A Primer of Real Analytic FunctionsIt 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. |
Contents
Elementary Properties | 1 |
Multivariable Calculus of Real Analytic Functions | 25 |
Classical Topics 67 | |
Some Questions of Hard Analysis 83 | |
Results Motivated by Partial Differential Equations 115 | |
Topics in Geometry 151 | |
Bibliography 187 | |
Index 203 | |
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algebraic analysis analytic variety apply assume Bk+1 bundle C₁ Cauchy-Kowalewsky theorem common factor compact complex analytic connected component convex coordinate Corollary Definition denote derivatives dimension embedding estimate Euclidean exist f is real follows formula Fourier transform func function defined function f Gevrey class go(y Hensel's lemma ho(y holds holomorphic functions implicit function theorem induction infinitely differentiable inverse function theorem k₁ Lemma Let f linear multiindex multiindices neighborhood nonnegative integers notation obtain open interval open set open subset partial differential equations polynomial positive integer power series Proposition prove Puiseux's theorem quasi-analytic radius of convergence real analytic functions real analytic manifold real analytic submanifold real numbers real roots result semianalytic series converges smooth space subanalytic Suppose tangent tempered distribution tion vanish identically vector Weierstrass preparation theorem zero set ΣΣ дул дуг дуп