Gamma: Exploring Euler's Constant

✍ Scribed by Julian Havil; Freeman Dyson

• Publisher: Princeton University Press
• Year: 2010
• Tongue: English
• Leaves: 290
• Series: Princeton Science Library; 53
• Edition: Course Book
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents
Foreword
Acknowledgements
Introduction
Chapter One. The Logarithmic Cradle
Chapter Two. The Harmonic Series
Chapter Three. Sub-Harmonic Series
Chapter Four. Zeta Functions
Chapter Five. Gamma’s Birthplace
Chapter Six. The Gamma Function
Chapter Seven. Euler’s Wonderful Identity
Chapter Eight. A Promise Fulfilled
Chapter Nine. What Is Gamma . . . Exactly?
Chapter Ten. Gamma as a Decimal
Chapter Eleven. Gamma as a Fraction
Chapter Twelve. Where Is Gamma?
Chapter Thirteen. It’s a Harmonic World
Chapter Fourteen. It’s a Logarithmic World
Chapter Fifteen. Problems with Primes
Chapter Sixteen. The Riemann Initiative
Appendix A. The Greek Alphabet
Appendix B. Big Oh Notation
Appendix C. Taylor Expansions
Appendix D. Complex Function Theory
Appendix E. Application to the Zeta Function
Name Index
Subject Index