| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| + | [[Category:MA598RSummer2009pweigel]] | ||
| + | [[Category:MA598]] | ||
| + | [[Category:math]] | ||
| + | [[Category:problem solving]] | ||
| + | [[Category:real analysis]] | ||
| + | |||
| + | == Problem #7.3, MA598R, Summer 2009, Weigel == | ||
| + | |||
Back to [[The_Pirate's_Booty]] | Back to [[The_Pirate's_Booty]] | ||
| Line 16: | Line 24: | ||
--[[User:Rlalvare|Rlalvare]] 13:24, 27 July 2009 (UTC) | --[[User:Rlalvare|Rlalvare]] 13:24, 27 July 2009 (UTC) | ||
| + | ---- | ||
| + | [[The_Pirate%27s_Booty|Back to the Pirate's Booty]] | ||
| + | |||
| + | [[MA_598R_pweigel_Summer_2009_Lecture_7|Back to Assignment 7]] | ||
| + | |||
| + | [[MA598R_%28WeigelSummer2009%29|Back to MA598R Summer 2009]] | ||
Latest revision as of 05:57, 11 June 2013
Problem #7.3, MA598R, Summer 2009, Weigel
Back to The_Pirate's_Booty
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)
