Airy function

In the physical sciences, the Airy function (or Airy function of the first kind) Template:Math 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 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 Template:Math and exponential for Template:Math, the Airy functions are oscillatory for Template:Math and exponential for Template:Math. 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

Plot of the Airy function Template:Math in the complex plane from Template:Math to Template:Math 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

Plot of the derivative of the Airy function Template:Math in the complex plane from Template:Math to Template:Math with colors created with Mathematica 13.1 function ComplexPlot3D

Definitions

File:Airy Functions.svg

Plot of Template:Math in red and Template:Math in blue

For real values of Template:Mvar, the Airy function of the first kind can be defined by the 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 Template:Mvar there is a positive real number Template:Mvar 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>

Template:Math satisfies the Airy equation <math display="block">y - xy = 0.</math> This equation has two linearly independent solutions. Up to scalar multiplication, Template:Math is the solution subject to the condition Template:Math as Template:Math. 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 Template:Math as Template:Math which differs in phase by Template:Math:

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

Plot of the Airy function Template:Math in the complex plane from Template:Math to Template:Math 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

Plot of the derivative of the Airy function Template:Math in the complex plane from Template:Math to Template:Math with colors created with Mathematica 13.1 function ComplexPlot3D

Properties

The values of Template:Math and Template:Math and their derivatives at Template:Math 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, Template:Math denotes the Gamma function. It follows that the Wronskian of Template:Math and Template:Math is Template:Math.

When Template:Mvar is positive, Template:Math is positive, convex, and decreasing exponentially to zero, while Template:Math is positive, convex, and increasing exponentially. When Template:Mvar is negative, Template:Math and Template:Math 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>Template:Cite journal</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 Template:Math and its derivative Template:Math

Neither Template:Math nor its derivative Template:Math have positive real zeros. The "first" real zeros (i.e. nearest to x=0) are:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

• "first" zeros of Template:Math are at Template:Math
• "first" zeros of its derivative Template:Math are at Template:Math

Asymptotic formulae

File:Mplwp airyai asymptotic.svg

Ai(blue) and sinusoidal/exponential asymptotic form of Ai(magenta)

File:Mplwp airybi asymptotic.svg

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 functions. The asymptotic behaviour of the Airy functions as Template:Mvar goes to infinity at a constant value of Template:Math depends on Template:Math: this is called the Stokes phenomenon. For Template:Math we have the following asymptotic formula for Template:Math:<ref name=":0">Template:Harvtxt, 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>{{#invoke:citation/CS1|citation |CitationClass=web }}</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 Template:Math, but only applicable when Template:Math:

<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 Template:Math and a formula for Template:Math when Template:Math or, equivalently, for Template:Math and Template:Math when Template:Math but not zero, are:<ref name=":0" /><ref name=":1">Template:Harvtxt, 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 Template:Math these are good approximations but are not asymptotic because the ratio between Template:Math or Template:Math and the above approximation goes to infinity whenever the sine or cosine goes to zero. 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 Template:Math and Template:Math. Similarly to before, when Template:Math:<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 Template:Math 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 Template:Math and Template:Math when Template:Math 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 Template:Math and ending at the point at infinity with argument π/3. Alternatively, we can use the differential equation Template:Math to extend Template:Math and Template:Math to entire functions on the complex plane.

The asymptotic formula for Template:Math is still valid in the complex plane if the principal value of Template:Math is taken and Template:Mvar is bounded away from the negative real axis. The formula for Template:Math is valid provided Template:Mvar 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 Template:Math and Template:Math are valid if Template:Math 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 Template:Math and Template:Math have an infinity of zeros on the negative real axis. The function Template:Math has no other zeros in the complex plane, while the function Template:Math also has infinitely many zeros in the sector <math>z\in\C : \tfrac{\pi}{3} < \left|\arg(z)\right| < \tfrac{\pi}{2}.</math>

Plots

<math>\Re \left[ \operatorname{Ai} ( x + iy) \right] </math>	<math>\Im \left[ \operatorname{Ai} ( x + iy) \right] </math>	\operatorname{Ai} ( x + iy) \right| \, </math>	<math>\operatorname{arg} \left[ \operatorname{Ai} ( x + iy) \right] \, </math>
File:AiryAi Real Surface.png	File:AiryAi Imag Surface.png	File:AiryAi Abs Surface.png	File:AiryAi Arg Surface.png
File:AiryAi Real Contour.svg	File:AiryAi Imag Contour.svg	File:AiryAi Abs Contour.svg	File:AiryAi Arg Contour.svg

<math>\Re \left[ \operatorname{Bi} ( x + iy) \right] </math>	<math>\Im \left[ \operatorname{Bi} ( x + iy) \right] </math>	\operatorname{Bi} ( x + iy) \right| \, </math>	<math>\operatorname{arg} \left[ \operatorname{Bi} ( x + iy) \right] \, </math>
File:AiryBi Real Surface.png	File:AiryBi Imag Surface.png	File:AiryBi Abs Surface.png	File:AiryBi Arg Surface.png
File:AiryBi Real Contour.svg	File:AiryBi Imag Contour.svg	File:AiryBi Abs Contour.svg	File:AiryBi Arg Contour.svg

Relation to other special functions

For positive arguments, the Airy functions are related to the 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, Template:Math and Template:Math 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 Template:Math and Template:Math 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 modified Bessel functions)

For negative arguments, the Airy function are related to the Bessel functions: <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, Template:Math are solutions of <math display="block">x^2y + xy' + \left (x^2 - \frac{1}{9} \right )y = 0.</math>

The Scorer's functions Template:Math and Template:Math solve the equation Template:Math. 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 Template:Mvar to decay to zero fast enough; Template:Math 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 heterojunctions.

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, 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>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>Template:Cite journal</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>Template:Cite book</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

• Airy zeta function

References

• Template:AS ref
• Template:Citation
• Frank William John Olver (1974). Asymptotics and Special Functions, Chapter 11. Academic Press, New York.
• Template:Citation
• Template:Citation

External links

• Template:Springer
• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:AiryFunctions%7CAiryFunctions.html}} |title = Airy Functions |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• Wolfram function pages for Ai and Bi functions. Includes formulas, function evaluator, and plotting calculator.
• Template:Dlmf