ABSTRACT
This chapter begins with the study dynamical systems in the plane. It considers functions of a complex variable. The study, begun by the French mathematicians Gaston Julia and Pierre Fatou in the 1920's, received renewed impetus with the pioneering computer graphics work of Benoit Mandelbrot in 1980 and the subsequent mathematical investigations of Adrien Douady, John Hubbard, Dennis Sullivan, and others. An alternative method of describing points in the complex plane is the polar representation of a complex number. Geometrically, addition of complex numbers is given by the parallelogram for addition of vectors in the plane. The polar representation of complex numbers makes it particularly easy to visualize complex multiplication. As the original particle makes one full loop around the circle, the square roots each traverse a semicircle, since people halve the corresponding polar angle.
