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Math Help - Expanding a general term in the infinite product?

  1. #1
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    Expanding a general term in the infinite product?

    I have the product

    sin(x)/x
    = (1 - x^2/pi^2)(1 - x^2/4pi^2)(1 - x^2/9pi^2)(1 - x^2/16pi^2)......

    But i'm looking to expand the whole product and find only the term in x^k, for some general k. and find an expression for this general coefficient.

    I know I can equate it to the Taylor series, but I want to do it this way to get a different expression. Please help, thanks!
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  2. #2
    MHF Contributor chisigma's Avatar
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    The function sinc(*) is defined as...

     sinc (x) = \left\{\begin{array}{cc}\frac{\sin \pi x}{\pi x}, &\mbox{if } x\ne 0\\1, & \mbox{if } x=0\end{array}\right. (1)

    ... and, in my opinion, it is a powerful kay to open many 'doors' in the mathematical world. It admits the 'simplest existing' infinite product expansion...

    sinc (x) = \prod_{n=1}^{\infty} (1-\frac{x^{2}}{n^{2}}) (2)

    ... and from (2) we can derive many interesting results. As example let suppose to compute the log of the sinc(*). With easy steps we obtain...

    \ln sinc (x) = \sum_{n=1}^{\infty} \ln (1-\frac{x^{2}}{n^{2}}) = - \sum_{k=1}^{\infty} \frac{x^{2k}}{k}\cdot \sum_{n=1}^{\infty} \frac{1}{n^{2k}}= - \sum_{k=1}^{\infty} \frac{x^{2k}\cdot \zeta(2k)}{k} (3)

    ... where \zeta(*) is the 'Riemann Zeta Function'. The (3) suggests as alternative way for the computation of \zeta(2k) the following...

    \zeta(2k)= - \frac{k}{(2k)!} \cdot \frac{d^{2k}}{dx^{2k}} \ln sinc (x)_{x=0} (4)

    This is only one of the possibilities that the sinc (*) opens to You ...

    Kind regards

    \chi \sigma
    Last edited by chisigma; January 23rd 2010 at 11:50 PM. Reason: trivial error in (4) corrected...
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  3. #3
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    Hi, thanks for the answer. It's pretty much what I was looking for. Do u have any research related to this information? Or promising lines of attack to the Riemann hypothesis?
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