# Airy Function

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## Airy function

[¦er·ē ¦fəŋk·shən] (mathematics)

Either of the solutions of the Airy differential equation.

## Airy Function

either of the functions Ai(z) and Bi(z), which are solutions of the second-order differential equation

*W″* – _{z}*W* = 0

where *z* is the independent variable.

The Airy functions of the argument (–*z*) may be expressed in terms of Bessel functions of order *v* = ±⅓:

The asymptotic representations for large |z| are

Airy functions play an important role in the theory of asymptotic representations of various special functions; they have diverse applications in mathematical physics—for example, in the theory of the diffraction of radio waves at the earth’s surface. Airy functions were studied by J. R. Airy in 1911.

### REFERENCES

Lebedev, N. N.*Spetsial’nye funktsii i ikh prilozheniia*. 2nd ed. Moscow-Leningrad, 1963.