Analytic solutions of functional equations

Contents......Page 10
Preface......Page 6
1.1 An Example......Page 13
1.2 Basic Definitions......Page 14
1.3 Notes......Page 21
2.1 Lebesgue Summable Sequences......Page 23
2.2 Relatively Summable Sequences......Page 30
2.3 Uniformly Summable Sequences......Page 33
2.4.1 Common Sequences......Page 37
2.4.2 Convolution Products......Page 38
2.4.3 Algebraic Derivatives and Integrals......Page 44
2.4.4 Composition Products......Page 46
2.5 Properties of Bivariate Sequences......Page 54
2.6 Notes......Page 59
3.1 Univariate Power Series Functions......Page 61
3.2 Univariate Analytic Functions......Page 68
3.3 Bivariate Power Series Functions......Page 75
3.4 Bivariate Analytic Functions......Page 79
3.5 Multivariate Power Series and Analytic Functions......Page 80
3.6 Matrix Power Series and Analytic Functions......Page 83
3.7 Majorants......Page 84
3.8 Siegel's Lemma......Page 89
3.9 Notes......Page 94
4.1 Introduction......Page 95
4.2 Analytic Implicit Function Theorem......Page 98
4.3 Polynomial and Rational Functional Equations......Page 102
4.4.1 Equation I......Page 112
4.4.2 Equation II......Page 114
4.4.3 Equation III......Page 115
4.4.4 Equation IV......Page 117
4.4.5 Equation V......Page 119
4.4.6 Schr oder and Poincar e Equations......Page 122
4.5 Nonlinear Equations......Page 126
4.6 Notes......Page 133
5.1 Introduction......Page 135
5.2 Linear Systems......Page 136
5.3 Neutral Systems......Page 140
5.4 Nonlinear Equations......Page 145
5.5 Cauchy-Kowalewski Existence Theorem......Page 151
5.6 Functional Equations with First Order Derivatives......Page 153
5.6.1 Equation I......Page 154
5.6.2 Equation II......Page 155
5.6.3 Equation III......Page 157
5.6.4 Equation IV......Page 159
5.6.5 Equation V......Page 160
5.6.6 Equation VI......Page 162
5.7 Functional Equations with Higher Order Derivatives......Page 164
5.7.1 Equation I......Page 165
5.7.2 Equation II......Page 166
5.7.3 Equation III......Page 168
5.7.4 Equation IV......Page 178
5.8 Notes......Page 182
6.1 Equations without Derivatives......Page 187
6.1.1 Babbage Type Equations......Page 188
6.1.2 Equations Involving Several Iterates......Page 194
6.1.3.1 Equation I......Page 202
6.1.3.2 Equation II......Page 204
6.2 Equations with First Order Derivatives......Page 209
6.2.1 Equation I......Page 210
6.2.2 Equation II......Page 214
6.2.3 Equation III......Page 218
6.2.4 Equation IV......Page 224
6.2.5 First Order Neutral Equation......Page 226
6.3 Equations with Second Order Derivatives......Page 234
6.3.1 Equation I......Page 235
6.3.2 Equation II......Page 242
6.3.3 Equation III......Page 247
6.3.4 Equation IV......Page 252
6.4 Equations with Higher Order Derivatives......Page 256
6.4.1 Equation I......Page 259
6.4.2 Equation II......Page 261
6.5 Notes......Page 269
A.1 Common Sequences......Page 271
A.2 Sums and Products......Page 272
A.4 Algebraic Derivatives and Integrals......Page 273
A.5 Tranformations......Page 274
A.7 Operational Rules......Page 275
A.8 Knowledge Base......Page 278
A.10 Operations for Analytic Functions......Page 279
Bibliography......Page 283
Index......Page 295