复变函数论

前言Chapter I Complex Numbers and Functions1 Complex Numbers1.1 Complex Number Field1.2 Complex Plane1.3 Modulus， Conjugation， Argument， Polar Representation1.4 Powers and Roots of Complex NumbersExercises2 Regions in the Complex Plane2.1 Some Basic Concept2.2 Domain and Jordan CurveExercises3 Functions of a Complex Variable3.1 The Concept of Functions of a Complex Variable3.2 Limits and ContinuousExercises4 The Extended Complex Plane and the Point at Infinity4.1 The Spherical Representation， the Extended Complex Plane4.2 Some Concepts in the Extended Complex PlaneExercisesChapter II Analytic Functions1 The Concept of the Analytic Function1.1 The Derivative of the Functions of a Complex Variable1.2 Analytic FunctionsExercises2 Cauchy-Riemann EquationsExercises3 Elementary Functions3.1 The Exponential Function3.2 Trigonometric Functions3.3 Hyperbolic FunctionsExercises4 Multi-Valued Functions4.1 The Logarithmic Function4.2 Complex Power Functions4. 3 Inverse Trigonometric and Hyperbolic FunctionsExercisesChapter III Complex Integration1 The Concept of Contour Integrals1.1 Integral of a Complex Function over a Real Interval1.2 Contour IntegralsExercisesCauchy-Goursat Theorem2.1 Cauchy Theorem2.2 Cauchy Integral Formula2.3 Derivatives of Analytic Functions2.4 Liouville's Theorem and the Fundamental Theorem of AlgebraExercisesHarmonic FunctionsExercisesChapter IV Series1 Basic Properties of Series1.1 Convergence of Sequences1.2 Convergence of Series1.3 Uniform convergenceExercises2 Power SeriesExercises3 Taylor SeriesExercises4 Laurent SeriesExercises5 Zeros of an Analytic Functions and Uniquely Determined AnalyticFunctions5.1 Zeros of Analytic Functions5.2 Uniquely Determined Analytic Functions5.3 Maximum Modulus PrincipleExercises6 The Three Types of Isolated Singular Points at a Finite PointExercises7 The Three Types of Isolated Singular Points at a Infinite PointExercisesChapter V Calculus of Residues1 Residues1.1 Residues1.2 Cauchy's Residue Theorem1.3 The Calculus of ResidueExercises2 Applications of Residue2.1 The Type of Definite Integral □2.2 The Type of Improper Integral □2.3 The Type of Improper Integral □Exercises3 Argument PrincipleExercisesChapter VI Conformal Mappings1 Analytic Transformation1.1 Preservation of Domains of Analytic Transformation1.2 Conformality of Analytic TransformationExercises2 Rational Functions2.1 Polynomials2.2 Rational FunctionsExercises3 Fractional Linear TransformationsExercises4 Elementary Conformal MappingsExercises5 The Riemann Mapping TheoremExercisesAppendixAppendix 1Appendix 2AnswersBibliography

《复变函数论（第二版）/山东省双语示范课程教材》共6章，主要内容包括复数与复变函数、解析函数、复变函数的积分、级数、留数及其应用和共形映射等，较全面、系统地介绍了复变函数的基础知识。内容处理上重点突出、叙述简明，每节末附有适量习题供读者选用，适合高等师范院校数学系及普通综合性大学数学系高年级学生使用。