[.ca] Gamma: Exploring Euler's Constant (ISBN 0691099839)

Gamma finally joins the ranks of e, pi, i:
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!

Far-reaching, but not "popular math":
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.

This would make an excellent alternative "Calc III":
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.

Math as it should be:
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!

Wonderful Read:
This book was very interesting to read. I am very much into reading about mathematics and its history, which this book supplied both in great quantities. The author does a great job of describing the role that logarithms and harmonics play in fields like number theory and analysis and their deep connection with the mysertious Riemann Hypothesis.