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{{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)<ref>{{AS ref|10|448}}</ref> and Olver (1974).<ref>[[Frank_William_John_Olver]] (1974). ''Asymptotics and Special Functions,'' Chapter 11. Academic Press, New York.</ref>

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.<ref>{{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}}</ref> 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==
* {{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]]