A primer of real analytic functions

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface to the Second Edition......Page 10
Preface to the First Edition......Page 12
1.1 Basic Properties of Power Series......Page 16
1.2 Analytic Continuation......Page 26
1.3 The Formula of Fail di Bruno......Page 31
1.4 Composition of Real Analytic Functions......Page 33
1.5 Inverse Functions......Page 35
2.1 Power Series in Several Variables......Page 40
2.2 Real Analytic Functions of Several Variables......Page 44
2.3 The Implicit Function Theorem......Page 50
2.4 A Special Case of the Cauchy-Kowalewsky Theorem......Page 57
2.5 The Inverse Function Theorem......Page 62
2.6 Topologies on the Space of Real Analytic Functions......Page 65
2.7 Real Analytic Submanifolds......Page 69
2.7.1 Bundles over a Real Analytic Submanifold......Page 71
2.8 The General Cauchy-Kowalewsky Theorem......Page 76
3.0 Introductory Remarks......Page 82
3.1 The Theorem of Pringsheim and Boas......Page 83
3.2 Besicovitch's Theorem......Page 87
3.3 Whitney's Extension and Approximation Theorems......Page 90
3.4 The Theorem of S. Bernstein......Page 94
4.1 Quasi-analytic and Gevrey Classes......Page 98
4.2 Puiseux Series......Page 110
4.3 Separate Real Analyticity......Page 119
5.1 Division of Distributions I......Page 130
5.1.1 Projection of Polynomially Defined Sets......Page 132
5.2 Division of Distributions II......Page 141
5.3 The FBI Transform......Page 150
5.4 The Paley-Wiener Theorem......Page 159
6.1 The Weierstrass Preparation Theorem......Page 166
6.2 Resolution of Singularities......Page 171
6.3 Lojasiewicz's Structure Theorem for Real Analytic Varieties......Page 181
6.4 The Embedding of Real Analytic Manifolds......Page 186
6.5.1 Basic Definitions......Page 192
6.5.2 Facts Concerning Semianalytic and Subanalytic Sets......Page 194
6.5.3 Examples and Discussion......Page 196
6.5.4 Rectilinearization......Page 199
Bibliography......Page 202
Index......Page 218
Back Cover......Page 226