Airy function

In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(x) and the related function Bi(x), are linearly independent solutions to the differential equation

This article is about the Airy special function. For the Airy stress function employed in solid mechanics, see Stress functions. For the Airy disk function that describes the optics diffraction pattern through a circular aperture, see Airy disk. For generic Airy distribution arising from optical resonance between two mirrors, see Fabry–Pérot interferometer.

known as the Airy equation or the Stokes equation.

Because the solution of the linear differential equation

is oscillatory for k<0 and exponential for k>0, the Airy functions are oscillatory for x<0 and exponential for x>0. In fact, the Airy equation 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).

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