Complex Variables with Applications

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Complex Variables with Applications David Wunsch Copyright© 2005 Pearson Education, Inc ISBN 0-321-22322-5 Contents Introduction 1. Complex Numbers 1.1 Introduction 1.2 More Properties of Complex Numbers 1.3 Complex Numbers and the Argand Plane 1.4 Integer and Fractional Powers of Complex Numbers 1.5 Points, Sets, Loci, and Regions in the Complex Plane 2. The Complex Function and Its Derivative 2.1 Introduction 2.2 Limits and Continuity 2.3 The Complex Derivative 2.4 The Derivative and Analyticity 2.5 Harmonic Functions 2.6 Some Physical Applications of Harmonic Functions 3. The Basic Transcendental Functions 3.1 The Exponential Function 3.2 Trigonometric Functions 3.3 Hyperbolic Functions 3.4 The Logarithmic Function 3.5 Analyticity of the Logarithmic Function 3.6 Complex Exponentials 3.7 Inverse Trigonometric and Hyperbolic Functions 3.8 More on Branch Cuts and Branch Points Appendix to Chapter 3: Phasors 4. Integration in the Complex Plane 4.1 Introduction to Line Integration 4.2 Complex Line Integration 4.3 Contour Integration and Green's Theorem 4.4 Path Independence, Indefinite Integrals, Fundamental Theorem of Calculus in the Complex Plane 4.5 The Cauchy Integral Formula and Its Extension 4.6 Some Applications of the Cauchy Integral Formula 4.7 Introduction to Dirichlet Problems-The Poisson Integral Formulafor the Circle and Half Plane Appendix: Green's Theorem in the Plane 5. Infinite Series Involving a Complex Variable 5.1 Introduction and Review of Real Series 5.2 Complex Sequences and Convergence of Complex Series 5.3 Uniform Convergence of Series 5.4 Power Series and Taylor Series 5.5 Techniques for Obtaining Taylor Series Expansions 5.6 Laurent Series 5.7 Properties of Analytic Functions Related to Taylor Series: Isolation of Zeros, Analytic Continuation, Zeta Function, Reflection 5.8 The z Transformation Appendix: Fractals and the Mandelbrot Set 6. Residues and Their Use in Integration 6.1 Introduction and Definition of the Residue 6.2 Isolated Singularities 6.3 Finding the Residue 6.4 Evaluation of Real Integrals with Residue Calculus, I 6.5 Evaluation of Integrals, II 6.6 Evaluation of Integrals, III 6.7 Integrals Involving Indented Contours 6.8 Contour Integrations Involving Branch Points and Branch Cuts 6.9 Residue Calculus Applied to Fourier Transforms 6.10 The Hilbert Transform 6.11 Uniform Convergence of Integrals and the Gamma Function 6.12 Principle of the Argument 7. Laplace Transforms and Stability of Systems 7.1 Laplace Transforms and Their Inversion 7.2 Stability-An Introduction 7.3 The Nyquist Stability Criterion 7.4 Generalized Functions, Laplace Transforms, and Stability 8. Conformal Mapping and Some of Its Applications 8.1 Introduction 8.2 The Conformal Property 8.3 One-to-One Mappings and Mappings of Regions 8.4 The Bilinear Transformation 8.5 Conformal Mapping and Boundary Value Problems 8.6 More on Boundary Value Problems-Streamlines as Boundaries 8.7 Boundary Value Problems with Sources 8.8 The Schwarz-Christoffel Transformation Appendix: The Stream Function and Capacitance 9. Advanced Topics in Infinite Series and Products 9.1 The Use of Residues to Sum Certain Numerical Series 9.2 Partial Fraction Expansions of Functions with an Infinite Number of Poles 9.3 Introduction to Infinite Products 9.4 Expanding Functions in Infinite Products Solutions to Odd-Numbered Exercises Index