Dr. Euler’s Fabulous Formula Cures Many Mathematical Ills

Dr. Euler's Fabulous Formula,
by Paul Nahin.

I’ve decided that Leonhard Euler (1707-1783) is my favorite mathematician, and I’m mad at myself for not getting his name mentioned in the 5th edition of Intermediate Physics for Medicine and Biology. I’ve discussed Euler before in this blog, when I reviewed William Dunham’s book Euler: The Master of Us All. I’ve just finished another book, this one by Paul Nahin, about Dr. Euler’s Fabulous Formula. Of course, Nahin means Eq. 11.28 in IPMB,
e^iθ = cosθ + i sinθ .

I liked the book, in part because Nahin and I seem to have similar tastes: we both favor the illustrations of Norman Rockwell over the paintings of Jackson Pollock, we both like to quote Winston Churchill, and we both love limericks:

I used to think math was no fun,
‘Cause I couldn’t see how it was done.
    Now Euler’s my hero
    For I now see why zero,
Equals e^πi + 1.

Nahin’s book contains a lot of math, and I admit I didn’t go through it all in detail. A large chunk of the text talks about the Fourier series, which Russ Hobbie and I develop in Chapter 11 of IPMB. Nahin motivates the study of the Fourier series as a tool to solve the wave equation. We discuss the wave equation in Chapter 13 of IPMB, but never make the connection between the Fourier series and this equation, perhaps because biomedical applications don’t rely on such an analysis as heavily as, say, predicting how a plucked string vibrates.

Nahin delights in showing how interesting mathematical relationships arise from Fourier analysis. I will provide one example, closely related to a calculation in IPMB. In Section 11.5, we show that the Fourier series of the square wave (y(t) = 1 for t from 0 to T/2 and equal to -1 for t from T/2 to T) is

y(t) = Σ b_k cos(k2πt/T)

where the sum is over all odd values of k (k = 1, 3, 5, ....) and b_k = 4/(π k). Evaluate both expressions for y(t) at t = T/4. You get

π/4 = 1 – 1/3 + 1/5 – 1/7 +…

This lovely result is hidden in IPMB’s Eq. 11.36. Warning: this is not a particularly useful algorithm for calculating π, as it converges slowly; including ten terms in the sum gives π = 3.04, which is still over 3% off.

In Figure 11.17, Russ and I discuss the Gibbs phenomenon: spikes that occur in y(t) at discontinuities when the Fourier series includes only a finite number of terms. Nahin makes the same point with the periodic function y(t) = (π – t)/2 for t from 0 to 2π. He describes the history of the Gibbs phenomena, which arises from a series of published letters between Josiah Gibbs, Albert Michelson, A. E. H. Love, and Henri Poincare. Interestingly, the Gibbs phenomenon was discovered long before Gibbs by the English mathematician Henry Wilbraham.

Fourier series did not originate with Joseph Fourier. Euler, for example, was known to write such trigonometric series. Fourier transforms (the extension of Fourier series to nonperiodic functions), on the other hand, were first presented by Fourier. Nahin discusses many of the same topics that Russ and I cover, including the Dirac delta function, Parseval’s theorem, convolutions, and the autocorrelation.

Nahin concludes with a section about Euler the man and mathematical physicist. I found an interesting connection to biology and medicine: when hired in 1727 by the Imperial Russian Academy of Sciences, it was as a professor of physiology. Euler spent several months before he left for Russia studying physiology, so he would not be totally ignorant of the subject when he arrived in Saint Petersburg!

I will end with a funny story of my own. I was working at Vanderbilt University just as Nashville was enticing a professional football team to move there. One franchise that was looking to move was the Houston Oilers. Once the deal was done, folks in Nashville began debating what to call their new team. They wanted a name that would respect the team’s history, but would also be fitting for its new home. Nashville has always prided itself as the home of many colleges and universities, so a name out of academia seemed appropriate. Some professors in Vanderbilt’s Department of Mathematics came up with what I thought was the perfect choice: call the team the Nashville Eulers. Alas, the name didn’t catch on, but at least I never again was uncertain about how to pronounce Euler.