How would you introduce Frullani integral to students?

Question

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).

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.