Explorations in Complex Analysis

This book is written for mathematics students who have encountered basic complex analysis and want to explore more advanced project and/or research topics. It could be used as (a) a supplement for a standard undergraduate complex analysis course, allowing students in groups or as individuals to explore advanced topics, (b) a project resource for a senior capstone course for mathematics majors, (c) a guide for an advanced student or a small group of students to independently choose and explore an undergraduate research topic, or (d) a portal for the mathematically curious, a hands-on introduction to the beauties of complex analysis. 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. There are more than 15 Java applets that allow students to explore the research topics without the need for purchasing additional software.

This new offering from the MAA is a collection of six inducements or invitations to further research for undergraduates with some background in complex analysis. The book is flexible enough to be a source of enrichment material, a basis for research projects, the kernel of a capstone course, or just a tool to ignite the interest of the mathematically curious.

Although the authors of the six chapters vary, the style and approach is much the same throughout. The goal is a kind of guided research that is focused on fostering independent student investigation and discovery. Each chapter begins with a guided tour of the topic and then offers students several opportunities to investigate further. Interspersed throughout the text are examples, Java applets, exercises, explorations and a variety of potential projects. The exercises are integrated into the text, designed with clear goals, and identified as essential for comprehending the material. The "explorations" are less goal-directed and aimed more at getting students to find directions to investigate on their own. The projects are optional activities, large and small, that might last a few weeks or a whole term.

The book's six chapters offer what the authors call "current research topics," although some are more current than others. Topics come in a reasonable variety. For those favoring geometry, there are chapters on minimal surfaces ("Soap Films, Differential Geometry and Minimal Surfaces") and "Circle Packing" (configurations of circles with specified patterns of tangency) To those who are inclined to complex function theory, there are "Anamorphosis, Mapping Problems and Harmonic Univalent Functions" (perhaps the closest to a bona fide current research topic) and "Mappings to Polygonal Domains" (creating univalent functions from one such domain to another). For the application-minded, there is "Applications to Flow Problems," about two-dimension vector fields in, for example, electromagnetic or fluid dynamics. Finally there is "Complex Dynamics," which introduces chaos and fractals via iteration of complex analytic functions.

This is an attractive book that should have a lot of appeal to students. It offers a number of excellent avenues into research for undergraduates. --Bill Satzer, MAA Reviews

This book provides an informative, student-centered approach to several diverse applications of complex variables. Brilleslyper (US Air Force Academy) and colleagues use visual aspects and various Java applets to enhance the presentation and deepen understanding. Each of the six chapters contains a discussion of the requisite applets. The text begins with "Complex Dynamics," including a discussion of the Mandelbrot set. the following chapters cover connections to differential geometry, fluid flow, and mapping problems. The final chapters are titled "Mappings to Polygonal Domains" and "Circle Packing." The book contains many examples and more than 320 exercises in addition to numerous "explorations" and both large and small projects. Readability is enhanced by over 210 figures, most in color. More than 100 references support the text. Two appendixes cover necessary background information and the Riemann sphere. The book would serve nicely for a senior undergraduate capstone course. --D.P. Turner CHOICE