Interactive Visualization in Complex Analysis

Analytic functions of a complex variable exhibit some of the most striking beauty found anywhere -but in the ages of black-on-white printed textbooks, this facet has been largely inaccessible to all except a few. Needham's recent text "Visual Complex Analysis" clearly demonstrates the power of a visual language as an organizing principle and as a useful tool to develop promising strategies for analytic arguments. But modern technology allows one to go much further: We discuss selected implementations in computer algebra systems and especially in JAVA applets whose ultimate interactivity transforms every learner into an experimenter and researcher! Selected examples include zooming into essential singularities, mappings of the complex plane, winding numbers, and convergence of Laurent series. We report how such tantalizing imagery transformed our own class, where amazing beauty led to inquiry and an urgent sense of "I want to know how/why that works". We contrast CAS-worksheets with model JAVA applets: On one side the user may modify and change everything -but the algebraic-symbolic language of CAS worksheets usually requires a nontrivial "manual". On the other side, well-designed JAVA applets ideally require no instructions at all. Moreover, by using the "mouse" for input, and a graphic language for output, they take advantage of tactile, kinaesthetic and visual pathways that arguably have been much underutilized in mathematics teaching in recent centuries.