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

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I'm stuck on this miscellaneous problem , please help! By the way Z^n is supposed to be lowered case.

Use the change of variable t = cos \theta

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.

http://en.wikipedia.org/wiki/Proofs_rel ... stribution
Check out the section "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. Whatever the case, this should be a clue as to whether you're on the right track or not if something like De Moivre's theorem fits in very nicely when changing to polar coordinates.