Let u be a solution of the differential equation
u″+Ru=0, where R is rational. Newton's
method of
finding the zeros of u consists of iterating the function
f(z)
=z−u(z)/u′(z).
With suitable hypotheses on R
and u, it is shown that the iterates of f converge on
an
open dense subset of the plane if they converge for
the zeros of R. The proof is based on the iteration theory of
meromorphic functions, and in particular on
the result that, if the family of K-quasiconformal deformations
of
a meromorphic function f depends on
only finitely many parameters, then every cycle of Baker domains of f
contains a singularity of f−1. This
result, together with classical results of Hille concerning the asymptotic
behaviour of solutions of the above
differential equations, is also used to study their value distribution.
For
example, it is shown that, if R is a rational function which satisfies
R(z)∼amzm
as z→∞ and has only k distinct zeros where
k<(m+2)/2,
then δ(0, u)[les ]k/(m+2−k)<1.