Talk:Gaussian integral

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I rewrote the entire page, which was exhausting work. I was a bit tired of being thorough by the time I got to the end, I was just trying to finish, so if anybody could polish the article up, it would be appreciated.

Also, any ideas for another section? And does anybody know any more references that could be added?

The original page included a generalization to much higher dimensions, which included a whole bunch of complicated mathematics. I took that out, but if somebody actually finds a good source for this, then that person could add the generalization. The original page also said that the gaussian integral could be replicated, via varaible substitution, by the gamma function as Gamma(1/2). I see how the two are equal with the transitive property, but I dont see how this could be shown with the variable changes and simple calculus. If anybody wishes to keep this in, he or she should include an explanation.

EulerGamma 18:46, 2 September 2006 (UTC)

The evaluation of Γ(1/2) by rewriting it as a Gaussian integral is standard textbook material. The substitution t = x2 is all there is to it; what do you need explained? The integral can also be calculated automatically by any computer algebra system. Fredrik Johansson 20:05, 2 September 2006 (UTC)
As for the generalization to higher dimensions, containing "a whole bunch of complicated mathematics" is not a good reason for deleting it. I know nothing about the topic, but simply googling for "n-dimensional gaussian integral" would reveal that it's a real and important concept. Fredrik Johansson 22:14, 2 September 2006 (UTC)
Okay, sorry for being such a nuisance with editing everybody's work. I included the n-dimensional parts back, but someone will have to find some more references. As for the gamma function and substitution, I can't believe I didn't see it, but I have not really been through any formal education past college algebra. Anything higher (like calculus) I learn from the web. I also put back the Gamma function section. I will try to look into things much more before I edit them in the future. EulerGamma 20:55, 3 September 2006 (UTC)

Actually, I am having second thoughts once again about the other two sections. (dimensional cases with matrices...) This, to me, is gibberish. Not that that means it shouldnt be included, but that I do not think the average reader of the Gaussian Integral would be able to comprehend such. Not only this, but there is no source that I see, and it definetely needs a source because it would take some advanced mathematician to be able to verify this or say that he/she has seen it before. EulerGamma 04:44, 6 September 2006 (UTC)

Why don't you just add proper tag ({{unreferenced}} or {{technical}}) instead of deleting the section? That you don't understand it doesn't mean someone else cannot. Samohyl Jan 06:11, 7 September 2006 (UTC)
just a comment. seems to me EulerGamma made his edits in good faith. since he says he'd be more careful in the future, a honest mistake like that is forgivable. :-) Mct mht 06:52, 7 September 2006 (UTC)

Yes, the sections on the multidimensional case look correct. They are most definitely not gibberish (though they could, like much of Wikipedia, use some editing). If you're looking for references, a place to start would be Gradshteyn and Ryzhik or books of tables of probability distributions.

If you're wondering how to derive the formulas, they come from the observation that exp(-x^2-y^2)=exp(-x^2)*exp(-y^2) along with a change of variable (a rotation). That is,

\int \int exp(-x^2-y^2)\,dxdy=\int exp(-x^2)\,dx \int exp(-y^2)\,dy,

and any symmetric matrix A has the decomposition A = QTΛQ where Q is an orthogonal matrix (a rotation with maybe some reflections); change Λ1 / 2Qx = y (note that det Q=1).

Cheers, Lunch 02:00, 8 September 2006 (UTC)

To Lunch: You should put some restriction on Λ or your "decomposition" could be trivial, i.e. Q = the identity matrix and Λ = A. Perhaps you want Λ to be diagonal? JRSpriggs 03:20, 8 September 2006 (UTC)
I was thinking, but didn't say it, that Λ is a diagonal matrix with positive entries on the diagonal. The article requires an A that is symmetric positive definite, and it is a convention in linear algebra that Λ is used to denote diagonal matrices. What I wrote was meant as more of a "get you started hint" than a detailed derivation. Lunch 20:02, 9 September 2006 (UTC)

I'll use those templates in the future. EulerGamma 23:46, 8 September 2006 (UTC)

Contents

[edit] typesetting conventions

Wrong:

\int \int exp(-x^2-y^2)\,dxdy=\int exp(-x^2)\,dx \int exp(-y^2)\,dy,

Right:

\int \int \exp(-x^2-y^2)\,dx\,dy=\int \exp(-x^2)\,dx \int \exp(-y^2)\,dy,

In the second display, (1) the comma is INSIDE the math tags; this prevents misalignment; (2) \exp rather than exp ; this not only prevents italicization but provides proper spacing; (3) \, between dx and dy. Michael Hardy 00:39, 10 September 2006 (UTC)

jeeez. all i was trying to do was give a helpful hint to someone to figure out a formula they had trouble with. and if you have a beef with the way the Holder condition page was laid out, please note i didn't write most of that page; i just added one sentence. Lunch 16:34, 10 September 2006 (UTC)

[edit] Nice article

Very nice article. You should try to nominate it for something. Just add some information about who did it first, when and why, etc., that is, some more than just the mathematical proof. Add some extra references perhaps. Success! Shinobu 19:35, 1 June 2007 (UTC)

[edit] alternative proof

I've added an alternative way to prove the Gauss integral. I think it looks easier. Maybe a little make up is needed, I'm not an expert wikipedian, yet... :-) Carlo

[edit] Similar Forms

I am not sure of the convention regarding when an anonymous user makes a change I believe to be incorrect. Normally, I would message the user. In this case though, I have undone his change after verifying that it doesn't make sense since the variable "a" is not in one side of the equality and I checked the original equation with my source (back cover of Griffiths' Quantum Mechanics). I believe there is another form involving the double-factorial (2n-1)!! which Anonymous may have been going for, but I cannot recall it at the moment. Hope this helps. Scott.medling (talk) 09:17, 10 January 2008 (UTC)