Dr. Euler's Fabulous Formula

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.

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'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.

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.