Talk:Improper integral

message: 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)
thread_title:  Structure 

message: There is now a free-standing Cauchy principal value page: example here -> there?
Charles Matthews 10:45, 14 May 2004 (UTC)
thread_title:  Move to Cauchy principal value? 

message: 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)
thread_title:  A suggestion 

message: 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],
: 
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)
replies: text: 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)

text: 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)
thread_title:  sigh.... 

message: 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)
replies: text: 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)
thread_title:  Much improved 

message: 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)
thread_title:  weird identity 

message: 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)
replies: text: I should also add that the above identity is true for suitable inputs ƒ, 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 , 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)

text: 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 ƒ).  71.182.247.220 (talk) 12:08, 27 October 2009 (UTC)
replies: text:  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)
replies: text:  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)

text: 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)
thread_title:  Identities 

message: 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)
thread_title:  missing the point 

message: 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)
replies: text: 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)

text: 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)
thread_title:  Last sentence in the multivariable section 

message: 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 . 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)
thread_title:  Improper Henstock–Kurzweil integrals