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Difficult integral involving exponential

  1. Apr 13, 2009 #1
    1. The problem statement, all variables and given/known data
    I'm trying to verify the Fourier transform but am getting stuck on the integration. Here is the pair:
    [tex]f(x) = e^{-ax^2}[/tex]
    [tex]\hat{f}(k) = \frac{1}{\sqrt{2a}}e^{-k^2/4a}[/tex]
    [tex]a>0[/tex]

    2. Relevant equations

    I know that
    [tex]\hat{f}(k)=\int_{-\infty}^{\infty}f(x)e^{ikx}dx[/tex]

    3. The attempt at a solution

    So I have
    [tex]\hat{f}(k)=\int_{-\infty}^{\infty}e^{-ax^2}e^{ikx}dx[/tex]
    [tex]\hat{f}(k)=\int_{-\infty}^{\infty}e^{-ax^2+ikx}dx[/tex]

    I tried using integration by parts and I'm not sure that's the right way to go. If it is I'm not sure how to go about it without getting a more complicated integral.
     
  2. jcsd
  3. Apr 14, 2009 #2

    Cyosis

    User Avatar
    Homework Helper

    You need to complete the square, which means:

    [tex]-(ax^2+ikx)=-\left[(\alpha x+\beta)^2+\gamma \right][/tex].

    Find [itex]\alpha,\beta[/itex] and [itex]\gamma[/itex].

    Edit1: it seems you have either listed [itex]\hat{f}(k)[/itex] wrong or the book where you got [itex]\hat{f}(k)[/itex] from is wrong, because the answer should be:

    [tex]
    \hat{f}(k) = \sqrt{\frac{\pi}{a}}e^{-k^2/4a}
    [/tex]

    edit2: While making no difference to the final answer in this case, shouldn't it be [tex]\hat{f}(k)=\int_{-\infty}^{\infty}f(x)e^{-ikx}dx[/tex], note the minus sign.
     
    Last edited: Apr 14, 2009
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