Integration Using Schwinger Parametrization

I recently read the terrific article Integration Using Schwinger Parametrization, by David M. Bradley, Albert Natian, and Sean M. Stewart in the American Mathematical Monthly. I won’t reproduce the entire article here, but I’ll hit a couple of early highlights.

The basic premise of the article is that a complicated integral can become tractable by changing it into an apparently more complicated double integral. The idea stems from the gamma integral

\Gamma(p) = \displaystyle \int_0^\infty t^{p-1} e^{-t} \, dt,

where $\Gamma(p) = (p-1)!$ if p is a positive integer. If we perform the substitution t = \phi u in the above integral, where \phi is a quantity independent of t, we obtain

\Gamma(p) = \displaystyle \int_0^\infty (\phi u)^{p-1} e^{-\phi t} \phi \, du = \displaystyle \int_0^\infty \phi^p u^{p-1} e^{-\phi u} \, du,

which may be rewritten as

\displaystyle \frac{1}{\phi^p} = \displaystyle \frac{1}{\Gamma(p)} \int_0^\infty t^{p-1} e^{-\phi t} \, dt

after changing the dummy variable back to t.

A simple (!) application of this method is the famous Dirichlet integral

I = \displaystyle \int_0^\infty \frac{\sin x}{x} \, dx

which is pretty much unsolvable using techniques from freshman calculus. However, by substituting \phi = x and p=1 in the above gamma equation, and using the fact that \Gamma(1) = 0! = 1, we obtain

I = \displaystyle \int_0^\infty \sin x \int_0^\infty e^{-xt} \, dt \, dx

= \displaystyle \int_0^\infty \int_0^\infty e^{-xt} \sin x \, dx \, dt

after interchanging the order of integration. The inner integral can be found by integration by parts and is often included in tables of integrals:

I = \displaystyle \int_0^\infty -\left[ \frac{e^{-xt} (\cos x + t \sin x)}{1+t^2} \right]_{x=0}^{x=\infty} \, dt

= \displaystyle \int_0^\infty \left[0 +\frac{e^{0} (\cos 0 + t \sin 0)}{1+t^2} \right] \, dt

= \displaystyle \int_0^\infty \frac{1}{1+t^2} \, dt.

At this point, the integral is now a standard one from freshman calculus:

I = \displaystyle \left[ \tan^{-1} t \right]_0^\infty = \displaystyle \frac{\pi}{2} - 0 = \displaystyle \frac{\pi}{2}.

In the article, the authors give many more applications of this method to other integrals, thus illustrating the famous quote, “An idea which can be used only once is a trick. If one can use it more than once it becomes a method.” The authors also add, “We present some examples to illustrate the utility of this technique in the hope that by doing so we may convince the reader that it makes a valuable addition to one’s integration toolkit.” I’m sold.

A vivid illustration of a discontinuous function

The essay Singular Limits in the May 2002 issue of Physics Today has a vivid illustration of a discontinuous function F(x) which measures the ickiness one feels after eating an apple but observing that proportion x of a maggot is still inside the apple. For this function, \displaystyle \lim_{x \to 0^+} F(x) \ne F(0).

Biting into an apple and finding a maggot is unpleasant enough, but finding half a maggot is worse. Discovering one-third of a maggot would be more distressing still: The less you find, the more you might have eaten. Extrapolating to the limit, an encounter with no maggot at all should be the ultimate bad-apple experience. This remorseless logic fails, however, because the limit is singular: A very small maggot fraction (f \ll 1) is qualitatively different from no maggot (f=0). 

High School Students Finding New Proofs of Old Theorems (Part 2): Pythagorean theorem

This is a new favorite story to share with students: two high school students recently figured out multiple new proofs of the Pythagorean theorem.

Professional article in the American Mathematical Monthly (requires a subscription): https://maa.tandfonline.com/doi/full/10.1080/00029890.2024.2370240

Video describing one of their five ideas:

Interview in MAA Focus: http://digitaleditions.walsworthprintgroup.com/publication/?i=836749&p=14&view=issueViewer

Interview by 60 Minutes:

https://www.youtube.com/watch?v=VHeWndnHuQs

Praise from Michelle Obama: https://www.facebook.com/michelleobama/posts/i-just-love-this-story-about-two-high-school-students-calcea-johnson-and-nekiya-/750580956432311/

High School Students Finding New Proofs of Old Theorems (Part 1): Dividing a line segment with straightedge and compass

This is one of my all-time favorite stories to share with students: how a couple of ninth graders in 1995 played with Geometer’s Sketchpad and stumbled upon a brand-new way of using only a straightedge and compass to divide a line segment into any number of equal-sized parts. This article was published in 1997 and made quite a media sensation at the time.

1997_litchfieldd._euclidfibonaccisketchpadDownload

Higher derivatives in ordinary speech

Just about every calculus student is taught that the first derivative is useful for finding the slope of a curve and finding velocity from position, and that the second derivative is useful for finding the concavity of a curve and finding acceleration from position.

I recently came across a couple of quotes that, taken literally, are statements about third and fifth derivatives.

Per Wikipedia, President Nixon announced in 1972 that the rate of increase of inflation was decreasing. Taken literally, this claims that “the second derivative of inflation is negative, and so the third derivative of purchasing power [since inflation is the derivative of purchasing power] is negative.” As dryly stated in the Notices of the American Mathematical Society, “[t]his was the first time a sitting president used the third derivative to advance his case for reelection”; the article then ponders the implications of the abuse of mathematics.

More recently, the popular blog Math With Bad Drawings had some fun analyzing a clause that appeared in a 2013 op-ed piece: “As the rate of acceleration of innovation increases…” Taken literally, the words rate, innovation and increases all refer to a first derivative (innovation would be the rate at which technology changes), while the word acceleration refers to a second derivative. Therefore, taken literally and not rhetorically (which was clearly the authors’ intent), this brief clause is a claim that the fifth derivative of technology is positive.

Horrible False Analogy

I had forgotten the precise assumptions on uniform convergence that guarantees that an infinite series can be differentiated term by term, so that one can safely conclude

\displaystyle \frac{d}{dx} \sum_{n=1}^\infty f_n(x) = \sum_{n=1}^\infty f_n'(x).

This was part of my studies in real analysis as a student, so I remembered there was a theorem but I had forgotten the details.

So, like just about everyone else on the planet, I went to Google to refresh my memory even though I knew that searching for mathematical results on Google can be iffy at best.

And I was not disappointed. Behold this laughably horrible false analogy (and even worse graphic) that I found on chegg.com:

Suppose Arti has to plan a birthday party and has lots of work to do like arranging stuff for decorations, planning venue for the party, arranging catering for the party, etc. All these tasks can not be done in one go and so need to be planned. Once the order of the tasks is decided, they are executed step by step so that all the arrangements are made in time and the party is a success.

Similarly, in Mathematics when a long expression needs to be differentiated or integrated, the calculation becomes cumbersome if the expression is considered as a whole but if it is broken down into small expressions, both differentiation and the integration become easy.

Pedagogically, I’m all for using whatever technique an instructor might deem necessary to to “sell” abstract mathematical concepts to students. Nevertheless, I’m pretty sure that this particular party-planning analogy has no potency for students who have progressed far enough to rigorously study infinite series.