Airy function

This article is about the Airy special function. For the Airy stress function employed in solid mechanics, see Stress functions.
In the physical sciences, the Airy function Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–92). The function Ai(x) and the related function Bi(x), which is also called the Airy function, but sometimes referred to as the "Bairy" function, are solutions to the differential equation
known as the Airy equation or the Stokes equation. This is the simplest secondorder linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential).
The Airy function is the solution to Schrödinger's equation for a particle confined within a triangular potential well and for a particle in a onedimensional constant force field. For the same reason, it also serves to provide uniform semiclasssical approximations near a turning point in the WKB method, 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 many semiconductor devices.
The Airy function also underlies the form of the intensity near an optical directional caustic, such as that of the rainbow. Historically, this was the mathematical problem that led Airy to develop this special function.
Contents
Definitions
For real values of x, the Airy function is defined by the improper integral
which converges because the positive and negative parts of the rapid oscillations tend to cancel one another out (as can be checked by integration by parts).
By differentiating under the integration sign, we find that y = Ai(x) satisfies the differential equation
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 π/2:
Properties
The values of Ai(x) and Bi(x) and their derivatives at x = 0 are given by
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 everincreasing frequency and everdecreasing amplitude. This is supported by the asymptotic formulae below for the Airy functions.
The Airy functions are orthogonal^{[1]} in the sense that
Asymptotic formulae
The asymptotic behaviour of the Airy functions as x goes to +∞ is given by the following asymptotic formulae:^{[2]}
For the limit in the negative direction we have^{[3]}
Asymptotic expansions for these limits are also available. These are listed in (Abramowitz and Stegun, 1954) and (Olver, 1974).
Complex arguments
We can extend the definition of the Airy function to the complex plane by
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 x^{2/3} 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 formulae 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}.
Plots
Relation to other special functions
For positive arguments, the Airy functions are related to the modified Bessel functions:
Here, I_{±1/3} and K_{1/3} are solutions of x^{2}y'' + xy' − (x^{2} + 1 / 9)y = 0.
The first derivative of Airy function is
Functions K_{1 / 3} and K_{2 / 3} can be represented in terms of rapidly converged integrals^{[4]} (see also modified Bessel functions )
For negative arguments, the Airy function are related to the Bessel functions:
Here, J_{±1/3} are solutions of x^{2}y'' + xy' + (x^{2} − 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:
Fourier transform
Using the definition of the Airy function Ai(x), it is straightforward to show its Fourier transform is given by
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.
Notes
 ^ David E. Aspnes, Physical Review, 147, 554 (1966)
 ^ Abramowitz & Stegun (1970), Eqns 10.4.59 and 10.4.63
 ^ Abramowitz & Stegun (1970), Eqns 10.4.60 and 10.4.64
 ^ M.Kh.Khokonov. Cascade Processes of Energy Loss by Emission of Hard Photons // JETP, V.99, No.4, pp. 690707 \ (2004).
References
 Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 10", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 446, ISBN 9780486612720, MR0167642, http://www.math.sfu.ca/~cbm/aands/page_446.htm.
 Airy (1838), "On the intensity of light in the neighbourhood of a caustic", Transactions of the Cambridge Philosophical Society (University Press) 6: 379–402, http://books.google.com/?id=yI8AAAAMAAJ&dq=Transactions+of+the+Cambridge+Philosophical+Society+1838
 Olver (1974). Asymptotics and Special Functions, Chapter 11. Academic Press, New York.
 Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.6.3. Airy Functions", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 9780521880688, http://apps.nrbook.com/empanel/index.html#pg=289
 Vallée, Olivier; Soares, Manuel (2004), Airy functions and applications to physics, London: Imperial College Press, ISBN 9781860944789, MR2114198, http://www.worldscibooks.com/physics/p345.html
External links
 Weisstein, Eric W., "Airy Functions" from MathWorld.
 Wolfram function pages for Ai and Bi functions. Includes formulas, function evaluator, and plotting calculator.
 Olver, F. W. J. (2010), "Airy and related functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 9780521192255, MR2723248, http://dlmf.nist.gov/9
Categories: Special functions
 Special hypergeometric functions
 Ordinary differential equations
Wikimedia Foundation. 2010.
Look at other dictionaries:
Airy — may refer to:*Sir George Biddell Airy, British astronomer who is the eponym of craters located on the Moon and Mars * Airy (lunar crater) * Airy (crater on Mars), the smaller crater Airy 0 within Airy defines the prime meridian of the planet *… … Wikipedia
AiryFunktion — Die Airy Funktion Ai(x) bezeichnet eine spezielle Funktion in der Mathematik. Die Funktion Ai(x) und die verwandte Funktion Bi(x), die ebenfalls Airy Funktion genannt wird, sind Lösungen der linearen Differentialgleichung auch bekannt als Airy… … Deutsch Wikipedia
Airy disk — Computer generated image of an Airy disk. The gray scale intensities have been adjusted to enhance the brightness of the outer rings of the Airy pattern … Wikipedia
Airy wave theory — In fluid dynamics, Airy wave theory (often referred to as linear wave theory) gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform… … Wikipedia
Point spread function — The point spread function (PSF) describes the response of an imaging system to a point source or point object. A related but more general term for the PSF is a system s impulse response. The PSF in many contexts can be thought of as the extended… … Wikipedia
George Biddell Airy — Infobox Scientist box width = 300px name = Sir George Biddell Airy, FRS image width = 240px birth date = birth date18010727 birth place = Alnwick, Northumberland, England death date = death date and age1892010218010727 death place =… … Wikipedia
Dirac delta function — Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention… … Wikipedia
Scorer's function — In mathematics, the Scorer s functions are special functions denoted Gi( x ) and Hi( x ). They can be defined as::mathrm{Gi}(x) = frac{1}{pi} int 0^infty sinleft(frac{t^3}{3} + xt ight), dt:mathrm{Hi}(x) = frac{1}{pi} int 0^infty expleft(… … Wikipedia
Gaussian function — In mathematics, a Gaussian function (named after Carl Friedrich Gauss) is a function of the form::f(x) = a e^{ { (x b)^2 over 2 c^2 } }for some real constants a > 0, b , c > 0, and e ≈ 2.718281828 (Euler s number).The graph of a Gaussian is a… … Wikipedia
Confluent hypergeometric function — In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular… … Wikipedia