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Circle method for Laplace transform.

  1. Apr 2, 2007 #1


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    If we have the Laplace transform:

    [tex] \int_{0}^{\infty}dtf(t)exp(-st) = \sum_{n=0}^{\infty}(f(n)-f(n-1))\frac{exp(-sn)}{s}=g(s) [/tex]

    (We have used Abel sum formula on the right hand)

    then making Z=exp(s) we find the Z-transform:

    [tex] \sum_{n=0}^{\infty}(f(n)-f(n-1))Z^{-n} [/tex]

    which can be inverted to get:

    [tex] 2\pi i (f(n)-f(n-1))=\oint g(lnZ) (lnZ)Z^{n-1} [/tex]

    my quetion is if using 'Circle method' you can get an asymptotic expansion for: [tex] f(n)-f(n-1) [/tex] n big, also another question if we have that:

    [tex] f(n)-f(n-1) \sim h(n) [/tex] then ??? [tex] f(n) \sim \int h(n)dn [/tex]
    Last edited: Apr 2, 2007
  2. jcsd
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