Airy function

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In mathematics, the Airy function Ai(x) is a special function named after the British astronomer George Biddell Airy. The function Ai(x) and the related function Bi(x), which is also called an Airy function, are solutions to the differential equation

y'' - xy = 0 , \,\!

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).

The Airy function describes the appearance of a star — a point source of light — as it appears in a telescope. The ideal point image becomes a series of concentric ripples because of the limited aperture and the wave nature of light (Suiter 1994). It is also the solution to Schrödinger's equation for a particle confined within a triangular potential well.

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[edit] Definitions

Plot of Ai(x) in red and Bi(x) in blue.
Plot of Ai(x) in red and Bi(x) in blue.

For real values of x, the Airy function is defined by the integral

\mathrm{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + xt\right)\, dt.

Although the function is not strictly integrable (the integrand does not decay as t goes to +∞), the improper integral converges because of the positive and negative parts of the rapid oscillations tend to cancel one another out (this can be checked by integration by parts).

By differentiating under the integration sign, we find that y = Ai(x) satisfies the differential equation

y'' - xy = 0 . \,\!

This equation has two linearly independent solutions. 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 goes to −∞ which differs in phase by (1/2)π:

\mathrm{Bi}(x) = \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + xt\right) + \sin\left(\frac{t^3}{3} + xt\right)\,dt

[edit] Properties

The values of Ai(x) and Bi(x) and their derivatives at x = 0 are given by

\begin{align} \mathrm{Ai}(0) &{}= \frac{1}{3^{2/3}\Gamma(\frac23)}, & \quad \mathrm{Ai}'(0) &{}= -\frac{1}{3^{1/3}\Gamma(\frac13)}, \\ \mathrm{Bi}(0) &{}= \frac{1}{3^{1/6}\Gamma(\frac23)}, & \quad \mathrm{Bi}'(0) &{}= \frac{3^{1/6}}{\Gamma(\frac13)}. \end{align}

Here, Γ denotes the Gamma function. It follows that the Wronskian of Ai(x) and Bi(x) is 1/π.

When x is positive, Ai(x) is positive, 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 formulas below for the Airy functions.

[edit] Asymptotic formulas

The asymptotic behaviour of the Airy functions as x goes to +∞ is given by

\begin{align} \mathrm{Ai}(x) &{}\sim \frac{e^{-\frac23x^{3/2}}}{2\sqrt\pi\,x^{1/4}} \\ \mathrm{Bi}(x) &{}\sim \frac{e^{\frac23x^{3/2}}}{\sqrt\pi\,x^{1/4}}. \end{align}

For the limit in the negative direction we have

\begin{align} \mathrm{Ai}(-x) &{}\sim \frac{\sin(\frac23x^{3/2}+\frac14\pi)}{\sqrt\pi\,x^{1/4}} \\ \mathrm{Bi}(-x) &{}\sim \frac{\cos(\frac23x^{3/2}+\frac14\pi)}{\sqrt\pi\,x^{1/4}}. \end{align}

Asymptotic expansions for these limits are also available. These are listed in (Abramowitz and Stegun, 1954) and (Olver, 1974).

[edit] Complex arguments

We can extend the definition of the Airy function to the complex plane by

\mathrm{Ai}(z) = \frac{1}{2\pi i} \int_{C} \exp\left(\frac{t^3}{3} - zt\right)\, dt,

where the integral is over a path C starting at the point at infinity with argument -(1/3)π and ending at the point at infinity with argument (1/3)π. Alternatively, we can use the differential equation y'' − xy = 0 to extend Ai(x) and Bi(x) to entire functions on the complex plane.

The asymptotic formula for Ai(x) is still valid in the complex plane if the principal value of x2/3 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| < (1/3)π−δ} for some positive δ. Finally, the formulas for Ai(−x) and Bi(−x) are valid if x is in the sector {x∈C : |arg x| < (2/3)π−δ}.

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 : (1/3)π < |arg z| < (1/2)π}.

[edit] Plots

\Re \left[ \mathrm{Ai} ( x + iy) \right] \Im \left[ \mathrm{Ai} ( x + iy) \right] | \mathrm{Ai} ( x + iy) | \, \mathrm{arg} \left[ \mathrm{Ai} ( x + iy) \right] \,


\Re \left[ \mathrm{Bi} ( x + iy) \right] \Im \left[ \mathrm{Bi} ( x + iy) \right] | \mathrm{Bi} ( x + iy) | \, \mathrm{arg} \left[ \mathrm{Bi} ( x + iy) \right] \,

[edit] Relation to other special functions

For positive arguments, the Airy functions are related to the modified Bessel functions:

\begin{align} \mathrm{Ai}(x) &{}= \frac1\pi \sqrt{\frac13 x} \, K_{1/3}\left(\frac23 x^{3/2}\right), \\ \mathrm{Bi}(x) &{}= \sqrt{\frac13 x} \left(I_{1/3}\left(\frac23 x^{3/2}\right) + I_{-1/3}\left(\frac23 x^{3/2}\right)\right). \end{align}

Here, I±1/3 and K1/3 are solutions of x2y'' + xy' − (x2 + 1 / 9)y = 0.

For negative arguments, the Airy function are related to the Bessel functions:

\begin{align} \mathrm{Ai}(-x) &{}= \frac13 \sqrt{x} \left(J_{1/3}\left(\frac23 x^{3/2}\right) + J_{-1/3}\left(\frac23 x^{3/2}\right)\right), \\ \mathrm{Bi}(-x) &{}= \sqrt{\frac13 x} \left(J_{-1/3}\left(\frac23 x^{3/2}\right) - J_{1/3}\left(\frac23 x^{3/2}\right)\right). \end{align}

Here, J±1/3 are solutions of x2y'' + xy' + (x2 − 1 / 9)y = 0.

The Scorer's functions solve the equation y'' − xy = 1 / π. They can also be expressed in terms of the Airy functions:

\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, \\ \mathrm{Hi}(x) &{}= \mathrm{Bi}(x) \int_{-\infty}^x \mathrm{Ai}(t) \, dt - \mathrm{Ai}(x) \int_{-\infty}^x \mathrm{Bi}(t) \, dt. \end{align}

[edit] History

The Airy function is named after the British astronomer George Biddell Airy, who encountered it in his study of optics (Airy 1838). The notation Ai(x) was introduced by Harold Jeffreys.

[edit] External links

[edit] References

  • Milton Abramowitz and Irene A. Stegun (1954). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (See §10.4). National Bureau of Standards.
  • Airy (1838). On the intensity of light in the neighbourhood of a caustic. Transactions of the Cambridge Philosophical Society, 6, 379–402.
  • Olver (1974). Asymptotics and Special Functions, Chapter 11. Academic Press, New York.
  • Harold Richard Suiter (1994). Star Testing Astronomical Telescopes: A Manual for Optical Evaluation and Adjustment. Richmond, VA: Willmann-Bell. ISBN 978-0-943396-44-6.  (with many example images)