04/26/2008, 02:12 AM
Bo, when you make the first step, please, plot the asymptotics in the complex plane, and we compare our results. It will be the second step.
In such a way, my theorem is wrong at b < e^(1/e); and, perhaps, at b=e^(1/e); so, it should be reformulated: the equirement b > e^(1/e) should be included into the conditions of the Theorem.
bo198214 Wrote:\( b=\sqrt{2} \) is indeed only of interest because it has a simple (integer) fixed point 2. So that is our standard reference (on this forum) base to compare two different methods of computing a tetration.At 1<b<e^(1/e), there are two real fixed points; each of them should correspond to the analytic tetration. Now I try to plot them both; then hope to provide the algorithm for the precize evaluation. Then I shall run it at b=sqrt(2).
In such a way, my theorem is wrong at b < e^(1/e); and, perhaps, at b=e^(1/e); so, it should be reformulated: the equirement b > e^(1/e) should be included into the conditions of the Theorem.
bo198214 Wrote:As for example real regular iteration/tetration is no more possible for \( b>e^{1/e} \) because there is no real fixed point.I am not sure if I understand you well. At b=2 and b=e, tetration F(z) looks pretty regular (except \( z\le-2 \)), and it is real at z>-2. Complex fixed points are easy to work with.