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Math Help - Integration using residue

  1. #1
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    Integration using residue

    Please, help to evaluate the integral (using residue)

     \int_{0}^{\infty} \frac{1}{(x^{2}+1)^n}dx,
    n - natural.

    Thx.
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  2. #2
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    Oh, a chance to do a little CA. Let's see. I will do the case -inf..inf and divide by 2.

    \int_{-\infty}^{\infty}\frac{1}{(x^{2}+1)^{n}}dx

    There is a pole of order n at z=i. The residue is

    \frac{1}{(n-1)!}\left[\frac{d^{n-1}}{dz^{2}}\left(\frac{1}{(z+i)^{n}}\right)\right]

    =\frac{1}{(n-1)!}\left[\frac{(-n)(-n-1)......(-n-n+2)}{(z+i)^{2n-1}}\right]

    =\frac{(-1)^{n-1}n(n+1)....(2n-2)(-1)^{n}i}{n^{2n-1}(n-1)!}


    Almost there, Can you take over?.
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  3. #3
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    ok thanks a lot
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