Gamma: Exploring Euler’s Constant (Princeton Science Library, 53) by Julian Havil (PDF)

Description

Among the many constants that appear in mathematics, π, e, and i are the most familiar. Following closely behind is y, or gamma, a constant that arises in many mathematical areas yet maintains a profound sense of mystery. In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma’s place in mathematics. Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + . . . Up to 1/n, minus the natural logarithm of n–the numerical value being 0.5772156. . . . But unlike its more celebrated colleagues π and e, the exact nature of gamma remains a mystery–we don’t even know if gamma can be expressed as a fraction. Among the numerous topics that arise during this historical odyssey into fundamental mathematical ideas are the Prime Number Theorem and the most important open problem in mathematics today–the Riemann Hypothesis (though no proof of either is offered!). Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.

User’s Reviews

Editorial Reviews: Review “[A] wonderful book. . . . Havil’s emphasis on historical context and his conversational style make this a pleasure to read. . . . Gamma is a gold mine of irresistible mathematical nuggets. Anyone with a serious interest in maths will find it richly rewarding.”—Ben Longstaff, New Scientist”This book is a joy from start to finish.”—Gerry Leversha, Mathematical Gazette”Wonderful. . . . Havil’s emphasis on historical context and his conversational style make this a pleasure to read. . . .Gammais a gold mine of irresistible mathematical nuggets. Anyone with a serious interest in math will find it richly rewarding.”–New Scientist”A joy from start to finish.”–Mathematical Gazette”[Gamma] is not a book about mathematics, but a book of mathematics. . . . [It] is something like a picaresque novel; the hero, Euler’s constantg, serves as the unifying motif through a wide range of mathematical adventures.” ― Notices of the American Mathematical Society”[Gamma] is enjoyable for many reasons. Here are just two. First, the explanations are not only complete, but they have the right amount of generality. . . . Second, the pleasure Havil has in contemplating this material is infectious.” ― MAA Online”It is only fitting that someone should write a book about gamma, or Euler’s constant. Havil takes on this task and does an excellent job.” ― Choice”Mathematics is presented throughout as something connected to reality. . . . Many readers will find in [Gamma] exactly what they have been missing.”—Mohammad Akbar, Plus Magazine”This book is written in an informal, engaging, and often amusing style. The author takes pains to make the mathematics clear. He writes about the mathematical geniuses of the past with reverence and awe. It is especially nice that the mathematical topics are discussed within a historical context.”—Ward R. Stewart, Mathematics Teacher Review “I like this book very much. So much, in fact, that I found myself muttering ‘neat stuff!’ all the way through. While it is about an important topic, there isn’t a single competitor. This amazing oversight by past authors is presumably the result of the topic requiring an author with a pretty sophisticated mathematical personality. Havil clearly has that. His skillful weaving of mathematics and history makes the book a ‘fun’ read. Many instructors will surely find the book attractive.”―Paul J. Nahin, author of Duelling Idiots and Other Probability Puzzlers and An Imaginary Tale”This is an excellent book, mathematically as well as historically. It represents a significant contribution to the literature on mathematics and its history at the upper undergraduate and graduate levels. Julian Havil injects genuine excitement into the topic.”―Eli Maor, author of e: The Story of a Number From the Back Cover “I like this book very much. So much, in fact, that I found myself muttering ‘neat stuff!’ all the way through. While it is about an important topic, there isn’t a single competitor. This amazing oversight by past authors is presumably the result of the topic requiring an author with a pretty sophisticated mathematical personality. Havil clearly has that. His skillful weaving of mathematics and history makes the book a ‘fun’ read. Many instructors will surely find the book attractive.”–Paul J. Nahin, author of Duelling Idiots and Other Probability Puzzlers and An Imaginary Tale”This is an excellent book, mathematically as well as historically. It represents a significant contribution to the literature on mathematics and its history at the upper undergraduate and graduate levels. Julian Havil injects genuine excitement into the topic.”–Eli Maor, author of e: The Story of a Number About the Author Julian Havil is a retired former master at Winchester College, England, where he taught mathematics for thirty-three years. He received a Ph.D. in mathematics from Oxford University. Freeman Dyson is professor emeritus of physics at the Institute for Advanced Study in Princeton. He is the author of several books, including Disturbing the Universe and Origins of Life. Read more

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

⭐I have always found Euler’s constant interesting because I would like to be able to say that it is one of the 3 great transcendental numbers along with pi and e. The number e is what Kurt Mahler called an S number; perhaps pi and Euler’s constant also are. My interest started about 1968 and I was soon led to the gamma and Riemann zeta functions. I am pleased to see that Havil confirms the path I followed.This is a very interesting book about a relatively unpublicized development of calculus. Admittedly this book requires good skills in mathematics, but let’s be candid: a good understanding of mathematics calls for experience in working things out.Now I must express a disappointment. I bought the book at least in part because I wanted to know the history of attempts at proving (or disproving) the transcendence of Euler’s constant. I have found almost nothing. There is a copious history of attempts at proving Fermat’s last theorem. Perhaps Euler’s constant has just not attracted so very much effort. I do not know of any financial reward that has been offered for Euler’s constant. Hilbert in 1900 did not specify this number in his seventh problem. I have not seen this problem emphasized as one for the 21st century.For some years I thought of questions that might lead to a proof or disproof of transcendence. How long is the shortest proof that Euler’s constant is transcendental or not? Can the question be answered in finitely many steps? Is there a number related to Euler’s constant in the way that e is related to pi? I did not find these questions mentioned in this book.It is my hope that some high school student (or even grade school student) will read this book and, perhaps after some 40 years, become the Andrew Wiles of Euler’s constant.

⭐Per the foreword, this book is “aimed at students of mathematics, be they eager high school students or undergraduates”. As a summa cum laude graduate math major (of some years ago) I expected to enjoy an romp thru some beautiful mathematical ideas. Well, the ideas are there, and Havil is to be commended for gathering some unusual and interesting topics. And much of the extensive mathematical notation is supported with nice numeric examples. However, much of it is not. All too often there are pages of integrals, sums, and products that go happily on without a clue to some of the beautiful things that are happening. The most frustrating example is the “proof” of Euler’s zeta function formula, one of the prettiest pieces of mathematics. I still cannot understand Havil’s presentation. (It was thrilling to read the same proof in “Prime Obsession” by Derbyshire so I know it can be explained with simple algebra.) Also, “Gamma” appears to be intended to be read in one sitting since it is rarely possible to begin at an advanced chapter. It is assumed that you remember definitions and notations which have appeared long before. To the author’s credit, there are occasional backward references by page number, but then, about half of these are frustratingly wrong. Finally, it would be nice to see a copy of the errata for this book. I hope this book appears in a 2nd edition where the level of its presentation is made much more consistent.

⭐An excellent mathematical book! It’s actuality less about Euler’s Gamma constant than about logarithmic functions and related. Prime Numbers and their relation to Riemann’s Zeta Function and the Li Function are covered and constitute a significant portion of the book. Much of this territory parallels mathematical portion of John Derbyshire’s “Prime Obsession”; another excellent book! Havil does not delve much into the personalities behind the mathematics, as Derbyshire does. Havil’s approach is also more mathematical; at a minimum the reader should have a good familiarity with Calculus. For a math hobbyist, the book gives lots of hints that can be used for calculating by computer esoteric functions such as the prime counting function Pi(n). Other topics are covered as well, for example Benford’s law which posits that the first significant digit of numbers seen in everyday life — such as street addresses — are counter-intuitively not uniformly distributed but instead distributed — you guessed it — in a way related to logarithms. Really a wide ranging discourse on everything logarithmic, Euler’s Gamma being one of them! A great read for the mathematically inclined!

⭐Although I say “accessible” , be warned this is a book for someone with say single subject A level maths, and one needs a pen & paper to work through it: the book is not a popularization, but a serious attempt to explain number theory , Gamma, and in the end the Riemann Hypothesis, to a wider audience.From introductory calculus, the integral of 1/x is the natural logarithm of x. Imagining the graph of 1/x divided into strips of width one, the graph is bounded above and below by these strips 1/n : Gamma is the limiting number, as n gets larger, of the difference (log(n)- (1+1/2+1/3+…+1/n)). It’s about 0.577,, No one knows whether it is the solution of an equation or is not, like pi or e.The book successfully answers the question “just what is it about the the complex zeros of the Riemann zeta function that makes them relevant for the distribution of prime numbers and the Prime Number Theorem?”Hence in some sense this book could be regarded as a follow on to John Derbyshire’s book

⭐Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

⭐A brilliant book and one you’ll return to again and again!Highly recommended for anyone with an interest in maths, an impressive and illuminating work.

⭐well written book on the Gamma Function

⭐Freeman Dyson is not a co-author, he merely wrote the foreword.I think the attribution on Amazon’s website is therefore misleading.

⭐Very nice and informative reading on Euler’constant gamma but has also much material on logarithms, the harmonic series and zeta functions. As the author makes clear at the begining, you really need pencil and paper sometimes. The last two chapters on the Prime Number Theorem and Riemann Hypothesis, respectively, are particularly harder than the rest of the book. In general, the book fails sometimes at not giving explicit references for various informations but the really curious reader can find these elsewhere as he decides to go deeper.

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