Rating: 
Summary: Gamma finally joins the ranks of e, pi, i
Review: After reading Paul Nahin's lovely book on i, "An Imaginary Tale" (also published by Princeton University Press), I could not believe it when the same publisher produced a book on gamma. Gamma seems to always have been one of the neglected constants in mathematics (by the general public). e, pi, and i seem to capture the imagination more, my guess is because the mathematics required to understand them are more elementary (I use the word "elementary" completely tongue in cheek), and you can quickly see the dazzling results they are associated with.Gamma is different. While you can understand the theory presented in Julian Havil's book if you stayed awake during second semester calculus, you definitely have to work at it. The requisite analytic number theory presented may turn away the average reader if they are not prepared to make the commitment to stay on the roller coaster for the full ride. You will be rewarded if you can break through the initial 2 or 3 chapters introducing us to the logarithm and the harmonic series. To be fair, as a previous reviewer has noted, the material on Napier and the logarithm has been done in a more satisfactory manner by Eli Maor in his book on e. But this is only a minor drawback. As long as you are comfortable with the natural logarithm, you can omit Chapter 1 with no loss. Chapter 4 starts off with the zeta function, arguably the most enticing and mysterious function in all of mathematics, despite approximately 150 years of analysis by the world's best mathematicians. This one function alone could arguably be said to be the genesis of analytic number theory (even though Dirichlet's work on primes in arithmetic progressions has typically been given credit for that role). All the familiar material is presented, including Euler's product formula, the "trivial" divisors of the zeta function, the infinitude of primes, Euler's evaluation of the zeta function for positive even integer powers, etc. Of course, the gamma function makes its obligatory appearance. After having read Nahin's book on i, I was initiated into the math connecting the gamma and zeta functions. But Nahin of course could not use Euler-Maclaurin summation or the familiar inequality arguments as this would have taken him too far afield. After having read the traditional fare, such as Hardy-Wright, Apostol, Hua, et al., it was nice to see a more conversational approach to the material. I literally felt like I was sitting in Havil's office while he dissected the material for me, on a level I could comprehend. My last comments on this book are the extras. As expected, Riemann's hypothesis and complex analysis make extended appearances. I appreciated the fact the Havil resisted the temptation to take the Riemann Hypothesis beyond the traditional mathematical lore and float off into the ethereal. This happened with John Derbyshire's otherwise excellent book "Prime Obsession", which devoted a little too much time to the psychoanalysis of Riemann, who after all, only scratched the surface of this problem. Derbyshire's book is highly recommended though for more material on the Prime Number Theorem, and some of its uses to formulate modern permutations of the Riemann Hypothesis. He presents the usual anecdotes on Riemann and Hardy (who had a major love affair with the Riemann Hypothesis), but these are sidelines only, as they should be. Also, the material on residue integration and analytic continuation in the appendices is enormously helpful to understand the post Riemann attacks on the problem. In addition, well, it's just pretty mathematics. The introduction by Freeman Dyson is quite impressive. How many books of popular mathematics get endorsements like that from world-class physicists? The praise is well deserved. This book belongs on every math enthusiast's bookshelf!
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Summary: Gamma: Exploring Euler's Constant, A Very Good Book
Review: Dr. Julian Havil's book Gamma: Exploring Euler's constant is a marvelous book. With verity and clarity it takes you through Euler's Constant, ... encountering the harmonic series, ... and on through the Riemann Zeta Function, and a lot more -- all done masterfully. Gamma is entertaining as well as edifying. Like other great mathematical works, a few typographical errors crept in, but in nowise diminish from the importance of the book:
1. Page 11, 1st paragraph, 7th line, "that" should be "than."
2. Page 12 at the bottom of the page, the left integral should have a parenthesis around "x+y."
3. Page 13 at the top of the page, in the integral on the right, "x" should be "a."
4. Page 13 in the integral beneath the first sentence, "x" should be "2a" in the integrand, and the upper limit should be "pi," not "2pi."
As an idiot (but not a blithering idiot) I realize I could be mistaken about the above errata.
Rating:
Summary: Excellent - highly recomended
Review: Gamma seemed like an unusual subject for a book, but the author really makes it hang together. There is so much mathematics in here that I didn't know, that the author presents clearly and with enthusiasm. His enthusiasm is infectious and I really enjoyed it. The book assumes you are prepared to put in some effort, but the reward is worth it. The material on the harmonic series and logs was particularly good. I even understood the Riemann Hypothesis sections!. I don't know the author (I think it's his first book) but I'll be at the front of the queue for his next. Highly recommended.
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Summary: This would make an excellent alternative "Calc III"
Review: I agree wholeheartedly with all the positive comments and enthusiasm that other reviewers have shown. This is a remarkable book, and there should be more like it. I am astounded at how much and what range of mathematics there is in a book of this length and level of accessbility. Which raises a very good point: This would be a superb book for "Calc III". It's unfortunate that many students end their study of mathematics slugging through integration by parts, partial fractions, sequences and series, the logarithm as integral, etc., the traditional hodge-podge of topics called Calculus II. And the ones who progress end up going straight into multivariable calculus with its div, grad, curl, and all that. There is never really any reward for all the work in hacking through Calc II. This book, however, would tie so much of it together, it would all suddenly seem so mysteriously connected and beautiful, and the reader (I hope) would want to go on to Complex Analysis. Thank you, Prof. Havil! I hope you find the proof to the Riemann Hypothesis.
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Summary: Brave and Successful
Review: I agree with the previous reviewers, Havil has used Gamma as a means to introduce a wealth of fascinating mathematics. His chosen historical approach succeeds in adding interest whilst at the same time tempering some quite difficult subject matter. The cleverest thing for me is his ability to introduce a wide variety of material in such a natural way; it all flows very smoothly and he explains the ideas with crystaline clarity. There needed to be a book on Gamma and this, for me, is exactly the right book to write on the topic; it fits well into the PUP stable having e and i already in it. Its good to learn about new mathematics and to learn about it in context; now I am starting to read more about Continued Fractions, about which I knew very little. The Riemann Hypothesis is much in vogue at the present, with two new books about it but written for the lay reader; Havil approaches it as a mathematician and takes the bull by the horns with great success. He needed Analytic Continuation and sensibly did not side-step that need. This is a book which will last-and be on the shelves of many a mathematics student. Good for him and I too look forward to his next book.
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Summary: Far-reaching, but not "popular math"
Review: I debated for a while whether this book deserved four stars or five. There's a lot of very interesting material here: if there's one thing this book does--perhaps better than any book I've read in quite some time--is show just how interrelated far-flung mathematical concepts can be (how are the prime numbers related to pi, for example?). My one complaint about the book--and the reason for giving it four stars instead of five--is that there are times when the formulae and notation get so dense that it's extremely difficult to follow the author's train of thought: I can think of a number of places where diagrams would have helped immensely. Likewise, since there's no list of symbols or formulae, it's not a book that you can simply browse through, in the sense that you can browse through, say, "A Brief History of Time." Finally, let me reiterate that this book assumes that you already know a fair amount of math: if you don't know what a capital pi means, for example, you're probably going to have a hard time understanding this book. But if you *do* know what that symbol means, though, then by all means, give this book a try.
Rating:
Summary: A hard but worthwhile read
Review: I did not get past the first chapter as I kept feeling that I had read this stuff before. Then I found Eli Maor's wonderful book "e: The Story of a Number" on our bookshelf and started to re-read it. Maor's book was published in 1994. "Gamma" has just been published and its account on Napier's discovery of logarithms is a badly paraphrased version of what Maor wrote 9 years earlier. This book does improve with persistent reading. The writing is still not strong but the mathematics is glorious and ultimately triumphs. Julian Havel has attempted something very difficult: explaining of fairly advanced maths to the general reader. He is uniquely brave in attempting this and generally succeeds.
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Summary: Levitt is wrong
Review: I have just read the recent review of a man called Levitt and cannot rest without countering it. I believe the book to be quite superb and to dismiss it in this uninformed way is facile. We must all be free to express our own opinions but I reckon that they should be based on some sort of sound knowledge-and Levitt seems not to have that. Describing the chapter on logs as a rehash of Maor's work is complete nonense, which is not a matter of opinion but of fact. To understand Gamma you have to have some mathematical training and and beyond that, interest. It is excellently written and filled a yawning gap in the literature. To suggest that Dyson's Preface is irrelevent is insulting to the great man; in putting his name to the work Dyson gives his own judgement of it...believe him instead!
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Summary: Nobody's perfect
Review: I recommend this book to anyone who has learned calculus. The book is full of mathematical gems set in historical context. For example, one of the Bernoullis showed that the integral of 1/x^x from x=0 to 1 is the sum of 1/n^n from n=1 to infinity. If you appreciate that result, then this book is for you. I have found only one error in the first 45 pages (which is as far as I have read). On page 13 the author is trying to compute the average distance of a planet from the sun, assuming an elliptical trajectory, and tacitly assuming a uniform distribution in the angle that astronomers call true anomaly. For this distribution, despite the assertion at the top of page 13, it is easy to show that if the ellipse is not a circle, then the average distance from the sun is less than the semimajor axis of the ellipse. So the author is human after all.
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Summary: Math as it should be
Review: The spectrum of 'popular math' books extends from those which describe the ideas, without enmeshing themselves in the gritty detail, to those which deal with that detail, but avoid being text books; Havil has written one of the latter. In the Introduction he says that the reader will judge the level of success of the project, which was to 'explain interesting mathematics interestingly'; that's an honest statement which naturally provokes a firm answer either way; for me he has done a fantasic job. I am trained in mathematics but sold my soul to commerce (some years ago) and miss the excitement and beauty of a subject that I found frustratingly addictive. A book such as this rejuvinates the wonder, challenges the intellect, informs and entertains. I reckon that it's a gamble on the part of PUP to continue to develop a series of books which ask more than most of the reader, but I admire their commitment-and I guess that these sort of books sell, otherwise the axe would fall. So, Gamma is well thought-out, beautifully structured and incredibly wide in its content; it's impossible not to learn from it and not to be fascinated by it, whether it be a historical or technical nugget that appeals. Havil's love of Euler is very evident-and very appealing (and justified). I have seen several reviews of the book in respected places (MAA online and New Scientist for example) and they have been unqualified in their admiration; I am too (for what it is worth!) I admire those who work hard enough to give us enlightening books to read, it must take years to produce the finished product, and here we have an example of its kind which is simply exemplary. If you love maths, buy it-but only if you love symbols too- he doesn't avoid the technical details, he revels in them!
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