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

[Gottfried]

then the limit can be given as
\( \lim_{\delta \to 0+} b[4](\delta -2) - \log_b(\delta)=\log_b\left(\frac{\text{sexp}'(0)}{\ln(b)}\right) \)

Well, thanks! Looks good...
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? And I think to see the chainrule in application,

The derivation at -1 can be derived from the value at 0 by:
\( \text{sexp}(x+1)' = \exp_b(\text{sexp}_b(x))' = \ln(b)\text{sexp}(x+1)\text{sexp}_b(x)' \)
hence for \( x=-1 \)
\( \text{sexp}'(0)=\ln(b)c \)

but if the height-parameter is the variable, how can we extract one instance of exp from sexp in context with the chainrule? (Maybe I'm misreading, though) I'll have another look into the faq/ref today.

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?

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... ), but yesterday I saw, that it seems to work in the sense of height-differences, so
sexp(x1+1) = exp(sexp(x1)) // <diagonalization>
whatever
sexp(x1) //<diagonalization>
may be... , and the situation focuses on the problem of finding the norm-parameter for the schröder/eigenvector-function which is still too difficult for me due to divergence of the complex series. (Ithink I'll put another note about this into the "matrix-operator-method" thread later)

Gottfried