Airy function

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).

For real values of x, the Airy function of the first kind can be defined by the improper Riemann integral:

The convergence of the integral on this interval can be proven by Dirichlet's test after substitution

Up to scalar multiplication, Ai(x) is the solution subject to the condition y → 0 as x → ∞.

The standard choice for the other solution is the Airy function of the second kind, denoted Bi(x).

It is defined as the solution with the same amplitude of oscillation as Ai(x) as x → −∞ which differs in phase by π/2:

When x is negative, Ai(x) and Bi(x) oscillate around zero with ever-increasing frequency and ever-decreasing amplitude.

The asymptotic behaviour of the Airy functions as |z| goes to infinity at a constant value of arg(z) depends on arg(z): this is called the Stokes phenomenon.

One is also able to obtain asymptotic expressions for the derivatives Ai'(z) and Bi'(z).

Similarly, an expression for Ai'(−z) and Bi'(−z) when |arg(z)| < 2π/3 but not zero, are[5]

We can extend the definition of the Airy function to the complex plane by

Alternatively, we can use the differential equation y′′ − xy = 0 to extend Ai(x) and Bi(x) to entire functions on the complex plane.

The asymptotic formula for Ai(x) is still valid in the complex plane if the principal value of x2/3 is taken and x is bounded away from the negative real axis.

Finally, the formulae for Ai(−x) and Bi(−x) are valid if x is in the sector

It follows from the asymptotic behaviour of the Airy functions that both Ai(x) and Bi(x) have an infinity of zeros on the negative real axis.

The Scorer's functions Hi(x) and -Gi(x) solve the equation y′′ − xy = 1/π.

Using the definition of the Airy function Ai(x), it is straightforward to show that its Fourier transform is given by

This can be obtained by taking the Fourier transform of the Airy equation.

There is only one dimension of solutions because the Fourier transform requires y to decay to zero fast enough; Bi grows to infinity exponentially fast, so it cannot be obtained via a Fourier transform.

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.

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.

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.

[7] In the mid-1980s, the Airy function was found to be intimately connected to Chernoff's distribution.

[8] 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 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.