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Math Help - residues, improper integral

  1. #1
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    residues, improper integral

    Use residues to evaluate the improper integral.
    <br />
\int^{\infty}_{-\infty} \frac{\cos x dx}{(x+a)^2 + b^2} ( b>0).

    I do not see how to evaluate this integral now. In this section, we use M_R and C_R. I do not see how to use this here. I need hints on how to do this. Thanks in advance.
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  2. #2
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    Quote Originally Posted by canberra1454 View Post
    Use residues to evaluate the improper integral.
    <br />
\int^{\infty}_{-\infty} \frac{\cos x dx}{(x+a)^2 + b^2} ( b>0).

    I do not see how to evaluate this integral now. In this section, we use M_R and C_R. I do not see how to use this here. I need hints on how to do this. Thanks in advance.
    You do understand, don't you, that " M_R" and " C_R" are not universal symbols? I have no idea what you mean by that.

    But what I would do is set up a contour that runs along the real axis from -R to R, then along the semi-circle in the upper complex half-plane back to R.

    It should be easy to show that, as R goes to infinity, the the integral around the semi-circle goes to 0 so the integral on the real axis is equal to the residues in the upper complex half-plane. And in that half-plane your integrand is analytic everywhere except where (x-a)^2+ b= 0; in other words at x= a+ i\sqrt{b}. Find the residue at that point.
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