Symbolic Integration I: Transcendental Functions, Second Edition (Algorithms and Computation in Mathematics)

Cover......Page 1
Algorithms and Computation in Mathematics, Volume 1......Page 2
Symbolic Integration I: Transcendental Functions (Second edition)......Page 4
3540214933......Page 5
Foreword......Page 6
Preface to the Second Edition......Page 8
Preface to the First Edition......Page 10
Contents......Page 14
1.1 Groups, Rings and Fields......Page 18
1.2 Euclidean Division and Pseudo-Division......Page 25
1.3 The Euclidean Algorithm......Page 27
1.4 Resultants and Subresultants......Page 35
1.5 Polynomial Remainder Sequences......Page 38
1.6 Primitive Polynomials......Page 42
1.7 Squarefree Factorization......Page 45
Exercises......Page 49
2 Integration of Rational Functions......Page 52
2.1 The Bernoulli Algorithm......Page 53
2.2 The Hermite Reduction......Page 56
2.3 The Horowitz-Ostrogradsky Algorithm......Page 62
2.4 The Rothstein-Trager Algorithm......Page 64
2.5 The Lazard-Rioboo-Trager Algorithm......Page 66
2.6 The Czichowski Algorithm......Page 70
2.7 Newton-Leibniz-Bernoulli Revisited......Page 71
2.8 Rioboo's Algorithm for Real Rational Functions......Page 76
2.9 In-Field Integration......Page 87
Exercises......Page 89
3.1 Derivations......Page 92
3.2 Differential Extensions......Page 96
3.3 Constants and Extensions......Page 102
3.4 Monomial Extensions......Page 107
3.5 The Canonical Representation......Page 116
Exercises......Page 121
4.1 Basic Properties......Page 124
4.2 Localizations......Page 127
4.3 The Order at Infinity......Page 132
4.4 Residues and the Rothstein-Trager Resultant......Page 135
Exercises......Page 143
5.1 Elementary and Liouvillian Extensions......Page 146
5.2 Outline and Scope of the Integration Algorithm......Page 151
5.3 The Hermite Reduction......Page 155
5.4 The Polynomial Reduction......Page 157
5.5 Liouville's Theorem......Page 159
5.6 The Residue Criterion......Page 164
5.7 Integration of Reduced Functions......Page 171
5.8 The Primitive Case......Page 174
5.9 The Hyperexponential Case......Page 177
5.10 The Hypertangent Case......Page 180
5.11 The Nonlinear Case with no Specials......Page 189
5.12 In-Field Integration......Page 192
Exercises......Page 195
6.1 The Normal Part of the Denominator......Page 198
6.2 The Special Part of the Denominator......Page 203
6.3 Degree Bounds......Page 210
6.4 The SPDE Algorithm......Page 219
6.5 The Non-Cancellation Cases......Page 223
6.6 The Cancellation Cases......Page 228
Exercises......Page 233
7.1 The Parametric Risch Differential Equation......Page 234
7.2 The Limited Integration Problem......Page 262
7.3 The Parametric Logarithmic Derivative Problem......Page 267
Exercises......Page 272
8 The Coupled Differential System......Page 274
8.1 The Primitive Case......Page 276
8.2 The Hyperexponential Case......Page 278
8.3 The Nonlinear Case......Page 280
8.4 The Hypertangent Case......Page 281
9.1 The Module of Differentials......Page 286
9.2 Rosenlicht's Theorem......Page 293
9.3 The Risch Structure Theorems......Page 299
9.4 The Rothstein-Caviness Structure Theorem......Page 310
Exercises......Page 313
10 Parallel Integration......Page 314
10.1 Derivations of Polynomial Rings......Page 315
10.2 Structure of Elementary Antiderivatives......Page 318
10.3 The Integration Method......Page 322
10.4 Simple Differential Fields......Page 328
References......Page 334
Index......Page 340