Dr. Euler's Fabulous Formula

Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (New in Paper)

Paul J. Nahin
Copyright Date: May 2011
Pages: 432
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  • Book Info
    Dr. Euler's Fabulous Formula
    Book Description:

    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."

    eISBN: 978-1-4008-3847-9
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-xii)
  3. Preface to the Paperback Edition
    (pp. xiii-xxiv)
    Paul J. Nahin
  4. What This Book Is About, What You Need to Know to Read It, and WHY You Should Read It
    (pp. xxv-xxviii)
  5. Preface “When Did Math Become Sexy?”
    (pp. xxix-xxxiv)
  6. Introduction
    (pp. 1-12)

    The nineteenth-century Harvard mathematician Benjamin Peirce (1809–1880) made a tremendous impression on his students. As one of them wrote many years after Peirce’s death, “The appearance of Professor Benjamin Peirce, whose long gray hair, straggling grizzled beard and unusually bright eyes sparkling under a soft felt hat, as he walked briskly but rather ungracefully across the college yard, fitted very well with the opinion current among us that we were looking upon a real live genius, who had a touch of the prophet in his make-up.”² That same former student went on to recall that during one lecture “he...

  7. Chapter 1 Complex Numbers (an assortment of essays beyond the elementary involving complex numbers)
    (pp. 13-67)

    Many years ago a distinguished mathematician wrote the following words, words that may strike some readers as somewhat surprising:

    I met a man recently who told me that, so far from believing in the square root of minus one, he did not even believe in minus one. This is at any rate a consistent attitude. There are certainly many people who regard $ \sqrt 2 $ as something perfectly obvious, but jib at $ \sqrt { - 1} $ . This is because they think they can visualize the former as something in physical space, but not the latter. Actually $ \sqrt { - 1} $ is a much simpler concept.¹

    I say...

  8. Chapter 2 Vector Trips (some complex plane problems in which direction matters)
    (pp. 68-91)

    As discussed in the opening section of chapter 1, one of the fundamental intellectual breakthroughs in the historical understanding of just what $ i = \sqrt { - 1} $ means, physically, came with the insight that multiplication by a complex number is associated with a rotation in the complex plane. That is, multiplying the vector of a complex number by the complex exponential $ e^{i\theta } $ rotates that vector counterclockwise through angle θ. This is worth some additional explanation, as this elegant property of complex exponentials often pays big dividends by giving us a way to formulate, in an elementary way, seemingly very difficult problems. In An Imaginary...

  9. Chapter 3 The Irrationality of π² (“higher” math at the sophomore level)
    (pp. 92-113)

    The search for ever more digits of π is many centuries old, but the question of its irrationality seems to date only from the time of Euler. It wasn’t until the 1761 proof by the Swiss mathematician Johann Lambert (1728–1777) that π was finally shown to be, in fact, irrational. Lambert’s proof is based on the fact that tan(x) is irrational if x = 0 is rational. Since tan(π/4) = 1 is not irrational, then π/4 cannot be rational, i.e., π/4 is irrational, and so then π too must be irrational. Lambert, who for a while was a colleague...

  10. Chapter 4 Fourier Series (named after Fourier but Euler was there first—but he was, alas, partially WRONG!)
    (pp. 114-187)

    This entire chapter is devoted to those trigonometric series satisfying certain conditions that go under the general title of Fourier series—named after the French mathematician Joseph Fourier (1768–1830)—but the story of these series begins well before Fourier’s birth.¹ And, as you must suspect by now, Euler’s formula plays a prominent role in that story. The prelude to the tale begins with a fundamental question: what is a function?

    Modern analysts answer that question by saying that a function f(t) is simply a rule that assigns a value to f that is determined by the value of t, that...

  11. Chapter 5 Fourier Integrals (what happens as the period of a periodic function becomes infinite, and other neat stuff)
    (pp. 188-274)

    In this fairly short introductory section I want to take a break from Fourier and jump ahead a century to Paul Dirac, the English mathematical physicist I mentioned way back in the Preface. His name today is synonymous with the concept of the impulse function (often called the Dirac delta function), which will be of great use to us—as much as will Euler’s formula—in the next section on the Fourier transform. The impulse (I’ll define it soon) is one of the most important technical tools a physicist or engineer has; Dirac himself was originally trained as an electrical...

  12. Chapter 6 Electronics and $ \sqrt { - 1} $ (technological applications of complex numbers that Euler, who was a practical fellow himself, would have loved)
    (pp. 275-323)

    Up to now this book has treated mathematical topics for the sake of the mathematics, alone. However, since my writing was strongly motivated by an admiration for Euler the engineering physicist nearly as deep as my admiration for him as a mathematician, it seems appropriate to end with a chapter on some technological uses of the complex number mathematics he helped develop. Euler’s large body of applied work shows that he took to heart, very seriously, the motto of the Berlin Academy of Sciences (of which he was a member from 1741 to 1766): “theoria cum praxi,” which my tiny...

  13. Euler: The Man and the Mathematical Physicist
    (pp. 324-346)

    While there is a steady stream of biographies treating famous (or, even better from an entertainment point of view, infamous) persons in popular culture, there is still not even one book-length biography, in English, of Euler. There are a German-language biography (1929) and a French-language one (1927), as well as two more recent (1948, 1982) non-English works, but all are obsolete by virtue of the vast Eulerian scholarship that has occurred since they were written. Euler himself wrote so prodigiously that it would be a huge undertaking for a biographer to write, with true understanding, of what he actually did....

  14. Notes
    (pp. 347-374)
  15. Acknowledgments
    (pp. 375-376)
  16. Index
    (pp. 377-380)
  17. Back Matter
    (pp. 381-381)