09/06/2009, 06:54 AM
(09/06/2009, 04:56 AM)Tetratophile Wrote: Does this prove the uniqueness for all tetration? i.e. that all tetrations are equal where they are invertible, since they are supposed to be analytic, equal to 1 at 0, and is a superfunction of exponentiation, ex hypothesis.
No, this is particularly about Ansus' construction, i.e. all tetrations that satisfy (4) are equal. There are so far no numerical testing on the other tetrations, I will see what I can do about this in the next time.
Quote:bo198214 Wrote:if no convergence issues arise.
I am confused on exactly what the problem you tetration guys are trying to solve with tetration.
I am guessing that it may be convergence or holomorphicity.
Most methods compute coefficients of powerseries as limits of some computations.
To be on solid ground we need to prove that:
1. The limit for each coefficient exists.
2. The resulting series has non-zero convergence radius.
Nothing of that is proved neither for the matrix power method, the intuitive method nor Ansus' extended sum method.
Though numerically everything looks good.
3. Also it would be nice to prove that the corresponding super-exponential is holomorphic on C without (-oo,-2]. Which also looks good numerically.