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Complex analysis - electron screening

  1. Jun 29, 2011 #1
    I have to understand how this integral is evaluated (it is taken from Fetter - Quantum theory of many particle systems)(14.24):

    http://dl.dropbox.com/u/158338/fis/fetter.pdf" [Broken]

    in particular, i don't know how the log brach cuts are defined..
    as far as I know, log branch cuts can be define as the regions at fixed log argument.. (i.e. -inf to 0 for [-pi;pi[).
    Gnuplot says that the "natural" branch cut ([-pi;pi[) is the discontiunous line in this pic:

    http://dl.dropbox.com/u/158338/fis/arg-rminus.png" [Broken]

    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Jul 2, 2011 #2
    sorry, the right pdf link is http://dl.dropbox.com/u/158338/fis/fetter2.pdf" [Broken]

    also, I can't understand how the integral contour is chosen.

    thanks :)
    Last edited by a moderator: May 5, 2017
  4. Jul 2, 2011 #3
    I had a quick look. The log branch cuts are obviously taken directly upwards (as in the figure). They are using Cauchy's Theorem applied to a contour which goes from -infinity to +infinity along the real axis (the original contour for the integral), plus a semi-circle into the upper half plane. However, since you can't really pass through the cuts, you have to slit the semi-circle along the cuts.

    By Cauchy's Theorem, that is equal to the residue at the pole along the imaginary axis.
    Last edited by a moderator: May 5, 2017
  5. Jul 5, 2011 #4
    so is it a sort of jordan lemma? the integral along the semi circle is not 0, but it's the integral along C1 and C2.
    In other words:

    integral + C1 + C2 = residue of the pole
    integral = residue of the pole - C1 - C2

    is it correct?

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