I have created a Newton fractal (below) using the Newton-Raphson method to find the five solutions of
f = (z^5-1)
The convergence theorem of Newtons method say "Suppose that f is smooth and that f(x) = 0 and f'(x) =/= 0. Then, there exists epsilon > 0 so that Newton’s method converges to the root x of f if the initial guess x 0 ∈ [x − epsilon, x + epsilon]."
However, I am struggling to see how the epsilon relates to the plot I have made. How could I determine an epsilon value for this? And is it possible to have epsilon as a complex number? Would this still create a circle surrounding x0?