Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Princeton Science Library)

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'm a fan of Prof. Nahin's writing. I think he's one of the best writers on "pop mathematics" for those who actually know a little math. Unlike many pop-sci and pop-math writers pitching to a more general audience who, when a difficult subject comes along, either punt, write airy nothings, or say something WRONG, Nahin explains well and accurately. Enlightening proofs and insights abound. This book has a "big tent" subject: the ubiquity of Euler's formula. And hence, the book is a bit of a "grab bag" with a very loose unity. You could almost read any chapter without reference to the others. But, as usual, it's well-written, full of historical facts and side-stories, and witty. As with most books coming out of modern publishing houses, it's not well proof-read. But there are no glaring inaccuracies. I'd recommend it to anyone with some basic calculus under his belt. It will be particularly enlightening and entertaining to scientists and engineers who have seen most of the math in here, but perhaps never quite as presented here. Even if you have a strong background in Fourier analysis, you'll still see something in a new light between these covers.

I bought this book when my daughter was home from college, she is an electrical engineering major and minoring in physics. At the time she was starting to learn Fourier, and she paged through this book and it took me 8 months to get it back. This provided an excellent study resource in Fourier series.My point is that this book is not really a companion to his other book, which is more of a historical text which discusses the math, but is really a popular math book.This book is a study of simply the math, and it is instructional on how euler is applied from the standpoint of e^piI. So it ends up being an excellent resource on Fourier, which encompasses most of the text. It is useful to study away from the typical textbooks, which are simply using it as a solution, and not as a goal of instruction.This is not light reading, so it is for math types, and it the author as always writes in a clear style.

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.

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.

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^i\pi + 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,

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.

同じ著者によるAn Imaginary Tale The Story of √-１のいわば続編ともいうべきもの。オイラーの公式の証明は出てこない。証明されたものとしてこれを使う数学となっている（物理学者ファインマンが１５歳のころの手書きの数学ノートの写真があり、そこにサイン・コサインの無限級数展開による「証明」が書かれているのみ）。この本をひとことで紹介するとすれば、「オイラーの公式を使って初等数学（大学１・２年次の教養課程数学）では出てこない、複雑ではあるが答えがきれいな積分などの定理を、基本的には初等数学の知識を使って導く楽しさを提供する本」ということになる。知識の積み上げを前提としながら読まねばならない教科書的な本ではなく、各章ごとに独立している。ある章を最後まで理解しないと次の章がまったく理解できない、という構成にはなっていない。ただ、中心になる数学知識はフーリエ変換であり、これはかなり詳しく解説と「証明」が出てきてこの知識は繰り返し用いられ、その後の各章でこれを使って様々な知識を導く構成となっている。「初等数学の範囲を超える定理」ということは、「初等数学」はこの本を読む知識として必要になる。著者がペーパーバック版に書いた前書きによれば、To read this book you should have a mathematical background equivalent to what a beginning third-year college undergraduate in an engineering or physics program of study would have completed.つまり、理工系大学教養課程修了時点の数学知識が必要ということ。そういう意味ではちょっとハードルが高いかもしれない。加えて、著者は不要な知識とは言っているが、電気（「電子」でない）工学分野に用いられる数学に話が及ぶ。確かに内容は単純で、AMラジオの原理にフーリエ変換と√-1が関係している、その原理的な分野の数学ということなのだが、電気工学分野にアレルギーとは言わないが、あまり興味がない、いわゆる「数学おたく」の者にとっては、数学の難しさにチャレンジする気持がどうしても萎えてしまうため、第６章の最後の約３０ページはかなり難しい（私は途中でギブアップした）。とはいえ、よくできているいい本だと思う。まず、全体が６つの章で構成され、そのおのおのの章が全体としてあるテーマを追いかけた構成となっている。けっしてバラバラの知識のエッセイ的な本ではないのが「数学のしっかりした本」という読みがいを感じる理由の一つ。そして、途中（４章以降）からはフーリエ変換から導く数式や定理を著者が示す通りにたどっていくとき、結果として驚くようなきれいな定理や数式が出てくるのがもう一つの楽しみ。そして、残念ながら私には感じることはできなかったが、もし電気工学の興味がある者が読むときには、数学の新たな「地平」が見えるであろうということは大いに予想される。数学知識の前提条件を満たす者にとっては非常に面白い本といえる。なお、英語は数学を解説する部分については極めてやさしい。最終章のオイラーの伝記部分になると本格的な英語の文章になるので、辞書なしでは読みづらい。

I've started it. Enjoying it. If you don't have university maths, it'll be very difficult for most. I have uni maths from umpteen years ago and am slowly going through the ideas... but I enjoy that kind of thing. If you've struggled with A level maths, it'll be difficult to follow.

Das ist die unglaublichste Formel aller Zeiten, die schönste und unvergesslichste...!!Unbedingt lesen! Toll, profund, amüsant geschrieben von einem wirklichen Kenner!