Физикада қолданылатын Эйри функциясы

Special function in the physical sciences
the Airy special 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

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).

The Airy function is the solution to the time independent Schrödinger 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 electrons trapped in semiconductor heterojunctions.

A transversally asymmetric optical beam, where the electric field profile is given by the Airy function, has the interesting property that its maximum intensity accelerates towards one side instead of propagating in a straight line as is the case in symmetric beams. This is at expense of the low intensity tail being spread in the opposite direction, so the overall momentum of the beam is of course conserved.

The Airy function underlies the form of the intensity near an optical directional caustic, such as that of the rainbow (called supernumerary rainbow). Historically, this was the mathematical problem that led Airy to develop this special function. In 1841, William Hallowes Miller experimentally measured the analog to supernumerary rainbow by shining light through a thin cylinder of water, then observing through a telescope. He observed up to 30 bands.

In the mid 1980s, the Airy function was found to be intimately connected to Chernoff's distribution. The Airy function also appears in the definition of Tracy–Widom distribution which describes the law of largest eigenvalues in Random matrix. Due to the intimate connection of random matrix theory with the Kardar–Parisi–Zhang equation, there are central processes constructed in KPZ such as the Airy process.

The Airy function is named after the British astronomer and physicist George Biddell Airy (1801–1892), who encountered it in his early study of optics in physics (Airy 1838). The notation Ai(x) was introduced by Harold Jeffreys. Airy had become the British Astronomer Royal in 1835, and he held that post until his retirement in 1881.