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Evaluating this integral (Principal value)

  1. Oct 27, 2006 #1
    Let's suppose we wish to calculate:

    [tex] PV \int_{0}^{\infty}dx\frac{e^{-sx}}{1-x} [/tex] (1)

    I don't know how to do it, my idea is take the identity:

    [tex] \frac{1}{1-x}=1+x+x^{2}+x^{3}+............ |x|<1 [/tex]

    [tex] \frac{1}{1-x}=-x^{-1}-x^{-2}-x^{-3}+x^{3}+............ |x|>1 [/tex]

    So (1) using Cauchy's principal value, is the same as to calculate with a positive small epsilon:

    [tex] \int_{0}^{1-\epsilon}dxx^{k} e^{-sx} [/tex]

    [tex] \int_{1+\epsilon}^{\infty}dxx^{-k} e^{-sx} [/tex]

    and calculate term by term integration to get the Principal value of (1)
     
  2. jcsd
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