Contents
Preface vi i
Chapter 1. Leonhar d Eule r (1707-1783 ) 1
l.l.Introduction 1
1.2. Earl y lif e 5
1.3. Th e firs t sta y i n St . Petersburg : 1727-174 1 8
1.4. Th e Berli n years : 1741-176 6 11
1.5. Th e secon d St . Petersbur g sta y an d th e las t years : 1766-178 3 12
1.6. Oper a Omni a 13
1.7. Th e personalit y o f Euler 14
Notes an d reference s 15
Chapter 2 . Th e Universa l Mathematicia n 2 1
2.1. Introductio n 2
2.2. Calculu s 2
2.3. Ellipti c integral s 2 3
2.4. Calculu s o f variations 3 3
2.5. Numbe r theor y 3 7
Notes an d reference s 5 7
Chapter 3 . Zet a Value s 5 9
3.1. Summar y 5 9
3.2. Som e remarks o n infinit e serie s an d product s an d thei r value s 6 4
3.3. Evaluatio n o f £(2 ) an d £(4 ) 6 8
3.4. Infinit e product s fo r circula r an d hyperboli c function s 7 7
3.5. Th e infinit e partia l fraction s fo r (sinx)
- 1
an d cotx .
Evaluation o f £(2fc ) an d L(2k + 1) 8 7
3.6. Partia l fractio n expansion s a s integral s 9 4
3.7. Multizet a value s 105
Notes an d reference s 110
Chapter 4 . Euler-Maclauri n Su m Formul a 113
4.1. Forma l derivatio n 113
4.2. Th e cas e when th e functio n i s a polynomial 116
4.3. Summatio n formul a wit h remainde r term s 117
4.4. Application s 121
Notes an d reference s 124
Chapter 5 . Divergen t Serie s an d Integral s 125
5.1. Divergen t serie s an d Euler' s idea s abou t summin g the m 125
5.2. Euler' s derivatio n o f the functiona l equatio n o f the zet a functio n 131
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