---
tag: 1N5D4PN92W
kind: concept
name: Airy function
msc: [33C10, 34B30]
sources:
  - url: https://en.wikipedia.org/wiki/Airy_function
same_as:
  wikidata: Q409415
  wikipedia: Airy function
---

## Summary

The Airy functions $\operatorname{Ai}$ and $\operatorname{Bi}$ are two standard independent solutions of the [[Airy equation|PNESAD6R5T]] $y'' = x y$.
$\operatorname{Ai}$ 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.

## Explanations

### L1 human

The equation $y'' = x y$ acts like a spring on the left of $0$ and like exponential growth on the right of $0$.
The Airy function $\operatorname{Ai}$ is the solution that wiggles on the left and dies out quickly on the right.
Its graph looks like a sine wave whose waves get shorter and lower toward the left, then a single hump near $0$, then a fast decay.
The bright bands just inside a rainbow follow the square of $\operatorname{Ai}$.