Among the many constants that appear in mathematics, p, 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 p 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.
"[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, Millennium Mathematics Project, University of Cambridge "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
Foreword xvAcknowledgements xviiIntroduction xixChapter OneThe Logarithmic Cradle 11.1 A Mathematical Nightmare- and an Awakening 11.2 The Baron's Wonderful Canon 41.3 A Touch of Kepler 111.4 A Touch of Euler 131.5 Napier's Other Ideas 16Chapter TwoThe Harmonic Series 212.1 The Principle 212.2 Generating Function for Hn 212.3 Three Surprising Results 22Chapter ThreeSub-Harmonic Series 273.1 A Gentle Start 273.2 Harmonic Series of Primes 283.3 The Kempner Series 313.4 Madelung's Constants 33Chapter FourZeta Functions 374.1 Where n Is a Positive Integer 374.2 Where x Is a Real Number 424.3 Two Results to End With 44Chapter FiveGamma's Birthplace 475.1 Advent 475.2 Birth 49Chapter SixThe Gamma Function 536.1 Exotic Definitions 536.2 Yet Reasonable Definitions 566.3 Gamma Meets Gamma 576.4 Complement and Beauty 58Chapter SevenEuler's Wonderful Identity 617.1 The All-Important Formula 617.2 And a Hint of Its Usefulness 62Chapter EightA Promise Fulfilled 65Chapter NineWhat Is Gamma Exactly? 699.1 Gamma Exists 699.2 Gamma Is What Number? 739.3 A Surprisingly Good Improvement 759.4 The Germ of a Great Idea 78Chapter TenGamma as a Decimal 8110.1 Bernoulli Numbers 8110.2 Euler -Maclaurin Summation 8510.3 Two Examples 8610.4 The Implications for Gamma 88Chapter ElevenGamma as a Fraction 9111.1 A Mystery 9111.2 A Challenge 9111.3 An Answer 9311.4 Three Results 9511.5 Irrationals 9511.6 Pell's Equation Solved 9711.7 Filling the Gaps 9811.8 The Harmonic Alternative 98Chapter TwelveWhere Is Gamma? 10112.1 The Alternating Harmonic Series Revisited 10112.2 In Analysis 10512.3 In Number Theory 11212.4 In Conjecture 11612.5 In Generalization 116Chapter ThirteenIt's a Harmonic World 11913.1 Ways of Means 11913.2 Geometric Harmony 12113.3 Musical Harmony 12313.4 Setting Records 12513.5 Testing to Destruction 12613.6 Crossing the Desert 12713.7 Shuffiing Cards 12713.8 Quicksort 12813.9 Collecting a Complete Set 13013.10 A Putnam Prize Question 13113.11 Maximum Possible Overhang 13213.12 Worm on a Band 13313.13 Optimal Choice 134Chapter FourteenIt's a Logarithmic World 13914.1 A Measure of Uncertainty 13914.2 Benford's Law 14514.3 Continued-Fraction Behaviour 155Chapter FifteenProblems with Primes 16315.1 Some Hard Questions about Primes 16315.2 A Modest Start 16415.3 A Sort of Answer 16715.4 Picture the Problem 16915.5 The Sieve of Eratosthenes 17115.6 Heuristics 17215.7 A Letter 17415.8 The Harmonic Approximation 17915.9 Different-and Yet the Same 18015.10 There are Really Two Questions, Not Three 18215.11 Enter Chebychev with Some Good Ideas 18315.12 Enter Riemann, Followed by Proof(s)186Chapter SixteenThe Riemann Initiative 18916.1 Counting Primes the Riemann Way 18916.2 A New Mathematical Tool 19116.3 Analytic Continuation 19116.4 Riemann's Extension of the Zeta Function 19316.5 Zeta's Functional Equation 19316.6 The Zeros of Zeta 19316.7 The Evaluation of (x) and p(x)19616.8 Misleading Evidence 19716.9 The Von Mangoldt Explicit Formula-and How It Is Used to Prove the Prime Number Theorem 20016.10 The Riemann Hypothesis 20216.11 Why Is the Riemann Hypothesis Important? 20416.12 Real Alternatives 20616.13 A Back Route to Immortality-Partly Closed 20716.14 Incentives, Old and New 21016.15 Progress 213Appendix AThe Greek Alphabet 217Appendix BBig Oh Notation 219Appendix CTaylor Expansions 221C.1 Degree 1 221C.2 Degree 2 221C.3 Examples 223C.4 Convergence 223Appendix DComplex Function Theory 225D.1 Complex Differentiation 225D.2 Weierstrass Function 230D.3 Complex Logarithms 231D.4 Complex Integration 232D.5 A Useful Inequality 235D.6 The Indefinite Integral 235D.7 The Seminal Result 237D.8 An Astonishing Consequence 238D.9 Taylor Expansions-and an Important Consequence 239D.10 Laurent Expansions-and Another Important Consequence 242D.11 The Calculus of Residues 245D.12 Analytic Continuation 247Appendix EApplication to the Zeta Function 249E.1 Zeta Analytically Continued 249E.2 Zeta's Functional Relationship 253References 255Name Index 259Subject Index 263