On the Newton's method for transcendental functions

The family of polynomials P-n : C x C --> C; (lambda, z) --> lambda(1 + z/n)(n) converges uniformly on compact subsets of the complex plane to the family of the complex exponentials E : C x C --> C; (lambda, z) --> lambdae(z), as n tends to infinity. Due to this convergence certain dynamical properties of the polynomials P-n(lambda,(.)) carry over to the exponentials E(, (.)). 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 is an element of C do the Julia sets of the polynomials P-n(lambda, (.)) converge to the Julia set of E(lambda, (.))? (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 J(N-fm) --> J(N-f) and the Hausdorff convergence of hyperbolic components in the families {N-fm} to the hyperbolic components of the family {N-f}.