08/06/2022, 06:21 PM
Just to be complete, the same procedure as with the Schröder-Koenigs iteration can also done with the Abel function tan.
When using Abel functions though the fixed points need to be \(\pm\infty\) not 0 anymore as with the Schröder iteration, because:
\[ \alpha(z_0) = t + \alpha(z_0) \]
If one now takes \(\alpha(x) = \tan(x) \) then we get a parabolic iteration that is analytic at infinitely many fixed points:
When using Abel functions though the fixed points need to be \(\pm\infty\) not 0 anymore as with the Schröder iteration, because:
\[ \alpha(z_0) = t + \alpha(z_0) \]
If one now takes \(\alpha(x) = \tan(x) \) then we get a parabolic iteration that is analytic at infinitely many fixed points:
