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Math Help - complex integral

  1. #1
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    complex integral

    Hi. I wish help to solve this complex integral:
    \int \limits_{-\infty}^{+\infty} {e^{-\pi x^2} \cdot e^{-2 \pi \cdot i \cdot x y} \ dx}
    Thanks everyone!
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  2. #2
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    Quote Originally Posted by enricokr View Post
    Hi. I wish help to solve this complex integral:
    \int \limits_{-\infty}^{+\infty} {e^{-\pi x^2} \cdot e^{-2 \pi \cdot i \cdot x y} \ dx}
    Thanks everyone!
    It is a known result from complex analysis (I can derive it if you wish) that:
    \int \limits_{-\infty}^{+\infty} {e^{-t^2} \cdot \cos( \alpha t) \ dt}=\sqrt{\pi} \cdot e^{-\alpha^2/4}<br />
    If you write your integral out it becomes,
    \int \limits_{-\infty}^{\infty} e^{-\pi x^2} \cos (2\pi yx) dx + i\int \limits_{-\infty}^{\infty} e^{-\pi x^2} \sin (2\pi yx) dx
    However, the second integral vanishes!

    Now use substitution x\to x\sqrt{\pi} to bring that integral into the form above.
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