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Publisher:

Princeton University Press

Publication Date:

2006

Number of Pages:

380

Format:

Hardcover

Price:

29.95

ISBN:

0691118221

Category:

General

[Reviewed by , on ]

Henry Ricardo

10/10/2006

What a treasure of a book this is! This is the fourth enthusiastic, informative, and delightful book Paul Nahin has written about the beauties of various areas of mathematics (search on “Nahin” for the others). The “fabulous formula” of the title is *e*^{iπ} + 1 = 0 (regarded by many to be the most beautiful mathematical formula) — or, more generally, *e*^{iθ} = cos θ + *i*sin θ. In part, this book may be regarded as an attempt to provide information that is missing from his earlier book, An Imaginary Tale: The Story of √–1.

Like the reviewer of one of Nahin’s previous works, When Least is Best , I had anticipated skimming the chapters for this review, but instead got seduced by the Preface (“When Did Math Become Sexy?”) and the Introduction into reading much more. This is not a book for the general reader with an appreciation of mathematics: It requires the mathematical maturity and substance of a “third year college undergraduate in an engineering or physics program of study.” The pages are packed with equations and mathematical diagrams. A perusal of the Table of Contents gives some idea of the richness of this book. As may be expected by mathematically savvy readers, there are discussions of vectors and matrices, infinite series, cyclotomic polynomials, number theory, Fourier analysis, and applications to electronics. By training Paul Nahin is an electrical engineer. He is a practical man who has an appreciation of mathematical beauty and respect for the rigor of mathematical derivations. Who better to guide us through the mathematics of complex exponentials and display their marvelous applications?

I enjoyed the author’s use of the Cayley-Hamilton theorem to prove DeMoivre’s theorem and the subsequent derivation of various trigonometric identities involving binomial coefficients. There is a nice derivation and application of the integral form of the Cauchy-Schwarz inequality to the question of the average speed of a falling body with respect to both time and distance. Chapter 2, “Vector Trips”, is devoted to interesting expositions of various pursuit problems.

In later chapters, we have a proof of the irrationality of the square of pi, the history of the Gibbs phenomenon, the application of Fourier series to Gauss’s quadratic sum, a discussion of synchronous radio receivers, and the mathematical details involved in making a speech scrambler. The last part of the book is an overview of Euler’s life and work.

Throughout, there are delightful anecdotes — some familiar, others fresh — and marvelous insights, both pure and applied. The back of the book consists of more than 150 notes arranged by chapter, notes referring to a wide range of disciplines and media.

At the time of this review, plans for celebrating the three hundredth anniversary of Euler’s birth in 2007 are being made. This book is a marvelous tribute to Euler’s genius and those who built upon it and would make a great present for students of mathematics, physics, and engineering and their professors. Paul Nahin’s name has been added to my list of those with whom I wouldn’t mind being stranded on a desert island — not only would he be informative and entertaining, but he would probably be able to rig a signaling device from sea water and materials strewn along the beach.

Henry Ricardo (henry@mec.cuny.edu) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.

*What This Book Is About, What You Need to Know
to Read It, and WHY You Should Read It*

*Preface*

"When Did Math Become Sexy"

*Introduction*

- concept of mathematical beauty
- equations, identities, and theorems
- mathematical ugliness
- beauty redux

*Chapter 1. Complex Numbers
(an assortment of essays beyond the elementary involving complex numbers)*

1.1 The "mystery" of √–1

1.2 The Cayley-Hamilton and De Moivre theorems

1.3 Ramanujan sums a series

1.4 Rotating vectors and negative frequencies

1.5 The Cauchy-Schwarz inequality and falling rocks

1.6 Regular n-gons and primes

1.7 Fermat's last theorem, and factoring complex numbers

1.8 Dirichlet's discontinuous integral

*Chapter 2. Vector Trips
(some complex plane problems in which direction matters)*

2.1 The generalized harmonic walk

2.2 Birds flying in the wind

2.3 Parallel races

2.4 Cat-and-mouse pursuit

2.5 Solution to the running dog problem

*Chapter 3. The Irrationality of π ^{2}
('higher' math at the sophomore level)*

3.1 The irrationality of π

3.2 The R(x) = B(x)ex + A(x) equation, D-operators, inverse operators, and operator commutativity

3.3 Solving for A(x) and B(x)

3.4 The value of R(πi)

3.5 The last step (at last!)

*Chapter 5. Fourier Integrals
(what happens as the period of a periodic function becomes infinite, and other neat stuff)*

5.1 Dirac's impulse "function"

5.2 Fourier's integral theorem

5.3 Rayleigh's energy formula, convolution, and the autocorrelation function

5.4 Some curious spectra

5.5 Poisson summation

5.6 Reciprocal spreading and the uncertainty principle

5.7 Hardy and Schuster, and their optical integral

*Chapter 6. Electronics and √–1
(technological applications of complex numbers that Euler, who was a practical fellow himself, would have loved)*

6.1 Why this Chapter is in this book

6.2 Linear, time-invariant systems, convolution (again), transfer functions, and causality

6.3 The modulation theorem, synchronous radio receivers, and how to make a speech scrambler

6.4 The sampling theorem, and multiplying< by sampling and filtering

6.5 More neat tricks with Fourier transforms and filters

6.6 Single-sided transforms, the analytic signal, and single-sideband radio

*Euler: The Man and the Mathematical Physicist*

*Notes*

*Acknowledgments*

*Index*

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