Airy function

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The Airy function describes a special function in mathematics. The function and its related function , also called the Airy function, are solutions to the linear differential equation

also known as the Airy equation. Among other things, it describes the solution of the Schrödinger equation for a linear potential well .

The Airy function is named after the British astronomer George Biddell Airy , who used this function in his work in optics (Airy 1838). The term was introduced by Harold Jeffreys .


Airy plot.svg

For real values , the Airy function is defined as a parameter integral:

A second, linearly independent solution of the differential equation is the Airy function of the second kind :


Asymptotic behavior

For to be and using the WKB approximation approximate:

The following relationships apply to against :


The Airy functions only have zeros on the negative real axis. The approximate position follows from the asymptotic behavior for to

Special values

The Airy functions and their derivatives have the following values:

This is the gamma function . It follows that the Wronsky determinant of and is equal .

Fourier transform

Its Fourier transform follows directly from the definition of the Airy function (see above) .

Note the symmetrical variant of the Fourier transform used here.

Further representations

  • Another infinite integral representation for is
  • There are the series representations

Complex arguments

and are whole functions . So they can be continued analytically on the entire complex level.

AiryAi Real Surface.png AiryAi Imag Surface.png AiryAi Abs Surface.png AiryAi Arg Surface.png
AiryAi Real Contour.svg AiryAi Imag Contour.svg AiryAi Abs Contour.svg AiryAi Arg Contour.svg

AiryBi Real Surface.png AiryBi Imag Surface.png AiryBi Abs Surface.png AiryBi Arg Surface.png
AiryBi Real Contour.svg AiryBi Imag Contour.svg AiryBi Abs Contour.svg AiryBi Arg Contour.svg

Related functions

Airy zeta function

Analogous to the other zeta functions, the Airy zeta function can be defined for the Airy function as

where the sum goes over the real (negative) zeros of .

Scorer functions

Function graphs of and .

Sometimes the two other functions and are added to the Airy functions. The integral definitions are

They can also be represented by the functions and .


Web links

Commons : Airy function  - collection of pictures, videos and audio files

Individual evidence

  1. Eric W. Weisstein : Airy Function Zeros . In: MathWorld (English).
  2. C. Banderier, P. Flajolet, G. Schaeffer, M. Soria: Planar Maps and Airy Phenomena. In Automata, Languages ​​and Programming. Proceedings of the 27th International Colloquium (ICALP 2000) held at the University of Geneva , Geneva, 9. – 15. July 2000 (Ed. U. Montanari, JDP Rolim, E. Welzl). Berlin: Springer, pp. 388-402, 2000
  3. Eric W. Weisstein : Airy Zeta Function . In: MathWorld (English).
  4. Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1954, page 447