(06/12/2010, 04:42 AM)bo198214 Wrote: I am referring to the regular Abel function at the primary fixed point.
The inverse function (regular superfunction) is entire. But the Abel function has singularities on the real line at \( \exp^{[n]}(0) \), \( n\in \mathbb{N} \).
Does this mean that there are no values of \( z \) such that \( \mathrm{reg}_F\left[\exp^z\right](u) = {}^n e = \exp^n(1) = \exp^{n+1}(0) \) for \( n = 0, 1, 2, ... \)? (here \( F \) is the fixpoint, \( u \) is the starting point, and \( reg \) means regular iteration) I.e. the regular iteration super-function does not contain the sequence of integer-height tetrations of e? That makes no sense, since an entire function must take on every complex value with at most one exception. Or are these singularities only on some "branches" of the Abel function, so there's other branches in which there are no singularities there, thus yielding values for where the integer-height tetration sequence occurs? (which would make more sense)