Julia sets of Newton's method for a class of complex-exponential function F(z)=P(z)eQ(z)

摘要

In this paper, we analyze the theory of the Julia set (J set) of Newton's method, construct the Julia sets of Newton's method of function F(z)=ze~(zw) (w ∈ ?) through iteration method, and analyze the attracting region of the two fixed points 0 and ∞ when w are different values. Consequently, we draw the following conclusions: (1) When the judge conditions for the iterative algorithm are changed to |N(z_n)-z_n| ≤EOF, the properties of the figures in our experiments are contrary to the conclusions in (Wegner and Peterson, Fractal Creations, pp. 168-231, 1991); (2) The attracting regions of the fixed points 0 and ∞ for w=2n (n=0,±2,±4, ...) are symmetrical about x-axis and y-axis; select the main argument to be in [-π,π), for arbitrary w=α (α ∈ ?), the attracting regions of the fixed points 0 and ∞ are symmetrical about the x-axis; (3) The attracting regions of the two fixed points 0 and ∞ of J set for w=±η have rotational symmetry of η times; (4) If w=-4.7, k=0.8, then the attracting regions of different magnifications display a startling similarity, J set holds infinite self-similar structures; (5) When w is a complex number, because the selection of main argument θ z in the negative x-axis is not continuous, the fault and rupture of the attracting regions of the two fixed points 0 and ∞ appear only in the negative x-axis.