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Contour integral example from "QFT for the gifted amatueur"

  1. Oct 26, 2015 #1
    Hi,

    I've never studied compex analysis before but I am trying to understand this example from "QFT for the gifted amatur".
    I don't understand why the residue at the pole is e-iEp(t-t')e-e(t-t'). How did the find e-e(t-t')?

    Thanks.

    Immagine.jpg
     
  2. jcsd
  3. Oct 26, 2015 #2

    stevendaryl

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    Staff Emeritus
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    Well, he (or she) starts with the function:

    [itex]\dfrac{i e^{-i E(t - t')}}{E - E_p + i \epsilon}[/itex]

    The residue of a function of the form [itex]\dfrac{A(E)}{E-K}[/itex] is just [itex]A(K)[/itex]. In this particular case, [itex]K = E_p - i \epsilon[/itex], and [itex]A(E) = i e^{-i E(t-t')}[/itex], so the residue is [itex]i e^{-i (E_p - i \epsilon) (t - t')} = i e^{-i E_p (t-t')} e^{- \epsilon (t-t')}[/itex]
     
  4. Oct 26, 2015 #3
    Thanks!
     
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