Preface Chapter 1. Complex Numbers and the Complex Plane 1.1 Complex Numbers and Their Properties 1.2 Complex Plane 1.3 Polar Form of Complex Numbers 1.4 Powers and Roots 1.5 Sets of Points in the Complex Plane 1.6 Applications Chapter 1 Review Quiz Chapter 2. Complex Functions and Mappings 2.1 Complex Functions 2.2 Complex Functions as Mappings 2.3 Linear Mappings 2.4 Special Power Functions 2.4.1 The Power Function z 2.4.2 The Power Function zl 2.5 Reciprocal Function 2.6 Applications Chapter 2 Review Quiz Chapter 3. Analytic Functions 3.1 Limits and Continuity 3.1.1 Limits 3.1.2 Continuity 3.2 Differentiability and Analyticity 3.3 Cauchy-Riemann Equations 3.4 Harmonic Functions 3.5 Applications Chapter 3 Review Quiz Chapter 4. Elementary Functions 4.1 Exponential and Logarithmic Functions 4.1.1 Complex Exponential Function 4.1.2 Complex Logarithmic Function 4.2 Complex Powers 4.3 Trigonometric and Hyperbolic Functions 4.3.1 Complex Trigonometric Functions 4.3.2 Complex Hyperbolic Functions 4.4 Inverse Trigonometric and Hyperbolic Functions 4.5 Applications Chapter 4 Review Quiz Chapter 5. Integration in the Complex Plane 5.1 Real Integrals 5.2 Complex Integrals 5.3 Cauchy-Goursat Theorem 5.4 Independence of Path 5.5 Cauchy's Integral Formulas and Their Consequences 5.5.1 Cauchy's Two Integral Formulas 5.5.2 Some Consequences of the Integral Formulas 5.6 Applications
Chapter 5 Review Quiz Chapter S. Series and Residues 6.1 Sequences and Series 6.2 Taylor Series 6.3 Laurent Series 6.4 Zeros and Poles 6.5 Residues and Residue Theorem 6.6 Some Consequences of the Residue Theorem 6.6.1 Evaluation of Real Trigonometric Integrals 6.6.2 Evaluation of Real Improper Integrals 6.6.3 Integration along a Branch Cut 6.6.4 The Argument Principle and Rouch6's Theorem 6.6.5 Summing Infinite Series 6.7 Applications Chapter 6 Review Quiz Chapter 7. Conformal Mappings 7.1 Conformal Mapping 7.2 Linear Fractional Transformations 7.3 Schwarz-Christoffel Transformations 7.4 Poisson Integral Formulas 7.5 Applications 7.5.1 Boundary-Value Problems 7.5.2 Fluid Flow Chapter 7 Review Quiz Appendixes: I Proof of Theorem 3. I II Proof of the Cauchy-Goursat Theorem III Table of Conformal Mappings Answers to Selected Odd-Numbered Problems Symbol Index Word Index
[