📂Functions

The Airy Function

Definition

The function below is referred to as the Airy function.

$$ \begin{align*} \operatorname{Ai}(x) &= \frac{1}{\pi}\sqrt{\frac{x}{3}}K_{1/3}\left( \frac{2}{3}x^{2/3} \right) \\ \operatorname{Bi}(x) &= \sqrt{\frac{x}{3}}\left[ I_{-1/3}\left( \frac{2}{3}x^{3/2} \right) + I_{1/3} \left( \frac{2}{3}x^{2/3} \right) \right] \end{align*} $$

Here, $I_{\nu}$ and $K_{\nu}$ are modified Bessel functions.

Description

The Airy function represents the solution to the Airy differential equation using Bessel functions.

Integral Form

The Airy function has the following integral form:

$$ \begin{align*} \operatorname{Ai}(x) &= \frac{1}{\pi} \int_{0}^{\infty} \cos (t^{3}/3 + xt) dt \\ \operatorname{Bi}(x) &= \frac{1}{\pi}\int_{0}^{\infty} \left[ \exp \left( -\frac{1}{3}t^{3}+xt \right)+\sin\left( \frac{1}{3}t^{3}+xt \right) \right]dt \end{align*} $$