# How would you introduce Frullani integral to students?

Some integration techniques are just "tricks", while some integrals are analytically significant in that they connect different fields of math or they embody higher level concepts.

In the commonly stated form, the Frullani's integral is a definite integral that encompass a nontrivial family of integrand.

One can apply it directly as a power-move, out of necessity, or more subtly in analysis.

With some effort, it can be extended one way or the other. For example, see the answer by fedja in this post, along with other valuable expositions on the derivation of Frullani integral. A certain generalization from the reals to complex can also be made.

My Questions

1. How would you introduce the Frullani integral to freshmen calculus students?

2. How would you re-introduce Frullani integral to junior or senior math majors who presumably care about more than just getting the answers?

The issue with presenting it as a "trick" is that students might learn only the bare minimum (merely the mechanical maneuver). Students at various levels inevitably sometimes wonder if "tricks" are not worth learning at all. How does one carefully make the distinction between tricks and more substantial things (here specifically the Frullani integral).

• 1. I wouldn't. Your author profile says nothing about what country you're in (that I could find), but in a U.S. freshman calculus class something like this (except possibly in a simple special case) would be highly inappropriate. 2. In my case, since I recognize having seen this in many (probably several dozen) advanced calculus texts, I would probably look it up in the dozen or so advanced calculus texts I have (and maybe some real analysis texts as well) and see what I could find. Also, I would look at similar books in a university library and google a lot. – Dave L Renfro Mar 9 '18 at 18:06
• For what you're looking for, I recommend creating your own reference file, such as I discuss in my answer to What are power series used for? (a reference request). As an example, this 11 September 2009 sci.math post on the Gudermannian function arose from my folder of stuff related to integration of the secant function (which is usually considered a trick integral with little independent interest). – Dave L Renfro Mar 9 '18 at 19:40
• Probably no one noticed this (or cares, for that matter), but that 11 September 2009 post of mine was in the ap-calculus discussion group (not Usenet), and not in the sci.math Usenet discussion group. – Dave L Renfro Mar 9 '18 at 21:31
• Frullani --> I discovered this only today. This is seriously being taught to first year university students?!?!?!!! – BCLC Mar 12 '18 at 17:19
• I think the '"how" question is hypothetical. Other recent post by questioner says he has left academia. Basically, the guy loves this integral and wants to spread it, thus the question. But one should ask why the rest of the free world has not seen the need to spread Frullani. – guest Mar 13 '18 at 6:10

I think it is hard to motivate this sort of issue at a typical U.S.-undergrad level. In my own recollection, I did not see such an integral/issue occur at all except in the discussion of the Gamma function in Whittaker-and-Watson, as a deus-ex-machina.

A saner context might be in discussion of principal-value integrals as distributions or tempered distributions. (The latter is more accessible, since the topology is easier to describe: Frechet rather than "LF"...) E.g., it is a reasonable exercise to prove that $f \to \lim_{\varepsilon\to 0^+} \int_{|x|\ge \varepsilon} {f(x)\over x}\;dx$ is a tempered distribution.

In that context, considering $f\to \int {f(ax)-f(bx)\over x}\;dx$ as a tempered distribution, explicitly noting which seminorms enter in proving its continuity, would have some value, I think.

• Thanks for the input. This is more along the lines of what I have been thinking. Indeed it's hard to motivate in lower division courses, that's partly why I posted the question. I wonder if it's possible to skillfully provide some flavor of real analysis at different stages. – Lee David Chung Lin Mar 13 '18 at 5:19

I almost certainly would not introduce that integral to freshmen. I probably wouldn't introduce it to sophomores or juniors either; not without some particular application in mind. (I don't know of one, but certainly it must have come up somewhere. Do you have an application in mind?) Still, playing along with a counterfactual world where I would do it, I'd do it using differentiation under the integral and elide questions of uniform convergence until after I'm done presenting the basic idea. Let $$\phi(a,b) = \int_0^\infty \frac{ f(ax)-f(bx) } {x} dx$$ Then \begin{align} \frac{\partial}{\partial a} \phi(a,b) &= \int_0^\infty \frac{\partial}{\partial a} \frac{ f(ax)-f(bx) } {x} dx \\ &= \int_0^\infty f'(ax) dx \\ &= \int_0^\infty \frac{1}{a} f'(u) du \\ &= -\frac{1}{a} \left( f(0)-f(\infty) \right) \\ \end{align} Similarly, one can find $$\frac{\partial}{\partial b} \phi(a,b) = \frac{1}{b} \left( f(0)-f(\infty) \right)$$ Now you can use standard methods to solve those differential equations and get the result, $$\phi(a,b) = \left(f(0)-f(\infty)\right)\ln\left(\frac{b}{a}\right)$$ This is a technique found in Advanced Calculus by Woods, which I found out about by reading people who were writing about Feynmann's book Surely you're joking Mr. Feynmann.

• I see now that my method is very similar to the method shown in one of the links. I'm still unclear where such an integral would naturally appear for an undergraduate. That's not meant to be dismissive, I'd love to have a justification/excuse to introduce differentiation under the integral! – Adam Mar 10 '18 at 14:41
• Thank you for participating in the discussion! Yes, I understand you're concern, and I've been looking for the examples I saw over the years from applications in science and statistics. If one doesn't mind doing integration just for the sake of integrations, then there are some basic examples. The links I included in the post above are intentionally more advanced to showcase the analytic side of it instead of just for calculation. – Lee David Chung Lin Mar 10 '18 at 15:11

The question asks "How would you introduce Frullani integral to students?" To answer this, it's necessary to first ask a different question: "Why do you want to introduce Frullani integral to students?"

I don't mean this just as a rhetorical question. If you want to introduce any new topic, the first thing you have to be clear on is: Why is this worth teaching? The universe of things you could be teaching is always larger than the number of opportunities you'll have to teach something, so every topic that gets included carries with it opportunity costs. Presumably if this is something you want to teach, it's because you think it's worth knowing.

So start there: What makes it worth knowing? What problems does this technique solve for you? What questions does it answer? If it's not useful, is it at least interesting? (Related: See https://matheducators.stackexchange.com/a/11410/29.) What makes it so? Is it interesting because it seems surprising? If so, why -- what expectation does it contradict? Is it interesting because it reveals some underlying connection between other ideas? If so, what are they? Once you answer those questions, you will know how to introduce this topic to your students.

Just show them it as one more trick and emphasize that it has large scope of application. I think that is enough. Also, not clear what else you all are supposed to be working on (what course) and if spending extra time on this integral is worth it. And if so, how worth it...like please don't skip other "tricks" that have large scope of application (e.g. partial fractions).

Whole question seems to read like "I love this integral and want to push it into the curriculum" not really "how to teach". But in terms of the literal question how to teach, you have your answer already. Show it as one more trick, drill it a bit, test it. And mention it's large extent of applications. Is there really a teaching methodology conundrum? Or a desire to justify the content to us or to colleagues?

And if your students aren't in real analysis yet, than all you can really do is say something like "it's useful in this higher math course see, where we do theory and such to rigorously prove things". But I wouldn't bother showing them WHERE it is used in RA if they are not in RA at all yet.

Also, I would only emphasize it in a regular calc class if they are a bunch of MIT, CalTech types who are taking some super enriched course anyways.

• I'm not sure why you think "Whole question seems to read like ... want to push it into the curriculum and not really how to teach". I added a paragraph at the end to see if it changes your mind. At the same time, I get what you're saying. With a more practical (cynical?) attitude, one should already be thankful if students learn anything at all. – Lee David Chung Lin Mar 9 '18 at 15:30