Talk:Gaussian Integral

I agree on the merge suggestion. --Alec (talk) 19:47, 21 March 2011 (CDT)

New Definition

Regarding:

$\ds \map \phi {-\infty, +\infty} = \int_{\to -\infty}^{\to +\infty} \frac 1 {\sqrt {2 \pi} } \map \exp {-\frac 1 2 x} \rd x = 1$

where $\phi$ is as defined in Definition:Gaussian Integral/Two Variables.

With the new definitions I'm putting it up, it would be useful to have that theorem as what is currently this page, and then I would like to rename this page's theorem to... something else I haven't thought of. In particular, I need that result to prove $\phi$ is a probability measure. Is swapping names like that a good idea? Or should I make one a corollary of the other? --GFauxPas (talk) 13:12, 12 February 2013 (UTC)

There are indeed a few standard definitions for "Gaussian". Yours is usual from the perspective of measure theory. I'm more used to the definition currently given, which arises in association with the Gamma function. --Lord_Farin (talk) 14:40, 12 February 2013 (UTC)
So, disambiguate "Gaussian"? If so, we'd need (arbitrarily chosen, perhaps) more specific names for each kind. Also, I'm limited in that my only sources for this stuff are probability and statistics texts. --GFauxPas (talk) 15:23, 12 February 2013 (UTC)
If the two usages of "Gaussian" basically describe variants of the same concept, then rather than disambiguation we are better off by transcluding. --prime mover (talk) 15:30, 12 February 2013 (UTC)
Agreed. --Lord_Farin (talk) 15:38, 12 February 2013 (UTC)
So, as to my original question...? --GFauxPas (talk) 15:56, 12 February 2013 (UTC)
't Is a bit of a difficult call. I think my preference is to have a transclusion of subpages (where the second subpage possibly uses the first in its proof, for brevity). --Lord_Farin (talk) 17:09, 12 February 2013 (UTC)

How 'bout Gaussian Integral (Variant 1) and so on? --GFauxPas (talk) 20:04, 12 February 2013 (UTC)

IMHO such "numbering" should be restrained in full page names and be relegated to subpages instead. As for linking to these subpages there's perhaps a need for good names, but we can always just give the full link like with most "General Result" sections. --Lord_Farin (talk) 21:11, 12 February 2013 (UTC)
I started constructing, any contributor should feel free to stop what I'm doing at any point to suggest a cleaner route as I go forward. --GFauxPas (talk) 13:46, 14 February 2013 (UTC)

In my opinion this whole formulation is just unnecessarily complicated. I think the version in February 2013, here was perfectly sufficient, and that the theorem statement can just be condensed to:

$\ds \int_{-\infty}^\infty e^{-x^2} \rd x = \sqrt \pi$

Introducing the extra notation just makes things cumbersome.

As to the second part: My idea, since we have to really do this for probability distributions eventually, was to have a page named for example Gaussian Distribution is Probability Distribution, (unsure on this wording. Gaussian pdf is pdf?) in which it would be established that, more generally:

$\ds \dfrac 1 {\sigma \sqrt {2 \pi} } \int_{-\infty}^\infty \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} } \rd x = 1$

Along with establishing non-negativity and so on. This would cover the statement here setting $\mu = 0$, $\sigma = 1$. I don't think it makes sense to couple these two theorems like this. I'm not sure if this is the same as establishing $\phi$ as defined here is a probability measure. (I have done very very limited reading in measure theory)

As to the name - I thought that "Gaussian integral" was used to specifically refer to the integral from $-\infty$ and $\infty$, so we could rename this to simply "Gaussian Integral" in line with, for instance, Dirichlet Integral or Frullani's Integral. However if there's a source I'm probably wrong in this.

No worries if this is the preferred way to formulate this, I'll go about cleaning up as it is if this is the case. (the proof is missing quite a lot) Caliburn (talk) 12:28, 2 July 2019 (EDT)

It occurs to me that I did another page with practically exactly this result while documenting Laplace Transforms from one of Spiegel's books.
Certainly, just calling it "Gaussian Integral" works for me -- but I would suggest we referenced a source, if you can find one so as to make sure we get it "right" (preferably hardcopy, I have reservations about the accuracy of the various online sources we use, MathWorld and PlanetMath having shown themselves to be particularly suspect recently).
Feel free to tidy this up, you have a free hand. --prime mover (talk) 14:38, 2 July 2019 (EDT)