Author	Krantz, Steven G. author
Title	A Primer of Real Analytic Functions [electronic resource] / by Steven G. Krantz, Harold R. Parks
Imprint	Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2002
Edition	Second Edition
Connect to	http://dx.doi.org/10.1007/978-0-8176-8134-0
Descript	XIII, 209 p. online resource

It is a pleasure and a privilege to write this new edition of A Primer 0/ Real Anaยญ lytic Functions. The theory of real analytic functions is the wellspring of matheยญ matical analysis. It is remarkable that this is the first book on the subject, and we want to keep it up to date and as correct as possible. With these thoughts in mind, we have utilized helpful remarks and criticisms from many readers and have thereby made numerous emendations. We have also added material. There is a now a treatment of the Weierstrass preparation theorem, a new argument to establish Hensel's lemma and Puiseux's theorem, a new treatยญ ment of Faa di Bruno's forrnula, a thorough discussion of topologies on spaces of real analytic functions, and a second independent argument for the implicit funcยญ tion theorem. We trust that these new topics will make the book more complete, and hence a more useful reference. It is a pleasure to thank our editor, Ann Kostant of Birkhรคuser Boston, for makยญ ing the publishing process as smooth and trouble-free as possible. We are grateful for useful communications from the readers of our first edition, and we look forยญ ward to further constructive feedback

1 Elementary Properties -- 1.1 Basic Properties of Power Series -- 1.2 Analytic Continuation -- 1.3 The Formula of Faร  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 Besicovitchโ{128}{153}s Theorem -- 3.3 Whitneyโ{128}{153}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 Lojasiewiczโ{128}{153}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