Iterations of the Newton Map of tan(z)

Abstract

The dynamical systems of trigonometric functions are explored, with a focus on t(z)=tan⁡(z) and the fractal image created by iterating the Newton map, F_t (z), of t(z). As a point of reference we present Newton’s method applied to polynomials and the iterations of families of trigonometric functions. The basins of attraction created from iterating F_t (z) are analyzed and, in an effort to determine the fate of each seed value, bounds are placed within the primary basins of attraction. We further prove x and y-axis symmetry of the function, and explore the infinite nature of the fractal images. Lastly, Newton iterations of the family〖 z〗^k tan⁡(z) are explored in comparison with F_t (z) and Householder’s methods are discussed.