Because they pop up all over mathematics, physics, computer science, economics, statistics, engineering, etc.  and are very useful? 

Dear Rogier Brussee:

Improper integrals are simply a 1-dimensional conceptual approach to convergence/divergence. To the extent they ever "pop up", they do so in ways that are readily related to convergence in general or other integrals that can deal with definite and/or indefinite integration of functions that Riemann integration can't. Also, as they are limited to 1D, they don't even really "pop up" as integration but as convergent series (yes, these are very related concepts, but as one can teach the latter without ever addressing the former and get more out of such  treatment, why bother treating what "pops up" in various fields simply because we have defined into existence an integral that slightly extends the Riemann integral in non-intuitive and largely conceptually divergent ways when we have vastly superior integrals that do all improper integrals could possibly do, do them better, and can do more. Also, as improper integrals require a 1D space and are dangerous guides/tools if they do not converge absolutely, while we have integrals for which none of these shortcomings are true, how are they "very useful"?

Dear Andrew,

There is a particular area where we employ analytic continuation via dilation analytic operators, where integrals which do not exist have a unique meaningful value. See e.g. section 3.2 in the enclosed pdf file. I do not know at which level this would be appropriate to include.

I would think that principal value integrals in connection with dispersion relations should be useful in teaching as well.

It seems to me that the vast majority of the students are never going to learn any other integral, and even those who will learn the Lebesgue integral (math, physics students) may do so *after* they need to come to terms with integrals over non-compact intervals.

For instance, in Statistics lectures they may hear about density functions and expectations of continuous random variables. The lecturer is not going to stop and explain what `integral from -infinite to +infinite' means.

Dear Pedro Train:

Thanks for your answer! One issue, though, is that I don't think that (to the extent improper integrals are integrals) these are either the only integrals students learn nor are they how students conceptualize integration. The vast majority of students learn what is essentially a combination of the fundamental theorem of calculus and Riemann integrals. Improper integrals are not Riemann integrals:

"Since the Riemann integral is restricted to bounded functions defined on bounded intervals, it is necessary to make special definitions in order to allow unbounded functions or unbounded intervals. These extensions, sometimes called improper integrals, were first carried out by Cauchy and we weill refer to the extensions as Cauchy-Riemann integrals." (italics in original)

Kurtz, D. S., & Swartz, C. W. (2004). Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and Mcshane (Series in Real Analysis Vol. 9). World Scientific.

There is a textbook I have used when tutoring students or as a supplement to course material (tutoring, as I do it as a side-job, allows me freedom in many ways to use what material I wish while I have much less freedom when teaching courses). One advantage is that it is entirely free. Another is it's approach to elementary integration:

"For all of the 18th century and a good bit of the 19th century integration theory...was simply the subject of antidifferentiation. Thus what we would call the fundamental theorem of the calculus would have been considered a tautology: that is how an integral is defined...This is often expressed by modern analysts by claiming that the Newton integral of a function f: [a,b]->R is defined as

[this is followed by a standard expression of the notation for the fundamental theorem: the integral from a to b dx= F(b)- F(a)]

The technical justification for this definition of the Newton integral is nothing more than the mean-value theorem of the calculus. Thus it is ideally suited for teaching integration theory to beginning students of the calculus. Indeed, it would be a reasonable bet that most students of the calculus drift eventually into a hazy world of little-remembered lectures and eventually think that this is exactly what an integral is anyway. Certainly it is the only method that they have used to compute integrals.

For these reasons we have called it the calculus integral. But none of us teach the calculus integral. Instead we teach the Riemann integral. Then, when the necessity of integrating unbounded functions arise, we teach the improper Riemann integral. When the student is more advanced we sheepishly let them know that the integration theory that they have learned is just a moldy 19th century concept that was replaced in all serious studies a full century ago.

We do not apologize for the fact that we have misled them; indeed we likely will not even mention the fact that the improper Riemann integral and the Lebesgue integral are quite distinct; most students accept the mantra that the Lebesgue integral is better and they take it for granted that it includes what they learned. We also do not point out just how awkward and misleading the Riemann theory is: we just drop the subject entirely.” (italics in original; emphases added)

Instead of a citation I can refer you to the freely (legally) available textbook The Calculus Integral (see attached link).

The authors are a little extreme here in my view, and indeed another book I like to use for supplementary material (Inside Interesting Integrals) not only simply presents the Lebesgue integral only to say that it will not be used, but defends this by citing Bartle’s famous paper “Return to the Riemann Integral.” However, what “the Riemann integral” is for Bartle isn’t what any undergraduate students would recognize, and often goes by another name. In Bartle’s textbook A Modern Theory of Integration (Graduate Studies in Mathematics 32) we are informed that he "shall employ a limiting process that was recently introduced by the Czech mathematician Jaroslave Kurzweil (b. 1926) and the English mathematician Ralph Henstock (b. 1923). This method is slightly more complicated than the Riemann process, yet it yields an integral that is considerably more general and easier to use than the ordinary Riemann integral.” (emphasis added).

I think that you are completely correct in that a lecturer won't "stop and explain what" it means to integrate anything in courses like probability, statistics, physics courses, etc. In fact, just what integration and integrals are (or how they should be defined) is an area of some contention in mathematics. The point, though, is to try to teach what it means to integrate from this to that in such a way that minimizes the student's difficulty, maximizes their understanding, and minimizes the amount of work needed to obtain this understanding. Throwing a curve ball like improper integrals just confuses things, as they aren't introduced in terms of integration theory (and shouldn't be, as this is too complex for that level), but simply introduced as “techniques of integration” or something similar, introduced before sequences & series, and thus just when the student has hopefully solved enough integration problems to get the hang of it, suddenly integrals can have this weird property of “convergence” or “divergence”, which is typically not defined other than what we call it when we can’t integrate something using these “new” integrals.

http://classicalrealanalysis.info/documents/T-CalculusIntegral-AllChapters-Portrait.pdf

I think everybody would agree with your main points (improper integrals are not Riemann integrals, they do not contribute to a conceptual understanding of integration, they are confusing to the students and there is the whole business of absolute convergence, things might be done differently at little cost e.g. Henstock-Kurzweil, students imagine integration as an anti-derivative).

I expunged improper integrals from my brain once I learned Lebesgue integration, yet I can say I'm fine with having been made to go through it. For instance, the limit in the weak law of large numbers can be the Cauchy principal value even when the strong law of large numbers doesn't hold (absence of absolute convergence again). I would have been infinitely perplexed by that had the principal value not been explained explicitly to me.

Having said that, what I imagine people objects to is the notion of having students attend other subjects and raise their hands saying 'That thing with infinities in the integral sign, I don't know what it is'.

I think what is discussed here has a lot to do with fundamental defects in the infinite related area of our mathematics.

Just see following divergent proof of newly discovered Harmonic Series Paradox example (given by Oresme in about 1360, very elementary and important, can be found in many current higher mathematical books written in all kinds of languages):

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                 （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．     （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．              （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                          （4）

Paradox is there whether or not we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite infinitesimal items in Harmonic Series by “brackets-placing rule" to change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity.

What is infinite?

What is number?

What is limit theory （treating technique for infinite related numbers）?

Dear Erkki J. Brändas:

I seem to have missed your reply, for which I apologize.

I was particularly interested in this statement: "I would think that principal value integrals in connection with dispersion relations should be useful in teaching as well."

Useful? Absolutely. Possible? To illustrate what I mean via a comparison, I think that introducing students to integration through e.g., measure theory and the Lebesgue integral would be fantastic...were it not for the fact that we are talking about students who are learning improper integrals in an elementary calculus class, and I can't imagine that any but a tiny number would be able to grasp such comparatively difficult concepts.

It may be possible for a good teacher to introduce students to improper integrals as a class of functions or something for which the integration methods they know are not adequate (that is, introduce improper integrals as "improper integration" rather than as an "integral", because they are not sufficiently familiar with integration to understand it as anything other using integrals, which up until they are introduced to improper integrals they usually regard as the Riemann integral which they can thankfully side-step using the "fundamental theorem(s)" of calculus).

The problem, however, is related to what I believe the reason we teach improper integrals the way we do: we wish to introduce integration theory but most of the extensions of the Riemann integral (not to mention the actual Riemann integral) are too advanced. So we compromise. First, we barely introduce the Riemann integral before announcing to a grateful students that one can sort of “sneak past” both the conceptual and computational difficulties of the Riemann integral by finding antiderivatives. Then we have students use this method many, many times. However, in the hopes of enabling the student not only to understand more of the nuances of integration and broaden their understanding of integrals in relation to types of function (among other things), we introduce improper integrals. I haven’t really ever seen this work. Mostly, this is because students have it in their heads that integration is something like the use of the Riemann integral (which they don’t usually understand because a) there are only so many functions an elementary calculus student can integrate using Riemann’s integral without using numerical approximations or more advanced mathematics). Integration is somehow this infinite summation that can be bypassed thanks to the fundamental theorem(s) of calculus.

So, when we introduce improper integrals, we don’t have the foundations set for the students to be anything but confused by these. Why are integrals suddenly converging? Many students are explicitly told that integration contrasts with differentiation because finding the derivative of a function gives you a new function (one in which you can simply plug values in to obtain the derivatives at a point), while integration gives you values. This isn’t, of course, true, but there is some truth in it as derivatives have that general form almost all students hate (the difference quotient) which all the functions students differentiate could be use in theory (rather than use differentiation rules).. Nothing comparable exists for integrals, which for students up until improper integrals sort of “act on” functions to obtain specific answers. Suddenly, the student is presented not with a more complicated integral but (although they do not know it) an entirely different way of understanding integration. However, this isn’t what happens. Improper integrals are not introduced as extending integration theory (but rather something that allows them to integrate certain functions they previously couldn’t for reasons they don’t usually understand), nor are improper integrals used as a way to connect integration with a much needed deeper understanding of limits, the vital importance of notions like l.u.b./supremum or closure,  etc.

This is why I think that, to the extent improper integrals should be covered at all in elementary calculus, they should be introduced within the context of sequences and series. It’s true we do find them again here, only now things can be even more confusing. These weird integrals are now part of a “test” for convergence, but were introduced in terms of convergence, and…[insert here many other complaints, issues, conceptual difficulties, and so forth I’ve found students struggling with].

Introducing improper integrals after learning about convergence can provide an ideal way not just to connect Riemann integrals to sequences & series but also introduce improper integrals and connect them to their first exposure to integration as infinite summation as well as to integration theory in general via reinforcing how absolutely fundamental and nuanced things like limits are and what the foundations that underlie calculus are.

Dear Geng Ouyang:

Thanks for the repy! Regarding your statement:

"I think what is discussed here has a lot to do with fundamental defects in the infinite related area of our mathematics."

I think that certainly may be part of it. It's definitely true that ideas about the infinite are hard to grasp, so hard that when Cantor sent a draft of one of his own proofs about uncountabley infinite sets he included the comment "I see it, but I don't believe it!" ("Je le vois, mais je ne le crois pas!").

In my view, however, the problem has more to do with the general nature of the way mathematics in general and calculus in particular are taught. Even before going to university, most students understand mathematics as nothing more than rote application of rules. In short, the tendency is to teach mathematics as algorithms as if students were calculators. While absolutely essential for most of pre-college mathematics, the few years before a student goes to college need not be a continuation of such an approach. But it is. And perhaps just as problematic, so much of what is taught is to prepare for calculus and is useless if a student never takes calculus.

Worse, a student who takes elementary calculus as it is taught is given a set of tools that can be used to solve problems they will never confront. It is as if we provide hammers, screwdrivers, etc., but no screws or nails. And what is perhaps worst of all, elementary calculus courses continue the algorithmic approach of having students procedurally solve carefully crafted problem sets using rote application of rules in the hopes that, after 2 or 3 semesters of this, students will somehow have learned implicitly concepts never taught. The extensive literature on the failure of too many students to adequately transition into higher-level mathematics courses (and in some cases transition into courses designed to help students make that tradition) seems to clearly indicate that whatever is being learned implicitly isn’t enough. Also, many a B.S. degree requires only calculus I, and as one can see from looking at the problem sets in dozens of calculus I textbooks that, whatever benefit exists in such a course, it is less than that of one in elementary probability, statistics, logic, etc.

Because elementary calculus introduces calculus topics with such little depth, these courses consist mainly of the application of skills learned in pre-calculus courses. I still remember reading the definition of a limit for the first time and thinking “why don’t they just use quantifiers?” I was lucky enough to have taken logic first, and the nuances of e.g., negating quantifiers was, for me, old hat.

Nor do such problems stop at Calculus I. When calculus students learn about vectors it is usually in terms of I, j, & k unit vectors, and all operations with these continually reinforce these ideas about thinking about vectors in terms of 2D or 3D space. If the student then takes a course in linear algebra, these vectors have disappeared and in their place we find the far superior notation e1, e2, e3,…I say superior because not only can we use those three vectors I just listed in place of the I, j, k unit vectors of multivariable calculus, but they are naturally generalizable to n-dimensional space, the student who continues in her or his mathematical education will almost certainly see unit vectors expressed this way or in some equivalent, equally generalized way  The fact that the same letter is used doesn’t reinforce associating each vector with the x, y, or z axes, while the use of three unit vectors does.

I’ve always found interesting a comparison between linear algebra and elementary calculus courses. The former rely heavily on pre-calculus mathematics, and most of the difficulties result from pre-calculus mathematics. Take, for example, limits (generally introduced in the 2nd chapter of a calculus textbook; Spivak’s magnificient text is a major exception). Limits are conceptually challenging. It took several centuries before limits were sufficiently rigorously defined despite attempts by some of the greatest mathematicians in history. So, rather than ensure students really understand (or just have a good understanding of) limits, we throw limit problems at them which require complicated algebraic manipulations, knowing trigonometric identities, etc., because the student is used to working with these and it is hoped will get a decent enough understanding of limits by practicing skills learned in pre-calculus courses. In other words, the problems are computationally harder than need be in order to avoid conceptual difficulties.

One can’t do this in linear algebra. Even before one is confronted with spaces that the humans cannot visualize, the abstract nature of linear transformations, change of basis, etc., mere matrix multiplication is already a challenge because it is an operation unlike any they’ve seen and even once one gets used to matrix multiplication such that they don’t keep trying to treat it as if it were the quite intuitive way matrices are added, there’s the computational load (i.e., just 3 x 3 matrices involve a lot of computations that ensure plenty of opportunity for error.). So linear algebra is taught using matrices with dimensions one will never encounter in application, with entries that are usually small, whole numbers to make the computations easy, and in general the entire subject is taught using problems designed to make the computations easy and as a result totally unrealistic. Why? Because the computations don’t matter. I’m not about to sit down and try to find eigenvalues and eigenvectors to satisfy some linear transformation in 10 dimensional space, let alone 10,000. That’s why I have MATLAB. What I need is to understand the concepts, and unlike with calculus there is no way to hope students will grasp these by throwing a bunch of problems that are conceptually simple but computationally more challenging.

One final point: Spivak’s calculus text begins with adding, subtracting, and multiplying. That’s chapter 1. Two chapters later we get to functions, which is how many calculus textbooks begin. It takes two more chapters (with 4 appendices) before we get to limits. What’s interesting is that the actual page in which the text begins to deal with limits is close to that most calculus textbooks do. Even more interesting is that chapter 1, the chapter that is simple arithmetic with variables (so simple I hesitate to call it simple algebra), is often challenging. This is because Spivak uses the only formal system that students are sure to know to begin to introduce them to mathematical structure, proof, and most importantly concepts rather than rules. Trivial problems using only addition/subtraction are made non-trivial by forcing the student to justify each step with one of a few postulates (for example, showing that a*0=0, which Spivak proves in 9 steps). This is the kind of text that, as with most abstract mathematics, one cannot ace by being good at computations. One must understand. Not just understand notions relating to the infinite, but why certain things are true or not, rather than told that they are.

Thanks Andrew for your kind reply!

For me personally, I realize how different the situation, studying math., is today compared the fifties and sixties. I guess it all depends on what you will need in your future profession. Since I have been attracted to many cross disciplinary adventures, I can see that there is a dire need for new and novel ideas in teaching math.., to avoid misinterpretations.

One reason is that quantum mechanics is emerging more and more, viz, philosphy, quantum computing, cryptology, neurology etc. In this respect I believe the Riemann- and the Stieltjes integrala are a must, with Lebesque and measure theory becoming particularly important in modern chaos theory and statistical mechanics.

The problem here may be that many math teachers are for natural reasons not experts in all these domains and therefore additional courses must rely on the competence on the teachers in the subject under study. Although this may not be a bad solution, it may still be a bit unsatisfactory for others.

Here I think you and other devoted teachers have an important mission!

Dear Erkki J. Brändas:

The main reason I am loathe to give up on measure theory and the Lebesgue theory is because I took the time to learn them. This is, I freely admit, a terrible justification. So while I remain prejudiced, I have at least not let that wholly blind me. I mentioned in an earlier response that I’ve used material from the book Interesting Integrals: A Collection of Sneaky Tricks, Sly Substitutions, and Numerous Other Stupendously Clever, Awesomely Wicked, and Devilishly Seductive Maneuvers for Computing Nearly 200 Perplexing Definite Integrals From Physics, Engineering, and Mathematics, though for reasons I think clear I didn’t give the subtitle in that reply. I did, however, note that the author “not only simply presents the Lebesgue integral only to say that it will not be used, but defends this by citing Bartle’s famous paper “Return to the Riemann Integral.”” Another favorite textbook by Hubbard and Hubbard (a combined linear algebra/multivariable calculus text), does cover Lebesgue integration but as they say in their preface, “we emphasize computationally effective algorithms, and we prove theorems by showing that these algorithms work.”

Most multivariable calculus textbooks (whether they are textbooks that include first semester calculus or not) introduce integration via multiple integrals and connecting them as much as possible with the kind of integration first semester students are familiar with. In their opening page to their chapter on integration, Hubbard & Hubbard write that because “most integrals can be systematically computed (by hand) only as antiderivatives, students often take this to be the definition. This is misleading: the definition of an integral is given by a Riemann sum”.  They do not stop here, of course, but their main approach to integration is extending the Riemann integral. Before they get to any math, they introduce the integral (sect. 4.1) with a few example descriptions, my favorite being the last two: “We will define such multiple integrals in this chapter. But you should always remember that the preceding example is too simple. We might want to understand the total rainfall in Britain, whose coastline is a very complicated boundary. (A celebrated article analyzes that coastline as a fractal, with infinite length). Or we might want to understand the total potential energy stored in the surface tension of a foam.”

So, while the authors are mathematicians (actually, I am not sure if Barbara Hubbard is) and naturally think measure theory important, their emphasis on diverse application and “effective algorithms” is part of what makes their text so idiomatically brilliant. I will say that it did not prepare me much for understanding foams, and even watching the bulk of two of Cambridge’s INI seminars “Foams and Minimal Surfaces”, a volume from the series Polymeric Forms (Polymeric Foams: Mechanisms and Materials), A. J. Wilson (Ed.)’s volume Foams: Physics, Chemistry and Structure, and several other sources that cover foams, I think it is safe to say that as much as learned, a good part of it was learning how much more there was to learn and how comparatively little I knew/know.

I am not a teacher; I teach and I tutor and have for years on everything from classes to pass standardized college entrance aptitude exams like the SATs to Latin, but this has always been a side job, often to make extra money (though I do enjoy it). I mention this because you brought up the increased attention to quantum physics and its relevance/applications in various fields. A long time ago, I had two undergraduate majors and at that point no minor- one (a combined psychology & sociology major) because I had intended to go into clinical psychology, and the other (Ancient Greek & Latin) because I hate reading things in translations rather than the original languages. For the past several years, my work has been on quantum physics, complex systems, and computational neuroscience (I might add here that even though I agree with Mermin that “[M]athematicians tend to despise Dirac notation, because it can prevent them from making important distinctions, but physicists love it, because they are always forgetting that such distinctions exist and the notation liberates them from having to remember”, I hated Dirac notation first because I already had a perfectly good notation to represent vectors, functional spaces, etc., and learning another one was almost a bit like learning linear algebra again).

My point is that despite a certain amount of cross-disciplinary research and interests, my main objection here is almost a retreat (if not a signal of defeat). I am focusing on one, tiny portion of the mathematical curricula because I can ask it clearly, rather than what really bothers me: the entire mathematical curricula from a few years before college (high school in the USA) through most standard calculus courses. I think it is ridiculous to turn so many students off of math by having them study topics the way that they do rather than teach logic, statistics, probability, set theory, and other courses that would not only lay the foundations for “real” mathematics but provide topics of immediate application instead of e.g., rationalizing complex algebraic expressions which is covered so that when the student takes calculus (which is often never) they will be able to find limits without understanding what these are.

At the end of the day, any student who continues long enough in their mathematical education as undergraduates and/or as graduate students will typically develop an understanding of integration not in terms of “integrals” so much as ways in which functions and more can be integrated. Eventually, one comes across enough integrals that are named like the “Riemann integral” is, e.g., the Fresnel integral or Feyman’s Path integral, to mistake these for methods of integration rather than specific integrals or a specific application of integration. Whatever misunderstandings or other problems encountered will be overcome, indeed forgotten. My concern is with the number of people who give up before finding out that they even enjoy (at least some) mathematical fields, let alone those who do enjoy it but are thrown roadblock after roadblock until they quit. Improper integrals was just the simplest, most concise example I could come up with that I see as being completely unnecessary (as taught) and wanted to be challenged. To that end, many thanks for the insightful remarks to you and all others!

Dear George Stoica:

Can you give me an example of an application of improper integrals as learned in elementary calculus (or as learned in a later course which requires the elementary calculus version)? As for specialists, improper integrals are, as I noted earlier, entirely different from what we teach in elementary calculus. Or rather, to the extent they are not entirely different the only way one can see how they relate is by knowing a great deal more about integration.

Also, as for applications, in most pre-calculus textbooks, in the chapter on systems of equations/linear systems, one section will be devoted to matrix operations and another to determinants. Undergraduates find matrix operations difficult and it usually takes a while for them to understand things like determinants, yet students with far less experience in mathematics are introduced to both matrix operations and determinants in two sections in one chapter because...?

Here, there is absolutely no question that both matrix operations and determinants have a VAST number of applications. Moreover, their presentation in a pre-calculus textbook and a college linear algebra textbook doesn't differ in terms of definitions (in fact, many presentations of both in such a chapter are quite like the first section of the relevant chapter in a linear algebra textbook). Yet there is absolutely nothing gained by teaching anybody about determinants in one section of one chapter in a textbook that is at best tangentially related to linear algebra. Same with matrix operations. Students never gain an understanding of what matrices are (still less determinants), and if they continue with mathematics in college (not just as math majors but those majoring in engineering, chemistry, physics, etc.) they will almost certainly be required to take a linear algebra course in which both topics will be covered from scratch.

Improper integrals do not have any such immediate applications, as can be seen merely by looking through calculus textbooks and seeing the "real world" problems in the problem sets. In fact, basically the entirety of elementary calculus lacks any useful application other than for later calculus courses. In the case of improper integrals, calculus students who learn about sequences and series will be reintroduced to improper integrals in a much more natural setting, and one that would probably be less difficult if they hadn't first learn impromper integrals in an UNnatural setting, namely that of pseudo-integration theory.

Dear Andrew,

Thanks for your exhaustive answers. I think, as an example of interesting and useful math, I recommend you a book (that is probably not so well-known), but is written by one of the founders of Quantum Chemistry, Per-Olov Löwdin (which also studied with Arne Beurling and Wolfgang Pauli).

The book is "Linear Algebra for Quantum Theory", Wiley Interscience 1998, containing even linear algebra for nondefinite metrics, a forgotten subject despite the popularity of Einstein's theory of special relativity and the Minkowski space.

Erkki J. Brändas:

Thank you for the recommendation, and for using "exhaustive" rather than the more accurate "exhausting"; when I start writing I inevitably think of things during the process and end up turning what should have been a line or two into behemoths like this reply or the one before it).

Unfortunately, I own the book. That said, it is so eclective that despite being a linear algebra approach I have used material from it for teaching/tutoring many different subjects. The treatment of everything from set theory in general to Gödel contains material I've used in "transition" courses (designed to prepare students used to rote manipulation of symbols for courses like real analysis, abstract algebra, topology, etc.). In addition the obvious (e.g., vector spaces), I've used material from it to teach what we mean by "mathematical structure" especially vis-à-vis mathematical spaces and how this affects our approach to certain mathematical "objects" like manifolds. Many quantum physis textbooks either assume one is familiar with infinite-dimensional spaces or they limit their treatment to Hilbert space. The chapter on binary spaces as well as the treatment of Banach spaces in addition to your standard vector spaces and so forth have proved quite useful instructional material for teaching students who aren't even studying physics, let alone quantum mechanics.

As for indefinite metrics, I always just chalked it up to a not uncommon overlap in similar fields within or among discplines combined with a comparatively limited range of applications (including those in which it is vital: In e.g., Walecka's Introduction to General Relativity the indefinite metric we find in 1.4 isn't described as such except in the footnote. Apparently, identifying it clearly as indefinite wasn't important. Finally, speaking of linear algebra, there's actually an entire book Indefinite Linear Algebra and its Applications which doesn't (directly) cover indefinite metrics at all.

What follows is something of a story of why I bought it, why & when I should have bought it, and a badly formulated instructional manual on how NOT to study. It can be skipped in its entirety.

I should have the book far earlier than I did. When I started to work with NMR technology and mathematical physics, and found the same subjects had different terms, notations, etc., I immediately spent more money than I could afford on 4 books, only one of which, Quantum Theory for Mathematicians (Graduate Texts in Mathematics) I found useful at that time. By the time I got around to Per-Olov Löwdin's text, it was too elementary, and all I could do was berate myself for not buying it instead of Essential Mathematical Methods for Physicists, Problems & Solutions in Quantum Mechanics,or Quantum Computation and Quantum Information (the other 3 books I bought. By the time I noticed  Linear Algebra for Quantum Theory (actually, I was probably eventually told by somebody during a rant about Dirac notation),  I had bought purchased and read several other books (as well as formally and informally auditing a few courses and attending some graduate seminars thanks to the kind permission of the professors mostly at my university but also some nearby). I'm not sure how graduate school works in Sweden, but in the US even if one is very lucky and not only attends graduate school for free but is awarded enough fellowships, grants, etc., to afford the cost of living (and even if one supplements this 'income" by working for companies that offer teaching and tutoring services; I didn't have enough experience to be a research consultant then), paying textbooks and other technical/academic books, monographs, or volumes gets expensive very quickly (especially if, like me, one has trouble reading e-books and even more trouble using the library rather than owning the book). 

**WARNING*** The following paragraph is is just a rant and may be skipped (and probably should be) ***

For example, I am almost positive that the following sample consists only ofl texts I purchased prior to buying Per-Olov Löwdin's text:  1,000 Solved Problems in Modern Physics (~$130),  Handbook of Quantum Logic and Quantum Structures (~$300), Mathematical Concepts of Quantum Mechanics (~$60), Mathematics for Theoretical Physics: Electrodynamics, Quantum Mechanics, General Relativity, and Fractals (I forget), Mathematical Foundations of Quantum Field Theory (~$100) not to mention the price of textbooks and materials for the graduate seminars and both undergraduate and graduate courses, well let's just say that even now I can only justify most of the academic sources I do by re-catogorizing "study" with "hobby", and main hobby at that: a month of krav maga costs less than many of my books and as much as many of the tactical/combatives courses/seminars I attend; even courses that run upwards of $1,000 like week-long High Risk Operations or dynamical entry courses at places like Academi (formerly Blackwater Training Center) are rare enough to be cheaper than the cost of the average number of books I buy in a week or two.

*** End of Rant***

I am always interested in texts with novel approaches (Spivak's Calculus, computational and statistical methods in musicology, pattern recognition in Biblical Studies, The Navajo Verb, or selected works for some topic such as quantum measurement, A Reader in Nineteenth Century Historical Indo-European Linguistics, The Historical Jesus: Critical Concepts in Religious Studies) that I saved and saved until I could afford this ~$2,000 dollar IN PRINT volume).

So whatever or whoever was responsible for alerting me to the existence of Per-Olov Löwdin's text, I can't say I bought it expecting to learn much. I bought it much for the same reason I've bought several other linear algebra books, actually (I was neurotic enough to BUY Linear Algebra Done Right ONLY so that I could see how accurate the title was). However, this was both a book on quantum physics and mathematical physics, but unlike other such books I have, this one wasn't a mathematical approach but specifically a linear algebra approach. And the only disappointment I experienced was that I didn't buy the book earlier.

Andrew,

I was fortunate to learn from this book, since early versions of the text started to circulate already in the early sixties and it was taught to generations of quantum chemists and theoretical physicists through the seventies, eighties until completion in the nineties.

However, modern quantum theory have taken an entirely different route in order to focus on interference techniques, quantum cryptology and quantum computing.

Nevertheless mathematical methods were always high on the agenda at the Quantum chemistry department in Uppsala, see e.g. the enclosed review.

Anyway, I am pleased that you did find Per-Olov Löwdin's book, but regret that he took such a long time to get it published.

Dear Erkki:

Thanks for the link!  Not just for its contents but for the context you gave. In any given field at any given time, it seems like there are a few places where, if one has particular interests, that's the place to be. Unfortunately, there isn't exactly a directory containing a list "cool institutes producing research  you'll really want to keep an eye on". One finds such places through networking or reading a good deal of research and realizing that much of the neatest stuff is all coming from e.g., the Santa Fe Institute, or the Isaac Newton Institute for Mathematical Sciences. Thanks to you I can add Uppsala University's "Department of Physical and Analytical Chemistry" to the list of departments, labs, centers, etc., producing research that I should follow because of their work and my interests. Much appreciated!

-Andrew

For example, an analytical solution for the stationary heat conduction  eq. (Laplace's eq.) in half-space is expressed as the Poisson integral, which is improper. And many other analytical solutions in math. physics. Also, while proving the existence theorems, where one managed to find the solution in the form of improper integral. The integral convergence proves the existence, if it diverges then not (assuming the uniqueness of the solution).

Andrew, sorry didn't see that you object teaching it in  elementary calculus. My example was 2D - not for the elementary calculus.  Though I remember that we had enormous home works/tests at MSU to calculate  improper integrals (without an online calculator!!!!) when we were only 17. And some of us entered the University even at 16. I tell students: it is easier to study at Caltech. If smb. likes my answers, please use a little green arrow.

Natalia, I was just informed by email of your response and had intended to bring up the very point you did: that my objection wasn't to teaching improper integrals per se but to teaching them first as another "integration" method and then reintroducing them when the so-called "integral test" for convergence arises. You beat me to the punch.

I alas did not have the experience of learning improper integrals via the traditional courses at first (or most of undergraduate mathematics) either through courses or "in order"). I learned it myself and then taught it and then had to take several of such courses. Essentially, I taught myself backwards until I reached a point I could go forwards, was then asked to teach, and then asked to (albeit only for certain courses) take what I had been teaching and could not get out of despite professors and students who were kind enough to protest on my behalf.

But that was some time ago, and I have had to take many courses since, many of which I will not feel comfortable teaching until I have published something in that topic.

That said, improper integrals are not defined in the 2-dimensional case. Perhaps you mean that expressions, functions, relations, etc., in R1 are graphically depicted in 2D space. Improper integrals aren't defined for 2D, but like the simplest equations in algebraic geometry/coordinate geometry can be graphed in 2D as they are essentially functions of a single variable (that is, the independent variable x is defined on the real number line).

In any event, I object to the way it is taught in elementary calculus. Namely, everything that is generally understood by undergrads in calculus courses (indeed, by pre-college students in advanced placement) concerning improper integrals is then reintroduced when the "integral test" for convergence if covered. Only

1) Virtually nothing need be taught about improper integrals here (at least as they are first introduced)

&

2) Much more can be gained by first introducing improper integrals in terms of series and the "integral test". In fact, I have increasingly become a believer in teaching (or at least emphasizing) limits, sequences, and series as much as possible and as early as possible, as these underlie the entirety of analysis.

Anyway, I thank your for your (as ever) valuable feedback/comments!

-Andrew

Of course the Lebesgue intergral and measure theory is in many ways superior, and in basic form not actually so difficult, but it takes some mathematical maturity to appreciate it. However, it is absurd to say that  you cannot do indefinite Riemann integrals in higher dimensions or have to go through ridiculous hoops:  just  stick to absolutely convergent integrals as you always must in higher dimensions for the simple reason that there is no natural increasing ordering of points (and as you must for the Lebesgue integral!).

Restricting to the two dimensional case for notational convenience:

If 

(*) lim_{R\to \infty) \int_{|x| < R} |f(x,y)| d(x,y) exists

then 

\int_{R^2} f(x,y) d(x,y) = lim_{R\to \infty) \int_{|x| < R} f(x,y) d(x,y)

exists and is called the improper Riemannian integral 

lemma:

if A_i  \subset A_{i + 1} \subset ... is a sequence of suitably nice subsets such that for all i, there are real numbers  L_i, U_i  tending to infinity with 

                  { | x| < L_i } \subset A_i \subset { |x| < U_i}

then 

lim_{ i \to \infty} \int_A_i |f(x,y)| d(x,y) exists

iff (*) and  

int_{R^2} f(x,y) d(x,y) = lim_{i \to \infty} \int_{A_i} f(x,y) d(x,y)

in other words any nice increasing family of sets will do to define (or compute) the integral.

Sometimes, the Fourier transform of a square integrable function on the real line is given by an improprer integral. So, the spectral analysis of a time dependant signal of finite energy may involve improper Riemannian integrals. Itcould be a reason to teach improper integrals, that are much popular among students than Lebesgue integrals.

In present traditional finite—infinite theory system, people have been creating many new “understandings， ideas” on “infinite”, “potential infinite” and “actual infinite” since Zeno’s time 2500 years ago. But it is difficult to solve those infinite related problems produced by the fundamental defects disclosed by the infinite related paradoxes since Zeno’s time, because within the present traditional finite—infinite theory system, “the infinite related problems” are strongly interlocked together with the foundation. So, though trying very hard willing to solve “some infinite related problems” with some new “understandings，ideas” within the present traditional finite—infinite theory system, but people finally discovered that nothing can be done because “everything is  so perfect” in present traditional finite—infinite theory system, many operations including all the operations in infinite related calculations (integrals) are “proper” because of no choices.

We know something is “improper” in present traditional finite—infinite theory system, but you have to say “it is the only proper way to do it”.

I know few people agree with me, but this is true.

"However, it is absurd to say that  you cannot do indefinite Riemann integrals in higher dimensions or have to go through ridiculous hoops"

I believe I said that improper integrals don't generalize to higher dimensions. If you equate improper integrals with Riemann integrals, than I would say your statement is wrong and your understanding wrong (but I don't believe this is true; I think it is simply a case of misunderstanding what was said, at least I hope).

If, in fact, you think that whatever "improper Riemannian integrals" may be can be generalized to higher-dimensional spaces, please indicate how. I know several professors of mathematics in more than one university, although I admit these mathematicians are only faculty in U.S. universities (Harvard, MIT, Brown, Umass Boston, BC, BU, Cornell, UCLA, and a few others). I'm not a mathematician by training (my doctoral project concerned physics & neuroscience). Nor have I ever taught a class in graduate level mathematics (and have tutored only a few graduate level mathematics students). So many mathematical fields are largely unfamiliar to me.

This isn't such a case. So if you actually believe improper integrals can be generalized to even arbitrary dimensions in Euclidean space, please demonstrate this. Thank you.

@Andrew Messing. I am afraid I don't understand your point. I gave a definition of an improper Riemann integral in two dimensions which as far a I know is bog standard, and which trivially generalizes to arbitrary finite dimensions:   do a Riemann integral of the  absolutely Riemann integrable function restricted to a ball of finite radius and take the limit for the radius going to infinity.  The limit is the indefinite Riemann integral iff it exists. The lemma just says that there is nothing special about balls: hypercubes

[-L,L]^n  \subset R^n

of side length 2L with L going to infinity (or in fact any increasing sequence of subsets with piecewise smooth boundary whose union is all of R^n  )  will do just as well. it also gives the same value as the Lebesgue integral (by the Lebesgue dominated convergence theorem and the fact that Riemann and Lebesgue integral on a compact subset of R^n are the same if both exist).  Mutatis mutandis you can similarly do improper absolutely convergent Riemannian integrals for functions that have singularities along piecewise smooth sets by taking the limit as epsilon -> 0 of the integral over the complement of a tubular neighborhood of radius epsilon. 

This is also the definition used in physics by the way. How do you think that the total mass of a mass distribution or the mass distribution with mass density going as 1/r in 3 dimensions? 

It would be more helpful if your LaTex code were clearer, but let's step back a bit (there's no point in debating whether or not a statement about integrals or integration is true if we are using different definition (and, even if we are using the same definitions but different notation, this can still prevent any constructive dialogue.

"For a function f on R we have the notion of improper Riemann integrability. This concept is of particular importance if f is Riemann integrable over R, while |f| is not...For the Riemann integral in Rn, with n >1, there is no useful analog of the concept of improper Riemann integrability. This is related to the fact that an unbounded set in Rn, with n > 1, can be approximated from within by compact sets in many different ways."

Duistermaat, J. J., & Kolk, J. A. C. (2004). Multidimensional Real Analysis II: Integration (Cambridge Studies in Advanced Mathematics No. 27). Cambridge University Press.

Personally I prefer to think of the improper integral as the Lebesgue integral in 1D, mostly because it can be proved equivalent but unlike the improper integral it "makes sense" higher dimensions:

"You may have seen improper integrals of unbounded  functions over unbounded domains. But this only works in dimension 1: improper integrals don't make sense in higher dimensions."

Hubbard, J. H., & Hubbard, B. B. (2009). Vector Calculus, Linear Algebra, and Differential Forms (4th Ed.). Matrix Editions.

Then there's my favorite, from a math text I was never able to teach from (although I have used it for supplementary material for students):

"For these reasons we have called it the calculus integral. But none of us teach the calculus integral. Instead we teach the Riemann integral. Then, when the necessity of integrating unbounded functions arise, we teach the improper Riemann integral. When the student is more advanced we sheepishly let them know that the integration theory that they have learned is just a moldy 19th century concept that was replaced in all serious studies a full century ago.

We do not apologize for the fact that we have misled them; indeed we likely will not even mention the fact that the improper Riemann integral and the Lebesgue integral are quite distinct; most students accept the mantra that the Lebesgue integral is better and they take it for granted that it includes what they learned. We also do not point out just how awkward and misleading the Riemann theory is: we just drop the subject entirely." (Thomson's The Calculus Integral; emphasis added, italics in original).

Of course, we can PROVE that the Lebesgue integral is equivalent (or equal) to the improper integral in 1D. So why does the author call hem quite distinct? Because is nonsensical in higher dimensional spaces, and the other isn't. .

Ok so I finally seem to understand your point.

The limit in a the integral over all of R^n of a not absolutely integrable function  does not make sense in higher dimensions without specifying how you take the limit. Of course it does not. That is why I said that you have to take  absolutely integrable functions. So you're point is there is no natural way to define the integral of a non absolutely integrable function in higher dimensions, which is true, and something that should be pointed out to students.

However, that really has very little to do with Riemann or Lebesgue integrals. 

Concretely: the integral $\int_R sin(x)/x$ does not exist as a Lebesgue integral. Of course the limit$ \lim_{R\to \infty} \int_{|x| < R} sin(x)/x = \pi$ does exist, obviously irrespective of whether you take the integral in the Riemann or Lebesgue sense. The improper Riemann integral takes this limit in the definition and therefore suppresses it in the notation, whereas the Lebesgue integral does not take it in the notation because the notation is already defined and taken.  IIRC Laurent Schwartz use the notation $\int_{\from}^{\to}$ to indicate taking the limit in his book on distribution theory.  

Since the Lebesgue integral requires absolute integrability anyway, the Lebesgue integral doesn't need a limit to define an integral over all of R^n, but it could be defined as a limit $\lim_{i \to \infty}  \int_{A_i} f \mu(dx)$ where $\{A_i\}_{i=1}^\infty$ is a $\sigma$-finite exhaustion, i.e. an increasing sequence of measurable sets of finite Lebesgue (\mu) measure such that $\union_i A_i = R^n$.   In this case one has to prove that the limit does not depend on the sequence of subsets which is again a direct consequence of the dominated convergence theorem which crucially depends on absolute integrability. In particular one can take a sequence of balls or hypercubes of increasing radius.  

Lebesgue integrals and more generally, measure theory are great, but they are mostly great because they make statements like the dominated convergence theorem and by extension, the completeness of $L_p$ true. Those are both practically and theoretically very useful tools.  However, I stand by my statement that there is really no problem to define Riemann integration in higher dimensions for suitably nice (e.g continuous) functions or functions with nice integrable singularities.  

Is it clear that the related question "Why should we teach convergent series ?" (and not only summable series) will produce the same answers ? I strongly doubt that it is the case, since teaching mathematics involves more than mathematics.

@ThieryFack That's a good point. The analogous question is  indeed why should we teach convergent series?  In the analogy, Andrew then states that we should not because it is misleading since convergent series over arbitrary countable sets (e.g sums over two indices) make no sense.  In analogy I then point out that convergent sums  (including not absolutely convergent ones) over ordered countable sets (i.e. essentially just i = 1,2,3,,..)  are useful, intuitive and crop up all over the place (in my very first reaction) . That convergent but not absolutely convergent sums do not work for arbitrary unordered countable sets is just the nature of things:  if your countable set is not ordered (so there is no well defined notion of going to infinity) the  naive "improper Riemann" definition:

take any exhaustion I_ 0 \subset I_1 \subset ..... \subset I 

where  #I_k < \infty and \union_k I_k =  I (equivalently choose an an ordering up to finite permutations  ) is countable and define 

        \sum_{ i\in I} a_i  = lim_{k \to \infty} \sum_{i \in I_k} a_i.

can give you any value you want (Riemann's series theorem .wikipedia.org/wiki/Riemann_series_theorem).  If the sum is absolutely convergent this works fine because it does not depend on the exhaustion as one easily proves.  If the sum is not absolutely convergent the naive definition by exhaustion  works fine and so absolute summability is certainly a natural condition. Moreover the definition by exhaustions gives the same answer as using the "Lebesgue" type definition:

write  a_i = a_i^+ - a_i^-   where a_i^\pm  = max(\pm a_i, 0) 

 and define

         \sum a_i = \sum_{i \in I} a_i^+  - \sum_{i \in I} a_i^-

where the sum over the nonnegative numbers b_i is defined as

 \sum_{i \in I} b_ i = sup_{ K \subset I,  #K < \infty} \sum_{ i \in K} b_i.

which is elegant if you know it but arguably requires more mathematical maturity.

So yes I agree that presentation has a lot to do with what you teach but in this case there is also a fairly basic mathematical reason why things are the way they are. 

In fact, we not only teach improper integrals but also have to stuff our students some other mysterious things as well.

Just see following divergent proof of Harmonic Series (given by Oresme in about 1360, very elementary and important, can be found in many current higher mathematical books written in all kinds of languages):

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                    （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．      （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．              （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                              （4）

We teach our students that we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite Un--->0 items in Harmonic Series by “brackets-placing rule" with modern limit theory and change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity.

 The more we try to explain the more doubts be aroused and we feel more helpless.

Isn’t it miserable?

the improper integral is very important in applications . I know that they use it in physics and chemistry

You are right Mr. Ramadan Sabra, just because the improper integral is very important in applications.

So, if people just take it as a useful tool, don’t ask “why” and care nothing about “conceptual break”, we will be in a peaceful world.

But we know it is impossible no to ask “why” and care nothing about “conceptual break”, that is why we are here in this thread discussing the tangled thing.

Regards, Geng

Sometimes we complain that students are not taught “correct ‘finite—infinite’ related mathematics”, but now I fully understand that at least since Zeno’s time, everyone in present traditional “finite--infinite” related science theory system have to teach our students all the theories in our text book. After all, we have no choices and so do our students. Sometimes we are helpless to teach our students something we are not agree in the bottom of our heart but sometimes we teach our students “wrong mathematical things” unconsciously because we ourselves are within the defected present traditional “finite--infinite” related science theory system------ some fallacious cases （the typical things are paradoxes）are produced logically by many reasons.

It is a long being suspended syndrome of infinite related fundamental defects (confusions): what are infinite, infinitesimal, 0, limit theory, infinite related numbers,…? Anyone working in present traditional infinite related science branches (mathematics, physics, …) is sure to be confused with infinite, infinitesimal, 0, limit theory, infinite related numbers,… And some fallacious cases many be produced naturally by this syndrome.

I think the thing really worries us scientists is: only a few people care how to avoid “purposely or unconsciously teaching our students wrong mathematical things” or how to solve the defects in present traditional “finite--infinite” related science theory system disclosed by the growing family members of “finite--infinite” related paradoxes.

“Applying mathematics” is only small branches of “theoretical mathematics (trunk)”, both “applying mathematics” and “theoretical mathematics” is needed in our science.

Something should be done sooner or later to get rid of those “finite--infinite” related fundamental defects in our science------ it is a huge project

You say "Lebesgue integrals (and others) will do all that improper integrals can and more."    This is not quite true.  The Lebesgue integral handles unbounded functions in a way that is different than that used for improper Riemann integrals.

Take   f(x)  as the derivative of the function   F(x) = x^2 \sin (1/x^2)   on [0,1].  This function is not Lebesgue integrable but it has an obvious value as an improper Riemann integral. 

Similarly the integral  of    x^{-1} \sin x   on  [0, \infty)  does not exist as a Lebesgue integral but does as a Riemann integral in the improper sense.

So perhaps one shouldn't skip this topic in the hopes that the student will pick up all the necessary ideas in a more senior course on integration theory.

An interesting perspective for a D.R.I.P proponent. BTW, I like your Theory of the Integral text. I've been recommending it for years.

But I never said (nor think) that improper integrals should be skipped, and said explicitly that they shouldn't. It's where they are covered and how that I am concerned with.

Thanks for the kind words Andrew.

To take up another of Andrew's themes: many or most students do indeed think of the integral exclusively as an antiderivative (just like all eighteenth century mathematicians did) in spite of our indoctrination of  Riemann integral  +  improper Riemann integral. 

Nearly all exercises in computing improper integrals can be done with the fundamental theorem of the calculus provided one is a bit sloppy with the conditions.  For a simple example, if one integrates    f(x) = x^{-1/2}  on  [0,1]  by using the primitive   F(x) = 2 x^{1/2}   the answer is easy enough, but will likely be graded "incorrect" since the student is expected to use the RIemann integral on  [t,1]   and then do an improper step taking   t -> 0+.   Pretty fussy.

An alternative is to skip the whole idea of "integrals as limits of sums" as the main theme and consider "integrals as inverting derivatives" as the main theme.  If you need a slogan: down with Leibnitz, up with Newton.   Follow the second theme and then later show the constructive steps (Riemann for some functions, improper Riemann for some, Lebesgue for others, Denjoy for more).    This is a theme that can be developed in   Rn  unlike improper integrals as Andrew points out.

So start with the Newton integral:  \int_a^b f(x) dx = F(b)-F(a)  now means that  F  is continuous and  F'(x)=f(x)  at all but finitely many points.   [There is a free text by Zakon that allows countably many exceptional points, but that would be for a later course.]

There are no "improper" integrals but all the same ideas remain:  instead of extending a Riemann integral one is checking whether a primitive can be extended to a continuous function.   The usual exercise of integrating  f(x) = x^{-p}  on  [0,1]  involves the same thinking as before but now nothing is improper--just integrable or not.

This site is inadequate to develop these ideas, but I did write an experimental "calculus" book that follows this fantasy development of giving students the Newton integral and its variants.  (Can be downloaded from classicalrealanalysis.com along with the Theory of the Integral text that Andrew mentioned.)

So, in terms of Andrew's question about where one might cover improper integrals, my unorthodox answer is roughly that the word could be abolished along with the idea that the Riemann integral is good starting point for calculus students.  All so-called improper integral problems are just then merely integrability problems.

Of course, none of this will come to pass.  The Riemann integral is here to stay. And if it stays then, since it treats only bounded functions, the improper Riemann integral is inevitable at the introductory level. 

Ooops.  Just read some of the earlier postings in this thread and Andrew has already made these points, even quoting me.   But to repeat (unnecessarily) I have always hated to use the phrase "improper integral" that somehow implies that any decent self-respecting integral would only treat of bounded functions and that unbounded functions represent some kind of pathology.  (Sorry-not used to the way this site works.)

One of the most important improper integrals are those involving Gamma and Beta functions. The relationship between these two functions, as well as the basic properties of Gamma function further lead to the well known important integrals on R from [x^(2n)*exp(-x^2)], and integral on [0, infinity) from [(x^p)*exp(-x^2)), p>=0. Passing to several variables, the most simple improper integrals are those on Cartesian products of unbounded intervals, involving the moments of the functions exp(-Sum x_j, j =1,...,n), exp( - Sum (x_j)*2). Here Fubini's theorem is the ingredient. Going back to the one variable case, the properties of the Gamma and related functions are basic in probability theory, statistics, physics. In the case of several variables, some functions mentioned above serve as weights or basic elements in various concrete problems, such as polynomial approximation on unbounded subsets, moment problems, etc. 

Nice Dear Octav Olteanu