Problem #7.3, MA598R, Summer 2009, Weigel
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Show that $ \int_{\mathbb{R}^{n}}e^{-|x|^{2}}d\vec{x} = \pi^{n/2} $
Proof by induction (by Robert the Pirate):
For $ n=1 $ it is an easy manipulation of Calculus 2 tricks. (I really don't feel like writing the whole thing out)
Now, assume that for $ n $ the equation is true. We just need to show that it holds for $ n+1 $
$ \int_{\mathbb{R}^{n+1}}e^{-|x|^{2}}dx_{1}\cdots dx_{n+1}= $ $ \int_{\mathbb{R}}e^{-x_{n+1}^{2}}(\int_{\mathbb{R}^{n}}e^{-|x|^{2}}dx_{1}\cdots dx_{n})dx_{n+1} $ $ =\pi^{1/2}\cdot \pi^{n/2} = \pi^{(n+1)/2} $ (By Fubini's Theorem, induction hypothesis, and properties of exp)
Q.E.D
--Rlalvare 13:24, 27 July 2009 (UTC)
