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

[Li(n)]

http://i43.tinypic.com/20793eq.gif

By the way , the Z^n part is supposed to be lowered case , sorry.

[Li(n)]

What if I Use the change of variable t = cos \theta

[Li(n)]

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.

[Li(n)]

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.

[D H]

What is your question?