A nice series for b^^h , base sqrt(2), by diagonalization

[bo198214]

However, I must have missing the core discussion about the derivatives - how do we get to values of sexp(h)' at h=0 or elsewhere?

We dont get it, except from a specific tetration (while the formula I gave is universal to all tetrations).
However if we have the derivation at 0 then we have it also at -1 and all natural numbers (by the chain rule).

Anyway. The confirmation for the validity of the formula is also one for the appropriateness for the guess about the limit behaviour/tendency of the series and makes me more confident, that with that series (and the underlying diagonalization) we are on the right track - don't you think so?

I dont even know what series you are talking about, I just saw a bunch of floating numbers that are said to have something to do with the matrix/diagonalization approach.

I would really appreciate if you could be more clear in your terminology:
The matrix power approach makes use of the established method to obtain non-integer powers (and other analytic functions) of finite matrices via diagonalization.
This is applied to the truncations of the Carleman/Bell matrix.

The matrix power method can be applied at different development points.
If it is applied to fixed points then it is equal to the regular iteration.
Thatswhy the latter case is not really interesting for me, there are better limit methods available than expanding powerseries to compute regular iterations, and nearly everything is already known about regular iteration.

However different the personal interests are, I would appreciate if you clearly specifiy the development point for your matrix method application.
Particularly also because it is still unknown how the matrix power approach is dependent on the development point. It seems that it yields different results at different development points.

By the way, another bit: I had the unsolved problem, that the diagonalization with fixpointshift does not give even near approximations for fractional heights if the base>e^(1/e) and the fixpoint is complex; that's why I was dismissing the method for such cases (which are the majority of cases... ),

Hm, the regular iteration (=matrix power at fixed point) at the primary complex fixed point \( \lambda \) has singularities on the real axis, i.e. at \( \exp^{\circ n}(0) \), \( n=0,1,\dots \) but no singularities at the upper half plane or elsewhere on the real axis. The radius of convergence is \( |\lambda| \), i.e. the distance from the development point \( \lambda \) to 0, as \( \Re(\lambda)\approx 0.3 \) and 0 is closer to \( \lambda \) than 1.

So the only values on the real axis for which the powerseries converges is on the open interval \( (0,2\Re(\lambda)) \).