Sections denoted with an asterisk (*) can be either omitted or read independently. Preface PartⅠ Fundamentals and Techniques of Complex Function Theory 1 Complex Numbers and Elementary Functions 1.1 Complex Numbers and Their Properties 1.2 Elementary Functions and Stereographic Projections 1.2.1 Elementary Functions 1.2.2 Stereographic Projections 1.3 Limits， Continuity， and Complex Differentiation 1.4 Elementary Applications to Ordinary Differential Equations 2 Analytic Functions and Integration 2.1 Analytic Functions 2.1.1 The Cauchy-Riemann Equations 2.1.2 Ideal Fluid Flow 2.2 Multivalued Functions *2.3 More Complicated Multivalued Functions and Riemann Surfaces 2.4 Complex Integration 2.5 Cauchys Theorem 2.6 Cauchys Integral Formula， Its a Generalization and Consequences 2.6.1 Cauchys Integral Formula and Its Derivatives *2.6.2 Liouville， Morera， and Maximum-Modulus Theorems *2.6.3 Generalized Cauchy Formula and a Derivatives *2.7 Theoretical Developments 3 Sequences， Series， and Singularities of Complex Functions 3.1 Definitions and Basic Properties of Complex Sequences，Series 3.2 Taylor Series 3.3 Laurent Series *3.4 Theoretical Results for Sequences and Series 3.5 Singularities of Complex Functions 3.5.1 Analytic Continuation and Natural Barriers *3.6 Infinite Products and Mittag-Leffler Expansions *3.7 Differential Equations in the Complex Plane： Painleve Equations *3.8 Computational Methods *3.8.1 Laurent Series *3.8.2 Differential Equations 4 Residue Calculus and Applications of Contour Integration 4.1 Cauchy Residue Theorem 4.2 Evaluation of Certain Definite Integrals 4.3 Principal Value Integrals and Integrals with Branch Points 4.3.1 Principal Value Integrals 4.3.2 Integrals with Branch Points 4.4 The Argument Principle， Rouches Theorem *4.5 Fourier and Laplace Transforms *4.6 Applications of Transforms to Differential Equations PartⅡ Applications of Complex Function Theory 5 Conformal Mappings and Applications 5.1 Introduction 5.2 Conformal Transformations 5.3 Critical Points and Inverse Mappings 5.4 Physical Applications *5.5 Theoretical Considerations - Mapping Theorems 5.6 The Schwarz-Christoffel Transformation 5.7 Bilinear Transformations *5.8 Mappings Involving Circular Arcs 5.9 Other Considerations 5.9.1 Rational Functions of the Second Degree 5.9.2 The Modulus of a Quadrilateral *5.9.3 Computational Issues 6 Asymptotic Evaluation of Integrals 6.1 Introduction 6.1.1 Fundamental Concepts 6.1.2 Elementary Examples 6.2 Laplace Type Integrals 6.2.1 Integration by Parts 6.2.2 Watsons Lemma 6.2.3 Laplaces Method 6.3 Fourier Type Integrals 6.3.1 Integration by Parts 6.3.2 Analog of Watsons Lcmma 6.3.3 The Stationary Phase Method 6.4 The Method of Steepest Descent 6.4.1 Laplaces Method for Complex Contours 6.5 Applications 6.6 The Stokes Phenomenon *6.6.1 Smoothing of Stokes Discontinuities 6.7 Related Techniques *6.7.1 WKB Method *6.7.2 The Mellin Transform Method 7 Riemann-Hiibert Problems 7.1 Introduction 7.2 Cauchy Type Integrals 7.3 Scalar Riemann-Hilbert Problems 7.3.1 Closed Contours 7.3.2 Open Contours 7.3.3 Singular Integral Equations 7.4 Applications of Scalar Riemann-Hilbert Problems 7.4.1 Riemann-Hilbert Problems on the Real Axis 7.4.2 The Fourier Transform 7.4.3 The Radon Transform *7.5 Matrix Riemann-Hilbert Problems 7.5.1 The Riemann-Hilbert Problem for Rational Matrices 7.5.2 Inhomogeneous Riemann-Hilbert Problems and Singular Equations 7.5.3 The Riemann-Hilbert Problem for Triangular Matrices 7.5.4 Some Results on Zero Indices 7.6 The DBAR Problem 7.6.1 Generalized Analytic Functions *7.7 Applications of Matrix Riemann-Hilbert Problems and Problems Appendix A Answers to Odd-Numbered Exercises Bibliography Index
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