(a) Show that an integral representation of the Airy function Ai $(x)$ is given by
$$
\text { Ai }(x)=\frac{1}{2 \pi i} \int_c e^{x i-t^3 / 3} d t,
$$
where $C$ is a contour which originates at $\infty e^{-2 \pi / 3}$ and terminates at $\infty e^{2 \pi i / 3}$.
(b) Use this integral representation to show that the Taylor series expansion of $\mathrm{Ai}(x)$ about $x=0$ is as given in $(3.2 .1)$.
(c) Using the method of steepest descents, find the asymptotic behavior of $\mathrm{Ai}(x)$ as $x \rightarrow+\infty$.
(d) Extend the steepest-descent argument used in part (c) to show that the same asymptotic behavior is valid for $x \rightarrow \infty$ with $|\arg x|<\pi$ and that there is no Stokes phenomenon at $|\arg x|=\pi / 3$.
(e) Show that there is no Stokes phenomenon at $|\arg x|=\pi / 3$ in a different way. Transform the integral in $(a)$ to
$$
\operatorname{Ai}(x)=\frac{1}{\pi} e^{-2 x^{3 / 1 / 3}} \int_0^{\infty} e^{-\pi^{t / 2} t^2} e^{i t^{3 / 3}} d t
$$
and then use Laplace's method.
Clue: For real positive $x$, deform $C$ in $(a)$ into the straight-line contour connecting $-x-i \infty$ to $-x+i \infty$ and then allow $x$ to be complex with $\left|\arg x^{1 / 2}\right|<\pi / 2$.
(f) Find the leading behavior of $\mathrm{Ai}(x)$ as $x \rightarrow-\infty$.
Clue: Show that the steepest-descent contour connecting $\infty e^{-2 \pi i / 3}$ to $\infty e^{2 \pi i / 3}$ consists of two pieces, one passing through the saddle point at $t=-i \sqrt{-x}$ and one passing through the saddle point at $t=+i \sqrt{-x}$.