NO previous experience with computers is required. We shall be concerned with functions for which the input and output are both complex numbers. Such a function has almost magical properties. For example, if it has a first derivative near a point, then it has derivatives of all orders. If it is differentiable inside and near a circle, then its values inside the circle are completely determined by its values on the circle itself.
Applications are numerous in physics and engineering.
Emphasis will be on the intuitive and computational aspects, but we will state theorems precisely, and work through some elementary proofs when they help our understanding. This course will study mathematical models for systems which change with time according to some deterministic rule. The course will emphasize the basic dichotomy between stable behavior, which is easy to understand and to predict, and chaotic behavior, which is difficult to understand and impossible to predict far in advance. In particular, it will study highly simplified models which display this distinction. As time permits, it will study the fractal geometry which is often associated with chaotic systems, and the probablistic methods which are needed to understand them.