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Homework Help: How to evaluate this integral?

  1. Feb 9, 2012 #1
    1. The problem statement, all variables and given/known data

    [tex]\int_{-\infty}^{\infty} \sin {( 2 \pi y )} \cdot e^{- \frac{(x-y)^2}{4 \nu t}} dy[/tex]

    x is between 0 and 1.
    [itex]\nu[/itex] is a positive constant.
    t is positive.

    2. Relevant equations



    3. The attempt at a solution

    I tried substituting one variable for x-y, and integration by parts. So far, I've gotten nowhere. Can someone tell me which direction to go?

    Thanks.

    Edit: Also tried considering the Gaussian integral. But when I change the integration variable to get [itex]e^{u^2}[/itex], there's also a Sin term present. I tried to use integration by parts to get rid of that, but couldn't.
     
    Last edited: Feb 9, 2012
  2. jcsd
  3. Feb 9, 2012 #2

    vela

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    I haven't worked this out, but what I'd try is starting with u=x-y and expand the sin using a trig identity. Argue the sin u term will integrate to 0 and use
    $$\cos \theta = \frac{e^{i\theta}+e^{-i\theta}}{2}$$ on the other term.
     
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