Ifis a quadratic function with roots that are the solutions to
then the roots may be found using the Newton – Raphson iteration formula
The formulais rational unless
and can be extended to
giving the extended function
Ifis a simple zero of
then
and
so
so that
is a fixed point of
To classify it find
(since
)
Thus a simple zero ofis a super attracting fixed point for the Newton – Raphson function
If the functionhas distinct zeros at
and
then these zeros must be simple and super attracting fixed points of
There exist open discs around
and
in which points are attracted to
and
respectively under iteration by
Ifthe Newton – Raphson formula is
We can use the conjugating function(a Mobius function with extension to
mapping
to 0 and
to infinity and is such that
).
Ifand
for
then
for
By inductionfor
as
if
and
as
if
and
remains on the unit circle if
To deduce the behaviour ofnote that
and
Also
We deduce thatas
if
and
as
if
and that
remains on the extended line
if
is on the line.