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86 of 89 people found the following review helpful
5.0 out of 5 stars Sequel to An Imaginary Tale
The reviews of An Imaginary Tale capture much of what will be said of Dr. Euler's Fabulous Formula. I happen to like Paul Nahin's books very much ever since reading The Science of Radio, one of my favorite books of all time. If you didn't like Imaginary, you won't like Dr. Euler's . If you like the earlier book, this one is a must.

Chapter One starts with an...
Published on April 26, 2006 by T. J. Shortridge

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9 of 11 people found the following review helpful
3.0 out of 5 stars Mixed Treat
Euler's Fabulous Formula, by Paul J. Nahin (2006, with new preface in 2011).

The formula is e(exponent ipi) + 1 = 0. Who but Euler would have found the knot that joins e, pi, and i so concisely?

Nahin has written a marvelous book. The target readership has a modicum of university math plus a burning desire to understand how mathematicians sail their...
Published on January 5, 2012 by Anyday


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86 of 89 people found the following review helpful
5.0 out of 5 stars Sequel to An Imaginary Tale, April 26, 2006
The reviews of An Imaginary Tale capture much of what will be said of Dr. Euler's Fabulous Formula. I happen to like Paul Nahin's books very much ever since reading The Science of Radio, one of my favorite books of all time. If you didn't like Imaginary, you won't like Dr. Euler's . If you like the earlier book, this one is a must.

Chapter One starts with an introduction to complex numbers. This would make nice supplemental material for an introduction to complex numbers. The chapter is not the standard treatment. It gives a very clear introduction to Gauss' proof of the construction of the regular heptadecagon . The chapter goes on to factoring complex numbers in the context of Fermat's last theorem, with a very clear discussion of Lame's proof for n=7 . Earlier in the chapter Nahin uses the Cayley-Hamilton theorem to get De Moivre's theorem in matrix form without any mention of physical rotations.

Fourier series and integrals comprise most of the book which ends with applications to single side band radio. This last topic is a nice inclusion for folks like me who liked Nahin's early book The Science of Radio. There is a story about G.H. Hardy and Arthur Schuster, that I had never seen elsewhere.

I would recommend this book to anyone who likes undergraduate calculus and has some exposure to linear algebra, maybe a second or third year undergraduate. The material is idiosyncratic enough to be entertaining for anyone who has had courses in complex analysis and number theory. It is a good introduction and supplemental reading for such courses, but not as a primary text.
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67 of 71 people found the following review helpful
5.0 out of 5 stars Errata please, February 13, 2007
Like all of Paul Nahin's books, I really like this one.

However, as with so many books an Errata would help. Mathematical and mathematical finance books are getting so expensive, that unless authors or publishers have a URL for Errata, readers esp. of mathematical books will wait for [sometimes years] for a second corrected edition of books.

I could be wrong about these but it seems these are typos:

p. 30 lines 5 & 6 curly bracket should only be around the 2 * cos(x/2) term

p. 121 second equation should be t=(v+u)/(2*c)

p. 121 '* (1/(2*c)' missing at end of the line

p. 123 line 17, first word should be 'bother' not 'other'

p. 127 line 3 and 4, it seems that the 'icnPI/l' [not the ones in the cos() or sin() terms] term after the 'B' and before the '2*cos' respectively, should not be there. Or am I missing something ?

p. 128 4th line from bottom should be 1753 not 1733

p. 143 2nd line before last equation should be '... (x- i * y)...'

p. 144 equation under 'In summary, then...' cases are reversed

p. 216 seems 1/(2*PI) is missing from right side of first equation, i.e. from "...G(u)G(omega-u)...du"
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42 of 43 people found the following review helpful
5.0 out of 5 stars Another fabulous book from Paul Nahin, August 29, 2006
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Here is a book that is a delight to read. It is well-written and the text flows marvelously between each page and around the many formulas that are so carefully presented and worked out. I rate this book as 5-stars for presenting ever more mathematics relating to complex numbers in a clear and detailed manner.

The book is, as the author notes, a continuation of his book, An Imaginary Tale, where Nahin discusses the square root of -1. (If you haven't read that book, read it first because many of the footnotes refer to it.) In this book, we see more of complex numbers and, in particular, we see many applications of Euler's Identity that "e^{i theta} = cos(theta)+ i sin(theta)." This simple looking indentity is rich in applications and explorations. Nahin takes you on a journey to these topics and does so in an easy to follow way.

There are interesting stories as you go such as the one where we find the Gibbs did not, contrary to almost all textbooks, discover what is call Gibbs Phenomena. There are other stories and anecdotes but I'll let you enjoy them on your own.

That said, I must also say that the book assumes you have a good understanding of complex numbers and are comfortable manipulating them. A solid undergraduate understanding is all that's needed and if you have done graduate work, all the better. If you're considering the book at all, and have the math background, read it.

If you don't know anything about complex numbers, well, this book may not be as good as it could be for you.
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11 of 11 people found the following review helpful
5.0 out of 5 stars Excellent expository book, March 24, 2007
By 
Aristarchus (San Diego, CA United States) - See all my reviews
Paul Nahin's book, "Dr. Euler's Fabulous Formula," is an excellent expository treatment of Euler's formula (you say, "which one?") e^i*theta = cos(theta) + i*sin(theta) and its profound, and far-reaching, ramifications. Dr. Nahin also gives an extensive informal discussion of Fourier series, Fourier transforms, the Dirac Delta Function, and what electrical engineers would call "signals and systems theory." Some mathematical purists may criticize the lack of pure rigor. However, this book is an "expository" book, not a rigorous "textbook." Ideally, I recommend that you read Dr. Nahin's book in conjunction with your standard college textbook. That way, you will get the best of both worlds. Your textbook will give you the disciplined rigor. Dr. Nahin's book will give you the "Aha... insight!" I read Dr. Nahin's book before taking a graduate level course in electrical engineering (EE) Signals and Systems. I breezed through the EE course with perfect scores on my exams, and I give a lot of credit to Dr. Nahin. When you study mathematics, you really need BOTH disciplined mathematical rigor AND intuitive insight and understanding. Beware, however, that this book has LOTS of mathematics in it. The book is loaded with serious mathematics. Don't read this book if you want something for the intelligent layperson. Read this book if you love mathematics, if you are an engineering or mathematics student, or if you like industrial-strength mathematics. Paul Nahin may single-handedly save Americans from mathematical illiteracy. He does something that the mathematical community does not do well... "market and sell" mathematics.
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9 of 9 people found the following review helpful
4.0 out of 5 stars excellent for fourrier series and fourrier transform exposition, March 29, 2007
By 
Arzi (L'Isle d'Abeau, France) - See all my reviews
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A very readable book. Many concepts developed around Euler's magic formula are clearly explained. Including a lucid exposition on the calculus of the sum of classical series such as the value of zeta function for several positive integer values of its argument. Paul Nahin excels in describing the origin and the development of fourrier series and fourrier integrals from Bernoulli to Fourrier and more. Anyone interested in this field will find something interesting in this book to learn. The reason I didn't rank it five stars is that I found explanations often too lengthy while the addition of a chapter on distribution theory could fill the gaps in mathematical rigor and make the transition from fourrier series to fourrier integrals more logical. I should add that the lack of rigor in transition from fourrier series to fourrier integrals, as described by P. Nahin, is inherent to the more fundamental problem of transition from discrete to continuous. Indeed, in mathematics, this is a very slippery terrain. In functional analysis, mathematicians go round this problem by introducing distribution theory. P. Nahin mentions only the name of distribution theory without any decription. I think a chapter on this theory would make the book a must have.
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7 of 7 people found the following review helpful
5.0 out of 5 stars Good clear explanation of Fourier series, April 10, 2007
By 
CR (Virginia) - See all my reviews
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Dr Eulers fabulous formula fits a niche between books for non mathematicians (too simple) and books only understood by mathematicians. It provides the best explanation of Fourier series and integrals that I have read. Its explanation of imaginary numbers is excellent, but not as good as Feynman in his lectures on physics. I reccomend it for those who want to understand how Fourier series work.
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9 of 11 people found the following review helpful
3.0 out of 5 stars Mixed Treat, January 5, 2012
Euler's Fabulous Formula, by Paul J. Nahin (2006, with new preface in 2011).

The formula is e(exponent ipi) + 1 = 0. Who but Euler would have found the knot that joins e, pi, and i so concisely?

Nahin has written a marvelous book. The target readership has a modicum of university math plus a burning desire to understand how mathematicians sail their vessel. It is as though we are invited back stage by the Wizard of Oz to see his manipulations. Throughout the book, Nahin's enthusiasm draws us forward, and his clarity of expression permits us to see how creative talent plus hard work have answered problems that appeared intractable.

The core of this volume is a glimpse into the uses of Euler's famous equation. But another vista is into Nahin himself. The chapters state an objective, but Nahin allows his pleasures to guide the train of exposition. As a result, we aren't following merely lines of logic but also paths of joy and insight. These twin channels, entwined of course, are what make the book a delight.

The upside is clarity. I never had to turn a page to find a diagram. Nor did I scratch my head over the starting point for a series of mathematical derivations. Nahin always starts his mathematical expositions from precisely where they left off; if you aren't an aficionado of math books, you have no idea how rare this is. Moreover, Nahin only manipulates a couple of segments of his equations at a time. His math is easy to follow. The secret, I believe, is that Nahin has an overarching concept of the narrative of his book, including the math ingredients. He doesn't hesitate to include math in the vector of his imagination. As a result, the math never bogs down the reader. To the contrary, the math carries you forward.

The downside is an occasional lapse in theme. Nahin sometimes takes us on sidetrips that are the equivalent of a naturalist leading a group down a narrow pathway in the everglades, because he has heard the call of an interesting bird. The path isn't in the line marked out on the brochure, but if you can steel yourself to enjoy a bit of serendipity, you won't mind the deviation. Towards the end especially, however, Nahim permits himself to wander. The frequent comments about electrical engineering eventually weary the reader: yes, yes, the reader says to himself, get on with the book and stop preening. We come to know, more than we'd like perhaps, the themes of Nahin's other books.

And yet this is one of the best mathematical books I've ever read. There were segments in which I kept repeating to myself the mantra: a man's reach must exceed his grasp. And yet I never entirely lost track of where Nahin was going. There is a wide audience capable of following Nahin in this journey towards knowing how Euler's formula has benefited the world. You don't emerge from the book a baptized mathematician, but you emerge with a great respect for Euler and some concept of how mathematicians sail their vessel.
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7 of 9 people found the following review helpful
5.0 out of 5 stars Great book on Fourier Series and Fourier Integrals, November 9, 2006
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This is a great book covering Fourier Series and Fourier Integrals. The author makes the material very accessible and I'd recommend this book to anyone currently studying or having studied the subject of Fourier Series/Integrals. I found the chapters covering pure math to be pure joy since the author's way of presenting the material is so pedagogical. If you enjoy books on the history of math and are keen on following along intersting derivations on the way to beautiful results, then this book is for you.
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2 of 2 people found the following review helpful
3.0 out of 5 stars Poor display of mathematical formulae, June 18, 2012
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This review is from: Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Kindle Edition)
I think that the Kindle needs its own version of LaTex in order to display the formulas clearly. The book has a lot of mathematical formulas that are displayed as images and not the best quality ones. The book looks a lot better when read in a PC monitor using the web reader or the Kindle App for PC. Otherwise I found the book very interesting the low score is because of the formula display.
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4 of 5 people found the following review helpful
5.0 out of 5 stars An excellence introductory book on advanced mathematics such as Euler's Identity, Irrationalioty, Fourier Series, September 21, 2007
By 
Man Kam Tam (Calexico, CA USA) - See all my reviews
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The primary topic of Nahin's "Dr. Euler's Fabulous Formula" is the complex number or more appropriately the Euler's identity: e power to (it) = cos(t) + isin(t). Nahin called this book the second half of his complex number series. The first book in the series is named "An Imaginary Tale: The Story of square root of minus one." The second book is called "Dr. Euler's Fabulous Formula." The primary topics of the second book are: Fourier series, which is covered on Chapter 4; Fourier Integrals on Chapter 5; the application of complex numbers on electronics Chapter 6.

The book has six chapters, which contains both pure and applied mathematics materials. Other than the three chapters mentioned above, the other three chapters are (i) Complex Numbers, (ii) Vector Trips, and (iii) The Irrationality of pi square. Chapter one is about the assortment of non elementary complex numbers such as applying complex number on obtaining the sum of a real series. Chapter three provides a detail proof of the irrationality of the number pi square using Euler's Identity. On the applied side: Chapter two demonstrates the application of complex number on mathematical modeling. Since Nahin is an eminent electrical engineering professor, his book also provides plenty of material on (a) partial differential equations (PDE) such as wave equation on chapter four, and (b) electrical engineering material such as baseband, carrying frequencies, antennas, radio receivers and speech scrambler on chapter six.

This is an excellence introductory book not only on pure complex numbers usage in mathematics such as summing a series but also on the usage of PDE, Fourier series, and Fourier Integral in physics and engineering.
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