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Math Help - Fourier transformation of e^(-|x|)

  1. #1
    Junior Member AlvinCY's Avatar
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    Fourier transformation of e^(-|x|)

    I'm currently stuck on a problem (it's a Fourier transform), I'm not sure it this is the right forum section to post it on, as it falls under several categories...

    \frac{1}{2\pi}\int\limits_{-\infty}^{\infty}e^{-|x|}e^{i\omega x}dx<br />

    I've broken the integral into two:

    \frac{1}{2\pi}[\int\limits_{-\infty}^{0}e^{x}e^{i\omega x}dx+\int\limits_{0}^{\infty}e^{-x}e^{i\omega x}dx]<br />

    Then combined the exponentials:

    \frac{1}{2\pi}[\int\limits_{-\infty}^{0}e^{x+i\omega x}dx+\int\limits_{0}^{\infty}e^{-x+i\omega x}dx]<br />

    But not sure where to go from here, I know the solution is meant to be:

    \frac{1}{\pi}\frac{1}{\omega^2+1}

    Any help you can offer me would be great!
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  2. #2
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    Quote Originally Posted by AlvinCY View Post
    I'm currently stuck on a problem (it's a Fourier transform), I'm not sure it this is the right forum section to post it on, as it falls under several categories...

    \frac{1}{2\pi}\int\limits_{-\infty}^{\infty}e^{-|x|}e^{i\omega x}dx<br />

    I've broken the integral into two:

    \frac{1}{2\pi}[\int\limits_{-\infty}^{0}e^{x}e^{i\omega x}dx+\int\limits_{0}^{\infty}e^{-x}e^{i\omega x}dx]<br />

    Then combined the exponentials:

    \frac{1}{2\pi}[\int\limits_{-\infty}^{0}e^{x+i\omega x}dx+\int\limits_{0}^{\infty}e^{-x+i\omega x}dx]<br />

    But not sure where to go from here, I know the solution is meant to be:

    \frac{1}{\pi}\frac{1}{\omega^2+1}

    Any help you can offer me would be great!
    = \frac{1}{2\pi} \left[\int\limits_{-\infty}^{0}e^{x(1+i\omega)}dx + \int\limits_{0}^{\infty}e^{-x(1-i\omega)} dx \right] = \frac{1}{2 \pi} \left[ \frac{1}{1 + i \omega} + \frac{1}{1 - i \omega} \right] = ....
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