front cover -- copyright page -- title page -- Contents -- Introduction -- Acknowledgements -- Using Java Applets -- Using Links in the Electronic Book -- Complex Dynamics: Chaos, Fractals, the Mandelbrot Set, and More -- Introduction -- Newton's Method -- Iteration of an Analytic Function -- Critical Points and Critical Orbits -- Exploring the Mandelbrot Set -- Transcendental Dynamics -- The Mandelbrot Set is Universal -- Concluding Remarks and New Directions -- Additional Exercises -- Bibliography -- Appendix: Definitions and Properties of the Julia and Fatou Sets -- Appendix: Global Conjugation and Möbius Map Dynamics -- Appendix: Code for Drawing Random Dynamics Pictures -- Soap Films, Differential Geometry, and Minimal Surfaces -- Introduction -- Differential geometry -- Minimal surfaces -- Weierstrass representation -- The Gauss map, G, and height differential, dh -- Minimal surfaces and harmonic univalent mappings -- Conclusion -- Additional exercises -- Bibliography -- Applications to Flow Problems -- Introduction -- Background and Fundamental Results -- Complex Functions and Vector Fields -- Complex Potential Functions -- Uniform Flows in the Plane and other Regions -- Sources and Sinks -- Flow in a Channel -- Flows in Other Regions -- Flows inside the Disk -- Interval Sources and Sinks -- Steady State Temperature Problems -- Flows with Source and Sinks not on the Boundary -- Vector Fields with Other Types of Singularities -- Bibliography -- Anamorphosis, Mapping Problems, and Harmonic Univalent Functions -- Introduction -- Anamorphosis and Möbius Maps -- The Family S of Analytic, Normalized, Univalent Functions -- The Family S_H of Normalized, Harmonic, Univalent Functions -- The Shearing Technique -- Properties of the Dilatation -- Harmonic Linear Combinations -- Convolutions -- Conclusion -- Additional Exercises -- Bibliography.

Mappings to Polygonal Domains -- Introduction -- Schwarz-Christoffel Maps -- The Poisson Integral Formula -- Harmonic Function Theory -- Rado-Kneser-Choquet Theorem -- Star Mappings -- Dilatations of Polygonal Maps are Blaschke Products -- An Important Univalence Theorem -- The Dilatation for Star Mappings -- Open Questions -- Bibliography -- Circle Packing -- First Impressions -- Basics of Circle Packing -- Circle Packing Manipulations -- Discrete Function Theory -- Function Construction -- Convergence -- Wrapup -- Additional Exercises -- Bibliography -- Background -- Functions of a Complex Variable as Mappings -- Continuity and Analyticity in C -- Contour Integration -- Taylor Series and Laurent Series -- Key Theorems -- More Advanced Results -- Bibliography -- The Riemann Sphere -- Stereographic Projection and Spherical Geometry -- The Spherical Metric -- Topology in C and C -- Continuity on the Riemann Sphere -- Analyticity on the Riemann Sphere -- Bibliography -- Index -- Index of Notation -- Index of Terms -- About the Authors.