Dr. Euler’s fabulous formula: cures many mathematical ills

Paul J. Nahin0691118221, 9780691118222

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[pi] i+1
–Paul Nahin, electrical engineer

In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler’s Fabulous Formula shares the fascinating story of this groundbreaking formula–long regarded as the gold standard for mathematical beauty–and shows why it still lies at the heart of complex number theory.

This book is the sequel to Paul Nahin’s An Imaginary Tale: The Story of I [the square root of -1] , which chronicled the events leading up to the discovery of one of mathematics’ most elusive numbers, the square root of minus one. Unlike the earlier book, which devoted a significant amount of space to the historical development of complex numbers, Dr. Euler begins with discussions of many sophisticated applications of complex numbers in pure and applied mathematics, and to electronic technology. The topics covered span a huge range, from a never-before-told tale of an encounter between the famous mathematician G. H. Hardy and the physicist Arthur Schuster, to a discussion of the theoretical basis for single-sideband AM radio, to the design of chase-and-escape problems.

The book is accessible to any reader with the equivalent of the first two years of college mathematics (calculus and differential equations), and it promises to inspire new applications for years to come. Or as Nahin writes in the book’s preface: To mathematicians ten thousand years hence, “Euler’s formula will still be beautiful and stunning and untarnished by time.”

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Table of contents :
What This Book Is About……Page 13
Preface. “When Did Math Become Sexy?”……Page 17
Introduction……Page 21
1.1. The “mystery” of $sqrt{-1}$……Page 33
1.2. The Cayley-Hamilton and De Moivre theorems……Page 39
1.3. Ramanujan sums a series……Page 47
1.4. Rotating vectors and negative frequencies……Page 53
1.5. The Cauchy-Schwarz inequality and falling rocks……Page 58
1.6. Regular n-gons and primes……Page 63
1.7. Fermat’s last theorem, and factoring complex numbers……Page 73
1.8. Dirichlet’s discontinuous integral……Page 83
2.1. The generalized harmonic walk……Page 88
2.2. Birds flying in the wind……Page 91
2.3. Parallel races……Page 94
2.4. Cat-and-mouse pursuit……Page 104
2.5. Solution to the running dog problem……Page 109
3.1. The irrationality of $pi$……Page 112
3.2. The $R(x)=B(x)e^x+A(x)$ equation, D-operators, inverse operators, and operator commutativity……Page 115
3.3. Solving for A(x) and B(x)……Page 122
3.4. The value of $R(pi i)$……Page 126
3.5. The last step (at last!)……Page 132
4.1. Functions, vibrating strings, and the wave equation……Page 134
4.2. Periodic functions and Euler’s sum……Page 148
4.3. Fourier’s theorem for periodic functions and Parseval’s theorem……Page 159
4.4. Discontinuous functions, the Gibbs phenomenon, and Henry Wilbraham……Page 183
4.5. Dirichlet’s evaluation of Gauss’s quadratic sum……Page 193
4.6. Hurwitz and the isoperimetric inequality……Page 201
5.1. Dirac’s impulse “function”……Page 208
5.2. Fourier’s integral theorem……Page 220
5.3. Rayleigh’s energy formula, convolution, and the autocorrelation function……Page 226
5.4. Some curious spectra……Page 246
5.5. Poisson summation……Page 266
5.6. Reciprocal spreading and the uncertainty principle……Page 273
5.7. Hardy and Schuster, and their optical integral……Page 283
6.2. Linear, time-invariant systems, convolution (again), transfer functions, and causality……Page 295
6.3. The modulation theorem, synchronous radio receivers, and how to make a speech scrambler……Page 309
6.4. The sampling theorem, and multiplying by sampling and filtering……Page 322
6.5. More neat tricks with Fourier transforms and filters……Page 325
6.6. Single-sided transforms, the analytic signal, and single-sideband radio……Page 329
Euler: The Man and the Mathematical Physicist……Page 344
Notes……Page 367
Acknowledgments……Page 395
Index……Page 397