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Fourier transform of the linear function

  1. Mar 1, 2014 #1
    Hello,
    I was wondering if one can give meaning to the Fourier transform of the linear function:

    [tex] \int_{-\infty}^{+\infty} x e^{ikx}\, dx [/tex]

    I found that it is [tex] \frac{\delta(k)}{ik} [/tex], does this make sense?
     
  2. jcsd
  3. Mar 2, 2014 #2

    vanhees71

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    This expression doesn't make sense since it's intrinsically undefined. A handwaving way is
    [tex]\int_{\mathbb{R}} \mathrm{d} x x \exp(\mathrm{i} k x)=-\mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \int_{\mathbb{R}} \mathrm{d} x \exp(\mathrm{i} k x)=-2 \pi \mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \delta(k).[/tex]
     
  4. Mar 2, 2014 #3
    Hmm.. seems to make sense. Why is there a minus sign popping up?
     
  5. Mar 2, 2014 #4

    mathman

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    d/dk(exp(ikx)) = ixexp(ikx). you need -i to get 1 for the original integral.
     
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