Iterations of the newton map of tan(z)
Date
2013-05
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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.
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Keywords
Newton's method, Dynamics of trigonometric functions, Tan(z)