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商品簡介
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 appropr本書是僅存的一部專門講述實解析函數的專著。全書分為六大部分，包括豐富的參考文獻和索引，可以說是一部數學分析思想的源泉。學習本書不僅可以學習該科目，也能很好地瞭解這門學科發展歷史。從基礎開始講述了本學科涉及的方方面面，對初學者和更高層次的科研人員同樣適用。目次：基本性質；實解析函數的多變量微積分；經典話題；Hard分析的一些問題；偏微分方程的一些結果；幾何知識。
讀者對象：數學專業的本科生、研究生、教師和科研人員。
讀者對象：數學專業的本科生、研究生、教師和科研人員。
目次
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
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
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