04/30/2013, 09:33 PM
@Balarka
to question 1)
No it does not always converge.
That is because almost always the half-iterate has singularities.
The half-iterate of exp has singularies.
More precisely EVERY half-iterate of exp has singularities.
See : http://math.eretrandre.org/tetrationforu...hp?tid=544
Since it does not always converge , we have a limit radius of convergeance WHEREEVER we expand.
In particular the fixpoints have singularities.
to question 2)
clearly related to question 1)
within its radius of convergeance the function is indeed analytic.
even better , analytic continuation is possible.
notice that the taylor series is EXACTLY bounded where there are singularies so when the expansion point is not near a singularity we have a nonzero radius of convergeance !
NOTE that the singularities are determined by the fixpoints of g and the function t(x).
regards
tommy1729
to question 1)
No it does not always converge.
That is because almost always the half-iterate has singularities.
The half-iterate of exp has singularies.
More precisely EVERY half-iterate of exp has singularities.
See : http://math.eretrandre.org/tetrationforu...hp?tid=544
Since it does not always converge , we have a limit radius of convergeance WHEREEVER we expand.
In particular the fixpoints have singularities.
to question 2)
clearly related to question 1)
within its radius of convergeance the function is indeed analytic.
even better , analytic continuation is possible.
notice that the taylor series is EXACTLY bounded where there are singularies so when the expansion point is not near a singularity we have a nonzero radius of convergeance !
NOTE that the singularities are determined by the fixpoints of g and the function t(x).
regards
tommy1729