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Can Borel resummation be applied to integrals ?

  1. Jul 23, 2007 #1
    The question is can we obtain the 'Borel sum' of an integral of f(x) from 0 to infinity as the Laplace transform of

    [tex] \int_{0}^{\infty}dx \frac{f(x)}{\Gamma(x+1+u)t^{x+u} [/tex]

    where 'alpha' is a real or Complex number
  2. jcsd
  3. Jul 23, 2007 #2
    i meant (but the nasty latex does not work)

    [tex] \int_{0}^{\infty} dx \frac{f(x)}{G(x+1+u)}t^{x+u} [/tex]

    then the Laplace transform evaluated at s=1 is the 'Borel sum' of the integral

    [tex] \int_{0}^{\infty} dx f(x) [/tex]

    Of course G(x) is the 'Gamma function' generalization of factorial n!
    Last edited: Jul 23, 2007
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