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the Airy function Ai(x)
In mathematics, the Airy function Ai(x) is a special function named after the British
astronomer George Biddell Airy. The function Ai(x) and the related function Bi(x),
which is also called an Airy function, are solutions to the differential equation
known as the Airy equation or the Stokes equation. This is the simplest secondorder linear differential equation with a turning point (a point where the character of
the solutions changes from oscillatory to exponential).
The Airy function describes the appearance of a star — a point source of light — as it
appears in a telescope. The ideal point image becomes a series of concentric ripples
because of the limited aperture and the wave nature of light (Suiter 1994). It is also
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 real values of x, the Airy function is defined by the integral
Although the function is not strictly integrable (the integrand does not decay as t goes
to +∞), the improper integral converges because of the positive and negative parts of
the rapid oscillations tend to cancel one another out (this can be checked by
integration by parts).
By differentiating under the integration sign, we find that y = Ai(x) satisfies the
differential equation
This equation has two linearly independent solutions. 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 goes to −∞ which
differs in phase by (1/2)π:
The values of Ai(x) and Bi(x) and their derivatives at x = 0 are given by
Here, Γ denotes the Gamma function. It follows that the Wronskian of Ai(x) and Bi(x)
is 1/π.
When x is positive, Ai(x) is positive, convex, and decreasing exponentially to zero,
while Bi(x) is positive, convex, and increasing exponentially. When x is negative,
Ai(x) and Bi(x) oscillate around zero with ever-increasing frequency and everdecreasing amplitude. This is supported by the asymptotic formulas below for the
Airy functions.
Asymptotic formulas
The asymptotic behaviour of the Airy functions as x goes to +∞ is given by
For the limit in the negative direction we have
Asymptotic expansions for these limits are also available