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Fourier Transform

  1. Nov 10, 2015 #1
    1.
    Find the fourier series of :
    $$g(t)=\frac{t+4}{(t^2+8t+25)^2}$$

    2. I have been trying to write the function to match the formula $$\mathcal{F} [\frac{1}{1+t^2}] = \pi e^{-\mid(\omega)\mid}$$


    3.
    I have simplified the function to
    $$(t+4)(\frac{1}{9}(\frac{1}{1+\frac{(t+4)^2}{9}})^2)$$
    Cant really see where to go from here.
     
  2. jcsd
  3. Nov 10, 2015 #2
    Try doing a partial fraction decomposition on g.
     
  4. Nov 10, 2015 #3
    I may be misleading you here but if you complete the square and then do a variable substitution you get what I would say is an easier problem.
     
  5. Nov 10, 2015 #4

    Ray Vickson

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    You can write the Fourier transform as ##{\cal F}(g) (w) = \int_{-\infty}^{\infty} e^{-i w t} g(t) \, dt##, then apply integration by parts: ##\int u \, dv = uv - \int v \, du##, with ##u = e^{-iwt}## and ##dv = g(t) \, dt##. By a change-of-variables, the integral ##\int g\, dt## is do-able, and afterwards you will be left with a form in which your previous FT formula can now be used.

    BTW: you are NOT doing Fourier series, you are doing the Fourier transform. These two concepts are very different and should never be mixed up.
     
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