The subject of real analytic functions is one of the oldest in
mathematical analysis. Today it is encountered early in one''s
mathematical training： the first taste usually comes rn calculus.
While most working mathematicians use real analytic functions from
time to time in their WOfk， the vast lore of real analytic
functions remains obscure and buried in the literature. It is
remarkable that the most accessible treatment of Puiseux''s thcorem
is in Lefschetz''s quute old Algebraic Geometry， that the clearest
discussion of resolution of singularities for real analytic
manifolds is in a book review by Michael Atiyah， that there is no
compre hensive discussion in print of the embedding problem for
real analytic manifolds.
We have had occasion in our collaborative research to become
acquainted with both the history and the scope of the theory of
real analytic functions. It seems both appropriate and timely for
us to gather together this information in a single volume. The
material presented here is of three kinds. The elementary topics，
covered in Chapter 1， are presented in great detail. Even results
like a real analytic inverse function theorem are difficult to find
in the literature， and we take pains here to present such topics
carefully. Topics of middling difficulty， such as separate real
analyticity， Puiseux series， the FBI transform， and related ideas
（Chapters 2-4）， are covered thoroughly but rather more
briskly. Finally there are some truly deep and difficult topics：
embedding of real analytic manifolds， sub and semi-analytic sets，
the structure theorem for real analytic varieties， and resolution
of singularities are disc，ussed and described. But thorough proofs
in these areas could not possibly be provided in a volume of modest
length.
目錄：
Prethce to the Second Edition
Preface to the First Edition
1 Elementary Propertles
1.1 Basic Properties of Power Series
1.2 Analytic Continuation
1.3 The Formula of Faa di Bruno
1.4 Composition of ReaI Analytic Functions
1.5 Inverse Functions .
2 Multivariable Calculus of ReaI Analytic Functions
2.1 Power Series in Several Variables
2.2 ReaI Analytic Functions of SeveraI Variables
2.3 Thelmplicit Function Theorem
2.4A Special Case of the Cauchy-Kowalewsky Theorem
2.5 The lnverse Function Theorem
2.6Topologies on the Space of Real Analytic Functions
2.7 ReaI Analytic Submarufolds
2.7.1Bundles over a Real Analytic Submanifold
2.8 The GeneraI Cauchy-Kowalewsky Theorem
4Some Questions of Hard Analysis
4.1 Quasi-analytic and Gevrey Classes
4.2 PuiseuxSeries
4.3 Separate Real Analyticity
5 Results Motivated by Partial DifferentiaI Equations
5.1 Division of Distributionsl
5.1.1Projection of Polynomially Defined Sets
5.2 DMsion of Distributionsll
5.3 The FBI Transform
5.4 The Paley-Wiener Theorem
6 Topics in Geometry
6.1 The Weierstrass Preparation Theorem
6.2 Resolution of Singularities
6.3 Lojasiewicz''s Structure Theorem for Real Analytic
Varieties
6.4 The Embedding of Real Analytic Manifolds
6.5 Semianalytic and Subanalytic Sets
6.5.1 Basic Definitions
6.5.2 Facts Concerning Semianalytic and Subanalytic Sets
6.5.3 Examples and Discussion
6.5.4 Rectilinearization
Blbliography
Index
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