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Evaluate this complex Integral

  1. May 6, 2012 #1
    1. The problem statement, all variables and given/known data
    Use Cauchy's Integral Theorem to evaluate the following integral

    ##\int_0^{\infty} \frac{x^2+1}{(x^2+9)^2} dx##

    2. Relevant equations

    Res ##f(z)_{z=z_0} = Res_{z=z_0} \frac{p(z)}{q(z)}=\frac{p(z_0)}{q'(z_0)}##

    3. The attempt at a solution

    I determine the roots of the denominator to be ##x=\pm 3i##.
    How do I convert these into polar form. I know ##z=re^{i\theta}##

    Do I need to convert these into ##z=e^{f(i\theta)}##?
     
  2. jcsd
  3. May 6, 2012 #2
    I wouldn't convert this to polar form. You need to figure out how to write [itex]f[/itex] as a rational function like [itex]p/q[/itex] that you have above. Where [itex]p[/itex] is analytic and non-zero at [itex]z_0[/itex].
     
  4. May 7, 2012 #3
    I realise the pole is second order therefore using the residue forumla for second order pole ##z=3i## we get

    ##=lim_{z \to 3i} \frac{d}{dz} \frac{ z^2+1}{(z^2+9)^2} = \frac{-2z(z^2-7)}{(z^2+9)^3} |_{z=3i}##

    but we are going to get 0 in the denominator here..?
     
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