Airy function

The Airy function refers to a special function in mathematics. The function and the related function, which is also called Airy function, are solutions of the linear equation

Also known as the Airy equation. Describes, among other things, the solution of the Schrödinger equation for a linear potential well.

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

  • 6.1 Airy zeta function
  • 6.2 Scorersche functions

Definition

Real values ​​for the Airy function is defined as the integral:

The second linearly independent solution of the differential equation is the Airy function of the second kind Bi ( x):

Properties

Asymptotic behavior

For against can be and using the WKB approximation, approximate:

For against the relationships:

Zeros

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

Special values

The Airy functions and their derivatives have the following values:

Here Γ denotes the gamma function. It follows that the Wronski determinant and is equal to.

Fourier transform

Directly from the definition of the Airy function (see above) followed by the Fourier transform.

Note the symmetric version used here, the Fourier transform.

Further illustrations

  • Using the hypergeometric function
  • For it can also be presented with the modified Bessel function of the first kind as follows:
  • Another infinite integral representation for Ai is
  • There are the series representations

Complex arguments

Ai ( ) and Bi are entire functions. They can be so analytically continued to the entire complex plane.

Related Functions

Airy zeta function

For the Airy function can be analogously define the Airy zeta function to the other zeta functions as

Wherein the sum of the real (negative) zero points is Ai.

Scorersche functions

Sometimes the other two functions are included as well and to the Airy functions. The integral definitions denominated

They can also be represented by the functions Ai and Bi.

Wronskian List of zeta functions Milton Abramowitz Irene Stegun Abramowitz and Stegun National Institute of Standards and Technology Wolfram Research
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