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Math Help - why is this integral finite

  1. #1
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    why is this integral finite

    Hello,

    i want to show, that \int_{\left||x\right||\leq 1} \left|P(x)\right|^{-q}*\left|x\right|^{-M}dx <\infty

    Here P is a polynomial of order m in \mathbb{R}^n and M,q are some constants.
    We know that \int_{\left||x\right||\leq 1} \left|P(x)\right|^{-q} <\infty for some constant q.
    But why is the first integral finite?
    Regards
    Last edited by Plato; July 23rd 2011 at 04:36 AM. Reason: LaTeX fix
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  2. #2
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    Re: why is this integral finite

    I canīt read your post.
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  3. #3
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    Re: why is this integral finite

    I hope, you can see it now.

    I have a idea, how to solve this problem. But i'm not sure, whether this way is correct.

    We know that \int_{\left||x\right||\leq 1} \left|P(x)\right|^{-q}*\left|x\right|^{-M}dx < \infty
    Can i conclude, that the set S=\{x\in \mathbb{R}^n : |P(x)\right|^{-q}*\left|x\right|^{-M}<C\} has finite measure? Why is this correct?

    Regards
    Last edited by Sogan; July 23rd 2011 at 01:39 PM.
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  4. #4
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    Re: why is this integral finite

    Take a close look at P(x). and q. Consider what you have and when it would be a problem.

    If q >= 0, P(x) ^ q is just a polynomial (possibyl degenerate). That should be finite.
    If q < 0, there would be a problem only if P(a) = 0 for some 'a' in [-1,1].
    However, if it's going to be a problem for some 'a' in [-1,1], we have that last piece of information.

    Did we get anywhere?
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