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Fourier transform of e^(-ltl)

  1. Mar 1, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the fourier transform of f(t)=exp(-ltl)


    2. Relevant equations
    The expression for the fourier transform.

    3. The attempt at a solution
    Applying the fourier transform I get an expression, where I have to take the limit of t->-∞ of exp(-i[itex]\omega[/itex]t) - how do I do that?
     
  2. jcsd
  3. Mar 1, 2012 #2

    jbunniii

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    What is the expression you obtained?
     
  4. Mar 1, 2012 #3
    Yes I better post it - just tried to avoid it since I'm not good at latex. I get:
    1/√2π(∫-∞0exp(t-iωt)dt + ∫0exp(-t-iωt)dt)
    =
    1/√2π([1/(1-iω)exp(t-iωt)]-∞0+[-1/(1+iω)exp(-t-iωt)]0
     
    Last edited: Mar 1, 2012
  5. Mar 1, 2012 #4

    jbunniii

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    OK, that looks right. But look more carefully: the limit you need is not

    [tex]\lim_{t \rightarrow \infty} \exp(-i\omega t)[/tex]
    but rather
    [tex]\lim_{t \rightarrow \infty} \exp(-|t|)\exp(-i\omega t)[/tex]
     
  6. Mar 1, 2012 #5
    oh god yes :)

    But still: What if exp(-iwt) tends to infinity? Okay I don't think it does since its a sum of a real cos and an imaginary sin but still I want to know how to calculate the limit of the term with the complex exponential :)
     
  7. Mar 1, 2012 #6
    Either way I get:

    1/√2π *(1/(1+ω2)

    Do you also get that? :)
     
  8. Mar 1, 2012 #7
    nvm that was almost correct but got the right one now :)
     
  9. Mar 1, 2012 #8

    jbunniii

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    [itex]\exp(-i\omega t)[/itex] has no limit as [itex]t \rightarrow \infty[/itex]. It just spins endlessly around the unit circle in the complex plane.

    [itex]\exp(-|t|)\exp(-i\omega t)[/itex] spirals endlessly around the origin, but the [itex]\exp(-|t|)[/itex] factor pushes it closer and closer to the origin as t increases toward [itex]\infty[/itex] (or decreases toward [itex]-\infty[/itex]). This is why it has a limit even though one of its factors does not.

    You can see this more formally by looking at the magnitude:

    [tex]|\exp(-|t|)\exp(-i\omega t)| = |\exp(-|t|)|\cdot |\exp(-i\omega t)| = |\exp(-|t|)| \cdot 1 = |\exp(-|t|)| = \exp(-|t|)[/tex]

    which goes to zero as t goes to either [itex]+\infty[/itex] or [itex]-\infty[/itex]. And the magnitude of a function goes to zero if and only if the function itself goes to zero.
     
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