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Index
Half title
Title
Copyright
Dedication
Contents
Preface to the Paperback Edition
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
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 π2 (“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(π i)
3.5 The last step (at last!)
Chapter 4. Fourier Series (named after Fourier but Euler was there first——but he was, alas, partially WRONG!)
4.1 Functions, vibrating strings, and the wave equation
4.2 Periodic functions and Euler’s sum
4.3 Fourier’s theorem for periodic functions and Parseval’s theorem
4.4 Discontinuous functions, the Gibbs phenomenon, and Henry Wilbraham
4.5 Dirichlet’s evaluation of Gauss’s quadratic sum
4.6 Hurwitz and the isoperimetric inequality
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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