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Math Help - Doubly Infinite Integral

  1. #1
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    Doubly Infinite Integral

    Show that for any real numbers a, b, c with a < b < c we have

    PV \int^{\infty}_{-\infty} \frac{1}{(x-a)(x-b)(x-c)}dx =0.


    I know, in general that the Cauchy principal value of a doubly infinite integral of a function f is

    PV \int_{-\infty}^{\infty}f(x)dx=\lim_{R \rightarrow  \infty}\int_{-R}^Rf(x)dx.

    Also, here we have three consecutive real numbers. However, I do not see how we get the Cauchy principal value from all this. I need a bit of help on how this one.
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  2. #2
    Super Member
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    Aug 2008
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    Hi. We can rely on the Residue Theorem to show this by considering a closed contour with indentations around and over each pole along the real axis and then circling back about the upper half-plane. We then have:

    \oint f(z)dz=P.V. \int_{-\infty}^{\infty} f(z)dz-\pi i(r_1+r_2+r_3)=0

    (since we're only going around \pi at each pole and in the reverse direction). Then:

    P.V.\int_{-\infty}^{\infty} \frac{1}{(x-a)(x-b)(x-c)}dx=\pi i (r_1+r_2+r_3)

    where r_1,r_2,r_3 are the residues at the poles. I'll leave it to you to show that the sum of those three residues is zero and also should say something about how the integral over the remaining large semi-circle of the contour tends to zero as R\to\infty.
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