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Re{n} > -1/2. Prove that

  1. Dec 28, 2011 #1
    20793eq.gif

    By the way , the Z^n part is supposed to be lowered case , sorry.
     
    Last edited: Dec 28, 2011
  2. jcsd
  3. Dec 28, 2011 #2
    What if I Use the change of variable t = cos \theta
     
  4. Dec 30, 2011 #3
    Here are some thoughts : At a glance, the second last inequality contains the MGF for the chi-squared distribution, and just looking at the integrals, the change of coordinates for the normal distribution may be involved somewhere in there. Consider also that the chi-square distribution with n - 1 degrees of freedom is the limit distribution of a sum of n Z^2 random variables, where Z is the standard normal.

    I also recall seeing the gamma function in the proof of the symmetry of geometric brownian motion about the x axis, so that may be distantly related.
     
  5. Dec 31, 2011 #4
    http://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution
    I found something , "derivation of the pdf for k degrees of freedom":

    The rest is just a matter of changing to polar coordinates.

    I'm not too well-versed with complex transforms, though, since there's a complex number. I think this is a clue whether I'm on the right track or not if something like De Moivre's theorem fits in very nicely when changing to polar coordinates.
     
  6. Dec 31, 2011 #5

    D H

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    What is your question?
     
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