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Math Help - Integral Representation of Gamma Function

  1. #1
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    Integral Representation of Gamma Function

    Let \Gamma_N(s) = \frac{N^s(N-1)!}{s(s+1)\cdot \ldots \cdot (s+N-1)}.


    Show, for every N \in \mathbb{N} that

    \int_0^N \displaystyle \left(1 - \frac{t}{N}\right)^N t^{s-1}dt = N^s \sum_{k=0}^N \frac{(-1)^k {N \choose k}}{k+s} = \frac{N}{N+s}\Gamma_N(s).

    Hint: For the second inequality, divide both sides by N^s and compare the resulting rational funtions in terms of their poles.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by EinStone View Post
    Let \Gamma_N(s) = \frac{N^s(N-1)!}{s(s+1)\cdot \ldots \cdot (s+N-1)}.


    Show, for every N \in \mathbb{N} that

    \int_0^N \displaystyle \left(1 - \frac{t}{N}\right)^N t^{s-1}dt = N^s \sum_{k=0}^N \frac{(-1)^k {N \choose k}}{k+s} = \frac{N}{N+s}\Gamma_N(s).

    Hint: For the second inequality, divide both sides by N^s and compare the resulting rational funtions in terms of their poles.
    Here's a hint for the first equality:

     \left(1-\frac{t}{N}\right)^N = \sum_{k=0}^N {N\choose k} \left(-\frac{t}{N}\right)^k by the binomial expansion theorem.
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  3. #3
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    I don't see how this can help me... , can you give further help?

    EDIT: Ah lol, I just saw that I can pull everything out of the integral then its easy. Ok I'll now try the second equation, if you can give any help please do it .
    Last edited by EinStone; April 5th 2010 at 03:14 AM.
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