**Classroom Resource Materials**

Volume: 40;
2012;
373 pp;
Hardcover

Print ISBN: 978-0-88385-778-6

Product Code: CLRM/40

List Price: $62.00

AMS Member Price: $46.50

MAA member Price: $46.50

**Electronic ISBN: 978-1-61444-108-3
Product Code: CLRM/40.E**

List Price: $62.00

AMS Member Price: $46.50

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# Explorations in Complex Analysis

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*Michael A. Brilleslyper; Michael J. Dorff; Jane M. McDougall; James S. Rolf; Lisbeth E. Schaubroeck; Richard L. Stankewitz; Kenneth Stephenson*

MAA Press: An Imprint of the American Mathematical Society

Research topics in the book include complex dynamics, minimal surfaces, fluid flows, harmonic, conformal, and polygonal mappings, and discrete complex analysis via circle packing. The nature of this book is different from many mathematics texts: the focus is on student-driven and technology-enhanced investigation. Interlaced in the reading for each chapter are examples, exercises, explorations, and projects, nearly all linked explicitly with computer applets for visualization and hands-on manipulation.

# Table of Contents

## Explorations in Complex Analysis

- front cover cover11
- copyright page ii3
- title page iii4
- Contents vii8
- Introduction xi12
- Complex Dynamics: Chaos, Fractals, the Mandelbrot Set, and More 120
- Introduction 120
- Newton's Method 322
- Iteration of an Analytic Function 1938
- Critical Points and Critical Orbits 3554
- Exploring the Mandelbrot Set 3958
- Transcendental Dynamics 5372
- The Mandelbrot Set is Universal 6180
- Concluding Remarks and New Directions 6180
- Additional Exercises 6685
- Bibliography 7695
- Appendix: Definitions and Properties of the Julia and Fatou Sets 7897
- Appendix: Global Conjugation and Möbius Map Dynamics 8099
- Appendix: Code for Drawing Random Dynamics Pictures 81100

- Soap Films, Differential Geometry, and Minimal Surfaces 85104
- Applications to Flow Problems 161180
- Introduction 161180
- Background and Fundamental Results 164183
- Complex Functions and Vector Fields 167186
- Complex Potential Functions 172191
- Uniform Flows in the Plane and other Regions 174193
- Sources and Sinks 176195
- Flow in a Channel 179198
- Flows in Other Regions 180199
- Flows inside the Disk 182201
- Interval Sources and Sinks 183202
- Steady State Temperature Problems 186205
- Flows with Source and Sinks not on the Boundary 190209
- Vector Fields with Other Types of Singularities 192211
- Bibliography 195214

- Anamorphosis, Mapping Problems, and Harmonic Univalent Functions 197216
- Introduction 197216
- Anamorphosis and Möbius Maps 198217
- The Family S of Analytic, Normalized, Univalent Functions 208227
- The Family S_H of Normalized, Harmonic, Univalent Functions 213232
- The Shearing Technique 221240
- Properties of the Dilatation 233252
- Harmonic Linear Combinations 240259
- Convolutions 252271
- Conclusion 262281
- Additional Exercises 263282
- Bibliography 268287

- Mappings to Polygonal Domains 271290
- Introduction 271290
- Schwarz-Christoffel Maps 272291
- The Poisson Integral Formula 284303
- Harmonic Function Theory 291310
- Rado-Kneser-Choquet Theorem 296315
- Star Mappings 301320
- Dilatations of Polygonal Maps are Blaschke Products 304323
- An Important Univalence Theorem 309328
- The Dilatation for Star Mappings 311330
- Open Questions 314333
- Bibliography 314333

- Circle Packing 317336
- Background 345364
- The Riemann Sphere 357376
- Index 365384
- About the Authors 371390