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Math Help - Fourier Transform - not sure if I totally understand?

  1. #1
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    Fourier Transform - not sure if I totally understand?

    So the question states:
    Find the fourier transform of the function f defined by:
    f(x) = -e^(ax) x<0
    = e^(-ax) x>0

    I think I know what I'm doing, but this is totally new to me, and confirmation that I know what I'm doing would be awesome!!!!
    So my answer is:

    the transform of e^(-ax) is:
    f hat k = integral(0 to infinity) e^(-ax) e^(-ikx) dx
    = integral(0 to infinity) e^-(a+ik)x dx
    = 1/(a+ik)

    the transform of -e^(ax) is:
    f hat k = - integral(0 to infinity) e^(ax) e^(-ikx) dx
    = - integral(0 to infinity) e^(a-ik)x dx
    = - 1/(a-ik)

    so total answer is 1/(a+ik) - 1/(a-ik)
    So cross multiplying with congugates, gives:

    2ik/(a^2+k^2)

    Is this correct?
    Any help would be hugely appreciated! Thanks!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by MeganKnox View Post
    So the question states:
    Find the fourier transform of the function f defined by:
    f(x) = -e^(ax) x<0
    = e^(-ax) x>0

    !
    \mathfrak{F}f(\omega)=\int_{-\infty}^{\infty}f(x)e^{-i\omega x}\;dx <br />
 = \int_{-\infty}^{0}-e^{ax}e^{-i\omega x}\;dx+ \int_{0}^{\infty}e^{-ax}e^{-i\omega x}\;dx<br />

    CB
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