1. ## Airy Functions problem

Show that $y(z)=\frac{1}{2\pi i}\int_{C}{exp(zt-\frac{t^{3}}{3})dt}$ is a solution of $y''(z)=zy(z)$

2. I think you need an i in there huh? Like:

$A(s)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(z^3/3+sz)}dz$

then $A''-sA=0$

Differentiate under the integral sign and use Cauchy's Theorem to express the Airy integral in terms of a contour integral over a square region bordering the real axis of height say i. Then since the integrand is analytic, we can express the Airy integral in terms of the horizontal leg at $z=x+i$ since the integral over the two vertical legs tend to zero. Then:

$Ai(s)=\frac{1}{2\pi}\int_{\infty}^{-\infty} e^{i(z^3/3+sz)}dz,\quad z=x+i$

Now:

$A''-sA=-\frac{1}{2\pi}\int_C (z^2+s)e^{i(z^3/3+sz)}dz;\quad z=x+i$

but that integrand is the differential of $f(z)=ie^{i(z^3/3+sz)}$ and since the integrand is analytic, we can just evaluate it at the endpoints:

$\frac{1}{2\pi}\int_{\infty}^{-\infty}df$

and note if you substitute $z=x+i$ into $f$, it's order is $O(e^{-x^2})$ so we have:

$O(e^{-x^2})\biggr|_{\infty}^{-\infty}$

which is zero.