03/20/2010, 08:12 AM
(03/10/2010, 11:13 AM)bo198214 Wrote:(03/10/2010, 03:10 AM)mike3 Wrote: Hmm. This suggests the possibility there is an alternative solution instead of regular iteration for the bases in \( (1, e^{1/e}] \) (after all, if it's analytic at \( e^{1/e} \) and the regular isn't, the two can't be equal if they're both analytic in that interval.).
Oh, my explanation was misunderstandable, the bipolar Abel function is always holomorphic and injective on a sickel between the two fixed points. But it may not be real on the real axis, or perhaps not even defined on the real axis, in the case of two real fixed points. You know the sickel would be above or below the real axis or possibly even wind around the fixed points, so being defined on pieces of the real axis; and it is not clear whether it can be extended to the real axis between the fixed points.
On the other hand to obtain alternative solutions you can always build up linear combinations of the two regular Abel functions \( \alpha \), \( \beta \):
\( \gamma(z)=c\alpha(z)+(1-c)\beta(z) \) which is again an Abel function:
\( \gamma(f(z))=c(\alpha(z)+1)+(1-c)(\beta(z)+1)=\gamma(z)+1 \)
However, for it to be not real valued in \( (1, e^{1/e}] \) and real valued in \( (e^{1/e}, \infty) \) would imply there is a singularity/branchpoint at \( b = e^{1/e} \), hence not holomorphic there, eh?
