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Fourier sine and cosine tranformation, difficult problem, (for me)

  1. Jan 29, 2012 #1
    1. The problem statement, all variables and given/known data

    What are the Fourier sine and cosine transformations of exp(5t)?


    2. Relevant equations

    Fc (ω) = (√(2/∏))∫exp(5t)cos(ωt)dt , (between boundaries of infinity and zero)

    3. The attempt at a solution

    When I try to integrate by parts I just end up going round in circles.
    Or, if I first convert cos(ωt) into 0.5(exp(iωt) + exp(-iωt)) and then multiply this with exp(5t),
    calculating the boundary values then becomes difficult
     
  2. jcsd
  3. Jan 29, 2012 #2

    vela

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    Were you really given e+5t and not e-5t?
     
  4. Jan 29, 2012 #3
    Oops sorry! I forgot to mention that negative, but I still can't do the problem.
     
  5. Jan 29, 2012 #4

    vela

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    Either method you described will work. Can you show us what you've gotten so far?
     
  6. Jan 30, 2012 #5
    Well for the Fourier cosine transformation I end up with
    (1/√2∏) ( (exp((iω - 5)t)/(iω - 5)) + (exp((-iω - 5)t) / (-iω - 5)) ) evaluated over the range from 0 to ∞, but what do

    exp((iω - 5)t) and exp((-iω - 5)t) equal at t = ∞? .

    Similarly for the Fourier sine transformation.

    If there are plenty worked solutions to Fourier sine and cosine transformation problems, where are they?
    Please I am referring to worked solutions to problems finding the Fourier sine and cosine transformations of a difficult expression, not of some stupidly simple number like 1.
     
  7. Jan 30, 2012 #6
    Apparently the ultimate solution is

    Fc(ω) = (√(2/∏))(5/5squaredωsquared)

    but how?
     
  8. Jan 30, 2012 #7
    For the Fourier cosine transformation (and similarly for the sine) I got this far by firstly converting cos(ωt) into 0.5(exp(iωt) + exp(-iωt)) and then multiplying by exp(-5t),
     
    Last edited: Jan 30, 2012
  9. Jan 30, 2012 #8
    Is there an altogether better approach to this problem?
     
  10. Jan 30, 2012 #9
    I only know the simple answer to the problem I asked about, but not how to produce it, which is the real issue.
    If you are not going to help me out with this particular problem, which I can assure you is becoming a serious distraction from the rest of the subject, could you please describe the best general approach to Fourier sine and cosine transformations? The only worked solutions I can find are pretty unhelpful, as they are just for the Fourier sine and cosine transformations of the number 1, which largely translates into finding the integral of sin(ωt) or cos(ωt).
     
  11. Jan 30, 2012 #10
    Should I use frequency shifting?
    The fourier transform of f(t)cos(ωsub0t) being equal to 0.5(F(ω-ωsub0) + F(ω+ωsub0))
    and the fourier transform of exp(-5t) being 1 / (5 + iω)

    is the Fourier Transformation of exp(-5t)cos(ωsub0t) equal to 1 / (5 + iωsub0)?
    In that case the simple, final answer provided would be wrong.
     
  12. Jan 30, 2012 #11

    vela

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    Take, for example, your first term. The factor of eiωt has a magnitude of 1, regardless of what t equals, so
    $$\left|\frac{e^{(i\omega-5)t}}{i\omega-5}\right| = \left|\frac{e^{i\omega t}}{i\omega-5}\right| e^{-5t} = \frac{1}{\sqrt{\omega^2+5^2}}e^{-5t}.$$ When t goes to infinity, that quantity goes to 0, which means ##\frac{e^{(i\omega-5)t}}{i\omega-5}## goes to 0.
     
  13. Jan 30, 2012 #12
    Thanks.
    If |exp(iωt)| = 1 and |exp(-iωt)| = 1 for all t, how do sin(ωt) and cos(ωt) vary with t, when they are functions of exp(iωt) and exp(-iωt)?
     
  14. Jan 30, 2012 #13

    vela

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    Because |exp(iωt)| and exp(iωt) aren't the same thing. Sine and cosine aren't functions of |exp(iωt)|.
     
  15. Jan 30, 2012 #14
    Thanks again. Going off on a tangent for a moment, do exp(iωt) and exp(-iωt) each form a circle of unit radius on an argand diagram?

    When I substituted the number 1 for |exp(iωt)| and again the number 1 for |exp(-iωt)|, I
    ended up with Fc(ω) =(√(2/∏)) x (5 / (5squared PLUS ωsquared))

    Apparently it should actually be Fc(ω) =(√(2/∏)) x (5 / (5squared TIMES ωsquared))
     
  16. Jan 30, 2012 #15

    vela

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    Yes.

    The plus sign is correct. Another way you can look at the integral
    $$\int_0^\infty e^{-5t}\cos\omega t\,dt = \left.\int_0^\infty e^{-st}\cos\omega t\,dt\right|_{s=5}$$ is as the Laplace transform of cos ωt evaluated when s=5. If you look up the Laplace transform for cosine in a table, you'll see there's supposed to be a plus there.
     
  17. Jan 30, 2012 #16
    Thanks.
     
  18. Jan 31, 2012 #17
    Does |exp(-iωt)| = 1 aswell?
     
  19. Jan 31, 2012 #18

    vela

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    What does Euler's formula tell you?
     
  20. Jan 31, 2012 #19
    How should I manipulate Euler's formula ?
     
  21. Jan 31, 2012 #20
    Incorporated this post in the next one...
     
    Last edited: Jan 31, 2012
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