Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills

✍ Scribed by Paul J. Nahin

• Publisher: Princeton University Press
• Year: 2006
• Tongue: English
• Leaves: 416
• Edition: 0
• Category: Library

✦ Synopsis

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."

✦ Table of Contents

Cover......Page 1
Contents......Page 10
Preface to the Paperback Edition......Page 14
What This Book Is About, What You Need to Know to Read It, and WHY You Should Read It......Page 26
Preface: "When Did Math Become Sexy?"......Page 30
Introduction......Page 36
1.1 The "mystery" of (omitted)......Page 48
1.2 The Cayley-Hamilton and De Moivre theorems......Page 54
1.3 Ramanujan sums a series......Page 62
1.4 Rotating vectors and negative frequencies......Page 68
1.5 The Cauchy-Schwarz inequality and falling rocks......Page 73
1.6 Regular n-gons and primes......Page 78
1.7 Fermat's last theorem, and factoring complex numbers......Page 88
1.8 Dirichlet's discontinuous integral......Page 98
2.1 The generalized harmonic walk......Page 103
2.2 Birds flying in the wind......Page 106
2.3 Parallel races......Page 109
2.4 Cat-and-mouse pursuit......Page 119
2.5 Solution to the running dog problem......Page 124
3.1 The irrationality of π......Page 127
3.2 The R(x) = B(x)e[sup(x)] + A(x) equation, D-operators, inverse operators, and operator commutativity......Page 130
3.3 Solving for A(x) and B(x)......Page 137
3.4 The value of R(πi)......Page 141
3.5 The last step (at last!)......Page 147
4.1 Functions, vibrating strings, and the wave equation......Page 149
4.2 Periodic functions and Euler's sum......Page 163
4.3 Fourier's theorem for periodic functions and Parseval's theorem......Page 174
4.4 Discontinuous functions, the Gibbs phenomenon, and Henry Wilbraham......Page 198
4.5 Dirichlet's evaluation of Gauss's quadratic sum......Page 208
4.6 Hurwitz and the isoperimetric inequality......Page 216
5.1 Dirac's impulse "function"......Page 223
5.2 Fourier's integral theorem......Page 235
5.3 Rayleigh's energy formula, convolution, and the autocorrelation function......Page 241
5.4 Some curious spectra......Page 261
5.5 Poisson summation......Page 281
5.6 Reciprocal spreading and the uncertainty principle......Page 288
5.7 Hardy and Schuster, and their optical integral......Page 298
6.2 Linear, time-invariant systems, convolution (again), transfer functions, and causality......Page 310
6.3 The modulation theorem, synchronous radio receivers, and how to make a speech scrambler......Page 324
6.4 The sampling theorem, and multiplying by sampling and filtering......Page 337
6.5 More neat tricks with Fourier transforms and filters......Page 340
6.6 Single-sided transforms, the analytic signal, and single-sideband radio......Page 344
Euler: The Man and the Mathematical Physicist......Page 359
Notes......Page 382
Acknowledgments......Page 410
E......Page 412
K......Page 413
R......Page 414
Z......Page 415