This basic book on functions of a complex variable represents the irreducible minimum of what every scientist and engineer should know about this important subject. From a preliminary discussion of complex numbers and functions to key topics such as the Cauchy theory, power series, and residues, distinguished mathematical writer Richard Silverman presents the fundamentals of complex analysis in a concise manner designed not to overwhelm the beginner. The author's lively style and simplicity of approach enable the reader to grasp essential topics without being distracted by secondary issues.Contents include: Complex Numbers; Some Special Mapping; Limits in the Complex Plane; Multiple-Valued Functions' Complex Functions; Taylor Series; Differentiation in the Complex Plane; Laurent Series; Integration in the Complex Plane; Applications of Residues; Complex Series; Mapping of Polygonal Domains; Power Series; and Some Physical Applications.Abundant exercise material and examples, as well as section-by-section comments at the end of each chapter make this book especially valuable to students and anyone encountering complex analysis for the first time.

Table of Contents:
Preface 1. Complex Numbers 1.1. Basic Concepts 1.2. The Complex Plane 1.3. The Modulus and Argument 1.4. Inversion Comments Problems 2. Limits in the Complex Plane 2.1. The Principle of Nested Rectangles 2.2. Limit Points 2.3. Convergent Complex Sequences 2.4. The Riemann Sphere and the Extended Complex Plane Comments Problems 3. Complex Functions 3.1. Basic Concepts 3.2. Curves and Domains 3.3. Continuity of a Complex Function 3.4. Uniform Continuity Comments Problems 4. Differentiation in the Complex Plane 4.1. The Derivative of a Complex Function 4.2. The Cauchy-Riemann Equations 4.3. Conformal Mapping Comments Problems 5. Integration in the Complex Plane 5.1. The Integral of a Complex Function 5.2. Basic Properties of the Integral 5.3. Integrals along Polygonal Curves 5.4. Cauchy's Integral Theorem 5.5. Indefinite Complex Integrals 5.6. Cauchy's Integral Formula 5.7. Infinite Differentiability of Analytic Functions 5.8. Harmonic Functions Comments Problems 6. Complex Series 6.1. Convergence vs. Divergence 6.2. Absolute vs. Conditional Convergence 6.3. Uniform Convergence Comments Problems 7. Power Series 7.1. Basic Theory 7.2. Determination of the Radius of Convergence Comments Problems 8. Some Special Mappings 8.1. The Exponential and Related Functions 8.2. Fractional Linear Transformations Comments Problems 9. Multiple-Valued Functions 9.1. Domains of Univalence 9.2. Branches and Branch Points 9.3. Riemann Surfaces Comments Problems 10. Taylor Series 10.1. The Taylor Expansion of an Analytic Function 10.2. Uniqueness Theorems 10.3. The Maximum Modulus Principle and Its Implications Comments Problems 11. Laurent Series 11.1. The Laurent Expansion of an Analytic Function 11.2. Isolated Singular Points 11.3. Residues Comments Problems 12. Applications of Residues 12.1. Logarithmic Residues and the Argument Principle 12.2. Rouche's Theorem and Its Implications 12.3. Evaluation of Improper Real Integrals 12.4. Integrals Involving Multiple-Valued Functions Comments Problems 13. Further Theory 13.1. More on Harmonic Functions 13.2. The Dirichlet Problem 13.3. More Conformal Mapping 13.4. Analytic Continuation 13.5. The Symmetry Principle Comments Problems 14. Mapping of Polygonal Domains 14.1. The Schwarz-Christoffel Transformation 14.2. Examples Comments Examples 15. Some Physical Applications 15.1. Fluid Dynamics 15.2. Examples 15.3. Electrostatics Comments Problems Selcted Hints and Answers Bibliography Index