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A Primer of Real Analytic Functions (Birkhäuser Advanced Texts Basler Lehrbücher) Hardcover – 1 Jul 2002
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"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."
"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
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, Faa de 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 Faa de Bruno formula; Topologies on the space of real analytic functions; Alternative characterizations of real analytic functions; Surjectivity of partial differential operators; The Weierstrass Preparation Theorem
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