复分析 第4版 内容简介

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.

复分析 第4版 本书特色

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.
此书为英文版！

复分析 第4版 目录

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 Principle
CHAPTER 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 Formula
CHAPTER Ⅳ Winding Numbers and Cauchy's Theorem
CHAPTER Ⅴ Applications of Cauchy's Integral Formula
CHAPTER Ⅵ Calculus of Residues
CHAPTER Ⅶ Conformal Mappings
CHAPTER Ⅷ Harmonic Functions
PART TWO Geometric Function Theory
CHAPTER Ⅸ Schwarz Reflection
CHAPTER Ⅹ The Riemann Mapping Theorem
CHAPTER Ⅺ Analytic Continuation Along Curves
PART THREE Various Analytic Topics
CHAPTER Ⅻ Applications of the Maximum Modulus Principle and Jensen's Formula
CHAPTER ⅩⅢ Entire and Meromorphic Functions
CHAPTER ⅩⅣ Elliptic Functions
CHAPTER ⅩⅤ The Gamma and Zeta Functions
CHAPTER ⅩⅥ The Prime Number Theorem
Appendix
Bibliography
Index