1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Integral using CIF

  1. Oct 8, 2011 #1
    1. The problem statement, all variables and given/known data
    [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
    [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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted