(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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| − | Show that <math>\int_{\mathbb{R}^{n}}e^{-|x|^{2}}d\ | + | Show that <math>\int_{\mathbb{R}^{n}}e^{-|x|^{2}}d\vec{x} = \pi^{n/2}</math> |
| − | Proof by induction(by | + | Proof by induction (by Robert the Pirate): |
For <math>n=1</math> it is an easy manipulation of Calculus 2 tricks. (I really don't feel like writing the whole thing out) | For <math>n=1</math> it is an easy manipulation of Calculus 2 tricks. (I really don't feel like writing the whole thing out) | ||
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Now, assume that for <math>n</math> the equation is true. We just need to show that it holds for <math>n+1</math> | Now, assume that for <math>n</math> the equation is true. We just need to show that it holds for <math>n+1</math> | ||
| − | <math>\int_{\mathbb{R}^{n}}e^{-|x|^{2}}dx_{1}\ | + | <math>\int_{\mathbb{R}^{n+1}}e^{-|x|^{2}}dx_{1}\cdots dx_{n+1}=</math> |
| + | <math>\int_{\mathbb{R}}e^{-x_{n+1}^{2}}(\int_{\mathbb{R}^{n}}e^{-|x|^{2}}dx_{1}\cdots dx_{n})dx_{n+1}</math> | ||
| + | <math>=\pi^{1/2}\cdot \pi^{n/2} = \pi^{(n+1)/2}</math> (By Fubini's Theorem and properties of exp) | ||
| + | |||
| + | Q.E.D | ||
Revision as of 08:19, 27 July 2009
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 and properties of exp)
Q.E.D
