Talk:Improper integral

Structure
I think this page is reasonably 'mature' now in its content; but that the topics could usefully be re-ordered.

Charles Matthews 09:07, 5 Nov 2003 (UTC)

Move to Cauchy principal value?
There is now a free-standing Cauchy principal value page: example here -> there?

Charles Matthews 10:45, 14 May 2004 (UTC)

The term "improper integral" seems more widely known that "Cauchy principal value" or "principal value", so if the pages get merged, I think it would be better to keep this one and make the other one a redirect page. Michael Hardy 01:02, 20 Sep 2004 (UTC)

Introduction
The introduction on this article needs a lot of work. I can hardly understand it as it currently is, so somebody unfamiliar with the material could be hopelessly lost.
 * 1) The definition of an improper integral is vague and as stated and is either not well defined or a super class of all definite Riemann integrals. (depending on how you read specified real number)
 * 2) There are multiple topics addressed with little or no transition making reading it misleading (as just before the third integral)


 * 1) There's too many specific details that should be in the body of the article, not in the intro (as three fourths of it talks of Lebesgue integration) GromXXVII 19:51, 7 November 2007 (UTC)

You added a "definition" that referred to "with integrals of functions that could possibly have infinite area", and you complain of vagueness? How vague can you get?

Can you be specific about what it is that you can't understand about the definition? Michael Hardy 21:26, 7 November 2007 (UTC)


 * In calculus, an improper integral is the limit of a definite integral, as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.
 * Lack of explanation as to the concept, it jumps right into the explicit limit definition (without explicit definitions)
 * approaches either a specified real number could mean any real number, which would imply all definite integrals are improper integrals, or some unknown specified real number the article never tells the reader about.


 * Essentially, the statement doesn’t tell the reader what an improper integral is, or why we need it. Then it leads into a bunch of cases without an explicit definition.


 * I was trying to link the two types (unbounded function, unbounded interval) into one cohesive, concise expression. Then it can be further refined into the different types and explicit definitions using limits. Sure it wasn’t the best possible, but with the other changes it made the article more readable and explanatory than it currently is and can always be improved upon. GromXXVII 21:41, 7 November 2007 (UTC)

...OK, I've looked over the article from top to bottom. I've expanded the first sentence a bit to include some expressions in mathematical notation for anyone who finds that easier to follow than the words preceeding it. But you really do need to be specific about each of your complaints. (1) What specifically do you find hard to understand about the initial sentence? (2) In which cases is the concept "not well defined"? (3) Are you saying it is a "super class of all definite Riemann integrals", and you object to that, or that it is not a "super class of all definite Riemann integrals", and you object to that? (4) What sort of "transition" do you have in mind "just before the third integral"? The point of that sentence is that even in cases where an integral can be defined without taking a limit, nonetheless it can't quickly be computed without taking a limit, i.e., in one sense it may not need to be considered improper, but for purposes of computation one treats it as improper. It seems to me that that needs to be given prominence early in the article. (5) Which specific details are you saying should not be in the introduction. Michael Hardy 21:46, 7 November 2007 (UTC)


 * 1) “specified real number” is ambiguous because there are restrictions on what the "number" can be to yield an improper integral. For instance, it implies that $$\int_0^1 1\,dx = \lim_{t\to 1} \int_0^t 1\,dx$$ is an improper integral. It’s not standard to use a limit in such case, but is certainly possible to do.
 * This is if the reader assumes “specified real number” could be anything real.


 * 2) It’s not well defined if the reader assumes there are restrictions on the specified number, but they’re not stated. I didn’t mean well defined in the mathematical sense, I meant it as in “part of the definition is not available to the reader”


 * 3) I’m saying it’s not a super class of all Riemann integrals, but the intro makes it seem so. (Also note it certainly doesn’t need to address this, just to be written so that it doesn’t have that tone)


 * 4) I was referring to this, where the integral appears to be trying to help explain the paragraph, but doesn’t quite have that purpose
 * “In some cases, the integral from a to c is not even defined, because the integrals of the positive and negative parts of f(x) dx from a to c are both infinite, but nonetheless the limit may exist. Such cases are "properly improper" integrals, i.e. their values cannot be defined except as such limits.


 * This should be reworded, but also in a different section.


 * The integral
 * $$\int_0^\infty\frac{dx}{1+x^2}$$
 * can be interpreted as
 * $$\lim_{b\to\infty}\int_0^b\frac{dx}{1+x^2}=\lim_{b\to\infty}\arctan{b}=\frac{\pi}{2},$$”


 * 5) I think that everything after “(see Figures 1 and 2)” can be moved somewhere else.


 * Ultimately I think it’s most important for the introduction to give a clear idea of what an improper integral is (which would address that a limit or other advanced techniques are used because standard Riemann integration does not work), and when an expression is an improper integral (which would address the issues of the interval the function is integrated over, and the issues of the range of the function)
 * There are perhaps other things that merit prominence to be in the introduction, but most of what is there doesn’t, especially because there isn’t a clear description currently, so all the stuff about computation, Lesbegue integration, and properly improper integrals serve to confuse instead of inform - even though they should be in the article.


 * It seems necessary to me that the article quickly:
 * Motivate why improper integrals are used
 * Explain when they’re used


 * Define the symbols used for them (in at least the two simplest cases of an unbounded function and interval). I had four explicit definitions to cover the most common cases so I put them all in a different section.


 * I think that the version I had would be a better place to continue: with fixing the intro and figuring out what else should have a prominent role. (because as is the introduction seems to have no structure, coherence, and doesn’t give the reader a good idea of what an improper integral is; and most of what is there doesn’t help with that)
 * I’ll go ahead and work on that unless if you’re just going to revert everything again… —Preceding unsigned comment added by GromXXVII (talk • contribs) 23:00, 7 November 2007 (UTC)


 * I've created more sections. A standard way to work on articles is to take one section at a time and deal with its issues. Charles Matthews 07:15, 8 November 2007 (UTC)

A suggestion
The article is currently written so as to accomodate both the Lebesgue and Riemann integrals. This has the effect that it would be, I think, completely impenetrable to someone who is not familiar with the Lebesgue integral. Since the article begins "in calculus", I think this is undesirable.

At the same time, the article is arguably not general enough. Whoever wrote that the improper integral is "not a kind of integral" was not familiar with the Kurzweil-Henstock, or generalized Riemann integral. Among the virtues of this integral is that it integrates all Lebesgue integrable functions and improperly Riemann integrable functions "automatically". Coming back down to earth, it seems clear from linguistic use, interpretation in terms of area and so forth that the improper (Riemann) integral is "some kind" of integral, right??

I recommend that the body of the article be written explictly in terms of the Riemann integral. Here a key fact is that any Riemann integrable function on [a,b] is bounded, which explains why a new definition is needed when the function blows up at one of the endpoints. It seems more clear to discuss separately the case of an infinite interval and an unbounded function. More examples would also be nice.

There can then be a section discussing the difference between the improper Riemann integral, and the Lebesgue integral, and ideally a final section discussing the Kurzweil-Henstock integral.

Again, remember that this is an article that second semester calculus students will try to read. Even the brighest calculus students do not know measure theory... Plclark (talk) 11:35, 20 November 2007 (UTC)Plclark

sigh....
OK, I'm going to return to this article soon. Right now we find this sentence:
 * The Riemann integral as commonly defined in calculus texts is only defined for a continuous function $$f$$ on a closed and bounded interval $$[a,b]$$,
 * The Riemann integral as commonly defined in calculus texts is only defined for a continuous function $$f$$ on a closed and bounded interval $$[a,b]$$,

That is not true. What is true is that introductory calculus texts often consider only the case where the function is continuous. But the definition they state works for many discontinuous functions as well, including piecewise continuous functions of a sort often dealt with in calculus texts. I think it's appropriate to consider in the article the manner in which the concept is first introduced in calculus texts, but this continuity issue is really not the essence of the reason why there is such a concept as that of improper integrals. That's one of a number of reasons I've flagged this for attention. Michael Hardy (talk) 21:07, 4 June 2008 (UTC)


 * Actually, I changed the phrasing because I was prompted by your "Attention" flag. You are right that the statement is incorrect: it should say that the Riemann integral is commonly defined in calculus texts for continuous functions.  (Not that this definition only applies to continuous functions, as is suggested by the current wording.)  Unfortunately, there is no elementary characterization of Riemann integrable functions suitable for an introductory section.  A function is Riemann integrable iff it is bounded and continuous almost everywhere, but this is quite a bit to put in the first sentence of the motivation.   siℓℓy rabbit  (  talk  ) 00:36, 5 June 2008 (UTC)

Maybe it would be better to avoid any discussion of different definitions of integral (e.g. Riemann versus Lebesgue) in the early parts of the article. There seem to be two essential issues: (1) some integrals can't be defined except as limits as a bound approaches some limit, and (2) some integrals, even if they can be defined, cannot conveniently be computed except as limits as a bound approaches some limit. In either case, an improper integral is a limit as one or more of the bounds of integration approaches some limit. Michael Hardy (talk) 00:54, 5 June 2008 (UTC)


 * Yes, that sounds reasonable. In fact, it is probably easier to simply define an improper integral from the outset as the limit of an integral rather than to attempt to motivate it by considering unbounded functions/domains.  The motivation is currently more of a potential source of confusion than anything else.  The situations in which one would want to (or need to) use an improper integral should still be mentioned, but perhaps after a definition has been properly formulated.   siℓℓy rabbit  (  talk  ) 03:26, 5 June 2008 (UTC)

Much improved
The article looks enormously improved. Here are some further quick comments (because silly rabbit has worked so hard recently on the edits, I would rather suggest them than change them myself):

1) I am not wild about the "first kind / second kind" dichotomy. This is old-fashioned language that has been all but abandoned nowadays, probably because it is so opaque: speaking of an object of the Nth kind gives no clues as to what sort of X the author might have in mind.  Ask a working complex geometer what a differential of the third kind is, and if you are lucky, they will tell you, "Well, I think it's this, but maybe that's the second kind...I can't quite remember."

Also this dichotomy is too simple, because combinations of both types are possible.

Anyway, you don't seem to use the terminology in the remainder of the article.

2) Similarly, what you say in the first sentence as the definition of an improper integral is a little too simplified, because there could be multiple (maybe infinitely many) "improprieties." Admittedly it makes it quite challenging to summarize the situation in a single sentence that is not technically wrong; I think the only way to go here is to be a bit more vague.

3) You don't seem to say in the introduction that an improper integral [of a non-negative function...] is an area under a curve. Isn't this a good motivating intuition?

4) You speak of the (proper) Riemann integral "not converging" for unbounded functions. As far as I know, one instead says that the integral is "not defined" or the function is not "integrable."  (It would make perfectly good sense to say that it is not convergent, since a Riemann integral is a Moore-Smith limit, but people don't seem to say it...)

5) The issue of improper integrability on the real line is a pedagogical minefield; this is one of the few topics in first year calculus for which I longer even expect that my explanation will make sense to most students. At a slightly higher level though -- say that of undergraduate analysis -- I have found it helpful to introduce a Cauchy criterion for convergence on the real line: a function f: R -> R which is Riemann integrable on each closed bounded subinterval I of R is improperly Riemann integrable if: for every positive epsilon, there exists a closed bounded interval I such that for any closed bounded intervals J_1 and J_2 containing I, |int_{J_1} f - int_{J_2} f| < epsilon.  The point is that this definition looks (to a sufficiently sophisticated eye) natural rather than ad hoc.

6) The subject heading "Types of integrals" doesn't seem quite right to me, although I'm not sure exactly what alternative to suggest.

7) Riemann and Darboux integrals are equivalent, so why not just discuss them together, or even omit mention of Darboux entirely?

"The Lebesgue theory does not see this as a deficiency..." I don't think the theory of Lebesgue integration has an opinion on this, or on anything! Anyway, this sounds like propaganda: the important point (which you have made) is that there is no containment between the class of improperly Riemann integrable and the class of Lebesgue integrable functions. This is just a fact: it doesn't need to be "spun". (It reminds me of an algebraic topology class when my instructor said that one of the great merits of homology / homotopy groups is that they are invariants of the homotopy equivalence class of the space, suggesting that it would be somehow uncouth to want to distinguish between non-homeomorphic but homotopically equivalent spaces.)

Similarly, the bit about the Kurzweil-Henstock theory can be reworded to be less POV: in a purely mathematical sense that theory _is_ stronger than the other two theories, since it properly includes them.

8) Concerning summability: some reference to the (simpler and more widely known) theory of summability of series seems to be in order here.

Anyway, like I said, it is a good-looking article. Plclark (talk) 17:39, 19 June 2008 (UTC)Plclark


 * Thank you for the insightful feedback. These are all very good points, and definitely should be addressed.  Regards,  siℓℓy rabbit  (  talk  ) 11:30, 20 June 2008 (UTC)

weird identity
Why is the improper integral of f(x)dx the same as it is for f(x-(1/x))dx? Wouldn't this also mean it would still work for f(x-(2/x)), f(x-(3/x))... f(x-(x/x)) which equals f(x-1)? srn347 —Preceding unsigned comment added by 68.7.25.121 (talk) 21:00, 12 November 2008 (UTC)

In fact, this identity is true for f(x-c/x) where c is a positive real number. --27.32.122.196 (talk) 04:30, 10 January 2016 (UTC)

Identities
I removed the following because it does not improve the article:


 * Identities
 * $$\int_{-\infty}^{\infty} f(x) \, dx = \int_{-\infty}^{\infty} f(x - 1/x) \, dx$$

There also seems to be some consensus that the above "section" does not belong. Please discuss merits for inclusion here. 71.182.247.220 (talk) 11:17, 27 October 2009 (UTC)


 * I should also add that the above identity is true for suitable inputs &fnof;, but in spite of being true it is not suitable for the article (WP:WEIGHT and WP:INDISCRIMINATE being the policies that codify this impression). This integral follows essentially from an elementary change of variables u = x &minus; 1/x, while appropriately accounting for the branch of the inverse function.  So it really is nothing more than a "fun fact", of which Wikipedia is WP:NOT an indiscriminate collection. 71.182.247.220 (talk) 11:39, 27 October 2009 (UTC)


 * Actually, I can only rigorously prove this for "suitable inputs" consisting of absolutely integrable functions (Lebesgue integrable), or if the integral is taken in a Cauchy principal value sense. Since the whole point of improper integrals is to regularize non-integrable functions at infinity, this example is particularly unsuited for the article (especially given that no hypotheses are imposed on the function &fnof;).  71.182.247.220 (talk) 12:08, 27 October 2009 (UTC)
 * Concerning this "whole point": I agree except for the part about "at infinity". It's not only at infinity, but a finite points as well. Michael Hardy (talk) 15:47, 29 October 2009 (UTC)
 * Agreed. But what is needed in the transformed integrand in this case is regularization at infinity, not at finite points, which was my point. 71.182.210.215 (talk) 23:36, 29 October 2009 (UTC)
 * 71.182.247.220 asked me to comment here since I added the identity a year or so ago. If I recall correctly I added it as a "fun fact" as 71 suggested, but I have no problems with it being removed. Ben (talk) 15:23, 29 October 2009 (UTC)

At one point I deleted the proposed identity from the article. It seemed to me that it had been put into a part of the article where it was given great prominence at the expense of essential facts that needed to be there, distracting attention from those. Michael Hardy (talk) 15:50, 29 October 2009 (UTC)

Multivariable improper integrals?
This page seems to be talking only about one variable improper integrals. A section on multivariable improper integrals, such as $$\iint_{\mathbb{R}^2} e^{x^2+y^2} \, d(x,y)$$ would be nice. 75.165.3.78 (talk) 17:35, 24 December 2009 (UTC)
 * Not sure, but I think the specific integral you mention doesn't exist as an improper integral. It looks to me like its Cauchy principal value doesn't exist either. Am I missing something?--75.80.43.80 (talk) 05:25, 9 December 2010 (UTC)
 * Oh... did you mean $$\iint_{\mathbb{R}^2} e^{-(x^2+y^2)} \, d(x,y)$$? In that case, just do it as an iterated integral. As long as you keep the limit calculations straight, it is the same definition as in the article.--75.80.43.80 (talk) 05:37, 9 December 2010 (UTC)
 * Oops! Just noticed this was posted LAST December. Reading the year helps. Oh well.--75.80.43.80 (talk) 05:39, 9 December 2010 (UTC)


 * I just created such section, though it should be much longer... Boris Tsirelson (talk) 20:52, 22 June 2015 (UTC)


 * Thanks to Sławomir Biały this section is now quite informative. Boris Tsirelson (talk) 05:44, 30 June 2015 (UTC)


 * "A function on an arbitrary domain A in $$\mathbb R^n$$ is extended to a function on $$\mathbb R^n$$ in a standard way by multiplying by the indicator function of A." — Yes, but Riemann integrability may be lost (even for a constant function) if A is not Jordan measurable; either require Jordan measurability (which is stronger than boundedness), or use exhaustion, or alternatively, take the supremum over Riemann integrable functions under the product by the indicator function. Boris Tsirelson (talk) 05:56, 30 June 2015 (UTC)


 * Perhaps I was unclear that this is how the Riemann integral is defined for arbitrary domains, by extending by zero outside the domain. In particular, in your example of constant functions, a non-zero constant is integrable over a domain if and only if the domain is Jordan measurable.  I will try to clarify this presently.   Sławomir Biały  (talk) 11:19, 30 June 2015 (UTC)


 * Ah, really? I did not know. I would prefer the more permissive definition... Boris Tsirelson (talk) 11:49, 30 June 2015 (UTC)


 * One can also define an integral, in an extended sense, by taking a partition of unity with rectangular supports, integrating over the supports, and summing. This at least works for open sets, or manifolds with smooth charts.  But it doesn't deal with arbitrary subsets of $$\mathbb R^n$$.  This doesn't seem to be a very popular approach for dealing with the Riemann integral (unless dealing with manifolds).  Presumably, once one gets to the point of invoking partitions of unity, it probably makes more sense to use the Lebesgue integral anyway, which renders such quibbles moot.   Sławomir Biały  (talk) 12:06, 30 June 2015 (UTC)


 * Also interesting. But I wonder, do you mean finite partitions of unity, or countable?
 * In fact, I teach these things (and there OR is not prohibited).
 * I believe that improper Riemann integration is still useful for (quite arbitrary) open sets, but beyond sets that are open (modulo sets of locally zero Jordan measure) it is indeed better to use Lebesgue integral. Boris Tsirelson (talk) 12:54, 30 June 2015 (UTC)


 * I am thinking of countable partitions of unity. There are several different notions of integral that are in play here.  One is the usual Riemann integral over a bounded set A.  This is defined by extending by zero off A.  The other is an extended integral, defined by partitions of unity (which we might call the "de Rham integral").  Then there are improper integrals, where the set A is allowed to be unbounded.  The ordinary Riemann integral is not equivalent to the de Rham integral since, as you pointed out, constant functions are de Rham integrable over any bounded open set, but may not be Riemann integrable.   I'm not sure what the "correct" perspective is for the article, but I've used the Riemann version, thinking that to be the most familiar.  But, as you say, improper integrals of the de Rham type can also be considered.   Sławomir Biały  (talk) 14:10, 30 June 2015 (UTC)

This recently overwrote what I wrote "In elementary calculus, the term improper integral is usually taken to mean either the limit of an integral as at least one of the two bounds of integration approaches either + ∞  {\displaystyle +\infty }  +\infty  or − ∞   {\displaystyle -\infty }  -\infty  a as at least one of the two bounds approaches a number at which the function being integrated has a vertical asymptote." This is mathematically incorrect. You don't need and asymptote at at infinity. You don't even need an asymptote. If I change it back someone will just over write it again. So it's useless to fight this. You need higher level thinking to solve this issue globally. Solution 1): Hire a math Ph.D. to verify valuable changes (expensive) 2) when users go to a math or physics or tech page ask them if it was useful and if not ask them to compare it to a previous version and rate both (I like this one) or 3) Use deep learning (machine learning) to compare your definition to similar pages from other sources.

Good luck. 1 love wikipedia — Preceding unsigned comment added by JimFoit (talk • contribs) 22:03, 23 September 2016 (UTC)

missing the point
I did this edit because whoever wrote that sentence seemed to miss the point. "Bad behavior at infinity" is not involved, and what is involved was not mentioned. Michael Hardy (talk) 04:59, 24 September 2016 (UTC)

Last sentence in the multivariable section
The present phrasing "In order to exist in this sense, the improper integral necessarily converges absolutely," obfuscates the nature of the implication, to wit: https://www.quora.com/Why-is-the-multivariable-improper-integral-usually-defined-only-when-it-converges-absolutely-Is-there-a-consistent-notion-of-conditionally-convergent-improper-integrals-in-mathbb-R-n?share=2252ca03&srid=8TIx. Is there some reason not to write this simply as: "Thus" (or "Note that") "if the improper integral exists in this sense, then it converges absolutely: convergence implies absolute convergence"? OlyDLG (talk) 19:43, 5 September 2018 (UTC)


 * Obfuscates? I do not feel so. As for me, it says just the same (as your version). Anyway, I do not object to such rewording. But anyway, rewording does not really answer the (nontrivial) question asked on quora: why such seemingly bad definition? Boris Tsirelson (talk) 08:28, 6 September 2018 (UTC)
 * A short answer is in the article, see quotes in the notes: "We need to make this stronger definition of convergence in terms of |f(x)| because cancellation in the integrals can occur in so many different ways in higher dimensions", and "In fact, for improper integrals of such functions, unconditional convergence turns out to be equivalent to absolute convergence". A detailed explanation is available in my course, Sect. 1b "Dimension 2 is a different story". Boris Tsirelson (talk) 08:34, 6 September 2018 (UTC)

Improper Henstock–Kurzweil integrals
Is the section on Types of integral correct to say: "For the Henstock–Kurzweil integral, improper integration is not necessary"?

This seems to be true for finite bounds but not infinite ones; or else it would contradict the Henstock–Kurzweil integral article which implicitly defines it on a finite interval $[a, b]$. The same article goes on to say "To study an improper Henstock–Kurzweil integral with finite bounds would not be meaningful. However, it does make sense to consider improper Henstock–Kurzweil integrals with infinite bounds". NeilOnWiki (talk) 17:07, 2 January 2022 (UTC)