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Fourier Integrals and Division

  1. Mar 20, 2014 #1
    1. The problem statement, all variables and given/known data

    (a) Find the Fourier transform f(ω) of: f(x) = cos(x) between -pi/2 and pi/2
    (b) Find the Fourier transform g(ω) of: g(x) = sin(x) between = -pi/2 and pi/2
    (c) Without doing any integration, determine f(ω)/g(ω) and explain why it is so

    2. Relevant equations

    f(ω) = ∫f(x)e-iωx dx

    3. The attempt at a solution

    I was able to do parts (a) and (b) and verified my answers, however part (c) is giving me some problems. The division is straightforward for the two transformed equations. When I do the division, I get f(ω)/g(ω) = -1/(iω). I have verified this with fellow students as well, and we have all gotten the same thing. I am just confused as to the why . I feel like it might be some identity of Fourier integrals, but I cannot find it. I have looked through my textbook and I have been looking online, but I cannot really understand exactly why I get the answer I do.

    Any help would be greatly appreciated.
     
  2. jcsd
  3. Mar 21, 2014 #2

    vela

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    Hint: What linear operator can you apply to g(x) to get f(x)?
     
  4. Mar 21, 2014 #3

    vanhees71

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    Even easier: What linear operator can you use to come from [itex]f(x)[/itex] to [itex]g(x)[/itex]:biggrin:.
     
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