.

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BK

Providence : American Mathematical Society, [2015]

xviii, 641 stran : ilustrace ; 27 cm

ISBN 978-1-4704-1100-8 (vázáno)

Obsahuje bibliografii na stranách 591-622 a rejstříky

001458376

Contents // Preface to the Series xi // Preface to Part 2 // Chapter 1. Preliminaries 1 // §1.1. Notation and Terminology 1 // §1.2. Complex Numbers 3 // §1.3. Some Algebra, Mainly Linear 5 // §1.4. Calculus on R and Rn 8 // §1.5. Differentiable Manifolds 12 // §1.6. Riemann Metrics 18 // §1.7. Homotopy and Covering Spaces 21 // §1.8. Homology 24 // §1.9. Some Results from Real Analysis 26 // Chapter 2. The Cauchy Integral Theorem: Basics 29 // §2.1. Holomorphic Functions 30 // §2.2. Contour Integrals 40 // §2.3. Analytic Functions 49 // §2.4. The Goursat Argument 66 // §2.5. The CIT for Star-Shaped Regions 69 // §2.6. Holomorphically Simply Connected Regions, Logs, and // Fractional Powers 71 // §2.7. The Cauchy Integral Formula for Disks and Annuli 76 // vii // Contents // viii // Chapter 3. Consequences of the Cauchy Integral Formula 79 // §3.1. Analyticity and Cauchy Estimates 80 // §3.2. An Improved Cauchy Estimate 93 // §3.3. The Argument Principle and Winding Numbers 95 // §3.4. Local Behavior at Noncritical Points 104 // §3.5. Local Behavior at Critical Points 108 // §3.6. The Open Mapping and Maximum Principle 114 // §3.7. Laurent Series 120 // §3.8. The Classification of Isolated Singularities; // Casorati-Weierstrass Theorem 124 // §3.9. Meromorphic Functions 128 // §3.10. Periodic Analytic Functions 132 // Chapter 4. Chains and the Ultimate Cauchy Integral Theorem 137 // §4.1. Homologous Chains 139 // §4.2. Dixon’s Proof of the Ultimate CIT 142 // §4.3. The Ultimate Argument Principle 143 // §4.4. Mesh-Defined Chains 145 // §4.5. Simply Connected and Multiply Connected Regions 150 // §4.6. The Ultra Cauchy Integral Theorem and Formula 151 // §4.7. Runge’s Theorems 153 // §4.8. The Jordan Curve Theorem for Smooth Jordan Curves 161 // Chapter 5. More Consequences of the CIT 167 // §5.1. The Phragmén -Lindelôf Method 168 //

§5.2. The Three-Line Theorem and the Riesz-Thorin // Theorem 174 // §5.3. Poisson Representations 177 // §5.4. Harmonic Functions 183 // §5.5. The Reflection Principle 189 // §5.6. Reflection in Analytic Arcs; Continuity at Analytic // §5.7. Calculation of Definite Integrals 201 // Chapter 6. Spaces of Analytic Functions 227 // §6.1. Analytic Functions as a Fréchet Space 228 // §6.2. Montel’s and Vitali’s Theorems 234 // §6.3. Restatement of Runge’s Theorems 244 // §6.4. Hurwitz’s Theorem 245 // §6.5. Bonus Section: Normal Convergence of Meromorphic // Functions and Marty’s Theorem 247 // Chapter 7. Fractional Linear Transformations 255 // §7.1. The Concept of a Riemann Surface 256 // §7.2. The Riemann Sphere as a Complex Projective Space 267 // §7.3. PSL(2,C) 273 // §7.4. Self-Maps of the Disk 289 // §7.5. Bonus Section: Introduction to Continued Fractions // and the Schur Algorithm 295 // Chapter 8. Conformal Maps 309 // §8.1. The Riemann Mapping Theorem 310 // §8.2. Boundary Behavior of Riemann Maps 319 // §8.3. First Construction of the Elliptic Modular Function 325 // §8.4. Some Explicit Conformal Maps 336 // §8.5. Bonus Section: Covering Map for General Regions 353 // §8.6. Doubly Connected Regions 357 // §8.7. Bonus Section: The Uniformization Theorem 362 // §8.8. Ahlfors’ Function, Analytic Capacity and the Painlevé // Problem 371 // Chapter 9. Zeros of Analytic Functions and Product Formulae 381 // §9.1. Infinite Products 383 // §9.2. A Warmup: The Euler Product Formula 387 // §9.3. The Mittag-Leffler Theorem 399 // §9.4. The Weierstrass Product Theorem 401 // §9.5. General Regions 406 // §9.6. The Gamma Function: Basics 410 // §9.7. The Euler-Maclaurin Series and Stirling’s Approximation 430 // §9.8. Jensen’s Formula 448 // §9.9. Blaschke Products 451 // §9.10. Entire Functions of Finite Order and the Hadamard Product Formula 459 //

§9.10. Entire Functions of Finite Order and the Hadamard Product Formula 459 // Chapter 10. Elliptic Functions 475 // §10.1. A Warmup: Meromorphic Functions on C 480 // §10.2. Lattices and §L(2, Z) 481 // §10.3. Liouville’s Theorems, Abel’s Theorem, and Jacobi’s // Construction 491 // §10.4. Weierstrass Elliptic Functions 501 // §10.5. Bonus Section: Jacobi Elliptic Functions 522 // §10.6. The Elliptic Modular Function 542 // §10.7. The Equivalence Problem for Complex Tori 552 // Chapter 11. Selected Additional Topics 555 // §11.1. The Paley-Wiener Strategy 557 // §11.2. Global Analytic Functions 564 // §11.3. Picard’s Theorem via the Elliptic Modular Function 570 // §11.4. Bonus Section: Zalcman’s Lemma and Picard’s // Theorem 575 // §11.5. Two Results in Several Complex Variables: // Hartogs’ Theorem and a Theorem of Poincaré 580 // §11.6. Bonus Section: A First Glance at // Compact Riemann Surfaces 586 // Bibliography 591 // Symbol Index 623 // Subject Index 525 // Author Index 633 // Index of Capsule Biographies 641