Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills (Princeton Science Library, 52) by Paul J. Nahin (PDF)

Description

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. In some ways a sequel to Nahin’s An Imaginary Tale, this book examines the many applications of complex numbers alongside intriguing stories from the history of mathematics. Dr. Euler’s Fabulous Formula is accessible to any reader familiar with calculus and differential equations, and promises to inspire mathematicians for years to come.

User’s Reviews

Editorial Reviews: Review “Professional, Scholarly Cover/Jacket Award, New York Book Show””Nahin includes gems from all over mathematics, ranging from engineering applications to beautiful pure-mathematical identities. . . . It would be good to have more books like this.”—Timothy Gowers, Nature”Nahin’s tale of the formula e[pi] i+1=0, which links five of the most important numbers in mathematics, is remarkable. With a plethora of historical and anecdotal material and a knack for linking events and facts, he gives the reader a strong sense of what drove mathematicians like Euler.”—Matthew Killeya, , New Scientist”It is very difficult to sum up the greatness of Euler. . . . This excellent book goes a long way to explaining the kind of mathematician he really was.”—Steve Humble, Mathematics Today”What a treasure of a book this is! This is the fourth enthusiastic, informative, and delightful book Paul Nahin has written about the beauties of various areas of mathematics. . . . This book is a marvelous tribute to Euler’s genius and those who built upon it and would make a great present for students of mathematics, physics, and engineering and their professors.”—Henry Ricardo, , MAA Reviews”The heart and soul of the book are the final three chapters on Fourier series, Fourier integrals, and related engineering. One can recommend them to all applied math students for their historical development and sensible content.”—Robert E. O’Malley, Jr., SIAM Review”This is a book for mathematicians who enjoy historically motivated mathematical explanations on a high mathematical level.”—Eberhard Knobloch, Mathematical Reviews”It is a ‘popular’ book, written for a general reader with some mathematical background equivalent to a first-year undergraduate course in the UK.”—Robin Wilson, London Mathematical Society Newsletter Review “If you ever wondered about the beauties and powers of mathematics, this book is a treasure trove. Paul Nahin uses Euler’s formula as the magic key to unlock a wealth of surprising consequences, ranging from number theory to electronics, presented clearly, carefully, and with verve.”―Peter Pesic, St. John’s College”The range and variety of topics covered here is impressive. I found many little gems that I have never seen before in books of this type. Moreover, the writing is lively and enthusiastic and the book is highly readable.”―Des Higham, University of Strathclyde, Glasgow From the Back Cover “If you ever wondered about the beauties and powers of mathematics, this book is a treasure trove. Paul Nahin uses Euler’s formula as the magic key to unlock a wealth of surprising consequences, ranging from number theory to electronics, presented clearly, carefully, and with verve.”–Peter Pesic, St. John’s College”The range and variety of topics covered here is impressive. I found many little gems that I have never seen before in books of this type. Moreover, the writing is lively and enthusiastic and the book is highly readable.”–Des Higham, University of Strathclyde, Glasgow About the Author Paul J. Nahin is the author of many bestselling popular math books, including Mrs. Perkins’s Electric Quilt, In Praise of Simple Physics, and An Imaginary Tale (all Princeton). He is professor emeritus of electrical engineering at the University of New Hampshire. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The book is in good order. I look forward to reading it.

⭐I came to this book because I enjoyed The Story of the Square Root of Minus One, another book by Paul Nahin. This book is of a very different nature: unlike that other book, this one is light on concepts and heavy on calculations.I enjoyed it quite a bit, however, hence the 4 stars, because I like complicated-looking integrals, but let me be frank I could not help thinking throughout: what’s the point? (and do I deserve to be treated to so many typos?)What’s the point? It shows many uses of Euler’s formula, but without explaining why we should care. A couple of chapters are devoted to Fourier series and transforms: again, what’s the point? Towards the end, Nahin writes something to the effect that he has “avoided giving physical interpretations to the mathematical calculations” and that’s precisely the problem: until the end of the book where there are clear references to things like electricity and other waves, we are never told (or reminded) why these clever manipulations are important.It was shocking not to see any reference to the Riemann hypothesis and zeta function, which are perhaps the most beautiful example of the use of Euler’s formula.To Nahin’s credit, he goes through the calculations step by step, so that if you do care (for some reason) then you can follow pretty much the whole thing without breaking a sweat (Nahin did the hard work). But I will confess that I did skip a few pages here and there: my eyes and brains got tired and the nagging thought came, well, what’s the point?Thoroughly recommended, however.

⭐Well this time I don’t agree with reviewers above in the sense that if we liked An Imaginary Tale, then this book would like us too.Certainly I enjoyed a great deal An Imaginary Tale, but I hoped I would find much more in Dr Euler’s Formula, as I was really very impressed the first time I met -in my second year of electrical engineering- the most beautiful equation in mathematics, as professor Nahin has pointed out, but I really was very dissapointed, that in this new book I did not find anything about the fact that Dr. Euler’s Fabulous Formula is most remarkable because even with differentiation and integration the mathematical operations that represent change, Euler’s Identity remains with the same form, except for being affected by the square root of minus one, i.e., by a process of rotation.It is this remarkable property the one that permits”to reduce steady-state sinusoidal problems to forms which are identical to those for resistive networks.”and that made that Charles Steinmetz was called”the wizard who generated electricity from the square root of minus one”when the great historical struggle between AC and DC current was solved by that famous paper of Steinmetz.Yes, it was this remarkable property that made me think that Dr. Euler’s Formula could cure not only many mathematical ills, but physical ones such as those of deducing both the pendulum formula and the Complex Schrodinger’s wave equation, based in a complex metrics in which Euler’s identity plays the fundamental role, an exercise that I did many years ago and put somewhere at LANL.Of course, I highly recommend this book by professor Nahin, as you will find in it a real complement to Fourier series and Integrals and to the study of Dirac’s impulse function in chapters 4 and 5 and an important application to electronics in chapter 6.

⭐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.

⭐I purchased this book because I love Nahin’s clear and logical writing style and how technical material is presented. The book is a continuation of an earlier book (An Imaginary Tale) and like all Nahin books, the topics are extremely well thought out and superbly organized. Additionally, I have always enjoyed reading about the life and mathematical works of Euler who hold in the highest regard.

⭐It is difficult to rate this book because while in many respects fascinating, it requires the reader to possess a very advanced understanding of mathematics to follow the algebra presented. In the introduction the author suggests many readers with moderate undergraduate mathematics skills will have the necessary background to cope. I think this is seriously misleading and possibly an attempt not to discourage readers with less advanced maths from purchasing. That said, the book is actually very good, albeit extremely hard work. Do not expect a quick read. In some sections if you want to fully understand the arguments do not expect to complete more than a couple of pages in an evening.P.S. One further issue. On my early version Kindle many very detailed formulae did not scale, so many mathematical symbols were actually incredibly tiny. Tedious to need a strong magnifying glass to scrutinize much of the mathematics.

⭐I have not read all of this yet, only a part of it. That it is by an engineer would not appeal to the snobbish, the disciples of GH Hardy, for example (perhaps). It is so clearly brilliant! Formulas and proofs are what mathematics is about. They seem within my grasp wherever I open the book. I know that time spent here will be well spent.How interesting that Euler could recite the whole of the Aeneid. So could Prof AJ Aitken of Edinburgh, my first teacher there. Now I see why he bothered. I do not much care for it myself. And why Prof John Conway of Princeton could recite pie to 500 decimals (and more!) like Aitken. It is all homage to Euler (well, mostly).I have found the book very clear and it is full of wonders and very accessible. I am greatly indebted to Paul Nahin. He has written something very important. He is an enthusiast and a scholar who can explain anything clearly. He is, for example, in a different league altogether from someone like Prof Stewart of Warwick. Imagine if I had read this before going up? It is miles better than Hardy’s book. My best students would have been devouring it before they went up had it been available then.This is a very well published book by Princeton with a beautiful cover.

⭐Excellent book as I have come to expect from this writer.

⭐Arrived on time and exactly as described. Not read it yet but it looks exactly what I wanted.

⭐Im Vorwort zitiert Paul. J. Nahin einen Artikel des Boston Globe „When Did Math Become Sexy“, in dem der ‘Einzug’ echter Mathematik etwa in solche Theater Stücke wie ‘Copenhagen’ oder ‘Proof” und Filme, etwa ‘A Beautiful Mind’, beleuchtet wird; die Eulersche Formel dürfte eine ebensolche öffentlich Aufmerksamkeit wert sein.Mit „Dr, Eulers Fabulous Formula“ setzt der Autor seine Geschichte der komplexen Zahlen – „An Imaginary Tale“ (1998) – fort; zwar ist auch dieser zweite Teil nicht als Lehrbuch gedacht, setzt aber beim Leser einige einfache mathematischen Vorkenntnis (etwa Differential- und Integralrechnung und Lineare Algebra) voraus. Es enthält das fortgeschrittenere Material, das der Autor aus dem ersten Buch aussparen musste, um dessen Umfang nicht zu sprengen.In diesem Band werden in interessanter Art und Weise, neben den Grundlagen, Anwendungen komplexer Zahlen in der Zahlentheorie und bei der Beschreibung von ‘Vector Walks’, dem Beweis der Irrationalität von pi^2; ferner für Fouier Reihen und Integrale, und deren Anwendung in der Elektronik, betrachtet. Das Buch schließt mit einer kurzen Darstellung des Leben und Werks von Leonhard Euler, dem großen Schweizer Mathematiker, der ein Meisters des unbekümmerten Umgangs der Analysis des ‘Unendlichen’ war; die nach ihm benannte Formel, die diesem Buch den Titel gab, ist dabei nur ein Beispiel seines geschickten Jonglieren mit unendlichen Reihen.Die für sich genommenen schon höchst faszinierenden und oft trickreichen mathematischen Miniaturen illustrieren das zentrale Thema des Buches: Mathematische Schönheit. Eulers Formel e^ipi + 1 = 0 ist dafür ein Paradebeispiel, sie setzt die beiden transzendenten Konstanten pi und e, die aus zwei sehr verschieden mathematischen Gebieten stammen, über die imaginäre Einheit i miteinander in Beziehung. Es ist nicht selten, dass solche unerwarteten Berührungen Ausgangspunkt neuer Erkenntnisse oder sogar Anlass zum Entstehen neuer mathematischer Theorien sind. Schönheit liegt dabei natürlich im Auge des Betrachters, schöne mathematische Beziehungen haben aber – wenn man David Wells folgt – eigne Gemeinsamkeiten: sie sind einfach, kurz, wichtig und überraschend; und insofern ist die Eulersche Formel so eine Art Goldener Standard.Fazit: Paul. Nahins Buch ist ein beeindruckende Kombination von interessanten Anwendungen komplexer Zahlen in einer Vielzahl von konkreten Beispielen, mit historischen Bezügen und Hintergründen, etwas, das übliche Einführungs- Lehrbücher in der Regel ausklammern,

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