Continuing the previous post in a more general way: it is a somehow ironic outcome, that - after the introduction of the asum() as provider for the fractional iteration because it seemed to be independent from the fixpoint-problematic (which we encounter once we start to construct power series for fractional iteration), because we need only integer height iterations - we find now even two formulae which are depending on the fixpoints ...
Well, but leave this aside for a moment. The crucial aspect for the correctness of the representation of the iteration-series by one (or two) power series derived from the Neumann-series of the Carleman-matrices for the function and for the inverse, seems to be, that the fixpoints must be attracting, so we must center the function and the inverse around that specific fixpoint which makes it happen, that it becomes an attracting one.
If we want to generalize that whole concept to the cases of x beyond the upper fixpoint, we have then the interval for x from \( t_1 \ldots \infty \) where \( t_1 \) is still attracting for the function \( f^{\circ -1}(x) = \log_b(1+x) \) but the infinity is now attracting for f(x). Can we develop f(x) around \( \infty \)?
Or can we understand/interpret such x as iterations with complex heights (as I had proposed it for the "regular iteration" in other threads) for instance using Sheldon's power series?
In general, I'm beginning to look at the same principle but for other/simpler functions than exp(x)-1, for instance the linear function, polynomials and such, which even might lack any finite fixpoint and the alternating iteration series can still be expressed by power series - so that one might derive some common behave and thus some insight for the case of infinity as fixpoint from that simpler examples also for the case here in question ...
Gottfried
Well, but leave this aside for a moment. The crucial aspect for the correctness of the representation of the iteration-series by one (or two) power series derived from the Neumann-series of the Carleman-matrices for the function and for the inverse, seems to be, that the fixpoints must be attracting, so we must center the function and the inverse around that specific fixpoint which makes it happen, that it becomes an attracting one.
If we want to generalize that whole concept to the cases of x beyond the upper fixpoint, we have then the interval for x from \( t_1 \ldots \infty \) where \( t_1 \) is still attracting for the function \( f^{\circ -1}(x) = \log_b(1+x) \) but the infinity is now attracting for f(x). Can we develop f(x) around \( \infty \)?
Or can we understand/interpret such x as iterations with complex heights (as I had proposed it for the "regular iteration" in other threads) for instance using Sheldon's power series?
In general, I'm beginning to look at the same principle but for other/simpler functions than exp(x)-1, for instance the linear function, polynomials and such, which even might lack any finite fixpoint and the alternating iteration series can still be expressed by power series - so that one might derive some common behave and thus some insight for the case of infinity as fixpoint from that simpler examples also for the case here in question ...
Gottfried
Gottfried Helms, Kassel
