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Homework Help: Integral using CIF

  1. Oct 8, 2011 #1
    1. The problem statement, all variables and given/known data
    evaluate
    [tex] \int_{-\infty}^{\infty} \frac{dx}{x^2 - \sigma^2} [/tex]


    2. Relevant equations

    CIF: [tex] \int f(z) = 2 \pi i \times \sum Res f(z) |_a [/tex]

    3. The attempt at a solution
    let
    [tex] \int_c \frac {dz}{z^2 - \sigma^2} = \int_c \frac{1}{(z - \sigma)(z+ \sigma)} [/tex]
    with poles at [itex] z = \pm \sigma [/itex] but i take only those in the upper half plane, and thus the residue:
    [tex] \lim_{ z \to \sigma} (z-\sigma) \frac{1}{( z- \sigma)( z + \sigma)} = \frac{1}{\sigma + \sigma} = \frac{1}{2 \sigma} [/tex]
    with the integral becoming :
    [tex] 2 \pi i \sum Res f(z) = 2 \pi i \frac{1}{2 \sigma} = \frac{\pi i }{\sigma} [/tex]
    please correct me!
    im trying out some stuff in residues and their applications to evaluation of integrals, the question had these three parts:
    1. let [itex] \sigma \to \sigma +i \gamma [/itex]
    2. let [itex] \sigma \to \sigma - i \gamma [/itex]
    3. Take the cauchy principle values.
    i think i just did the third part, if i did it properly to begin with... but that aside, how do i begin on the other two parts?
     
  2. jcsd
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