Airy function 1

Airy functionIn the physical sciences, the Airy function Ai(x) is a special function named after the British astronomer GeorgeBiddell 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 second-order linear differential equationwith 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 welland for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniformsemiclassical approximations near a turning point in the WKB method, when the potential may be locallyapproximated by a linear function of position. The triangular potential well solution is directly relevant for theunderstanding of many semiconductor devices.The Airy function also underlies the form of the intensity near an optical directional caustic, such as that of therainbow. Historically, this was the mathematical problem that led Airy to develop this special function. The Airyfunction is also important in microscopy and astronomy; it describes the pattern, due to diffraction and interference,produced by a point source of light (one which is smaller than the resolution limit of a microscope or telescope).

Definitions

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

For real values of x, the Airy functionis defined by the improper integral

which converges because the positiveand negative parts of the rapidoscillations tend to cancel one anotherout (as can be checked by integrationby parts).

By differentiating under the integrationsign, we find that y = Ai(x) satisfies thedifferential equation

This equation has two linearlyindependent solutions. The standard choice for the other solution is the Airy function of the second kind, denotedBi(x). It is defined as the solution with the same amplitude of oscillation as Ai(x) as x goes to −∞ which differs inphase by π/2:

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PropertiesThe 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 ever-increasingfrequency and ever-decreasing amplitude. This is supported by the asymptotic formulae below for the Airyfunctions.The Airy functions are orthogonal[1] in the sense that

Asymptotic formulaeThe 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 argumentsWe can extend the definition of the Airy function to the complex plane by

where the integral is over a path starting at the point at infinity with argument -(1/3)π and ending at the point atinfinity with argument (1/3)π. Alternatively, we can use the differential equation to extend Ai(x) andBi(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 isbounded 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}.

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It follows from the asymptotic behaviour of the Airy functions that both Ai(x) and Bi(x) have an infinity of zeros onthe negative real axis. The function Ai(x) has no other zeros in the complex plane, while the function Bi(x) also hasinfinitely many zeros in the sector {z∈C : (1/3)π < |arg z| < (1/2)π}.

Plots

Relation to other special functionsFor positive arguments, the Airy functions are related to the modified Bessel functions:

Here, I±1/3 and K1/3 are solutions of The first derivative of Airy function is

Functions and can be represented in terms of rapidly converged integrals[4] (see also modified Besselfunctions )For negative arguments, the Airy function are related to the Bessel functions:

Here, J±1/3 are solutions of .

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

Fourier transformUsing the definition of the Airy function , it is straightforward to show its Fourier transform is given by

HistoryThe Airy function is named after the British astronomer and physicist George Biddell Airy (1801–1892), whoencountered it in his early study of optics in physics (Airy 1838). The notation Ai(x) was introduced by HaroldJeffreys. Airy had become the British Astronomer Royal in 1835, and he held that post until his retirement in 1881.

Notes[1] David E. Aspnes, Physical Review, 147, 554 (1966)[2] Abramowitz & Stegun (1970), Eqns 10.4.59 and 10.4.63[3] Abramowitz & Stegun (1970), Eqns 10.4.60 and 10.4.64[4][4] M.Kh.Khokonov. Cascade Processes of Energy Loss by Emission of Hard Photons // JETP, V.99, No.4, pp. 690-707 \ (2004).

References• Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 10" (http:/ / www. math. sfu. ca/ ~cbm/ aands/

page_446. htm), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, NewYork: Dover, pp. 446, ISBN 978-0486612720, MR0167642.

• Airy (1838), "On the intensity of light in the neighbourhood of a caustic" (http:/ / books. google. com/?id=-yI8AAAAMAAJ& dq=Transactions+ of+ the+ Cambridge+ Philosophical+ Society+ 1838), Transactions ofthe Cambridge Philosophical Society (University Press) 6: 379–402

• 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" (http:/ / apps.

nrbook. com/ empanel/ index. html#pg=289), Numerical Recipes: The Art of Scientific Computing (3rd ed.), NewYork: Cambridge University Press, ISBN 978-0-521-88068-8

• Vallée, Olivier; Soares, Manuel (2004), Airy functions and applications to physics (http:/ / www. worldscibooks.com/ physics/ p345. html), London: Imperial College Press, ISBN 978-1-86094-478-9, MR2114198

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External links• Weisstein, Eric W., " Airy Functions (http:/ / mathworld. wolfram. com/ AiryFunctions. html)" from MathWorld.• Wolfram function pages for Ai (http:/ / functions. wolfram. com/ Bessel-TypeFunctions/ AiryAi/ ) and Bi (http:/ /

functions. wolfram. com/ Bessel-TypeFunctions/ AiryBi/ ) functions. Includes formulas, function evaluator, andplotting calculator.

• Olver, F. W. J. (2010), "Airy and related functions" (http:/ / dlmf. nist. gov/ 9), in Olver, Frank W. J.; Lozier,Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press,ISBN 978-0521192255, MR2723248

Article Sources and Contributors 6

Article Sources and ContributorsAiry function  Source: http://en.wikipedia.org/w/index.php?oldid=504374353  Contributors: Alberto da Calvairate, Alejo2083, Arc de Ciel, Arthur Rubin, Charles Matthews, Creidieki, Cyp,Dchristle, Diegotorquemada, Dysprosia, El Roih, Electron0511, Emperorbma, Free0willy, Fuse809, Gaius Cornelius, Gareth Owen, Giftlite, Hannes Eder, Haseldon, Headbomb, Ichbin-dcw, JitseNiesen, Josuedavila, Jovianic, KSmrq, Krishnachandranvn, Linas, Michael Hardy, MichalKotowski, Nerissa-Marie, Netheril96, Njerseyguy, Olaf3142, PAR, PV=nRT, Papa November, R.e.b.,SUL, Schneelocke, ServiceAT, Sj, Stevvers, Tcnuk, Tobias Bergemann, 35 anonymous edits

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