"This is the second, improved edition of the only existing monograph devoted to real-analytic functions, whose theory is rightly considered in the preface 'the wellspring of mathematical analysis.' Organized in six parts, [with] a very rich bibliography and an index, this book is both a map of the subject and its history. Proceeding from the most elementary to the most advanced aspects, it is useful for both beginners and advanced researchers.

-MATHEMATICAL REVIEWS

"Bringing together results scattered in various journals or books and presenting them in a clear and systematic manner, the book is of interest first of all for analysts, but also for applied mathematicians and researchers in real algebraic geometry."

-ACTA APPLICANDAE MATHEMATICAE

"This well-organized and clearly written advanced textbook introduces students to real analytic functions of one or more real variables. Many historical remarks, examples and references to the literature encourage the beginner to study further this ample, valuable and exciting theory." (1st ed.) New to the 2nd ed.: A more revised and comprehensive treatment of the Faà di Bruno formula An alternative treatment of the implicit function theorem Topologies on the space of real analytic functions * The Weierstrass Preparation Theorem. Reference text for self-study or classroom.

Inhalt
1 Elementary Properties.- 1.1 Basic Properties of Power Series.- 1.2 Analytic Continuation.- 1.3 The Formula of Faà di Bruno.- 1.4 Composition of Real Analytic Functions.- 1.5 Inverse Functions.- 2 Multivariable Calculus of Real Analytic Functions.- 2.1 Power Series in Several Variables.- 2.2 Real Analytic Functions of Several Variables.- 2.3 The Implicit function Theorem.- 2.4 A Special Case of the Cauchy-Kowalewsky Theorem.- 2.5 The Inverse function Theorem.- 2.6 Topologies on the Space of Real Analytic Functions.- 2.7 Real Analytic Submanifolds.- 2.8 The General Cauchy-Kowalewsky Theorem.- 3 Classical Topics.- 3.0 Introductory Remarks.- 3.1 The Theorem ofPringsheim and Boas.- 3.2 Besicovitch's Theorem.- 3.3 Whitney's Extension and Approximation Theorems.- 3.4 The Theorem of S. Bernstein.- 4 Some Questions of Hard Analysis.- 4.1 Quasi-analytic and Gevrey Classes.- 4.2 Puiseux Series.- 4.3 Separate Real Analyticity.- 5 Results Motivated by Partial Differential Equations.- 5.1 Division of Distributions I.- 5.2 Division of Distributions II.- 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.