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Weird Integral

  1. Sep 18, 2006 #1

    dextercioby

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    It's formula #686 of Zwillinger's book "CRC Standard Mathematical Tables and Formulae" 31-st edition, 5-th chapter.

    He claims that

    [tex] \int_{0}^{1} \frac{dx}{\sqrt{\ln\left(\ln\frac{1}{x}\right)}} =\sqrt{\pi} [/tex].

    Is it correct...?:confused: And if so, how does one find/prove something like that...?

    Daniel.
     
  2. jcsd
  3. Sep 18, 2006 #2

    StatusX

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    That doesn't look right. As x goes from 0 to 1, ln(1/x) goes from infinity to 0, and ln(ln(1/x)) goes from infinity to -infinity, all monotonically. ln(ln(1/x))=0 at x=1/e and so the integral from 0 to 1/e is real and nonzero, while the integral from 1/e to 1 is imaginary and nonzero, so the answer should be complex.
     
    Last edited: Sep 18, 2006
  4. Sep 18, 2006 #3

    dextercioby

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    I could do that myself as well. It looked rather odd that i couldn't find the integral in the bibliographical resources...

    Daniel.
     
  5. Sep 18, 2006 #4

    George Jones

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    It seems that there are too many ln's, i.e.,

    [tex]
    \int_{0}^{1} \frac{dx}{\sqrt{\ln\frac{1}{x}}} = \Gamma \left( \frac{1}{2} \right) =\sqrt{\pi}
    [/tex]
     
  6. Sep 18, 2006 #5

    dextercioby

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    Wow, that can be it. It sounds very reasonable.

    Daniel.
     
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