Airy Function - Asymptotic Formulae

Asymptotic Formulae

The asymptotic behaviour of the Airy functions as x goes to +∞ is given by the following asymptotic formulae:

$\begin{align} \mathrm{Ai}(x) &{}\sim \frac{e^{-\frac{2}{3}x^{3/2}}}{2\sqrt\pi\,x^{1/4}} \\ \mathrm{Bi}(x) &{}\sim \frac{e^{\frac{2}{3}x^{3/2}}}{\sqrt\pi\,x^{1/4}}. \end{align}$

For the limit in the negative direction we have

$\begin{align} \mathrm{Ai}(-x) &{}\sim \frac{\sin \left(\frac23x^{3/2}+\frac14\pi \right)}{\sqrt\pi\,x^{1/4}} \\ \mathrm{Bi}(-x) &{}\sim \frac{\cos \left(\frac23x^{3/2}+\frac14\pi \right)}{\sqrt\pi\,x^{1/4}}. \end{align}$