Airy function explained

In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function, are linearly independent solutions to the differential equation$$\backslash frac\; -\; xy\; =\; 0,$$known as the Airy equation or the Stokes equation.

Because the solution of the linear differential equation$$\backslash frac\; -\; ky\; =\; 0$$is oscillatory for and exponential for, the Airy functions are oscillatory for and exponential for . 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).

Definitions

For real values of, the Airy function of the first kind can be defined by the improper Riemann integral:$$\backslash operatorname(x)\; =\; \backslash dfrac\backslash int\_0^\backslash infty\backslash cos\backslash left(\backslash dfrac\; +\; xt\backslash right)\backslash ,\; dt\backslash equiv\; \backslash dfrac\; \backslash lim\_\; \backslash int\_0^b\; \backslash cos\backslash left(\backslash dfrac\; +\; xt\backslash right)\backslash ,\; dt,$$which converges by Dirichlet's test. For any real number there is a positive real number such that function $\backslash tfrac3\; +\; xt$ is increasing, unbounded and convex with continuous and unbounded derivative on interval

[M, inf ty). The convergence of the integral on this interval can be proven by Dirichlet's test after substitution $u=\backslash tfrac3\; +\; xt.$

satisfies the Airy equation$$y$$ - xy = 0.This equation has two linearly independent solutions.Up to scalar multiplication, is the solution subject to the condition as .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 as which differs in phase by :$$\backslash operatorname(x)\; =\; \backslash frac\; \backslash int\_0^\backslash infty\; \backslash left[\backslash exp\backslash left(-\backslash tfrac\{t^3\}\{3\}\; +\; xt\backslash right)\; +\; \backslash sin\backslash left(\backslash tfrac\{t^3\}\{3\}\; +\; xt\backslash right)\backslash ,\backslash right]dt.$$

Properties

The values of and and their derivatives at are given byHere, denotes the Gamma function. It follows that the Wronskian of and is .

When is positive, is positive, convex, and decreasing exponentially to zero, while is positive, convex, and increasing exponentially. When is negative, and 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^[1] in the sense that$$\backslash int\_^\backslash infty\; \backslash operatorname(t+x)\; \backslash operatorname(t+y)\; dt\; =\; \backslash delta(x-y)$$again using an improper Riemann integral.

Real zeros of and its derivative Neither nor its derivative have positive real zeros. The "first" real zeros (i.e. nearest to x=0) are:^[2]

• "first" zeros of are at
• "first" zeros of its derivative are at

Asymptotic formulae

As explained below, the Airy functions can be extended to the complex plane, giving entire functions. The asymptotic behaviour of the Airy functions as goes to infinity at a constant value of depends on : this is called the Stokes phenomenon. For we have the following asymptotic formula for :^[3]

$$\backslash operatorname(z)\backslash sim\; \backslash dfrac\; \backslash exp\backslash left(-\backslash fracz^\backslash right)\; \backslash left[\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash dfrac\{(-1)^n\; \backslash ,\; \backslash Gamma\backslash !\backslash left(n+\backslash frac\{5\}\{6\}\backslash right)\; \backslash ,\; \backslash Gamma\backslash !\backslash left(n+\backslash frac\{1\}\{6\}\backslash right)\; \backslash left(\backslash frac\{3\}\{4\}\backslash right)^n\}\{2\backslash pi\; \backslash ,\; n!\; \backslash ,\; z^\{3n/2\}\}\; \backslash right].$$or$$\backslash operatorname(z)\backslash sim\; \backslash dfrac\; \backslash left[\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash dfrac\{\backslash Gamma\backslash !\backslash left(n+\backslash frac\{5\}\{6\}\backslash right)\; \backslash ,\; \backslash Gamma\backslash !\backslash left(n+\backslash frac\{1\}\{6\}\backslash right)\}\{n!\; (-2\backslash zeta)^n\}\; \backslash right].$$where

\zeta = \tfrac 23 z ^{3/2} . In particular, the first few terms are^[4] $$\backslash operatorname(z)\; =\; \backslash frac\backslash left(1\; -\; \backslash frac\; +\; \backslash frac\; +\; O(\backslash zeta^)\backslash right)$$There is a similar one for, but only applicable when :

$$\backslash operatorname(z)\backslash sim\; \backslash frac\backslash exp\backslash left(\backslash fracz^\backslash right)\backslash left[\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash dfrac\{\; \backslash Gamma\backslash !\backslash left(n+\backslash frac\{5\}\{6\}\backslash right)\; \backslash ,\; \backslash Gamma\backslash !\backslash left(n+\backslash frac\{1\}\{6\}\backslash right)\; \backslash left(\backslash frac\{3\}\{4\}\backslash right)^n\}\{2\backslash pi\; \backslash ,\; n!\; \backslash ,\; z^\{3n/2\}\}\; \backslash right].$$A more accurate formula for and a formula for when or, equivalently, for and when but not zero, are:^[5]

&-\frac \cos\left(\fracz^+\frac \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(-z) \sim& \frac \cos \left(\fracz^ + \frac \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 \sin\left(\fracz^ + \frac \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

When these are good approximations but are not asymptotic because the ratio between or 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 and . Similarly to before, when :

$$\backslash operatorname\text{'}(z)\backslash sim\; -\backslash dfrac\; \backslash exp\backslash left(-\backslash fracz^\backslash right)\backslash left[\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash frac\{1+6n\}\{1-6n\}\; \backslash dfrac\{(-1)^n\; \backslash ,\; \backslash Gamma\backslash !\backslash left(n\; +\; \backslash frac\{5\}\{6\}\backslash right)\; \backslash ,\; \backslash Gamma\backslash !\backslash left(n\; +\; \backslash frac\{1\}\{6\}\backslash right)\; \backslash left(\backslash frac\{3\}\{4\}\backslash right)^n\}\{2\backslash pi\; \backslash ,\; n!\; \backslash ,\; z^\{3n/2\}\}\; \backslash right].$$

When we have:

$$\backslash operatorname\text{'}(z)\backslash sim\; \backslash frac\backslash exp\backslash left(\backslash fracz^\backslash right)\backslash left[\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash frac\{1+6n\}\{1-6n\}\; \backslash dfrac\{\; \backslash Gamma\backslash !\backslash left(n\; +\; \backslash frac\{5\}\{6\}\backslash right)\; \backslash ,\; \backslash Gamma\backslash !\backslash left(n\; +\; \backslash frac\{1\}\{6\}\backslash right)\; \backslash left(\backslash frac\{3\}\{4\}\backslash right)^n\}\{2\backslash pi\; \backslash ,\; n!\; \backslash ,\; z^\{3n/2\}\}\; \backslash right].$$

Similarly, an expression for and when but not zero, are

&-\frac \sin\left(\fracz^ + \frac \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'(-z) \sim& \ \frac \sin\left(\fracz^ + \frac \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 \cos\left(\fracz^ + \frac \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

Complex arguments

We can extend the definition of the Airy function to the complex plane by$$\backslash operatorname(z)\; =\; \backslash frac\; \backslash int\_\; \backslash exp\backslash left(\backslash tfrac\; -\; zt\backslash right)\backslash ,\; dt,$$where the integral is over a path C starting at the point at infinity with argument and ending at the point at infinity with argument π/3. Alternatively, we can use the differential equation to extend and to entire functions on the complex plane.

The asymptotic formula for is still valid in the complex plane if the principal value of is taken and is bounded away from the negative real axis. The formula for is valid provided is in the sector

x\in\C : \left|\arg(x)\right| < \tfrac{\pi}{ 3 } - \delta for some positive δ. Finally, the formulae for and are valid if is in the sector x\in\C : \left|\arg(x)\right| < \tfrac{ 2 \pi}{ 3 } - \delta.

It follows from the asymptotic behaviour of the Airy functions that both and have an infinity of zeros on the negative real axis. The function has no other zeros in the complex plane, while the function also has infinitely many zeros in the sector

z\in\C : \tfrac{\pi}{ 3 } < \left|\arg(z)\right| < \tfrac{\pi}{ 2 }.

Plots

Relation to other special functions

For positive arguments, the Airy functions are related to the modified Bessel functions:Here, and are solutions of $$x^2y$$ + xy' - \left (x^2 + \tfrac \right)y = 0.

The first derivative of the Airy function is$$\backslash operatorname(x)\; =\; -\; \backslash frac\; \backslash ,\; K\_\backslash !\backslash left(\backslash frac\; x^\backslash right)\; .$$

Functions and can be represented in terms of rapidly convergent integrals^[6] (see also modified Bessel functions)

For negative arguments, the Airy function are related to the Bessel functions:Here, are solutions of$$x^2y$$ + xy' + \left (x^2 - \frac \right)y = 0.

The Scorer's functions and solve the equation . They can also be expressed in terms of the Airy functions:

Fourier transform

Using the definition of the Airy function Ai(x), it is straightforward to show its Fourier transform is given by$$\backslash mathcal(\backslash operatorname)(k)\; :=\; \backslash int\_^\; \backslash operatorname(x)\backslash \; e^\backslash ,dx\; =\; e^.$$This can be obtained by taking the Fourier transform of the Airy equation. Let $\backslash hat\; y\; =\; \backslash frac\backslash int\; y\; e^dx$, then

i\hat y' + k ^{2} \hat y = 0 , which then has solutions \hat y = C e ^{ik ^{3/3}} . There only one dimension of solutions because the Fourier transform requires to decay to zero fast enough, and grows to infinity exponentially fast, so it cannot be obtained via 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.^[7]

Probability

In the mid-1980s, the Airy function was found to be intimately connected to Chernoff's distribution.^[8]

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. ^[9]

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

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

External links

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

Notes and References

1. David E. Aspnes, Physical Review, 147, 554 (1966)
2. Web site: Airy and Related Function . dlmf.nist.gov . 9 October 2022.
3. , Eqns 10.4.59, 10.4.61
4. Web site: DLMF: §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions . 2023-05-11 . dlmf.nist.gov.
5. , Eqns 10.4.60 and 10.4.64
6. M.Kh.Khokonov. Cascade Processes of Energy Loss by Emission of Hard Photons // JETP, V.99, No.4, pp. 690-707 \ (2004).
7. [iarchive:transactionsofca07camb/page/n249/mode/2up|Miller, William Hallowes. "On spurious rainbows." ''Transactions of the Cambridge Philosophical Society'' 7 (1848): 277.]
8. Chernoff's distribution and differential equations of parabolic and Airy type. Groeneboom. Piet. Lalley. Steven. Temme. Nico. Journal of Mathematical Analysis and Applications. 423. 2. 1804–1824. 2015. 10.1016/j.jmaa.2014.10.051 . 119173815 . free. 1305.6053.
9. Book: Quastel. Jeremy. Remenik. Daniel. Topics in Percolative and Disordered Systems . Airy Processes and Variational Problems . Springer Proceedings in Mathematics & Statistics . 2014. 69 . 121–171 . 10.1007/978-1-4939-0339-9_5 . 1301.0750 . 978-1-4939-0338-2 . 118241762 . https://link.springer.com/chapter/10.1007/978-1-4939-0339-9_5.