The dynamics of newton's method on the exponential function in the complex plane

Newton's Method is an iterative procedure designed to locate the roots of a function. This paper studies the dynamics of Newton's method applied to the complex exponential function F(z) = P(z)e$\sp{Q(z)}$, where P and Q are polynomials. Of course, one need only apply the numerical algorithm to the polynomial P to determine the roots of F. However, investigation of the dynamics of this family of rational maps produces some unexpected and interesting results that differed considerably from results that follow from applying Newton's method to polynomials. Sutherland proved in his thesis that the immediate basins of attraction for the roots of polynomials are large and, in fact, have a lower bound for their width. The implications for the efficiency of this numerical method are that proper choice of initial values will guarantee convergence to the roots. In contrast, I show that the immediate basins of roots of $Pe\sp{Q}$ are small and in fact have finite area. The key to this result is the fact that infinity is a parabolic fixed point, as opposed to the case for polynomials where infinity is always a fixed repeller. The Fatou Flower Theorem provides a description of the local dynamics about the fixed point. The study begins with examination of a reduced problem, Newton's method on F(z) = $e\sp{z\sp{n}}$. Distinctions between the cases for n even and n odd are established. Symbolic dynamics is employed to describe the behavior of points in the Julia set under iteration. The proof of finite area of the basins rests on constructing sufficiently large attracting petals at infinity. Orbits of points in these petals tend to infinity under iteration of N and are thus in the basin of infinity. Moreover, the basins of the roots of F are squeezed between the petals at infinity. The union of the Julia set with the basins of the fixed points is the complement of the basin of infinity. Therefore, petals are constructed with an appropriately high order of tangency at infinity so that the complement of their union is a simply connected region with finite area. It follows that the set of all basins of the roots is a subset of this region and hence itself has finite area.