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Book Description

Complex variables provide powerful methods for attacking problems that can be very difficult to solve in any other way, and it is the aim of this book to provide a thorough grounding in these methods and their application. Part I of this text provides an introduction to the subject, including analytic functions, integration, series, and residue calculus and also includes transform methods, ODEs in the complex plane, and numerical methods. Part II contains conformal mappings, asymptotic expansions, and the study of Riemann-Hilbert problems. The authors provide an extensive array of applications, illustrative examples and homework exercises. This new edition has been improved throughout and is ideal for use in undergraduate and introductory graduate level courses in complex variables.

1. Cover
2. Half Title
3. Title Page
5. Contents
6. Preface
7. Part I: Fundamentals and Techniques of Complex Function Theory
1. 1. Complex Numbers and Elementary Functions
1. 1.1 Complex Numbers and Their Properties
2. 1.2 Elementary Functions and Stereographic Projections
3. 1.3 Limits, Continuity, and Complex Differentiation
2. 2. Analytic Functions and Integration
1. 2.1 Analytic Functions
2. 2.2 Multivalued Functions
3. 2.3 More Complicated Multivalued Functions and Riemann Surfaces
4. 2.4 Complex Integration
5. 2.5 Cauchy’s Theorem
6. 2.6 Cauchy’s Integral Formula, Its ∂[Overbar] Generalization and Consequences
7. 2.7 Theoretical Developments
3. 3. Sequences, Series, and Singularities of Complex Functions
1. 3.1 Definitions and Basic Properties of Complex Sequences, Series
2. 3.2 Taylor Series
3. 3.3 Laurent Series
4. 3.4 Theoretical Results for Sequences and Series
5. 3.5 Singularities of Complex Functions
6. 3.6 Infinite Products and Mittag–Leffler Expansions
7. 3.7 Differential Equations in the Complex Plane: Painlevé Equations
8. 3.8 Computational Methods
4. 4. Residue Calculus and Applications of Contour Integration
1. 4.1 Cauchy Residue Theorem
2. 4.2 Evaluation of Certain Definite Integrals
3. 4.3 Principal Value Integrals and Integrals with Branch Points
4. 4.4 The Argument Principle, Rouché’s Theorem
5. 4.5 Fourier and Laplace Transforms
6. 4.6 Applications of Transforms to Differential Equations
8. Part II: Applications of Complex Function Theory
1. 5. Conformal Mappings and Applications
1. 5.1 Introduction
2. 5.2 Conformal Transformations
3. 5.3 Critical Points and Inverse Mappings
4. 5.4 Physical Applications
5. 5.5 Theoretical Considerations – Mapping Theorems
6. 5.6 The Schwarz–Christoffel Transformation
7. 5.7 Bilinear Transformations
8. 5.8 Mappings Involving Circular Arcs
9. 5.9 Other Considerations
2. 6. Asymptotic Evaluation of Integrals
1. 6.1 Introduction
2. 6.2 Laplace Type Integrals
3. 6.3 Fourier Type Integrals
4. 6.4 The Method of Steepest Descent
5. 6.5 Applications
6. 6.6 The Stokes Phenomenon
7. 6.7 Related Techniques
3. 7. Riemann–Hilbert Problems
1. 7.1 Introduction
2. 7.2 Cauchy Type Integrals
3. 7.3 Scalar Riemann–Hilbert Problems
4. 7.4 Applications of Scalar Riemann–Hilbert Problems
5. 7.5 Matrix Riemann–Hilbert Problems
6. 7.6 The DBAR Problem
7. 7.7 Applications of Matrix Riemann–Hilbert Problems and ∂[Overbar] Problems
9. Appendix A: Answers to Odd-Numbered Exercises
10. Bibliography
11. Index