(New page: Show that <math>\int_{\mathbb{R}^{n}}e^{-|x|^{2}}d\bar{x} = \pi^{n/2}</math> Proof by induction(by Pirate Robert): For <math>n=1</math> it is an easy manipulation of Calculus 2 tricks. (...) |
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Revision as of 08:57, 27 July 2009
Show that $ \int_{\mathbb{R}^{n}}e^{-|x|^{2}}d\bar{x} = \pi^{n/2} $
Proof by induction(by Pirate Robert):
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}}e^{-|x|^{2}}dx_{1}\ldots dx_{n+1} $
