Foreword ...................................................... vii
Preface ........................................................ ix
Chapter 1 Introduction ......................................... 1
Chapter 2 Background ........................................... 3
2.1 Series ..................................................... 3
2.2 Feynman graphs as combinatorial objects .................... 3
2.3 Feynman graphs as physical objects ........................ 12
Chapter 3 Dyson-Schwinger equations ........................... 21
3.1 B+ ........................................................ 21
3.2 Dyson-Schwinger equations ................................. 24
3.3 Setup ..................................................... 29
Chapter 4 The first recursion ................................. 35
4.1 From the renormalization group equation ................... 35
4.2 From S * Y ................................................ 37
4.3 Properties ................................................ 41
Chapter 5 Reduction to one insertion place .................... 43
5.1 Colored insertion trees ................................... 43
5.2 Dyson-Schwinger equations with one insertion place ........ 46
Chapter 6 Reduction to geometric series ....................... 51
6.1 Single equations .......................................... 51
6.2 Systems ................................................... 53
Chapter 7 The second recursion ................................ 55
7.1 Single equations .......................................... 55
7.2 Systems ................................................... 56
7.3 Variants .................................................. 57
Chapter 8 The radius of convergence ........................... 59
8.1 Single equations .......................................... 59
8.2 Systems ................................................... 63
8.3 Possibly negative systems ................................. 65
Chapter 9 The second recursion as a differential equation ..... 67
9.1 P(x) = x .................................................. 68
9.2 QED as a single equation .................................. 70
9.3 Results on global solutions ............................... 71
9.4 The running coupling ...................................... 75
Bibliography ................................................... 79
Index .......................................................... 81
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