Airy function

Summary

The Airy functions $\operatorname{Ai}$ and $\operatorname{Bi}$ are two standard independent solutions of the Airy equation $y'' = x y$. $\operatorname{Ai}$ decays rapidly as $x \to +\infty$, and $\operatorname{Bi}$ grows; both oscillate as $x \to -\infty$. They describe every simple turning point, where a wave changes from oscillating to decaying. This makes them central in optics near caustics, in quantum mechanics near classical turning points, and in uniform asymptotics of other special functions.