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Cosine Fourier transform

  1. Dec 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the cosine Fourier transform of the function f(t)=e-at

    2. Relevant equations



    3. The attempt at a solution

    F(w)=(2/π)0.5∫dt e-atcos(wt)
    The integral is from 0 to +∞

    Using euler's formula I got the result

    F(w)=(2/π)0.5( eit(w-a)/i(w-a) - e-it(w+a)/i(w+a) )

    I don't know what to do from here... I can't just substitute!
    If anyone could point me in the right direction I'd appreciate!
     
  2. jcsd
  3. Dec 25, 2012 #2

    Simon Bridge

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    You need a common denominator don't you?
    If I understand you correctly - you converted the cosine in the integral into a sum of exponentials, then did the integration?

    That would give you something like:
    $$e^{-at}\cos(wt)=\frac{1}{2} e^{-at}\left ( e^{iwt}+e^{-iwt} \right ) = \frac{1}{2}\left (e^{(iw-a)t}+e^{-(iw+a)t}\right )$$ ...in the integrand - which does not look like it ends up looking like yours. Can you show the intermediate steps in your working?

    Alternatively:
    You could try using integration by parts instead (hint: twice).
     
  4. Dec 26, 2012 #3
    You're correct. I made a mistake! The actual result is:

    [itex]{\frac{1}{2\sqrt{2π}}\left[\frac{exp(t(iw-a))}{iw-a}-\frac{exp(-t(iw+a))}{iw+a}\right]}[/itex]

    The problem remains tho... I don't know what to do from here, since I'm integrating from 0 to ∞ !
     
  5. Dec 27, 2012 #4

    Simon Bridge

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    Applying the limits - ##t=0## case is easy. It is ##t\rightarrow \infty## in the first term that gives you the problem?

    To procede from here you need to convert the exponentials back into cosine and sine functions ... think in terms of a product of and exponential and trig functions.

    I still think it is easier to just follow the hint in post #2.
     
  6. Dec 27, 2012 #5
    I tought about using your hint, but I'd end up with cosine and sine functions, and neither of them converge to a value as x goes to infinity! That's why I avoided using it!
     
  7. Dec 27, 2012 #6

    Dick

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    I think there is probably an unstated assumption that a>0. So your trig functions don't converge but they are bounded. e^(-at) goes to zero as t->infinity. What happens?
     
  8. Dec 28, 2012 #7

    Simon Bridge

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    Thanks Dick - yes: that would be a problem if the trig functions were not multiplied by a decreasing exponential.
     
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