Editing Airy function

{{short description|Special function in the physical sciences}}
{{About|the Airy special function|the Airy stress function employed in solid mechanics|Stress functions|the Airy disk function that describes the optics diffraction pattern through a circular aperture|Airy disk|generic Airy distribution arising from optical resonance between two mirrors|Fabry–Pérot interferometer|the Airy equation as an example of a linear dispersive partial differential equation|Dispersive partial differential equation}}

In the physical sciences, the '''Airy function''' (or '''Airy function of the first kind''') {{math|'''Ai(''x'')'''}} is a [[special function]] named after the British astronomer [[George Biddell Airy]] (1801–1892). The function Ai(''x'') and the related function '''Bi(''x'')''', are [[Linear independence|linearly independent]] solutions to the [[differential equation]]
<math display="block">\frac{d^2y}{dx^2} - xy = 0 , </math>
known as the '''Airy equation''' or the '''Stokes equation'''.

Because the solution of the linear differential equation
<math display="block">\frac{d^2y}{dx^2} - ky = 0</math>
is oscillatory for {{math|''k''<0}} and exponential for {{math|''k''>0}}, the Airy functions are oscillatory for {{math|''x''<0}} and exponential for {{math|''x''>0}}.  In fact, the Airy equation 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).

[[File:Plot of the Airy function Ai(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the Airy function {{math|Ai(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the Airy function {{math|Ai(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]

[[File:Plot of the derivative of the Airy function Ai'(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the derivative of the Airy function {{math|Ai'(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the derivative of the Airy function {{math|Ai'(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]

==Definitions==
[[File:Airy Functions.svg|right|thumb|400px|Plot of {{math|Ai(''x'')}} in red and {{math|Bi(''x'')}} in blue]]
For real values of {{mvar|x}}, the Airy function of the first kind can be defined by the [[improper integral|improper]] [[Riemann integral]]:
<math display="block">\operatorname{Ai}(x) = \dfrac{1}{\pi}\int_0^\infty\cos\left(\dfrac{t^3}{3} + xt\right)\, dt\equiv \dfrac{1}{\pi} \lim_{b\to\infty} \int_0^b \cos\left(\dfrac{t^3}{3} + xt\right)\, dt,</math>
which converges by [[Dirichlet's test]]. For any [[real number]] {{mvar|x}} there is a positive real number {{mvar|M}} such that function <math display="inline">\tfrac{t^3}3 + xt</math> is increasing, unbounded and convex with continuous and unbounded derivative on interval <math>[M,\infty).</math> The convergence of the integral on this interval can be proven by Dirichlet's test after substitution <math display="inline">u=\tfrac{t^3}3 + xt.</math>

{{math|1=''y'' = Ai(''x'')}} satisfies the Airy equation
<math display="block">y'' - xy = 0.</math>
This equation has two [[linear independence|linearly independent]] solutions.
Up to scalar multiplication, {{math|Ai(''x'')}} is the solution subject to the condition {{math|''y'' → 0}} as {{math|''x'' → ∞}}.
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 {{math|Ai(''x'')}} as {{math|''x'' → −∞}} which differs in phase by {{math|''π''/2}}:
[[File:Plot of the Airy function Bi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the Airy function Bi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the Airy function {{math|Bi(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]
<math display="block">\operatorname{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>
[[File:Plot of the derivative of the Airy function Bi'(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the derivative of the Airy function Bi'(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the derivative of the Airy function {{math|Bi'(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]

==Properties==
The values of {{math|Ai(''x'')}} and {{math|Bi(''x'')}} and their derivatives at {{math|1=''x'' = 0}} are given by
<math display="block">\begin{align}
 \operatorname{Ai}(0) &{}= \frac{1}{3^{2/3} \, \Gamma\!\left(\frac{2}{3}\right)}, & \quad \operatorname{Ai}'(0) &{}= -\frac{1}{3^{1/3} \, \Gamma\!\left(\frac{1}{3}\right)}, \\
 \operatorname{Bi}(0) &{}= \frac{1}{3^{1/6} \, \Gamma\!\left(\frac{2}{3}\right)}, & \quad \operatorname{Bi}'(0) &{}= \frac{3^{1/6}}{\Gamma\!\left(\frac{1}{3}\right)}.
\end{align}</math>
Here, {{math|Γ}} denotes the [[Gamma function]]. It follows that the [[Wronskian]] of {{math|Ai(''x'')}} and {{math|Bi(''x'')}} is {{math|1/''π''}}.

When {{mvar|x}} is positive, {{math|Ai(''x'')}} is positive, [[convex function|convex]], and decreasing exponentially to zero, while {{math|Bi(''x'')}} is positive, convex, and increasing exponentially. When {{mvar|x}} is negative, {{math|Ai(''x'')}} and {{math|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.

The Airy functions are orthogonal<ref>{{cite journal | last=Aspnes | first=David E. | title=Electric-Field Effects on Optical Absorption near Thresholds in Solids | journal=Physical Review | volume=147 | issue=2 | date=1966 | issn=0031-899X | doi=10.1103/PhysRev.147.554 | pages=554–566}}</ref> in the sense that
<math display="block"> \int_{-\infty}^\infty \operatorname{Ai}(t+x) \operatorname{Ai}(t+y) dt = \delta(x-y)</math>
again using an improper Riemann integral.

;Real zeros of {{math|Ai(''x'')}} and its derivative {{math|Ai'(''x'')}}
Neither {{math|Ai(''x'')}} nor its [[derivative]] {{math|Ai'(''x'')}} have positive real zeros. The "first" real zeros (i.e. nearest to x=0) are:<ref>{{cite web |url=https://dlmf.nist.gov/9.9 |title=Airy and Related Function |website=dlmf.nist.gov |access-date=9 October 2022}}</ref>
* "first" zeros of {{math|Ai(''x'')}} are at {{math|''x'' ≈ −2.33811, −4.08795, −5.52056, −6.78671, ...}}
* "first" zeros of its derivative {{math|Ai'(''x'')}} are at {{math|''x'' ≈ −1.01879, −3.24820, −4.82010, −6.16331, ...}}

==Asymptotic formulae==
[[File:Mplwp airyai asymptotic.svg|thumb|320px|Ai(blue) and sinusoidal/exponential asymptotic form of Ai(magenta)]]
[[File:Mplwp airybi asymptotic.svg|thumb|320px|Bi(blue) and sinusoidal/exponential asymptotic form of Bi(magenta)]]
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 {{mvar|{{abs|z}}}} goes to infinity at a constant value of {{math|[[arg (mathematics)|arg]](''z'')}} depends on {{math|arg(''z'')}}: this is called the [[Stokes phenomenon]]. For {{math|{{abs|arg(''z'')}} < ''π''}} we have the following [[asymptotic formula]] for {{math|Ai(''z'')}}:<ref name=":0">{{harvtxt|Abramowitz|Stegun|1983|p=448|ignore-err=yes}}, Eqns 10.4.59, 10.4.61</ref>

<math display="block">
\operatorname{Ai}(z)\sim 
\dfrac{1}{2\sqrt\pi\,z^{1/4}} 
\exp\left(-\frac{2}{3}z^{3/2}\right) 
\left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(n+\frac{5}{6}\right) \, \Gamma\!\left(n+\frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].</math>
or<math display="block">
\operatorname{Ai}(z)\sim 
\dfrac{e^{-\zeta}}{4\pi^{3/2}\,z^{1/4}} 
\left[ \sum_{n=0}^{\infty} \dfrac{\Gamma\!\left(n+\frac{5}{6}\right) \, \Gamma\!\left(n+\frac{1}{6}\right)}{n! (-2\zeta)^n} \right].</math>
where <math>\zeta = \tfrac 23 z^{3/2}.</math> In particular, the first few terms are<ref>{{Cite web |title=DLMF: §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions |url=https://dlmf.nist.gov/9.7 |access-date=2023-05-11 |website=dlmf.nist.gov}}</ref><math display="block">\operatorname{Ai}(z) = \frac{e^{-\zeta}}{2\pi^{1/2}z^{1/4}}\left(1 - \frac{5}{72 \zeta} + \frac{385}{10368 \zeta^2} + O(\zeta^{-3})\right)
</math>
There is a similar one for {{math|Bi(''z'')}}, but only applicable when {{math|{{abs|arg(''z'')}} < ''π''/3}}:

<math display="block">
\operatorname{Bi}(z)\sim 
\frac{1}{\sqrt\pi\,z^{1/4}}
\exp\left(\frac{2}{3}z^{3/2}\right)
\left[ \sum_{n=0}^{\infty} \dfrac{ \Gamma\!\left(n+\frac{5}{6}\right) \, \Gamma\!\left(n+\frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].</math>
A more accurate formula for {{math|Ai(''z'')}} and a formula for {{math|Bi(''z'')}} when {{math|''π''/3 < {{abs|arg(''z'')}} < ''π''}} or, equivalently, for {{math|Ai(−''z'')}} and {{math|Bi(−''z'')}} when {{math|{{abs|arg(''z'')}} < 2''π''/3}} but not zero, are:<ref name=":0" /><ref name=":1">{{harvtxt|Abramowitz|Stegun|1983|p=448|ignore-err=yes}}, Eqns 10.4.60 and 10.4.64</ref><math display="block">\begin{align}
 \operatorname{Ai}(-z) \sim&{} \ 
 \frac{1}{\sqrt\pi\,z^{1/4}} 
 \sin\left( \frac{2}{3}z^{3/2} + \frac{\pi}{4} \right) 
 \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6} \right) \, \Gamma\!\left(2n+\frac{1}{6}\right) \left(\frac{3}{4} \right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt]

 &{}-\frac{1}{\sqrt\pi \, z^{1/4}} 
\cos\left(\frac{2}{3}z^{3/2}+\frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right] \\[6pt]
 
 \operatorname{Bi}(-z) \sim&{} 
 \frac{1}{\sqrt\pi \, z^{1/4}}
 \cos \left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6}\right) \, \Gamma\!\left(2n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt]
 
 &{}+ \frac{1}{\sqrt\pi\,z^{\frac{1}{4}}} 
 \sin\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right].
\end{align}</math>

When {{math|1={{abs|arg(''z'')}} = 0}} these are good approximations but are not asymptotic because the ratio between {{math|Ai(−''z'')}} or {{math|Bi(−''z'')}} and the above approximation goes to infinity whenever the sine or cosine goes to zero.
[[Asymptotic analysis|Asymptotic expansions]] for these limits are also available. These are listed in (Abramowitz and Stegun, 1983) and (Olver, 1974).

One is also able to obtain asymptotic expressions for the derivatives {{math|Ai'(z)}} and {{math|Bi'(z)}}. Similarly to before, when {{math|{{abs|arg(''z'')}} < ''π''}}:<ref name=":1" />

<math display="block">
\operatorname{Ai}'(z)\sim 
-\dfrac{z^{1/4}}{2\sqrt\pi\,} 
\exp\left(-\frac{2}{3}z^{3/2}\right)
\left[ \sum_{n=0}^{\infty} \frac{1+6n}{1-6n} \dfrac{(-1)^n \, \Gamma\!\left(n + \frac{5}{6}\right) \, \Gamma\!\left(n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].</math>

When {{math|{{abs|arg(''z'')}} < ''π''/3}} we have:<ref name=":1" />

<math display="block">
\operatorname{Bi}'(z)\sim 
\frac{z^{1/4}}{\sqrt\pi\,}
\exp\left(\frac{2}{3}z^{3/2}\right)
\left[ \sum_{n=0}^{\infty} \frac{1+6n}{1-6n} \dfrac{ \Gamma\!\left(n + \frac{5}{6}\right) \, \Gamma\!\left(n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].</math>

Similarly, an expression for {{math|Ai'(−''z'')}} and {{math|Bi'(−''z'')}} when {{math|{{abs|arg(''z'')}} < 2''π''/3}} but not zero, are<ref name=":1" />

<math display="block">\begin{align}
 \operatorname{Ai}'(-z) \sim&{} 
 -\frac{z^{1/4}}{\sqrt\pi\,} 
 \cos\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \frac{1+12n}{1-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6}\right) \, \Gamma\!\left(2n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt]

 &{}-\frac{z^{1/4}}{\sqrt\pi\,} 
 \sin\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \frac{7+12n}{-5-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right] \\[6pt]
 
 \operatorname{Bi}'(-z) \sim&{} \ 
 \frac{z^{1/4}}{\sqrt\pi\,} 
 \sin\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \frac{1+12n}{1-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6}\right) \, \Gamma\!\left(2n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt]
 
 &{}-\frac{z^{1/4}}{\sqrt\pi\,} 
 \cos\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \frac{7+12n}{-5-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right] \\
\end{align}</math>

==Complex arguments==
We can extend the definition of the Airy function to the complex plane by
<math display="block">\operatorname{Ai}(z) = \frac{1}{2\pi i} \int_{C} \exp\left(\tfrac{t^3}{3} - zt\right)\, dt,</math>
where the integral is over a path ''C'' starting at the point at infinity with argument {{math|−''π''/3}} and ending at the point at infinity with argument π/3. Alternatively, we can use the differential equation {{math|1=''y''′′ − ''xy'' = 0}} to extend {{math|Ai(''x'')}} and {{math|Bi(''x'')}} to [[entire function]]s on the complex plane.

The asymptotic formula for {{math|Ai(''x'')}} is still valid in the complex plane if the principal value of {{math|''x''<sup>2/3</sup>}} is taken and {{mvar|x}} is bounded away from the negative real axis. The formula for {{math|Bi(''x'')}} is valid provided {{mvar|x}} is in the sector <math>x\in\C : \left|\arg(x)\right| < \tfrac{\pi}{3} - \delta</math> for some positive δ. Finally, the formulae for {{math|Ai(−''x'')}} and {{math|Bi(−''x'')}} are valid if {{math|''x''}} is in the sector <math>x\in\C : \left|\arg(x)\right| < \tfrac{2\pi}{3} - \delta.</math>

It follows from the asymptotic behaviour of the Airy functions that both {{math|Ai(''x'')}} and {{math|Bi(''x'')}} have an infinity of zeros on the negative real axis. The function {{math|Ai(''x'')}} has no other zeros in the complex plane, while the function {{math|Bi(''x'')}} also has infinitely many zeros in the sector <math>z\in\C : \tfrac{\pi}{3} < \left|\arg(z)\right| < \tfrac{\pi}{2}.</math>

===Plots===

{| style="text-align:center" align=center
! <math>\Re \left[ \operatorname{Ai} ( x + iy) \right] </math>
! <math>\Im \left[ \operatorname{Ai} ( x + iy) \right] </math>
! <math>\left| \operatorname{Ai} ( x + iy) \right| \, </math>
! <math>\operatorname{arg} \left[ \operatorname{Ai} ( x + iy) \right] \, </math>
|-
|[[File:AiryAi Real Surface.png|200px]]
|[[File:AiryAi Imag Surface.png|200px]]
|[[File:AiryAi Abs Surface.png|200px]]
|[[File:AiryAi Arg Surface.png|200px]]
|-
|[[File:AiryAi Real Contour.svg|200px]]
|[[File:AiryAi Imag Contour.svg|200px]]
|[[File:AiryAi Abs Contour.svg|200px]]
|[[File:AiryAi Arg Contour.svg|200px]]
|}

{| style="text-align:center" align=center
! <math>\Re \left[ \operatorname{Bi} ( x + iy) \right] </math>
! <math>\Im \left[ \operatorname{Bi} ( x + iy) \right] </math>
! <math>\left| \operatorname{Bi} ( x + iy) \right| \, </math>
! <math>\operatorname{arg} \left[ \operatorname{Bi} ( x + iy) \right] \, </math>
|-
|[[File:AiryBi Real Surface.png|200px]]
|[[File:AiryBi Imag Surface.png|200px]]
|[[File:AiryBi Abs Surface.png|200px]]
|[[File:AiryBi Arg Surface.png|200px]]
|-
|[[File:AiryBi Real Contour.svg|200px]]
|[[File:AiryBi Imag Contour.svg|200px]]
|[[File:AiryBi Abs Contour.svg|200px]]
|[[File:AiryBi Arg Contour.svg|200px]]
|}

==Relation to other special functions==
For positive arguments, the Airy functions are related to the [[Bessel function#Modified Bessel functions|modified Bessel functions]]:
<math display="block">\begin{align}
 \operatorname{Ai}(x) &{}= \frac1\pi \sqrt{\frac{x}{3}} \, K_{1/3}\!\left(\frac{2}{3} x^{3/2}\right), \\
 \operatorname{Bi}(x) &{}= \sqrt{\frac{x}{3}} \left[I_{1/3}\!\left(\frac{2}{3} x^{3/2}\right) + I_{-1/3}\!\left(\frac{2}{3} x^{3/2}\right)\right].
\end{align}</math>
Here, {{math|''I''<sub>±1/3</sub>}} and {{math|''K''<sub>1/3</sub>}} are solutions of <math display="block">x^2y'' + xy' - \left (x^2 + \tfrac{1}{9} \right )y = 0.</math>

The first derivative of the Airy function is
<math display="block"> \operatorname{Ai'}(x) = - \frac{x} {\pi \sqrt{3}} \, K_{2/3}\!\left(\frac{2}{3} x^{3/2}\right) .</math>

Functions {{math|''K''<sub>1/3</sub>}} and {{math|''K''<sub>2/3</sub>}} can be represented in terms of rapidly convergent 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]])

For negative arguments, the Airy function are related to the [[Bessel function]]s:
<math display="block">\begin{align}
 \operatorname{Ai}(-x) &{}= \sqrt{\frac{x}{9}} \left[J_{1/3}\!\left(\frac{2}{3} x^{3/2}\right) + J_{-1/3}\!\left(\frac{2}{3} x^{3/2}\right)\right], \\
 \operatorname{Bi}(-x) &{}= \sqrt{\frac{x}{3}} \left[J_{-1/3}\!\left(\frac{2}{3 }x^{3/2}\right) - J_{1/3}\!\left(\frac23 x^{3/2}\right)\right].
\end{align}</math>
Here, {{math|''J''<sub>±1/3</sub>}} are solutions of
<math display="block">x^2y'' + xy' + \left (x^2 - \frac{1}{9} \right )y = 0.</math>

The [[Scorer's function]]s {{math|Hi(''x'')}} and {{math|-Gi(''x'')}} solve the equation {{math|1=''y''′′ − ''xy'' = 1/π}}. They can also be expressed in terms of the Airy functions:
<math display="block">\begin{align}
 \operatorname{Gi}(x) &{}= \operatorname{Bi}(x) \int_x^\infty \operatorname{Ai}(t) \, dt + \operatorname{Ai}(x) \int_0^x \operatorname{Bi}(t) \, dt, \\
 \operatorname{Hi}(x) &{}= \operatorname{Bi}(x) \int_{-\infty}^x \operatorname{Ai}(t) \, dt - \operatorname{Ai}(x) \int_{-\infty}^x \operatorname{Bi}(t) \, dt.
\end{align}</math>

==Fourier transform==
Using the definition of the Airy function Ai(''x''), it is straightforward to show that its [[Fourier transform]] is given by
<math display="block">\mathcal{F}(\operatorname{Ai})(k) := \int_{-\infty}^{\infty} \operatorname{Ai}(x)\ e^{- 2\pi i k x}\,dx = e^{\frac{i}{3} (2\pi k)^3}.</math>This can be obtained by taking the Fourier transform of the Airy equation. Let <math display=inline>\hat y = \frac{1}{2\pi i}\int y e^{-ikx}dx</math>. Then, <math>i\hat y' + k^2 \hat y = 0</math>, which then has solutions <math>\hat y = C e^{ik^3/3}.</math> There is only one dimension of solutions because the Fourier transform requires {{mvar|y}} to decay to zero fast enough; {{math|1=Bi}} grows to infinity exponentially fast, so it cannot be obtained via a Fourier transform.

==Applications==

=== Quantum mechanics ===
The Airy function is the solution to the [[time-independent Schrödinger 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 approximation]], 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 electrons trapped in semiconductor [[heterojunction]]s.

=== Optics ===

A transversally asymmetric optical beam, where the electric field profile is given by the Airy function, has the interesting property that its maximum intensity ''accelerates'' towards one side instead of propagating in a straight line as is the case in symmetric beams. This is at expense of the low-intensity tail being spread in the opposite direction, so the overall momentum of the beam is of course conserved.

=== Caustics ===

The Airy function underlies the form of the intensity near an optical directional [[caustic (optics)|caustic]], such as that of the [[rainbow]] (called supernumerary rainbow). Historically, this was the mathematical problem that led Airy to develop this special function. In 1841, [[William Hallowes Miller]] experimentally measured the analog to supernumerary rainbow by shining light through a thin cylinder of water, then observing through a telescope. He observed up to 30 bands.<ref>[[iarchive:transactionsofca07camb/page/n249/mode/2up|Miller, William Hallowes. "On spurious rainbows." ''Transactions of the Cambridge Philosophical Society'' 7 (1848): 277.]]</ref>

=== Probability ===

In the mid-1980s, the Airy function was found to be intimately connected to [[Chernoff's distribution]].<ref>{{cite journal|title=Chernoff's distribution and differential equations of parabolic and Airy type|last1=Groeneboom|first1=Piet|last2=Lalley|first2=Steven|last3=Temme|first3=Nico|journal=[[Journal of Mathematical Analysis and Applications]]|volume=423|issue=2|pages=1804–1824|year=2015|doi=10.1016/j.jmaa.2014.10.051 |s2cid=119173815 |doi-access=free|arxiv=1305.6053}}</ref>

The Airy function also appears in the definition of [[Tracy–Widom distribution]] which describes the law of largest eigenvalues in [[Random matrix]]. Due to the intimate connection of random matrix theory with the [[Kardar–Parisi–Zhang equation]], there are central processes constructed in KPZ such as the [[Airy process]].<ref>{{cite book|last1=Quastel|first1=Jeremy|last2=Remenik|first2=Daniel|title=Topics in Percolative and Disordered Systems |chapter=Airy Processes and Variational Problems |series=Springer Proceedings in Mathematics & Statistics |year= 2014|volume=69 |pages=121–171 |doi=10.1007/978-1-4939-0339-9_5 |arxiv=1301.0750 |isbn=978-1-4939-0338-2 |s2cid=118241762 |chapter-url=https://link.springer.com/chapter/10.1007/978-1-4939-0339-9_5}}</ref>

==History==
The Airy function is named after the 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.

==See also==
{{Portal|Mathematics|Physics}}
*[[Airy zeta function]]

==Notes==
{{Reflist}}

==References==
*{{AS ref|10|448}}
* {{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=https://books.google.com/books?id=-yI8AAAAMAAJ&q=Transactions+of+the+Cambridge+Philosophical+Society+1838|publisher=University Press|bibcode=1838TCaPS...6..379A}}
* [[Frank William John Olver]] (1974). ''Asymptotics and Special Functions,'' Chapter&nbsp;11. Academic Press, New York.
* {{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 | access-date=2011-08-09 | archive-date=2011-08-11 | archive-url=https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=289 | url-status=dead }}
* {{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 | access-date=2010-05-14 | archive-url=https://web.archive.org/web/20100113044654/http://worldscibooks.com/physics/p345.html | archive-date=2010-01-13 | url-status=dead }}

==External links==
* {{springer|title=Airy functions|id=p/a011210}}
* {{MathWorld | urlname=AiryFunctions | title=Airy Functions}}
* 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.
* {{dlmf|title= Airy and related functions |id=9|first=F. W. J.|last= Olver}}

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[[Category:Special functions]]
[[Category:Special hypergeometric functions]]
[[Category:Ordinary differential equations]]