Complex Tetration and Ackermann Function

[UVIR]

See http://tetration.org/tetration_net/tetra...torics.htm and http://tetration.org/tetration_net/tetra...omplex.htm for my solutions for complex tetration. The combinatorics information can be used for solutions for complex Ackermann function.

If I understand it right you use the uniqueness of continuous iteration of formal powerseries with a fixed point. To achieve this you need a fixed point of \( b^x \) and call it \( a_0 \).
Then you continuously iterate \( \exp_b(x)=b^x \) expanded at this fixed point and define then tetration as \( {}^xb=\exp_b^{\circ x}(1) \).
This is an interesting approach and makes me ponder about the possible fixed points, solutions of \( b^x=x \). Because \( \sqrt[x]{x} \) has its maximum at \( x=e \) we conclude that exactly for \( b=e^{1/e} \) (interestingly this base occurs also in Jayd's approach) there is exactly one real fixed point \( a_0=e \) of \( b^x \).

Yes this is striking. Do you have also some Mathematica/Maple/etc code prepared? Is convergence guarantied? It would be quite interesting to compare it with the solutions of Andrew and Jayd.
However because there is no base transform formula for tetration, it maybe that \( e^{1/e} \) is the only base with a certain uniqueness.

Note also that Kneser used a similar approach to define the continuous iterates of \( e^x \), he however used a complex fixed point (as we have seen for \( b=e \) there is no real fixed point). And the result was not real valued, so made some manipulations to make it real valued, see

[1] H. Kneser, Reelle analytische Lösungen der Gleichung \( \varphi(\varphi((x)) = e^x \) und verwandter Funktionalgleichungen, J. Reine Angew. Math. 187 (1949), 56–67 (German).

(Note: in the coming days I will make a post about continuous iteration of powerseries with fixed points.)

This method seems very similar to what this guy has in mind in this page.

I tried using his method to extract a Taylor expansion, but got nowehere. Can anybody make sense of what he means? Daniel?