Talk:Product integral

Filling a gap[edit]

This article was created to fill a gap I found amongst mathematicians and non-mathematicians (especially ecologists and biologists) alike. There is a need to add something about MATRIX PRODUCT INTEGRALS (especially with regard to Survival Analysis - see Richard Gill's webpage in references)

Daryl Williams 01:45, 19 June 2007 (UTC)[reply]

Indeed, if matrix-valued functions are not treated, doesn't this sort of integral degenerate into something of a triviality? Michael Hardy (talk) 18:14, 27 October 2010 (UTC)[reply]

Typography in note to The Fundamental Theorem[edit]

On my Firefox browser from Fedora Core 5, the prime on f'(x) in the note to "The Fundamental Theorem" does not show up, although I see it is there when I display the page in the "edit this page" mode. Tashiro (talk) 11:43, 7 January 2008 (UTC)[reply]

Type III[edit]

I've deleted Type III because it was simply incorrect equation: right sides of Type I and Type III were equal while left sides were different. Also note that notation used in this 'example' usually used for finite/descrete product. --MathFacts (talk) 11:25, 4 April 2009 (UTC)[reply]

Links?[edit]

I find it rather annoying and silly that this page contains multiple links which lead to the page itself. If there does not exist another page that discusses the topic in greater detail, then the link should be red. If this page contained enough information for each of these topics, then there would be no need for these links at all. If there is not enough information available for this page, then it should be amalgamated with other pages like it, in a page called "alternative calculi", though I am not sure of the existence of such a page. To summarize, this page needs to be fixed. — Preceding unsigned comment added by 67.5.154.76 (talk) 23:30, 1 September 2011 (UTC)[reply]

Fixed. I've stripped them all out. 67.198.37.16 (talk) 08:36, 27 May 2024 (UTC)[reply]

Bigeometric integral???[edit]

I was not sure where to post this comment: either in the multiplicative calculus page or here.

But considering the bigeometric derivative is $exp(xf'(x)/f(x))$ , one can recover $f(x)$ by solving for it. No one explicitly states:
$\exp \left(\int log(F(x))/xdx\right).$

Is this because the actual formula is more difficult, the bigeometric integral should be obvious, or no one has gotten around to it yet? — Preceding unsigned comment added by 67.1.255.84 (talk) 09:30, 7 August 2014 (UTC)[reply]

Whole books have been written about this subject. For example the book by Dollard and Friedman. All examples listed are boring. It gets more interesting in the non-commutative case, ie when we are not able to take logarithms and reduce the whole thing to ordinary integrals. I sometimes wonder if product integrals are good for anything. Maybe as an example of an idea that is promising but doesn't deliver. 00:21, 18 May 2024‎ User:Rhj2

You got that right. See below. The article seems to be a large amount of WP:OR written by someone unaware of non-commutative geometry, together with a very small amount of factually correct material. 67.198.37.16 (talk) 00:30, 27 May 2024 (UTC)[reply]

Logarithm???[edit]

Type-II "geometric integrals" have a logarithm appear out of thin air, with no explanation of how the logarithm is to be obtained. The formula given seems to be a variant of the famous $\det A=\exp {\text{trace}}\log A$ formula, but this formula is correct only in certain proscribed cases: A must be trace-class operator, or perhaps A is computed using the Berezinian or in assorted other cases where $\log f(x)$ can be meaningfully defined for some algebra-valued f. Indeed, when the trace can be understood as an integral, one has $\det A=\exp \textstyle \int \log A$ commonly seen in physics texts, and $\det A=\exp {\mathcal {P}}\textstyle \int \log A$ with ${\mathcal {P}}$ the path-ordering operator, to make sure that the product integral is being taken in the correct order. See, for example, Dyson series where these concerns are handled correctly.

This article plays fast-and-loose with such details. The entire section of "results" for geometric integrals can only work for the real numbers, and possibly for matrix fields, but this is hardly clear. Compare, for example, to the care taken in the articles on path-ordering and ordered exponential. Where's the time-ordering operator? Where's the path dependence? 67.198.37.16 (talk) 23:54, 26 May 2024 (UTC)[reply]

I resolved the above complaint by splitting the article into two distinct parts: the commutative and the non-commutative part. Hopefully, this will allow a cleaner development in the future. 67.198.37.16 (talk) 02:51, 27 May 2024 (UTC)[reply]

After rewriting the lede to the article to provide a clean distinction between the commutative and the non-commutative cases, I started looking at the various references. It appears that large parts of this article, currently appearing under the commutative section headings, appears to be WP:OR, written by multiple editors over the last 10-15 years, and not supported by the references. It seems that many had a poor grasp of the fact that, even for Volterra, the integrand was a matrix, and not the real numbers. As a result, many of the equations in the commutative subsections appear to be incorrect for any case other than the reals (i.e. they wouldn't even hold for commuting matrix fields). I don't have the patience to untangle that part; as almost all of the references are pay-walled. It is not connected up to the classical theory of field (mathematics), such as topological fields, or even differential Galois theory of linear differential equations, so who knows. The main open reference for this article, A. Slavík, Product integration, its history and applications does seem to get into this. Its just not reflected in this article. 67.198.37.16 (talk) 08:25, 27 May 2024 (UTC)[reply]