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Homework Help: Integration of exponential function

  1. May 7, 2009 #1
    1. The problem statement, all variables and given/known data
    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?

    2. Relevant 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)

    3. The attempt at a solution
  2. jcsd
  3. May 7, 2009 #2


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    Homework Helper

    First I have to rewrite it in nicer notation:
    [tex]\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)[/tex]
    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)
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