مقدمه‌ای بر تحلیل مختلط کلاسیک: جلد اول

Main subject categories: • Complex analysis • Curves • Connectedness • Convexivity • Complex derivative • Curvilinear integrals • Power series • Exponential functions • Cauchy-Goursat Theorem • Schwarz' Lemma • Convergent sequences of holomorphic functions • Polynomial and rational approximation ‒ Runge Theory • Riemann Mapping Theorem • Simple and double connectivity • Isolated singularities • Omitted values and normal familiesThe exposition of the book is excellent with outstanding clarity. An Introduction to Classical Complex Analysis......Page 4 Copyright Page......Page 5 Contents......Page 6 Preface......Page 10 § 1 Set Theory......Page 14 § 3 The Battlefield......Page 15 § 4 Metric Spaces......Page 16 § 5 Limsup and All That......Page 19 § 6 Continuous Functions......Page 21 § 7 Calculus......Page 22 § 1 Elementary Results on Connectedness......Page 23 § 2 Connectedness of Intervals, Curves and Convex Sets......Page 24 § 3 The Basic Connectedness Lemma......Page 29 § 4 Components and Compact Exhaustions......Page 30 § 5 Connectivity of a Set......Page 34 § 6 Extension Theorems......Page 38 Notes to Chapter I......Page 40 § 1 Holomorphic and Harmonic Functions......Page 42 § 2 Integrals along Curves......Page 45 § 3 Differentiating under the Integral......Page 48 § 4 A Useful Sufficient Condition for Differentiability......Page 50 Notes to Chapter II......Page 51 § 2 Power Series......Page 54 § 3 The Complex Exponential Function......Page 61 § 4 Bernoulli Polynomials, Numbers and Functions......Page 74 § 5 Cauchy's Theorem Adumbrated......Page 78 § 6 Holomorphic Logarithms Previewed......Page 79 Notes to Chapter III......Page 81 § 2 Curves Winding around Points......Page 84 § 3 Homotopy and the Index......Page 91 § 4 Existence of Continuous Logarithms......Page 93 § 5 The Jordan Curve Theorem......Page 103 § 6 Applications of the Foregoing Technology......Page 107 § 7 Continuous and Holomorphic Logarithms in Open Sets......Page 112 § 8 Simple Connectivity for Open Sets......Page 114 Notes to Chapter IV......Page 116 § 1 Goursat’s Lemma and Cauchy’s Theorem for Starlike Regions......Page 121 § 2 Maximum Principles......Page 128 § 3 The Dirichlet Problem for Disks......Page 135 § 4 Existence of Power Series Expansions......Page 145 § 5 Harmonic Majorization......Page 152 § 6 Uniqueness Theorems......Page 166 § 7 Local Theory......Page 173 Notes to Chapter V......Page 184 § 1 Schwarz’ Lemma and the Conformal Automorphisms of Disks......Page 192 § 2 Many-to-one Maps of Disks onto Disks......Page 198 § 3 Applications to Half-planes, Strips and Annuli......Page 199 § 4 The Theorem of Carathéodory, Julia, Wolff, et al.......Page 204 § 5 Subordination......Page 208 Notes to Chapter VI......Page 216 § 1 Convergence in H(U)......Page 219 § 2 Applications of the Convergence Theorems; Boundedness Criteria......Page 229 § 3 Prescribing Zeros......Page 238 § 4 Elementary Iteration Theory......Page 243 Notes to Chapter VII......Page 252 § 1 The Basic Integral Representation Theorem......Page 257 § 2 Applications to Approximation......Page 261 § 3 Other Applications of the Integral Representation......Page 266 § 4 Some Special Kinds of Approximation......Page 269 § 5 Carleman’s Approximation Theorem......Page 274 § 6 Harmonic Functions in a Half-plane......Page 277 Notes to Chapter VIII......Page 290 § 1 Introduction......Page 294 § 2 The Proof of Carathéodory and Koebe......Page 299 § 4 Boundary Behavior for Jordan Regions......Page 304 § 5 A Few Applications of the Osgood–Taylor–Carathéodory Theorem......Page 311 § 6 More on Jordan Regions and Boundary Behavior......Page 316 § 7 Harmonic Functions and the General Dirichlet Problem......Page 323 § 8 The Dirichlet Problem and the Riemann Mapping Theorem......Page 334 Notes to Chapter IX......Page 338 § 1 Simple Connectivity......Page 345 § 2 Double Connectivity......Page 349 Notes to Chapter X......Page 356 § 1 Laurent Series and Classification of Singularities......Page 360 § 2 Rational Functions......Page 367 § 3 Isolated Singularities on the Circle of Convergence......Page 376 § 4 The Residue Theorem and Some Applications......Page 378 § 5 Specifying Principal Parts—Mittag-Leffler’s Theorem......Page 391 § 6 Meromorphic Functions......Page 396 § 7 Poisson’s Formula in an Annulus and Isolated Singularities of Harmonic Functions......Page 399 Notes to Chapter XI......Page 407 § 1 Logarithmic Means and Jensen’s Inequality......Page 412 § 2 Miranda’s Theorem......Page 418 § 3 Immediate Applications of Miranda......Page 433 § 4 Normal Families and Julia’s Extension of Picard’s Great Theorem......Page 437 § 5 Sectorial Limit Theorems......Page 442 § 6 Applications to Iteration Theory......Page 451 § 7 Ostrowski’s Proof of Schottky’s Theorem......Page 452 Notes to Chapter XII......Page 457 Bibliography......Page 463 Name Index......Page 545 Subject Index......Page 555 Symbol Index......Page 569 Series Summed......Page 570 Integrals Evaluated......Page 571 This book is an attempt to cover some of the salient features of classical, one variable complex function theory. The approach is analytic, as opposed to geometric, but the methods of all three of the principal schools (those of Cauchy, Riemann and Weierstrass) are developed and exploited. The book goes deeply into several topics (e.g. convergence theory and plane topology), more than is customary in introductory texts, and extensive chapter notes give the sources of the results, trace lines of subsequent development, make connections with other topics, and offer suggestions for further reading. These are keyed to a bibliography of over 1,300 books and papers, for each of which volume and page numbers of a review in one of the major reviewing journals is cited. These notes and bibliography should be of considerable value to the expert as well as to the novice. For the latter there are many references to such thoroughly accessible journals as the American Mathematical Monthly and L'Enseignement Mathématique. Moreover, the actual prerequisites for reading the book are quite modest; for example, the exposition assumes no prior knowledge of manifold theory, and continuity of the Riemann map on the boundary is treated without measure theory. By Robert B. Burckel. North And South America Edition Published By Academic Press ... (pure And Applied Mathematics, A Series Of Monographs And Textbooks, Vol. 82) Includes Indexes. Bibliography: V. 1, P. 462-543. Covers some of the salient features of classical, one variable complex function theory. This book develops and exploits methods of all three of the principal schools (those of Cauchy, Riemann and Weierstrass).