Timeline for Explanation for gamma function in formula for $n$-ball volume

8 events

when toggle format	what		by	license	comment
Feb 28, 2022 at 9:05	comment	added	D.R.		It's certainly a nice property, but not entirely relevant/critical here, right? In fact I don't really know where exactly such a property is "powerful"/useful.
Feb 28, 2022 at 8:30	comment	added	S. Carnahan♦		@D.R. The Gaussian lets you reduce to a product of 1-dimensional integrals. This is incredibly powerful.
Feb 24, 2022 at 23:15	comment	added	D.R.		Isn't it true that given any $G(r)$ on $[0,\infty)$ such that $g(x_1,\ldots, x_n) := G(\sqrt{x_1^2+\ldots + x_n^2})$ is a rotationally symmetric probability distribution on $\mathbb R^n$, then the same computation holds, i.e. $1 = \int_{\mathbb R^n} g d x = \int_0^\infty \text{vol}(S^{n-1}(r)) G(r) d r$? In that case, why is the Gaussian the most natural choice? Perhaps it is the fact that the Gaussian is the only distribution that is rotational symmetric and has independent coordinates, so it's the only that gives a sort of "uniform" formula for all $n$?
S Oct 30, 2015 at 0:23	history	suggested	hbp	CC BY-SA 3.0	fix some LaTeX symbols.
Oct 30, 2015 at 0:05	review	Suggested edits
S Oct 30, 2015 at 0:23
Nov 28, 2009 at 3:52	history	edited	S. Carnahan♦	CC BY-SA 2.5	TeXified
Nov 19, 2009 at 20:27	history	edited	S. Carnahan♦	CC BY-SA 2.5	added 70 characters in body; added 4 characters in body
Nov 19, 2009 at 20:21	history	answered	S. Carnahan♦	CC BY-SA 2.5