Vtyjkyuk

I am given a set of independent solutions of the Airy equation

$$ y''(z) - z~ y (z) = 0,$$

as contour integrals

$$ y_1,2(z) = int_C_1,2 e^ileft( t^3/3 + z tright) dt, $$

where $C_1$ is the path along the real axes starting from negative infinity to the positive, and $C_2$ starts from infinity from direction $5pi/6$ passes through the point (0,0) and ends in the infinity in direction $3pi/2$. Note that these contours are not in typical regions for Airy function.

How do I show that these are indeed the solutions of the Airy equation and that they are independent?

If I just plug them into the equation
$$ y_1,2''(z) - z~ y_1,2 (z) = -
int_C_1,2 left( t^2 + z right)e^ileft( t^3/3 + z tright) dt
= i
int_C_1,2 fracddt e^ileft( t^3/3 + z tright) dt, $$
and try using the fundamental theorem of calculus, it does not seem to help me much.
How do I proceed?

$C_2$ is the easier of the two, and is where the Fundamental Theorem of Calculus comes in. Suppose $t = Re^5ipi/6$. What's $lim_Rrightarrowinftye^i(t^3/3+zt)$? What about when $t = Re^3ipi/2$?

For $C_1$, you should look at the wedge-shaped contours $se^5ipi/6 cup Re^ithetacup [-R, 0]$ and $[0, R]cup Re^itheta cup Rge sge 0$. The radial contours along $5pi/6$ and $pi/6$ can still be handled by FTC. For the circle contours, show that the integrand decays quickly with $R$, and thus the integral over the whole arc vanishes as $Rrightarrow infty$.