• 1.1 Complex Numbers and Their Properties
• 1.2 The Complex Plane
• 1.3 Polar Form of Complex Numbers
• 1.4 Powers and Roots
• 1.5 Sets in the Complex Plane
• 1.6 Applications
• Chapter 1 Review Quiz

• 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
• 2.4.2 The Power Function
• 2.5 Reciprocal Function
• 2.6 Limits and Continuity
• 2.6.1 Limits
• 2.6.2 Continuity
• 2.7 Applications
• Chapter 2 Review Quiz

• 3.1 Differentiability and Analyticity
• 3.2 Cauchy-Riemann Equations
• 3.3 Harmonic Functions
• 3.4 Applications
• Chapter 3 Review Quiz

• 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

• 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

• 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
• 6.6.5 Summing Infinite Series
• 6.7 Applications
• Chapter 6 Review Quiz

• 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 2.1
II Proof of the Cauchy-Goursat Theorem
III Table of Conformal Mappings