復(fù)分析

內(nèi)容概要

The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra reading material for students on their own. A large number of routine exercises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students.

書籍目錄

Foreword Prerequisites PART ONE Basic Theory CHAPTER Ⅰ Complex Numbers and Functions   1. Definition   2. Polar Form   3. Complex Valued Functions   4. Limits and Compact Sets   5. Complex Differentiability   6. The Cauchy-Riemann Equations   7. Angles Under Holomorphic Maps CHAPTER Ⅱ Power Series   1. Formal Power Series   2. Convergent Power Series   3. Relations Between Formal and Convergent Series   4. Analytic Functions  5. Differentiation of Power Series  6. The Inverse and Open Mapping Theorems  7. The Local Maximum Modulus PrincipleCHAPTER Ⅲ Cauchy's Theorem,First Part  1. Holomorphic Functions on Connected Sets  2. Integrals Over Paths  3. Local Primitive for a Holomorphic Function  4. Local Primitive for a Holomorphic Function  5. The Homotopy Form of Cauchy's Theorem  6. Existence of Global Primitives.Definition of the Logarithm  7. The Local Cauchy FormulaCHAPTER Ⅳ Winding Numbers and Cauchy's TheoremCHAPTER Ⅴ Applications of Cauchy's Integral FormulaCHAPTER Ⅵ Calculus of ResiduesCHAPTER Ⅶ Conformal MappingsCHAPTER Ⅷ Harmonic FunctionsPART TWO Geometric Function TheoryCHAPTER Ⅸ Schwarz ReflectionCHAPTER Ⅹ The Riemann Mapping TheoremCHAPTER Ⅺ Analytic Continuation Along CurvesPART THREE Various Analytic TopicsCHAPTER Ⅻ Applications of the Maximum Modulus Principle and Jensen's FormulaCHAPTER ⅩⅢ Entire and Meromorphic FunctionsCHAPTER ⅩⅣ Elliptic FunctionsCHAPTER ⅩⅤ The Gamma and Zeta FunctionsCHAPTER ⅩⅥ The Prime Number TheoremAppendixBibliographyIndex

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