Airy function

The Airy functions Ai(x) and Bi(x) are standard solutions of the differential equation y″ − xy = 0. They describe wave behavior near turning points and arise in asymptotic analysis.

Airy differential equation: The second-order linear equation y″ − xy = 0, whose independent solutions are Ai(x) and Bi(x). This equation defines the Airy functions and determines their behavior.
WKB approximation: A semiclassical method for approximating solutions of differential equations with slowly varying coefficients. Near WKB turning points, Airy functions replace the approximation that becomes singular.
George Biddell Airy: An English astronomer and mathematician who served as Astronomer Royal from 1835 to 1881. Airy introduced the function in his mathematical treatment of optical intensity near a caustic.
Bi function: The Airy function Bi(x), a solution of y″ − xy = 0 linearly independent of Ai(x). Bi grows exponentially for positive x, unlike the decaying Ai solution.
Stokes phenomenon: The abrupt change in an asymptotic expansion’s dominant exponential contributions as a complex variable crosses certain lines. Airy solutions exhibit changing exponential asymptotics across Stokes lines.
Geometric optics: The approximation of light propagation by rays, neglecting wave effects except where they become significant. Airy functions resolve wave behavior where ray-based predictions fail near caustics.
Airy’s theory of the rainbow: A wave-optical account of rainbow intensity that explains the bright primary bow and supernumerary bows. Airy’s analysis of rainbow diffraction gave rise to the function bearing his name.
Bessel function: A solution of Bessel’s differential equation, commonly arising in problems with circular or cylindrical symmetry. Bessel functions suit radial geometry, while Airy functions describe simple turning-point transitions.
Turning point (differential equations): A point where the leading coefficient or local wave number in a differential equation vanishes, changing solution behavior. The Airy equation models the transition between oscillation and exponential growth near a simple turning point.
Caustic (optics): A surface or curve where neighboring light rays concentrate, producing enhanced intensity. The Airy pattern describes the wave field near a fold caustic.

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