# Dr. Euler's Fabulous Formula

Nahin published this book originally in 2006. This copy is a reprint in the *Princeton Science Library* of the revised paperback edition of 2010. It is a sequel to the author's *An Imaginary Tale: The Story of √-1* (see here for a review). The series brings reprints in cheap paperback and eBook format of classics and bestsellers and makes them available for a new generation of a potential readership. It covers a broad spectrum of science books and among them many are about mathematics.

Euler's fabulous formula of the title is of course the extraordinary, and (think of it) amazing formula $e^{i\pi}+1=0$. Thus the square root of $-1$ is not far off. Complex numbers and complex functions being already introduces in his previous book, Nahin can concentrate on specific applications of this Euler formula. Although in principle all the material covered can be read and understood by mathematics or engineering students at an advanced undergraduate level and the material is covered in a leisurely almost pleasant discourse, there is a lot of serious mathematics that is covered. A good mathematical background is necessary. Big chunks of a (complex) analysis course are covered. The difference with a classical course is that in lecture notes one has a strict target list of concepts, properties and theorems that have to be covered. The reader is then guided through all these topics in most efficient and matter-of-fact kind of way. Here the same landscape is explored but there is no urgent target. The guide is the Euler formula and the reader is leisurely exploring some topics that are from far or near related to it without a strict travel plan or compelling arrival time.

Because the Euler formula is known as the most beautiful formula in mathematics, there is an introductory support act contemplating what it means when a mathematical formula or a proof is generally accepted to be "beautiful". During the main mathematical dish, of the subsequent chapters, often some historical background is given, if not in the text, then it is in the extensive list of notes at the end of the book. As a dessert we can read a biography of Euler in the last (unnumbered) chapter.

So what is then the main dish? There are five chapters. The first is about complex numbers, but it goes beyond the first elementary steps that were already in *An Imaginary Tale*. By interpreting multiplication with a unimodular complex number as a rotation that can be represented by a matrix multiplication, a Cayley-Hamilton theorem can be found. Furthermore formulas of De Moivre, Cauchy-Schwartz, infinite series, the construction of n-gons, and its relation with Fermat's last theorem, and Dirichlet's integral of the sinc function. The same mixture of exploring and digressing is maintained in the other chapters. The next chapter is called vector trips. The interpretation of complex numbers in the plane allows a geometric interpretation of summing power series, leading eventually to the solution of some differential equations. Another chapter proves the irrationality of $\pi^2$, and a thicker one introduces Fourier series. The idea already lingered in Euler's time where Euler, D'Alembert, and Daniel Bernoulli were contemplating the solution of the wave equation. The wiggles appearing in Fourier approximations near discontinuities is well known and nowadays identified as the Gibbs phenomenon. It is noteworthy that Nahin, as for most topics discussed in this book, gives the historical background of this phenomenon and discloses that it was actually discussed in a 1848 paper published fifty years before Gibbs by a forgotten Englishman Henry Wilbraham. An equally thick chapter is devoted to the Fourier integral and the continuous Fourier transform, including the Dirac delta function, the Poisson summation formula, the uncertainty principle, autocorrelation and convolution. The closing section here is about a difficult integral discussed by Arthur Schuster (1851-1934) in connection with optics. Hardy got interested and evaluated the integral, which is another instance where Hardy helped solving an applied problem, something he rejected in his *A Mathematician's Apology*. The final chapter is about applications in electronics: signal processing, linear time invariant systems, filters, and more.

It is clear from the interpretation, the wording, and the examples that Nahin's background is in electrical engineering. Not that this is diminishing the value of his treatment of all the mathematics in this book. There is however a bias. It is also a typical engineering hands-on attitude to check the validity of some formulas with a numerical simulation, even if they were mathematically proved already. The title of Euler's biography in the trailing chapter is *Euler: The Man and the Physicist*. Despite Hardy's attitude towards applied mathematics, one has to admit that historically mathematics has developed also, and probably mainly so, because of the applications. In this sense, the book stays close to the spirit of Euler's approach to mathematics who made no proper distinction between pure and applied mathematics, and therefore the whole book is also a tribute to Euler.

Let me give a quote from chapter 3 to illustrate the way Nahin tells his story. "Thus we have at last [some integral expression for $R(\pi i)$]. The reason I say *at last* is that we are not going to evaluate the integral. You probably have two reactions to this —first, relief (it is a pretty scary-looking thing) and, second shock (why did we go through all the work needed to derive it?). In fact, all we need for our proof that $\pi^2$ is irrational are the following two observations about $R(\pi i)$." But don't be mistaken, there is a lot of serious mathematics and formulas. If this book falls under "popular mathematics", it can only be popular for the readers literate in at least some more than elementary calculus. Many classical mathematical issues are discussed, but often using an original approach. This makes it also a recommendable read for professional mathematicians

**Submitted by Adhemar Bultheel |

**10 / Apr / 2017