Complex iterated radicals

Abstract:

We prove the convergence of the sequence $S$ defined by ${z_{n + 1}} = {({z_n} - c)^{1/2}},c$ real, for any choice of ${z_0}$. Let $k = |\tfrac {1}{4} - c{|^{1/2}}$. If $c < 0$ or $c = \tfrac {1}{4},S$ has only one fixed point $w = \tfrac {1}{2} + k$ and converges to $w$ for any ${z_0}$. If $0 \leqq c < \tfrac {1}{4},S$ has the fixed points ${w_1} = \tfrac {1}{2} + k$ and ${w_2} = \tfrac {1}{2} - k$, and for any ${z_0} \ne {w_2},S$ converges to ${w_1}$. If $c > \tfrac {1}{4},S$ has the fixed points ${w_1} = \tfrac {1}{2} + ik$ and ${w_2} = \tfrac {1}{2} - ik$ and converges to ${w_1}$ if $\operatorname {Re} ({z_0}) \geqq 0$ and to ${w_2}$ otherwise. We show that convergence is strictly monotone when the neighborhood system is the pencil of coaxial circles with ${w_1}$ and ${w_2}$ as limiting points, and give rates of convergence.