A Primer of Real Analytic Functions

by Krantz, Steven G.; Parks, Harold R.

• ISBN13: 9780817642648
• ISBN10: 0817642641
• Edition: 2nd
• Format: Hardcover
• Copyright: 2002-07-01
• Publisher: Birkhauser

Summary

The subject of real analytic functions is one of the oldest in modern mathematics and is the wellspring of the theory of analysis, both real and complex. To date, there is no comprehensive book on the subject, yet the tools of the theory are widely used by mathematicians today. Key topics in the theory of real analytic functions that are covered in this text and are rather difficult to pry out of the literature include: the real analytic implicit function theorem, resolution of singularities, the FBI transform, semi-analytic sets, Faà di Bruno's formula and its applications, zero sets of real analytic functions, Lojaciewicz's theorem, Puiseaux's theorem. New to this second edition are such topics as:* 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 This well organized and clearly written advanced textbook introduces students to real analytic functions of one or more real variables in a systematic fashion. The first part focuses on elementary properties and classical topics and the second part is devoted to more difficult topics. Many historical remarks, examples, references and an excellent index should encourage student and researcher alike to further study this valuable and exciting theory.

Table of Contents

Preface to the Second Edition	ix
Preface to the First Edition	xi
Elementary Properties	1	(24)
Basic Properties of Power Series	1	(10)
Analytic Continuation	11	(5)
The Formula of Faa di Bruno	16	(2)
Composition of Real Analytic Functions	18	(2)
Inverse Functions	20	(5)
Multivariable Calculus of Real Analytic Functions	25	(42)
Power Series in Several Variables	25	(4)
Real Analytic Functions of Several Variables	29	(6)
The Implicit Function Theorem	35	(7)
A Special Case of the Cauchy-Kowalewsky Theorem	42	(5)
The Inverse Function Theorem	47	(3)
Topologies on the Space of Real Analytic Functions	50	(4)
Real Analytic Submanifolds	54	(7)
Bundles over a Real Analytic Submanifold	56	(5)
The General Cauchy-Kowalewsky Theorem	61	(6)
Classical Topics	67	(16)
Introductory Remarks	67	(1)
The Theorem of Pringsheim and Boas	68	(4)
Besicovitch's Theorem	72	(3)
Whitney's Extension and Approximation Theorems	75	(4)
The Theorem of S. Bernstein	79	(4)
Some Questions of Hard Analysis	83	(32)
Quasi-analytic and Gevrey Classes	83	(12)
Puiseux Series	95	(9)
Separate Real Analyticity	104	(11)
Results Motivated by Partial Differential Equations	115	(36)
Division of Distributions I	115	(11)
Projection of Polynomially Defined Sets	117	(9)
Division of Distributions II	126	(9)
The FBI Transform	135	(9)
The Paley-Wiener Theorem	144	(7)
Topics in Geometry	151	(36)
The Weierstrass Preparation Theorem	151	(5)
Resolution of Singularities	156	(10)
Lojasiewicz's Structure Theorem for Real Analytic Varieties	166	(5)
The Embedding of Real Analytic Manifolds	171	(6)
Semianalytic and Subanalytic Sets	177	(10)
Basic Definitions	177	(2)
Facts Concerning Semianalytic and Subanalytic Sets	179	(2)
Examples and Discussion	181	(3)
Rectilinearization	184	(3)
Bibliography	187	(16)
Index	203