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Inequality involving abs. value of complex-valued multiple integral

  1. Mar 10, 2006 #1

    benorin

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    How do I show that

    [tex]0\leq \Re (s) +k\Rightarrow\left| (s)_k \int_0^1\cdots\int_0^1 (n+x_1+\cdots +x_k)^{-s-k}\, dx_1\cdots\, dx_k \right| \leq |(s)_k|n^{-\Re (s) -k}[/tex]​

    where k is a nongegative integer and [tex](s)_k:=s(s+1)\cdots (s+k-1)[/tex] is the Pochammer symbol (aka the rising factorial) ?

    If it helps, I know (and have previously proven) that

    [tex](s)_k\int_0^1\cdots\int_0^1 (n+x_1+\cdots +x_k)^{-s-k}\, dx_1\cdots\, dx_k = \sum_{m=0}^{k}(-1)^{m} \left(\begin{array}{c}k\\m\end{array}\right) (n+m)^{-s} =: \Delta ^k (n^{-s})[/tex]​

    where [tex]\Delta [/tex] is the forward difference operator.
     
    Last edited: Mar 10, 2006
  2. jcsd
  3. Mar 10, 2006 #2

    shmoe

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    You don't need your previous work, just use the simplest upper bound for your integral you can think of.
     
  4. Mar 10, 2006 #3

    benorin

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    I got it. Thanks shmoe.
     
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