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Math Help - Fourier integral representation

  1. #1
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    Fourier integral representation

    Having trouble with the following finding the fourier integral representation for the following:

    f(t) = te^-|t|

    Ive got answer but I need verfication
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  2. #2
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    Quote Originally Posted by taltas View Post
    Having trouble with the following finding the fourier integral representation for the following:

    f(t) = te^-|t|

    Ive got answer but I need verfication
    Showing your working and answer might expedite the process of helping you. You did realise that f(t) is odd, yes? That simplifies the calculation .....
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    I've been working on a similar problem myself, i've plotted the graph in MATLAB and see that it is an odd function, equating all A terms to 0.
    I'm having trouble doing the integral of t*exp(-abs(t))sin(wt) thought. Any help would be greatly appreciated!!
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    Quote Originally Posted by taltas View Post
    Having trouble with the following finding the fourier integral representation for the following:

    f(t) = te^-|t|

    Ive got answer but I need verfication
    You will have to sort out the constants and a few other things yourself, but we are interested in:

    F(\omega)=\int_{-\infty}^{\infty} t e^{|t|}e^{i \omega t} ~dt

    Split the integral into two parts:

    F(\omega)=\int_{0}^{\infty} t e^{t}e^{i \omega t} ~dt<br />
+\int_{-\infty}^{0} t e^{-t}e^{i \omega t} ~dt

    in the second integral change the variable \tau=-t:

    F(\omega)=\int_{0}^{\infty} t e^{t}e^{i \omega t} ~dt<br />
+\int_{\infty}^{0} \tau e^{\tau}e^{-i \omega \tau} ~d \tau

    Now the second integral is minus the complex conjugate of the first so we can write:

    F(\omega)=2 i {\rm{Im}} \left[ \int_{0}^{\infty} t e^{t}e^{i \omega t} ~dt\right]=<br />
2 i {\rm{Im}} \left[ \int_{0}^{\infty} t e^{(1+i \omega) t} ~dt\right]<br />

    and the integral inside the rightmost brackets can be done by parts.

    RonL
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