# $i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$

Here $i$ is complex number, $n$ is positive integer. Show that $$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$$

This question appears from Stein's book Introduction to Fourer Analysis on Eulidean Space, Chapter 4.4

It would be helpful if any hints are given. Thank you!

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if $k$ and $n$ are even, it seems trivial with using the Stirling formula, as $\Gamma(n) = n!$. – S4M Feb 6 '13 at 13:09
@S4M: $\Gamma{(n)} = (n-1)!$ – Ron Gordon Feb 6 '13 at 13:39
@rlgordonma: Doh! But it doesn't change anything to the rest. To the OP: as $\Gamma(x) > 0 \text{ for } x>0$, I don't see how this can be true if $i^{-k}\neq 1$, ie if $k\neq 0\text{ mod } 4$, since when that's not the case you will have something either negative either complex equivalent to a positive quantity. – S4M Feb 9 '13 at 9:44