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Math Help - Interpolation problem

  1. #1
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    Interpolation problem

    Show that B (m + 1, n) = (-1)^m \Delta^m \left(\frac 1n \right)where m is a positive integer, n > 0

    Some problem solving hint is required from any member of this forum.
    Last edited by Vinod; July 13th 2013 at 04:57 AM.
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  2. #2
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    Re: Interpolation problem

    Hey Vinod.

    What is the definition of delta?
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  3. #3
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    Re: Interpolation problem

    Hi Chiro,
    Thanks for your prompt response. Yesterday, I found difficult to understand the answer provided in the book.But today I revised interpolation topic. As a result, I understood the answer to the problem
    Ans: We have

     \int_0^{\infty}e^{-nx}dx=\frac1n

    \triangle^m \int_0^{\infty}e^{-nx}dx=\triangle^m\left(\frac1n\right)

    \rightarrow \int_0^{\infty}\triangle^m e^{-nx}dx=\triangle^m\left(\frac1n\right)
    But
    \triangle^m e^{-nx}=\triangle^{m-1} (e^{-(n+1)x}- e^{-nx})

    =\triangle^{m-1} e^{-nx}(e^{-x}-1)
    =e^{-nx}(e^{-x}-1)^m
    Hence
    \int_0^{\infty} e^{-nx}(e^{-x}-1)^m dx=\triangle^m\left(\frac1n\right)

    Putting z = e^{-x} ,we obtain

    (-1)^m \int_0^{\infty}z^{n-1}(1-z)^m dz=\triangle^m\left(\frac1n\right)

    \rightarrow. B(m+1,n)=(-1)^m\triangle^m\left(\frac1n\right)



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