## Book Review: Dr. Euler’s Fabulous Formula

Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills by Paul J. Nahin

In my review of Nahin’s The Logician and the Engineer I mentioned that it could almost have been a surrogate for my digital logic course when I was taking electrical engineering. Likewise, Dr. Euler’s Fabulous Formula would be the popular equivalent of my signals course.

The equation in question is famous $\displaystyle e^{i \theta} = \cos \theta + i \sin \theta$ and in particular the form when $\displaystyle \theta = \pi$ which gives the formula its most popular formulation $\displaystyle e^{i \pi} + 1 = 0$. This however is the starting point of the book, as Nahin considers Dr. Euler’s to be a sequel to his previous book An Imaginary Tale which I have only read a bit several years ago (I plan to read it in full soon and of course review it, so we’re going backwards here). So the prerequisites for enjoying this book are familiarity with complex numbers as well as calculus as the book is jam-packed with differentiations and integrals. I think you’d be fine with just freshman calculus (though there’s a tiny bit of double integration but that’s not a big deal).

The first half of the book is very playful, as it jumps around and applies the formula to various different problems (like a cat chasing a mouse or showing that $\displaystyle \pi^2$ is irrational). Along the way there are many infinite series summations which I always enjoy seeing.

It’s the latter half of the book though that made me remark about how it’s like a popular science version of a signals textbook. There’s a chapter of Fourier series, then one on Fourier integrals and the Fourier transform, and the final chapter is on the application of the Fourier transforms to problems in electronics, principally radio transmissions. There’s not really any electronics proper though, as in actually constructing devices out of resistors and capacitors and whatnot, since that would be outside the scope of the book. It is a mathematics book, but Nahin was an electrical engineer himself so he probably couldn’t resist showing how directly applicable Fourier transforms as a practical concern.

There’s a final small biographical sketch of Leonhard Euler to close out the book.