On the Newton’s method for transcendental functions

Abstract

The family of polynomials $P_{n}: \mathbb{C}\times \mathbb{C} \to \mathbb{C}; (\lambda , z) \mapsto \lambda (1 + z/n)^{n}$ converges uniformly on compact subsets of the complex plane to the family of the complex exponentials $E:\mathbb{C}\times \mathbb{C} \to \mathbb{C}; (\lambda , z) \mapsto \lambda e^{z}$, as $n$ tends to infinity. Due to this convergence certain dynamical properties of the polynomials $P_{n}(\lambda , \cdot )$ carry over to the exponentials $E(\lambda , \cdot)$. Thus it possible to study entire transcendental maps, the exponentials, by considering polynomials for which the theory is well-known. Two particular problems have received attraction: (1) For a fixed parameter $\lambda \in \mathbb{C}$ do the Julia sets of the polynomials $P_{n}(\lambda , \cdot )$ converge to the Julia set of $E(\lambda , \cdot)$? (2) Do the hyperbolic components in the parameter space of $P_{n}$ converge to hyperbolic components of the family $E$? In the present paper we study the Newton’s method associated with the entire transcendental functions $f(z) = p(z)e^{q(z)} + az + b$, with complex numbers $a$ and $b$, and complex polynomials $p$ and $q$. These functions $N_{f}$ can be approximated by the Newton’s method associated with $f_{m}(z) = p(z)(1+q(z)/m)^{m} +az +b$. In this paper we study the convergence of the Julia sets $\mathcal{J}(N_{f_{m}})\to \mathcal{J}(N_{f})$ and the Hausdorff convergence of hyperbolic components in the families $\{ N_{f_{m}}\}$ to the hyperbolic components of the family $\{ N_{f} \}$.