Gamma: Exploring Euler's Constant

290 Pages·2003·7.98 MB·English

by  Julian Havil

.') .\H I
LeonhardEuler(1707-1783)
ReadEuler, readEuler, heismasterofusal/,
Pierre-SimonLaplace(1749-1827)
Eulercalculatedwithouteffort,justasmenbreathe.
aseaglessustain fhemseh'esinfheair.
DominiqueFran~oisJeanArago(1786-1853)
ThestudyofEuler'sworksremainsthebestinstructioninthevariousareasof
mathematicsandcanbereplacedbynoother.
CarlFrederickGauss(1777-1855)
Gamma
EXPLORING EULER'S CONSTANT
Julian Havil
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright © 2003byPrincetonUniversityPress
PublishedbyPnncetonUniversityPress,4IWilliamStreet,
Princeton,NewJersey08540
IntheUnitedKingdom PrincetonUniversityPress,3MarketPlace,
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Allrightsreserved
LibraryofCongressCataloguing-in-PublicationData
Havil,Julian, 1952
Gamma.exploringEuler'sconstant/JulianHavil
p.cm.
Includesbibliographicalreferencesandindex
ISBN0-691-09983-9(acid-freepaper)
I.Mathematicalconstants.2.Euler,Leonhard, 1707-1783.1.Title
QA4I.H23 2003
5I3--dc2I 2002192453
BntishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBntishLibrary.
ThisbookhasbeencomposedinTimes.TypesetbyT&TProductionsLtd,London.
Printedonacid-freepaper.@
www.pupress.pnnceton.edu
PrintedintheUnitedStatesofAmerica
10987654
DEDICATEDTOSIMON ANDDANIEL,WHOWILLNEVERREADIT
BUTWHOWILLALWAYS BEPROUDTHATI WROTEIT, ANDTO
GRAEME,WHOSHAREDTHATPRIDE, AS HESHAREDEVERYTHING

Ihadafeeling onceaboutMathematics-thatIsawitall. Depthbeyonddepth
was revealed to me-the Byss andAbyss. Isaw-asone might see the transit
ofVenusoreventhe LordMayor'sShow-aquantitypassingthrough infinity
and changing its sign from plus to minus. Isaw exactly why it happened and
why thetergiversation was inevitablebutitwasafterdinnerand Iletitgo.
SirWinstonChurchill (1874-1965)

Contents
Foreword xv
Acknowledgements xvii
Introduction xix
CHAPTERONE
TheLogarithmicCradle 1
1.1 AMathematical Nightmare-andanAwakening 1
1.2 TheBaron'sWonderfulCanon 4
1.3 ATouchofKepler 11
1.4 ATouchofEuler 13
1.5 Napier'sOtherIdeas 16
CHAPTER1\vo
TheHarmonic Series 21
2.1 ThePrinciple 21
2.2 Generating Functionfor Hn 21
2.3 ThreeSurprisingResults 22
CHAPTERTHREE
Sub-HarmonicSeries 27
3.1 AGentleStart 27
3.2 HarmonicSeriesofPrimes 28
3.3 The KempnerSeries 31
3.4 Madelung'sConstants 33
CHAPTERFOUR
ZetaFunctions 37
4.1 Wheren IsaPositiveInteger 37
4.2 Wherex Isa Real Number 42
4.3 TwoResultsto EndWith 44
ix
CONTENTS
CHAPTERFIVE
Gamma'sBirthplace 47
5.1 Advent 47
5.2 Birth 49
CHAPTERSIX
TheGammaFunction 53
6.1 Exotic Definitions... 53
6.2 ... YetReasonable Definitions 56
6.3 GammaMeetsGamma 57
6.4 Complementand Beauty 58
CHAPTERSEVEN
Euler'sWonderful Identity 61
7.1 TheAll-ImportantFormula... 61
7.2 ... AndaHintofIts Usefulness 62
CHAPTEREIGHT
APromiseFulfilled 65
CHAPTERNINE
WhatIsGamma ... Exactly? 69
9.1 GammaExists 69
9.2 GammaIs... WhatNumber? 73
9.3 ASurprisinglyGood Improvement 75
9.4 TheGermofaGreatIdea 78
CHAPTERTEN
Gammaas aDecimal 81
10.1 Bernoulli Numbers 81
10.2 Euler-Maclaurin Summation 85
10.3 TwoExamples 86
10.4 The ImplicationsforGamma 88
CHAPTERELEVEN
Gammaas aFraction 91
11.1 AMystery 91
11.2 AChallenge 91
11.3 AnAnswer 93
11.4 ThreeResults 95
11.5 Irrationals 95
11.6 PelI's Equation Solved 97
x