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What is the Fourier Transform of t*f(t)

  1. Apr 16, 2012 #1
    1. The problem statement, all variables and given/known data
    f(t)=t*e^(-2t^2)
    Find the Fourier Transform F(w) of f(t).

    It is given that when f(t)=e^[(-at^2)/2] F(w)=√(2*pi/a)*e^[(-w^2)/2a]

    2. Relevant equations

    3. The attempt at a solution
    The transform of e^(-2t^2) is easily obtained from the given information, and I got stuck when I tried to do it with integrate by part.

    Is it related to the convolution? something like F(fg)=F(f)F(g) ??
    Is there any theorem for the Fourier Transform of t*f(t)?
    And also for Laplace Transform too?
     
  2. jcsd
  3. Apr 17, 2012 #2
    Hmm. It is a convolution integral of the transforms, was that what you where looking for so:
    [itex]\mathcal{F}t*f(t) = \int w*f(w-t)\,\text{d}t[/itex]
    But that makes the problem more complicated.
     
  4. Apr 17, 2012 #3
    Try completing the square in the argument of the exponential in the transform integral and then doing a change of variables. That should break up the integral into two parts. One of the parts will be easy to integrate, and for the other you can use the information given.
     
  5. Apr 18, 2012 #4
    So there isnt a theorem where I can simply put 2 transforms function together?!


    Hi Tachyon, is this what you mean?

    1. ∫t*e^-2t^2 e^-iwt dt
    2. -2t^2-iwt = -2(t^2+iwt/2) = -2[(t+iw/4)^2+w^2/16]=-2(t+iw/4)^2 -(w^2)/8
    as I let p=t+iw/4 therefore dp=dt
    3. ∫t*e^((-w^2)/8)dt +∫e^(-2p^2) dp

    Can I used the f(t)=e^[(-at^2)/2] F(w)=√(2*pi/a)*e^[(-w^2)/2a] information here? But isn't the Transform =∫f(p)e-iwp dp? or F(w)=∫f(t)e-iwt dt

    And one more stupid question from me....
    What happen if we need to do a transform of f(t) to F(w), with no limits of t given??
    for example after integration: F(w)= [(t^2)/2] limit ∞ to -∞(?!) + something...

    Or there must be a limit??
     
  6. Apr 18, 2012 #5
    Step 2. looks fine to me, but there is a mistake going into step 3. so take a look at that.

    You should have from [itex]p=t+i\omega /4[/itex] that [itex]dp=dt[/itex], but also that [itex]t=p-i\omega /4[/itex]. It is the second of these equations that will split your integral into a sum. One of these integrals will be of the form [itex]e^{-2p^2}[/itex] while the other will be of the form [itex]pe^{-2p^2}[/itex]. To evaluate the first of these, you will have to use the information given, and to evaluate the second one you can use basic integration techniques.

    I am not sure I understand your second question, but in general to use the Fourier Transform you are using right now, you will want t to be an unbounded variable, that is it should vary from -∞ to +∞. If you were given limits for t, so that for example a<t<b, then you should consider doing a Finite Fourier transform instead.
     
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