \mathrm{Ai}(x) &{}= \frac1\pi \sqrt{\frac{x}{3}} \, K_{\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right), \\
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| \mathrm{Bi}(x) &{}= \sqrt{\frac{x}{3}} \left(I_{\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right) + I_{-\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right)\right).
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| \end{align}</math>
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| Here, ''I''<sub>±1/3</sub> and ''K''<sub>1/3</sub> are solutions of
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| :<math>x^2y'' + xy' - \left (x^2 + \tfrac{1}{9} \right )y = 0.</math>
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| The first derivative of Airy function is
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| :<math> \mathrm{Ai'}(x) = - \frac{x} {\pi \sqrt{3}} \, K_{\frac{2}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right) .</math>
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| Functions ''K''<sub>1/3</sub> and ''K''<sub>2/3</sub> can be represented in terms of rapidly converged integrals<ref>M.Kh.Khokonov. Cascade Processes of Energy Loss by Emission of Hard Photons // JETP, V.99, No.4, pp. 690-707 \ (2004).</ref> (see also [[Bessel function#Modified Bessel functions|modified Bessel functions]] )
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| For negative arguments, the Airy function are related to the [[Bessel function]]s:
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| :<math>\begin{align}
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| \mathrm{Ai}(-x) &{}= \sqrt{\frac{x}{9}} \left(J_{\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right) + J_{-\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right)\right), \\
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| \mathrm{Bi}(-x) &{}= \sqrt{\frac{x}{3}} \left(J_{-\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right) - J_{\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right)\right).
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| \end{align}</math>
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| Here, ''J''<sub>±1/3</sub> are solutions of
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| :<math>x^2y'' + xy' + \left (x^2 - \tfrac{1}{9} \right )y = 0.</math>
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| The [[Scorer's function]]s solve the equation ''y''′′ − ''xy'' = 1/π. They can also be expressed in terms of the Airy functions:
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| :<math>\begin{align} | |
| \mathrm{Gi}(x) &{}= \mathrm{Bi}(x) \int_x^\infty \mathrm{Ai}(t) \, dt + \mathrm{Ai}(x) \int_0^x \mathrm{Bi}(t) \, dt, \\
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| \mathrm{Hi}(x) &{}= \mathrm{Bi}(x) \int_{-\infty}^x \mathrm{Ai}(t) \, dt - \mathrm{Ai}(x) \int_{-\infty}^x \mathrm{Bi}(t) \, dt.
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| \end{align}</math>
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| ==Fourier transform==
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| Using the definition of the Airy function Ai(''x''), it is straightforward to show its [[Fourier transform]] is given by
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| :<math>\mathcal{F}(\mathrm{Ai})(k) := \int_{-\infty}^{\infty} \mathrm{Ai}(x)\ e^{- 2\pi i k x}\,dx = e^{\frac{i}{3}(2\pi k)^3}.</math>
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| ==Fabry–Pérot interferometer Airy Function==
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| The transmittance function of a [[Fabry–Pérot interferometer]] is also referred to as the ''Airy Function'':<ref>{{cite book | first=Eugene|last=Hecht|year=1987|title=Optics|edition=2nd ed.|publisher=Addison Wesley|isbn=0-201-11609-X}} Sect. 9.6</ref>
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| :<math>T_e = \frac{1}{1+F\sin^2(\frac{\delta}{2})},</math>
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| where both surfaces have reflectance ''R'' and
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| :<math> F = \frac{4R}{{(1-R)^2}}</math>
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| is the ''coefficient of finesse''.
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| ==History==
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| The Airy function is named after the [[British (people)|British]] [[astronomer]] and [[physicist]] [[George Biddell Airy]] (1801–1892), who encountered it in his early study of [[optics]] in physics (Airy 1838). The notation Ai(''x'') was introduced by [[Harold Jeffreys]]. Airy had become the British [[Astronomer Royal]] in 1835, and he held that post until his retirement in 1881.
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| ==See also==
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| *The proof of [[Witten's conjecture]] used a matrix-valued generalization of the Airy function.
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| *[[Airy zeta function]]
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| ==Notes==
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| <references />
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| ==References==
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| *{{AS ref|10|446}}
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| * {{citation|last=Airy |year=1838|title= On the intensity of light in the neighbourhood of a caustic|journal=Transactions of the Cambridge Philosophical Society|volume=6|pages= 379–402|url=http://books.google.com/?id=-yI8AAAAMAAJ&dq=Transactions+of+the+Cambridge+Philosophical+Society+1838|publisher=University Press}}
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| * Olver (1974). ''Asymptotics and Special Functions,'' Chapter 11. Academic Press, New York.
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| * {{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 6.6.3. Airy Functions | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=289}}
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| * {{Citation | last1=Vallée | first1=Olivier | last2=Soares | first2=Manuel | title=Airy functions and applications to physics | url=http://www.worldscibooks.com/physics/p345.html | publisher=Imperial College Press | location=London | isbn=978-1-86094-478-9 | mr=2114198 | year=2004}}
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| ==External links==
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| * {{springer|title=Airy functions|id=p/a011210}}
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| * {{MathWorld | urlname=AiryFunctions | title=Airy Functions}}
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| * Wolfram function pages for [http://functions.wolfram.com/Bessel-TypeFunctions/AiryAi/ Ai] and [http://functions.wolfram.com/Bessel-TypeFunctions/AiryBi/ Bi] functions. Includes formulas, function evaluator, and plotting calculator.
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| * {{dlmf|title= Airy and related functions |id=9|first=F. W. J.|last= Olver}}
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| [[Category:Special functions]]
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| [[Category:Special hypergeometric functions]]
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| [[Category:Ordinary differential equations]]
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