Mystery of the rational iteration arising from relaxed Newton’s method

Newton’s iterative method for finding solutions to nonlinear equations leads to some beautiful images when applied to complex functions. These images represent the basins of attraction of roots of complex functions. The interesting point about the basins of attraction for the roots of a complex function F approximated by relaxed Newton’s method is that they are fractals called Julia sets of a rational function NF,h. The main aim of the present paper is to describe such fractal sets.