Paul J. Nahin - Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills

Dr. Eulers Fabulous Formula

In an entry made in one of his teenage notebooks in April 1933, just before his fifteenth birthday, the future physics Nobel Prize winner Richard Feynman (19181988) took notice of a major theme of this book. Notice the power series expansions for the exponential, sine, and cosine functions given just below the most remarkable formula in math. The next line is the start of the standard derivation of Eulers formula (also known as Eulers identity) eiu = cos(u) + i sin(u), of which the remarkable formula is the special case of u = . (Feynmans source, The Science History of the Universe, was a ten-volume reference set first published in 1909.) Although remembered today as a physicist, Feynman was also a talented mathematician, who wrote in his The Character of Physical Law (1965), To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty of nature.... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. Feynman would surely have agreed with one of the early working titles to this book: Complex Numbers Are Real! (Photograph courtesy of the Archives, California Institute of Technology)

Paul J. Nahin

With a new preface by the author

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

Copyright 2006 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock,
Oxfordshire OX20 1TW
press.princeton.edu
All Rights Reserved

Eighth printing, and first paperback printing,
with a new preface by the author, 2011
Paperback ISBN: 978-0-691-15037-6

The Library of Congress has cataloged the cloth edition of this book as follows

Nahin, Paul J.

Dr. Eulers fabulous formula : cures many mathematical ills / Paul J. Nahin.

p. cm.

Includes bibliographical references and index.
ISBN-13: 978-0-691-11822-2 (cl : acid-free paper)
ISBN-10: 0-691-11822-1 (cl : acid-free paper)

1. Numbers, Complex. 2. Eulers numbers. 3. MathematicsHistory. I. Title.
QA255.N339 2006
512.788dc22 2005056550

British Library Cataloging-in-Publication Data is available

This book has been composed in New Baskerville
Printed on acid-free paper.
Printed in the United States of America

9 10 8

For Patricia Ann,

who (like Eulers formula) is both complex and beautiful

May the God who watches over the right use of

mathematical symbols, in manuscript, print,

and on the blackboard, forgive me [my sins].

Hermann Weyl, professor of mathematics from 1933
to 1952 at the Institute for Advance Study, in his book
The Classical Groups, Princeton 1946, p. 289

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

concept of mathematical beauty

equations, identities, and theorems

mathematical ugliness

beauty redux

1.2 The Cayley-Hamilton and
De Moivre theorems

(named after Fourier but Euler was there firstbut he was, alas,
partially WRONG!)

(what happens as the period of a periodic function becomes infinite, and
other neat stuff)

(technological applications of complex numbers that Euler, who was a
practical fellow himself, would have loved)

Finally I meet my imaginary part!

Nobel Prize-winning (1989) experimental physicist Wolfgang Paul upon meeting the Nobel Prize-winning (1945) theoretical physicist Wolfgang Pauli (this joke makes sense once you remember the structure of a complex number and that

1 = ei,

Proves that Euler was a sly guy.

But (2)

Was totally new

And raised respect for him sky-high.

William C. Waterhouse, professor of mathematics at Penn State University

Dr. Euler is the sequel to my earlier book An Imaginary Tale: The Story of (Princeton University Press, 1998, 2007, 2010). Both books were written because I have, since high school, been completely and utterly fascinated by the mystery of and by the beautiful calculations that flow, seemingly without end, from complex numbers and functions of complex variables. I once came across a poem, titled The Happy Land, that nicely and exactly catches that fascination. Here it is, and if it resonates with you, too, then you are just the sort of reader for whom I wrote both of my books.

The Happy Land

I hear them speak of a Happy Land,

The chosen researchers a joyful band,

Professor, ah where is that radiant shore,

Shall I attain it and weep no more?