By the way , the Z^n part is supposed to be lowered case , sorry.
What if I Use the change of variable t = cos \theta
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.
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.
What is your question?
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