## ::Airy function

### ::concepts

{{#invoke:Hatnote|hatnote}}

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(**, which is also called the

*x*)**Airy function**, but sometimes referred to as the Bairy function, are solutions to the differential equation

- <math>\frac{d^2y}{dx^2} - xy = 0 , \,\!</math>

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 is the solution to Schrödinger's equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniform semiclassical approximations near a turning point in the WKB approximation, 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. The Airy function is also important in microscopy and astronomy; it describes the pattern, due to diffraction and interference, produced by a point source of light (one of which is smaller than the resolution limit of a microscope or telescope).

**Airy function sections**

Intro Definitions Properties Asymptotic formulae Complex arguments Relation to other special functions Fourier transform Other uses of the term Airy Function History See also Notes References External links

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{{#invoke:Hatnote|hatnote}}

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(**, which is also called the

*x*)**Airy function**, but sometimes referred to as the Bairy function, are solutions to the differential equation

- <math>\frac{d^2y}{dx^2} - xy = 0 , \,\!</math>

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 is the solution to Schrödinger's equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniform semiclassical approximations near a turning point in the WKB approximation, 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. The Airy function is also important in microscopy and astronomy; it describes the pattern, due to diffraction and interference, produced by a point source of light (one of which is smaller than the resolution limit of a microscope or telescope).

**Airy function sections**

Intro Definitions Properties Asymptotic formulae Complex arguments Relation to other special functions Fourier transform Other uses of the term Airy Function History See also Notes References External links

PREVIOUS: Intro | NEXT: Definitions |

<< | >> |