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B Limit using the Sohotski-Plemenj formula

  1. Mar 18, 2016 #1
    Hi All, I am desperate to understand a calculation presented in a paper by Sethna, "Elastic theory has zero radius of convergence", freely available online

    $$ lim_{\epsilon \to +0}Z(-P+i\epsilon) = lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \int_{0}^{\infty} \mathrm{d}y \exp \{ -\alpha y -\beta x^2 - y(-P+i \epsilon)x \}$$

    The author says use is to be made of the Theorem

    $$ \lim_{\epsilon \to +0} \int \mathrm{d}x f(x)/(x+i \epsilon) = P.V. \int \mathrm{d}x f(x)/x -i \pi f(0) $$

    but how to get there?

    My best attempt is to integrate on the y variable, obtaining

    $$ lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \exp \{ -\beta x^2 \} [- \frac{1}{-\alpha-(-P+i \epsilon)x} ]$$

    To use the Theorem I need the denominator to be of the form y + i*epsilon, which is not as in the attempt I try the variable of Integration is multiplied by epsilon so I am unable to proceed.

    I am even more puzzled because if I set epsilon to zero, the integral I wrote coincides with the P.V. integral reported in the paper as the P.V. contribution in the Sohotski-Plemelj formula.

    How to calculate that limit then?

    Any help would be so appreciated
  2. jcsd
  3. Mar 23, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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