《实解析函数入门(第2版)》编著者克兰兹。

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.

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

3 ClassicaI Toplcs

3.0 Introductory Remarks

3.1 TheTheorem ofPringsheim and Boas

3.2 Besicovitch'sTheorem

3.3 Whitney's Extension and Approximation Theorems

3.4 TheTheorem ofS.Bernstein

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