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Math Help - Fourier Transform Problem

  1. #1
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    Fourier Transform Problem

    I have got the below fourier problem:

    The Fourier transform of a function f(t) is defined as:

    F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^ \infty f(t) e^{-jwt}dt

    I have worked through it and got two different answers depending on whether I used a positive contour or negative contour.

    For the positive contour:
    j\sqrt{\frac{\pi}{2}} e^\omega

    For the negative contour:
    j\sqrt{\frac{\pi}{2}} e^{-\omega}

    I'm not sure whether these need to be combined or which one to use for the final answer. Looked at Wolfram Alpha and this gave something completely different.


    Can someone please point me in the right direction.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by jezzyjez View Post
    I have got the below fourier problem:

    The Fourier transform of a function f(t) is defined as:

    F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^ \infty f(t) e^{-jwt}dt

    I have worked through it and got two different answers depending on whether I used a positive contour or negative contour.

    For the positive contour:
    j\sqrt{\frac{\pi}{2}} e^\omega

    For the negative contour:
    j\sqrt{\frac{\pi}{2}} e^{-\omega}

    I'm not sure whether these need to be combined or which one to use for the final answer. Looked at Wolfram Alpha and this gave something completely different.


    Can someone please point me in the right direction.
    Just post the question as it is given please.

    CB
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    Just post the question as it is given please.

    CB
    The fourier transform of a function f(t) is defined as

    <br /> <br />
F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^ \infty f(t) e^{-jwt}dt<br />

    use this formula and residue calculus to compute the Fourier transform of the function

    f(t) = \frac{t}{1 + t^2}
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by jezzyjez View Post
    The fourier transform of a function f(t) is defined as

    <br /> <br />
F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^ \infty f(t) e^{-jwt}dt<br />

    use this formula and residue calculus to compute the Fourier transform of the function

    f(t) = \frac{t}{1 + t^2}
    Think about which way you traverse the two contours and what that means for the part of the integral along the real axis, also which contour you use depends on the sign of \omega to make the integral over the semicircular arc go to zero, see the very similar problem on the Wikipedia page.

    CB
    Last edited by CaptainBlack; May 10th 2010 at 06:20 AM.
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