Integration of exponential function

[jshw]

Homework Statement

Int(-infinity to +infinity) exp[i(t^3/3 + at^2 + bt)]dt = 2pi*exp[ia(2a^2/3 - b)]*Ai(b-a^2)
O.Vallee gives this formula in his book, "Airy Functions and Applications to Physics"
but there are no proof of this formula. I tried to prove this formula, but I failed.
Would you give some hints?

Homework Equations

Int(-infinity to +infinity) exp[i(t^3/3 + at^2 + bt)]dt = 2pi*exp[ia(2a^2/3 - b)]*Ai(b-a^2)

The Attempt at a Solution

 

[diazona]

First I have to rewrite it in nicer notation:
$$\int_{-\infty}^{+\infty} \exp\left[i\left(\frac{t^3}{3} + at^2 + bt\right)\right]\mathrm{d}t = 2\pi \exp\left[ia\left(\frac{2a^2}{3} - b\right)\right]\mathrm{Ai}(b-a^2)$$
Now that that's taken care of... surely you must have a definition of the Airy function to work with? What is it? (That's what you should have put in the "relevant equations" section)