Question
How do I show the equivalence of the following 2 definitions?
Definition 1:
$$A_1(x) :=\frac{1}{2 \pi i} \int_C \exp \left( \frac{1}{3}z^3 - xz \right) dz \tag{1}$$
Where $C$ is the path from $-3/\pi$ infinity to $3/\pi$ infinity in complex plane (Integral path).
Definition 2: $$A_2(x) := \frac{1}{\pi} \int_0^\infty \cos \left( \frac{t^3}{3} + xt \right) dt \tag{2}$$
What I tried
$$I(\theta, a \to b) := \frac{1}{2 \pi i} \int_a^b \exp \left( \frac{1}{3}(re^{i\theta})^3 - x (r e^{i\theta}) \right) e^{i\theta} dr $$
I understand:
$$A_1(x) = I(-\pi/3, -\infty \to 0) + I(\pi/3, 0 \to \infty)$$
$$A_2(x) = I(0, -\infty \to 0) + I(0, 0 \to \infty)$$
$$I(\pi/3, 0 \to \infty) = I(\theta, 0 \to \infty) \qquad \text{if} \quad \pi / 6 < \theta < 3\pi / 6$$
I can't show:
$$I(\pi/3, 0 \to \infty) = I(\pi / 2, 0 \to \infty)$$