Book Description
I used to think math was no fun
'Cause I couldn't see how it was done
Now Euler's my hero
For I now see why zero
Equals e
[pi] i+1
--Paul Nahin, electrical engineer
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.
This book is the sequel to Paul Nahin's An Imaginary Tale: The Story of I [the square root of -1], which chronicled the events leading up to the discovery of one of mathematics' most elusive numbers, the square root of minus one. Unlike the earlier book, which devoted a significant amount of space to the historical development of complex numbers, Dr. Euler begins with discussions of many sophisticated applications of complex numbers in pure and applied mathematics, and to electronic technology. The topics covered span a huge range, from a never-before-told tale of an encounter between the famous mathematician G. H. Hardy and the physicist Arthur Schuster, to a discussion of the theoretical basis for single-sideband AM radio, to the design of chase-and-escape problems.
The book is accessible to any reader with the equivalent of the first two years of college mathematics (calculus and differential equations), and it promises to inspire new applications for years to come. Or as Nahin writes in the book's preface: To mathematicians ten thousand years hence, "Euler's formula will still be beautiful and stunning and untarnished by time."
Customer Reviews:
An excellence introductory book on advanced mathematics such as Euler's Identity, Irrationalioty, Fourier Series.......2007-09-22
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.
Good clear explanation of Fourier series.......2007-04-11
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.
excellent for fourrier series and fourrier transform exposition.......2007-03-29
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.
Excellent expository book.......2007-03-25
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.
Errata please.......2007-02-14
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"
![An Imaginary Tale: The Story of i [the square root of minus one]](http://ec1.images-amazon.com/images/I/210UP%2BQ84GL.jpg)
Average customer rating:
- imaginary reality
- packed with math and worth the effort
- the complex clarified
- Good Work!
- A Little Tough
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An Imaginary Tale: The Story of i [the square root of minus one]
Paul J. Nahin
Manufacturer: Princeton University Press
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ASIN: 0691027951 |
Amazon.com
At the very beginning of his book on i, the square root of minus one, Paul Nahin warns his readers: "An Imaginary Tale has a very strong historical component to it, but that does not mean it is a mathematical lightweight. But don't read too much into that either. It is *not* a scholarly tome meant to be read only by some mythical, elite group.... Large chunks of this book can, in fact, be read and understood by a high school senior who has paid attention to his or her teachers in the standard fare of pre-college courses. Still, it will be most accessible to the million or so who each year complete a college course in freshman calculus.... But when I need to do an integral, let me assure you I have not fallen to my knees in dumbstruck horror. And neither should you."
Nahin is a professor of electrical engineering at the University of New Hampshire; he has also written a number of science fiction short stories. His style is far more lively and humane than a mathematics textbook while covering much of the same ground. Readers will end up with a good sense for the mathematics of i and for its applications in physics and engineering. --Mary Ellen Curtin
Book Description
Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them.
In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called "imaginary numbers"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times.
Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics.
Customer Reviews:
imaginary reality.......2007-04-04
The book fascinated me. It brought to life what I'd anticipated as dry, boring dates and theorems. I don't follow all the calculations in the book, yet. But it has stretched my brain with the reading.
packed with math and worth the effort.......2006-05-01
There is a lot to be learned from An Imaginary Tale. However, it will take some effort on part of the reader. Overall, I think it is a worthwhile read, but I do have the following criticisms.
1. In my opinion, the book is a bit deceptive about the level of mathematics required by the reader. It states that a freshman calculus student could follow the math, but even those students will probably get stuck on the problems the author presents that uses differential equations and multi-variable calculus. One section tackling a topic from electrical engineering was far too technical. Either you are already familiar with this material or you would do best to just skip it as I did.
2. I caught a few typos in the math equations that could confuse less astute readers and some (but not most) of the diagrams are crude, like they were drawn hastily. I found that a little insulting, considering the mathematical sophistication expected of the reader.
3. In the section entitled Wizard Mathematics, I became a little tired of seeing different ways to produce expressions with pi using complex numbers. The author gave at least one too many of these, in my opinion.
4. There are ALOT of equations and it would have been easier to follow everything if they had been numbered. In one derivation, he plugs an expression from about three pages back into an equation presently used, with no reference or warning, leaving the reader to wonder for a minute, where did that come from?
I doubt most people could passively read this book and fully appreciate it. You should be prepared to scribble calculations at some parts. I would not have appreciated this book nearly as much if I didn't have pencil and paper ready to make sure I understood the intermediate steps he left out in his various derivations. That's how I caught some typos. Fortunately, I was able fill in most of the blanks so the material was meaningful. For the few places where I couldn't figure out how he got from step A to step B, I was able to keep going without loss of continuity. Reading this book definitely sharpened my math skills and I learned some genuinely interesting uses for complex numbers so it was worth the time and effort.
the complex clarified.......2006-03-29
I really enjoyed this book even though it was quite a bit of work for me. I found for a book of this scope, this one takes three to four times longer to read. This is no fault of the author. The text is clear, interesting and very informative. Equations are typeset in a format suited to algebraic equations in contrast to some similar books where equations are embedded in sentences. The reason for the long read time is the amount of material presented in a condensed format. Literature teachers would appreciate the economy of it all. The intermediate steps left out of some proofs are to be either trusted or calculated by the reader. To truly experience this history and gain an appreciation of the math skills, one should work through these steps.
The author gives the reader an appreciation of many key mathematicians. Complex problems are solved. One of my favorite solutions is Gamow's problem of finding the treasure without the gallows for reference. I found the problems on spacetime physics, hyperspace, and Kepler's laws especially keen. But, the total scope of material is diverse. The author covers the zeta function, the gamma function, and the relationships between pi and i. There is so much more. The book feels deceptively light in your hands, it's content dense.
The last chapter is a real reward; and I really appreciate the author's approach on complex function theory which I would have had no hope of understanding on my own. The reader is guided in integration through the complex plane with all the required steps shown for some elementary functions. I have never read a better introduction to these fundamentals. Obviously, Nahin's goal is to educate.
This is the first book by Nahin that I have read. I expect the next one to be challenging and rewarding as well.
Good Work!.......2006-01-20
I just finished reading this book. The writer has used a very gripping approach to explain the evolution of the idea of complex numbers. However, at some points, the book does get involved too much into mathematicas and loses its historical touch. I would say it is 70-% history and 30% Maths.
Great work!
A Little Tough.......2005-12-24
The author does a great job explaining what the square root of -1 is used and how it is used. I was looking for a little more casual book. This book does get into some relativity complex math. Be prepared to work if you want to read it.
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Moduli Spaces of Curves, Mapping Class Groups and Field Theory
Manufacturer: American Mathematical Society
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ASIN: 0821831674 |
Book Description
This is a collection of articles that grew out of a workshop organized to discuss deep links among various topics that were previously considered unrelated. Rather than a typical workshop, this gathering was unique as it was structured more like a course for advanced graduate students and research mathematicians.
In the book, the authors present applications of moduli spaces of Riemann surfaces in theoretical physics and number theory and on Grothendieck's dessins d'enfants and their generalizations. Chapter 1 gives an introduction to Teichmüller space that is more concise than the popular textbooks, yet contains full proofs of many useful results which are often difficult to find in the literature. This chapter also contains an introduction to moduli spaces of curves, with a detailed description of the genus zero case, and in particular of the part at infinity. Chapter 2 takes up the subject of the genus zero moduli spaces and gives a complete description of their fundamental groupoids, based at tangential base points neighboring the part at infinity; the description relies on an identification of the structure of these groupoids with that of certain canonical subgroupoids of a free braided tensor category. It concludes with a study of the canonical Galois action on the fundamental groupoids, computed using Grothendieck-Teichmüller theory. Finally, Chapter 3 studies strict ribbon categories, which are closely related to braided tensor categories: Here they are used to construct invariants of 3-manifolds which in turn give rise to quantum field theories. The material is suitable for advanced graduate students and researchers interested in algebra, algebraic geometry, number theory, and geometry and topology.
Book Description
This is a modern introduction to the analytic techniques used in the investigation of zeta-function. Riemann introduced this function in connection with his study of prime numbers, and from this has developed the subject of analytic number theory. Since then, many other classes of "zeta-function" have been introduced and they are now some of the most intensively studied objects in number theory. Professor Patterson has emphasized central ideas of broad application, avoiding technical results and the customary function-theoretic approach.
Customer Reviews:
Analytic Number Theory through Zeta-Function.......2007-01-29
This introductory textbook gives a very good insight of analytic number theory through the special topic of Riemann Zeta-Function. It also includes of course an excellent exposition of its relationship with the Prime Number Theorem and with Riemann and Lindelöf Hypotheses. Moreover, seven appendices provide analytic technical complements needed in the core of the text. Many well-chosen exercises and problems accompany each chapter. I use this textbook in my course on Zeta-Function with great success.
An excellent resource for those interested in Riemann's.......2003-04-12
Zeta function. This book contains a lot of application, theory, and, to my surprise, several practice problems at the end of each section to maximize the learning experience. The chapters are concise and the mathematics is relatively easy follow for those with some experience in special functions.
That, however, is the major thing to note: one needs some experience with special functions in order to find this material accessible. (Obviously, right? Otherwise one wouldn't be buying this book! However, much of this material is beyond the grasp of the average mathematics student that stopped at a bachelor's degree.)
Although this book is called an introduction, I don't think that view is entirely appropriate. The material is quite extensive, and the historocity of the zeta function and its development were kept to a minimum. The precursors to the zeta function and its development by Euler and Riemann (especially Euler's original proof to the Basel Problem) are fantastic. If you're interested in Euler's role in the development, I would look to Dunham's Euler: The Master of Us All, and for Riemann, one should turn to Edwads' Riemann's Zeta Function to read Riemann's original paper.
If you're looking for depth, conciseness, and a broad view of Riemann's zeta function, this book should suit your purposes. If you want a more historical view, I would suggest either of the other books I've mentioned, and not this one.
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Nevanlinna's Theory of Value Distribution: The Second Main Theorem and its Error Terms (Springer Monographs in Mathematics)
William Cherry , and
Zhuan Ye
Manufacturer: Springer
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ASIN: 3540664165 |
Book Description
On the one hand, this monograph serves as a self-contained introduction to Nevanlinna's theory of value distribution because the authors only assume the reader is familiar with the basics of complex analysis. On the other hand, the monograph also serves as a valuable reference for the research specialist because the authors present, for the first time in book form, the most modern and refined versions of the Second Main Theorem with precise error terms, in both the geometric and logarithmic derivative based approaches. A unique feature of the monograph is its "number-theoretic digressions". These special sections assume no background in number theory and explore the exciting interconnections between Nevanlinna theory and the theory of Diophantine approximation.
Book Description
In its first six chapters, this text presents the basic ideas and properties of the Jacobi elliptic functions as a historical essay. Accordingly, it is based on the idea of inverting integrals which arise in the theory of differential equations and, in particular, the differential equation that describes the motion of a simple pendulum. The later chapters present a more conventional approach to the Weierstrass functions and to elliptic integrals, and the reader is introduced to the richly varied applications of the elliptic and related functions.
Book Description
L-functions associated to automorphic forms encode all classical number theoretic information. They are akin to elementary particles in physics. This book provides an entirely self-contained introduction to the theory of L-functions in a style accessible to graduate students with a basic knowledge of classical analysis, complex variable theory, and algebra. Also within the volume are many new results not yet found in the literature. The exposition provides complete detailed proofs of results in an easy to read format using many examples and without the need to know and remember many complex definitions. The main themes of the book are first worked out for GL(2,R) and GL(3,R), and then for the general case of GL(n,R). In an appendix to the book, a set of Mathematica functions is presented, designed to allow the reader to explore the theory from a computational point of view.
Book Description
This volume is the third of three in a series surveying the theory of theta functions which play a central role in the fields of complex analysis, algebraic geometry, number theory and most recently particle physics.
Based on lectures given by the author at the Tata Institute of Fundamental Research in Bombay, these volumes constitute a systematic exposition of theta functions, beginning with their historical roots as analytic functions in one variable (Volume I), touching on some of the beautiful ways they can be used to describe moduli spaces (Volume II), and culminating in a methodical comparison of theta functions in analysis, algebraic geometry, and representation theory (Volume III).
Researchers and graduate students in mathematics and physics will find these volumes to be valuable additions to their libraries.
Book Description
The first of a series of three volumes surveying the theory of theta functions and its significance in the fields of representation theory and algebraic geometry, this volume deals with the basic theory of theta functions in one and several variables, and some of its number theoretic applications.
Requiring no background in advanced algebraic geometry, the text serves as a modern introduction to the subject.
Book Description
This volume is a sequel to the author's Introduction to Analytic Number Theory (UTM 1976, 3rd Printing 1986). It presupposes an undergraduate background in number theory comparable to that provided in the first volume, together with a knowledge of the basic concepts of complex analysis. Most of this book is devoted to a classical treatment of elliptic and modular functions with some of their number-theoretic applications. Among the major topics covered are Rademacher's convergent series for the partition modular function, Lehner's congruences for the Fourier coefficients of the modular function j, and Hecke's theory of entire forms with multiplicative Fourier coefficients. The last chapter gives an account of Bohr's theory of equivalence of general Dirichlet series. In addition to the correction of misprints, minor changes in the exercises and an updated bibliography, this new edition includes an alternative treatment of the transformation formula for the Dedekind eta function, which appears as a five-page supplement to Chapter 3.
Customer Reviews:
Eminently Readable Introduction to Modular Forms.......2005-03-18
The contents of this text have been clearly refined by having been taught. As a result, the book is clear, well presented and easy to learn from. This is in marked contrast to most technical books, where one is often left scratching ones head by page five, thinking "what the heck does that symbol mean?" Not here: each new concept is clearly defined and articulated with the basic theorems and lemmas surrounding it. No notation is used without first having been carefully defined. In addition, the exercises contain a number of additional goodies (again, better than the usual fare).
If you want to learn the material, learn it quickly, learn it in a way that free from roadblocks and detours, this is the book. It is an excellent intro to modular forms, modular functions, the j-invariant, the Weierstrass elliptic functions and the Hecke operators, in the context of the modular group SL(2,Z).
The only criticisms would be:
-- It is far from being an exhaustive treatment of SL(2,Z)-related ideas
-- It fails to mention that the ideas of the j-invariant and modular forms can be generalized in many directions, e.g through fuchsian/kleinian groups
-- It contains absolutely no adelic/p-adic material.
But these are hardly faults: this book opens the doorway to these advanced topics.
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