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Math Help - Principle Value integral

  1. #1
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    Principle Value integral

    Prove that PV \int_{-\infty}^\infty e^{-ix}dx = 0.
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  2. #2
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    Re: Principle Value integral

    If I remember correctly, PV \int_{-\infty}^\infty e^{-ix}dx = \lim_{N\to\infty}\int_{-N}^N e^{-ix}dx.

    And to integrate, you treat -i as a constant \int e^{-ix}=\frac{1}{-i}e^{-ix} = ie^{-ix}

    so PV \int_{-\infty}^\infty e^{-ix}dx = \lim_{N\to\infty} ie^{-iN} -ie^{iN} = \lim_{N\to\infty} 2\sin{N}

    and it looks like it doesn't converge.

    - Hollywood
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