AiryFunktion

AiryFunktion, commonly called the Airy function in English, refers to a pair of special functions that solve the Airy equation. The standard form of the differential equation is y'' - x y = 0. The two linearly independent solutions are Ai(x) and Bi(x); together they form a fundamental set, so any solution is a linear combination C1 Ai(x) + C2 Bi(x). The functions are named after George Biddell Airy.

Ai(x) and Bi(x) are entire functions with markedly different asymptotic behavior. Ai(x) decays exponentially for large positive x while oscillating for large negative x. Bi(x) grows exponentially for large positive x and also oscillates for large negative x. These properties make Ai suitable for modeling behavior near turning points in differential equations and wave problems, whereas Bi is typically used in contexts that require a growing solution.

Integral and series representations provide practical definitions. Ai(x) has a well-known integral representation Ai(x) = (1/π) ∫_0^∞ cos(t^3/3 + x t) dt, among other forms. Both Ai and Bi can be expanded in power series about x = 0, and together with their derivatives they satisfy Ai''(x) = x Ai(x) and Bi''(x) = x Bi(x). The Wronskian W(Ai, Bi) is 1/π, confirming they are linearly independent.

Applications of the Airy functions span physics and engineering. They arise in quantum mechanics for linear potentials and near turning points, in optics and diffraction theory through the Airy pattern, and in semiclassical approximations and the study of caustics and wave propagation.