A Primer of Real Analytic Functions

It is a pleasure and a privilege to write this new edition of A Primer 0/ Real Ana lytic Functions. The theory of real analytic functions is the wellspring of mathe 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 treat 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 func 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 Birkhäuser Boston, for mak 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 for ward to further constructive feedback.

List of contents

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.

About the author

Steven Krantz, Ph.D., is Chairman of the Mathematics Department at Washington University in St. Louis. An award-winning teacher and author, Dr. Krantz has written more than 45 books on mathematics, including Calculus Demystified, another popular title in this series. He lives in St. Louis, Missouri.

Summary

It is a pleasure and a privilege to write this new edition of A Primer 0/ Real Ana­ lytic Functions. The theory of real analytic functions is the wellspring of mathe­ 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 treat­ 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 func­ 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 Birkhäuser Boston, for mak­ 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 for­ ward to further constructive feedback.

Additional text

"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. Names such as Cauchy-Kowalewsky (Kovalevskaya), Weierstrass, Borel, Hadamard, Puiseux, Pringsheim, Besicovitch, Bernstein, Denjoy-Carleman, Paley-Wiener, Whitney, Gevrey, Lojasiewicz, Grauert and many others are involved either by their results or by their concepts."

—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

Report

"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. Names such as Cauchy-Kowalewsky (Kovalevskaya), Weierstrass, Borel, Hadamard, Puiseux, Pringsheim, Besicovitch, Bernstein, Denjoy-Carleman, Paley-Wiener, Whitney, Gevrey, Lojasiewicz, Grauert and many others are involved either by their results or by their concepts."
-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

Product details

• Authors: Steven Krantz, Steven G Krantz, Steven G. Krantz, Harold R Parks, Harold R. Parks
• Publisher: Springer, Basel
• Languages: English
• Product format: Hardback
• Released: 05.08.2002
• EAN: 9780817642648
• ISBN: 978-0-8176-4264-8
• No. of pages: 209
• Dimensions: 160 mm x 244 mm x 19 mm
• Weight: 534 g
• Illustrations: XIII, 209 p.
• Series: Birkhäuser Advanced Texts; Basler Lehrbücher; Birkhäuser Advanced Texts Basler Lehrbücher; Birkhäuser Advanced Texts; Birkhäuser Advanced Texts Basler Lehrbücher; Basler Lehrbücher
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Analysis