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Evaluating the n-th difference

  1. Jul 17, 2007 #1
    hi, my problem is to evaluate the n-th difference to be able to approximate the n-th derivative of a function f(x) at the point x=0

    as for small 'h' then [tex] \frac{\Delta ^{n}}{h^{n} \sim \frac{d^{n}f(0)}{dx^{n}} [/tex]

    i think that the n-th difference (with step h) has the representation:

    [tex] \Delta ^{n} f(0) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty}ds \frac{f(s)}{(s(s-h)(s-2h)(s-3h).....(s-nh)}x^{s} [/tex]

    and that evaluating this we could obtain an (approximate) asymptotic expansion for n-->infinity.

    the intention is given a generating function

    [tex] f(x)= a(0)+a(1)x+a(2)x^{2}+..................... [/tex]

    then to calculate a(n) you need to know the n-th derivative, we can approximate this derivative by the n-th forward difference at x=0 with an small step 'h'
  2. jcsd
  3. Jul 18, 2007 #2


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