Gamma Exploring Euler's Constant

ISBN-10: 0691141339

ISBN-13: 9780691141336

Edition: 2009

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Book details

List price: $13.95
Copyright year: 2009
Publisher: Princeton University Press
Publication date: 7/26/2009
Binding: Paperback
Pages: 304
Size: 6.00" wide x 9.25" long x 1.00" tall
Weight: 0.880
Language: English

Foreword
Acknowledgements
Introduction
The Logarithmic Cradle
A Mathematical Nightmare- and an Awakening
The Baron''s Wonderful Canon
A Touch of Kepler
A Touch of Euler
Napier''s Other Ideas
The Harmonic Series
The Principle
Generating Function for Hn
Three Surprising Results
Sub-Harmonic Series
A Gentle Start
Harmonic Series of Primes
The Kempner Series
Madelung''s Constants
Zeta Functions
Where n Is a Positive Integer
Where x Is a Real Number
Two Results to End With
Gamma''s Birthplace
Advent
Birth
The Gamma Function
Exotic Definitions
Yet Reasonable Definitions
Gamma Meets Gamma
Complement and Beauty
Euler''s Wonderful Identity
The All-Important Formula
And a Hint of Its Usefulness
A Promise Fulfilled
What Is Gamma Exactly?
Gamma Exists
Gamma Is What Number?
A Surprisingly Good Improvement
The Germ of a Great Idea
Gamma as a Decimal
Bernoulli Numbers
Euler -Maclaurin Summation
Two Examples
The Implications for Gamma
Gamma as a Fraction
A Mystery
A Challenge
An Answer
Three Results
Irrationals
Pell''s Equation Solved
Filling the Gaps
The Harmonic Alternative
Where Is Gamma?
The Alternating Harmonic Series Revisited
In Analysis
In Number Theory
In Conjecture
In Generalization
It''s a Harmonic World
Ways of Means
Geometric Harmony
Musical Harmony
Setting Records
Testing to Destruction
Crossing the Desert
Shuffiing Cards
Quicksort
Collecting a Complete Set
A Putnam Prize Question
Maximum Possible Overhang
Worm on a Band
Optimal Choice
It''s a Logarithmic World
A Measure of Uncertainty
Benford''s Law
Continued-Fraction Behaviour
Problems with Primes
Some Hard Questions about Primes
A Modest Start
A Sort of Answer
Picture the Problem
The Sieve of Eratosthenes
Heuristics
A Letter
The Harmonic Approximation
Different-and Yet the Same
There are Really Two Questions, Not Three
Enter Chebychev with Some Good Ideas
Enter Riemann, Followed by Proof(s)
The Riemann Initiative
Counting Primes the Riemann Way
A New Mathematical Tool
Analytic Continuation
Riemann''s Extension of the Zeta Function
Zeta''s Functional Equation
The Zeros of Zeta
The Evaluation of (x) and p(x)
Misleading Evidence
The Von Mangoldt Explicit Formula-and How It Is Used to Prove the Prime Number Theorem
The Riemann Hypothesis
Why Is the Riemann Hypothesis Important?
Real Alternatives
A Back Route to Immortality-Partly Closed
Incentives, Old and New
Progress
The Greek Alphabet
Big Oh Notation
Taylor Expansions
Degree 1
Degree 2
Examples
Convergence
Complex Function Theory
Complex Differentiation
Weierstrass Function
Complex Logarithms
Complex Integration
A Useful Inequality
The Indefinite Integral
The Seminal Result
An Astonishing Consequence
Taylor Expansions-and an Important Consequence
Laurent Expansions-and Another Important Consequence
The Calculus of Residues
Analytic Continuation
Application to the Zeta Function
Zeta Analytically Continued
Zeta''s Functional Relationship
References
Name Index
Subject Index
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