Talk:An elegant rearrangement of a conditionally convergent iterated integral

First-time user, don't know how to enter equations, but I'd like to make a point verbally: this "example" fails. Bear with me, I'll address both the trig version and the quotient rule inspiration. I'll show only the dydx case.

On the trig version, if you express x and y in polar coordinates, differentiate, and manipulate away r keeping all other terms, you get dy=x*sec^2(theta)*dtheta+tan(theta)*dx. You have 3 variables constrained in how they co-vary: x, y, and theta. You can add a constraint to hold x constant, and that will zero out the tan(theta)*dx term everywhere except on the y-axis, where you cannot assume that infinity*0=0. So, the integral over y is 1/(1+x^2) everywhere except at x=0, for which the integral is undefined (but diverging to neg infinity). Thus, the integral over x is also undefined (but less than or equal to pi/4).

On the quotient rule inspiration, take d/dy y/(x^2+y^2), get (x^2-y^2)/(x^2+y^2)^2, infer that the indefinite integral w/r y is y/(x^2+y^2) + c. So far so good. But now the move to the definite integral is again only warranted when x is not 0. Where it is warranted, substitute 1 and 0 for y, subtract, get 1/(1+x^2), as before. But for x=0, the substitution produces division by zero, and thus an undefined definite integral over y, and therefore also an undefined integral over x.

kristo dot miettinen at us dot army dot mil

Is this really the best way to integrate $$\int \int \frac{x^2-y^2}{(x^2+y^2)^2} dxdy$$ ? Another way is to see that it is what is obtained from the quotient rule when evaluating $$\frac{d}{dx} \left( \frac{-x}{x^2+y^2} \right) = \frac{-(x^2+y^2)-(-x)(2x)}{(x^2+y^2)^2} = \frac{x^2-y^2}{(x^2+y^2)^2}$$. The squared term in the denominator is the give-away. This is a lot cleaner than the messy trig substitution used on this page. Mark.howison 06:51, 7 December 2005 (UTC)

I don't think this should be merged with Fubini's theorem. Examples in matheamatics are often worth separate articles, listed in the list of mathematical examples. Michael Hardy 21:06, 14 October 2005 (UTC)


 * I was thinking about the Proofs Project wich is structured like, e.g,:


 * Laplace operator/Proofs


 * and I tough something similar could be done with the examples, like this:


 * Fubini's theorem/Examples


 * You at least have to agree that Fubini's theorem/examples is shorter (and thus easier to link) than An elegant rearrangement of a conditionally convergent iterated integral. The same way you don't have separate proofs, you don't have separate examples. I'll be bold and merge the article in 72 hours if no one complains. User:Mdob | Talk 22:45, 14 October 2005 (UTC)
 * Agree with Mdob. The Fubini's theorem has an applications section with currently has only one example.--Banana04131 22:14, 15 October 2005 (UTC)

Merging in 24 hours
I've done a partial move to FT/Examples. If nobody complains I'm going to make a full merge in [FT, since the article is pretty empty and a separate page with examples, in this case, seems wasteful somewhat. --User:Mdob | Talk 20:38, 17 October 2005 (UTC)

subpage?
Some say there should be no subpages on Wikipedia, and my gut reaction is to agree. But "examples of Fubini's theorem" is not a good name for this article that gives only one example which, while clearly relevant to Fubini's theorem, is not really an example of Fubini's theorem (except the part where the fact that the two values of the iterated integral differ is used to infer that the integral of the absolute value is infinite). It's certainly not the kind of thing one thinks of when one hears the phrase "examples of Fubini's theorem". I'd suggest "examples relevant to Fubini's theorem", but two objections arise: (1) it's too much like the title that this page had originally ("an elegant rearrangement of a conditionally convergent integral) and would succumb to similar objections, and (2) one might not think of that title when searching. "Fubini's theorem/examples" seems immune to both objections. Michael Hardy 22:12, 18 January 2006 (UTC)

Currently there is the following line in the article


 * $$\int_0^1\int_0^1

\left|\frac{x^2-y^2}{(x^2+y^2)^2}\right|\,dx\,dy=\int_0^1\left[\int_0^y \frac{y^2-x^2}{(x^2+y^2)^2}\,dx+\int_y^1\frac{x^2-y^2}{(x^2+y^2)^2}\,dx\right]\,dy$$

Two questions:

1) Why is the leading sign in
 * $$\int_0^y

\frac{y^2-x^2}{(x^2+y^2)^2}\,dx$$ switched? That is, why isn't it
 * $$\int_0^y

\frac{x^2-y^2}{(x^2+y^2)^2}\,dx$$ ?

2) Is it allowed to freely split the integral at y, even if y may lie outside the space between 0 and 1? Thanks, --Abdull 16:12, 24 May 2006 (UTC)


 * (1) Because we're taking an absolute value. The absolute value of a &minus; b is b &minus; a if a < b.  In the first integral y is bigger than x, and y2 is bigger than x2, so the absolute value is y2 &minus; x2, and in the second integral, the order is reversed.


 * (2) Not necessarily--it may depend on integrability. But that has no relevance here, since it is clearly stated that y is between 0 and 1.


 * Michael Hardy 21:58, 24 May 2006 (UTC)

Why can't this just be merged into the Fubini's Theorem page, instead of giving it a series of awkward titles?


 * Just today I removed the "mergeto" notice that sat there for many months. I think this deserves a separate article.  Wikipedia is quite extensive; we don't need to make it cramped. Michael Hardy 02:06, 25 May 2006 (UTC)

Er...I just added that mergeto yesterday. Why should this deserve a separate article? Every page that links here could just as well link to Fubini's theorem. This article specifically relates to Fubini's theorem. --Stlemur 13:22, 25 May 2006 (UTC) (sorry I forgot to sign last time)


 * Every page that links here could just as well link to Fubini's theorem.


 * Not true. List of mathematical examples certainly could not just as well link to Fubini's theorem. Michael Hardy 00:10, 26 May 2006 (UTC)


 * ... and, generally, examples often deserve their own pages. Otherwise there could be no list of mathematical examples, which is listed at list of mathematics lists and clearly serves a useful purpose. Michael Hardy 00:16, 26 May 2006 (UTC)


 * Also: the "mergeto" notice was added to this page on October 14th, 2005. Michael Hardy 00:14, 26 May 2006 (UTC)

I disagree that examples deserve their own Wikipedia pages. Short examples should go in the originating articles. More complicated demonstrations should go in Wikibooks. In this specific instance, we have two short incomplete articles instead of a more-complete medium-length one. --Stlemur 01:30, 26 May 2006 (UTC)


 * Which of the items listed at list of mathematical examples would you delete or merge? What makes you think there's generally an "originating page"?  What if there are 20 different pages to which an example is equally relevant?  Michael Hardy 02:03, 26 May 2006 (UTC)

Why I just changed this article's title back to what it was originally
I have changed the title back to what it was originally, "an elegant rearrangement of a conditionally convergent iterated integral", because I think the misleading nature of any title that explicitly mentions Fubini's theorem is in large part the cause of Stlemur's misunderstanding. He actually thought that "Fubini's theorem" was this page's "originating page" and that because of that, the article should get merged into that one.

One could with equal plausibility say that absolute convergence is the "originating page", but the title mentioning Fubini's theorem obscured that.

Stlemur said that all of the pages that link here could just as well link to Fubini's theorem. That is wrong: the article titled absolute convergence could not "just as well link to Fubini's theorem.

I suspect Stlemur may have thought I was proposing that ALL examples in math articles should be in separate articles by themselves. I was not. Routine examples, e.g., illustrating how to use partial fractions to find a particular integral, should not normally be articles by themselves. But anyone who looks at the list of mathematical examples will find mathematical objects that may be relevant to multiple different articles, not just one, that link to it, some of which are unusually elegant or enlightening, some of which are substantial discoveries in their own right. Michael Hardy 22:58, 26 May 2006 (UTC)


 * PS: I'll take care of the links tomorrow (unless someone beats me to it)..... Michael Hardy 22:59, 26 May 2006 (UTC)

No, what I'm saying is, this article should be part of Fubini's theorem. I'm away this weekend but will get back to it later. In summary: I'd appreciate a third opinion on this. --Stlemur 00:27, 27 May 2006 (UTC)
 * 1) There is nothing that this article can do that could not be better-served, that is, have more information more readily accessible, in an extended Fubini's theorem article incorporating this one;
 * 2) "An elegant rearrangement..." is simply a bad article title: Unwieldy, unencyclopedic, and let's throw in POV too.


 * There is not now an article titled conditional convergence; that is just a redirect page to absolute convergence. If anyone were to create an article collecting examples of conditionally convergent sums and integrals whose values change when they get rearranged, this would fit right in, just as well as it would if anyone collected examples relevant to Fubini's theorem.  If this material is all moved into the Fubini's theorem article, then it would also have to get copied into the conditional convergence article as well. Michael Hardy 22:55, 29 May 2006 (UTC)

Is this encyclopedic?
My instinct is to say this belongs in a wikibook, with a link from the Fubini's theorem wp article. It's not as though this is a notorious example with a history. Or is it? Then I would like to see that in the article. 128.135.60.26 19:02, 24 September 2006 (UTC)


 * I think it's in some of the standard counterexamples books, such as (and, I suspect, including) Counterexamples in Analysis. Definitely I think a reasonably simple example such as this one should appear in Wikipedia, since we should be clear about when Fubini's theorem applies and when it doesn't. Michael Hardy 01:36, 25 September 2006 (UTC)