The function $w(z)$ satisfies the third-order differential equation

$$\frac{d^{3} w}{d z^{3}}-z w=0$$

subject to the conditions $w(z) \rightarrow 0$ as $z \rightarrow \pm i \infty$ and $w(0)=1$. Obtain an integral representation for $w(z)$ of the form

$$w(z)=\int_{\gamma} e^{z t} f(t) d t$$

and determine the function $f(t)$ and the contour $\gamma$.

Using the change of variable $t=z^{1 / 3} \tau$, or otherwise, compute the leading term in the asymptotic expansion of $w(z)$ as $z \rightarrow+\infty$.