The Airy function $\operatorname{Ai}(z)$ is defined by

$$\operatorname{Ai}(z)=\frac{1}{2 \pi i} \int_{C} \exp \left(-\frac{1}{3} t^{3}+z t\right) d t$$

where the contour $C$ begins at infinity along the ray $\arg (t)=4 \pi / 3$ and ends at infinity along the ray $\arg (t)=2 \pi / 3$. Restricting attention to the case where $z$ is real and positive, use the method of steepest descent to obtain the leading term in the asymptotic expansion for $\operatorname{Ai}(z)$ as $z \rightarrow \infty$ :

$$\operatorname{Ai}(z) \sim \frac{\exp \left(-\frac{2}{3} z^{3 / 2}\right)}{2 \pi^{1 / 2} z^{1 / 4}}$$

$\left[\right.$ Hint: put $\left.t=z^{1 / 2} \tau .\right]$