GFR Wrote:I think that the two real h's (of the t's) corresponding to the same b = sqrt(2), this time only with the + sign (sorry, Bo, ... it's the age), i.e. h = 2 and h = 4 must be different indeed. Any ... serious serial development should indeed show this situation. We cannot think that we may start from b = sqrt(2) and than, ... bingo! ... we suddely have two different values. The "strange object" that we may call h or t is the result of the application of a "two-valued function". But, two-valued "functions" are not politically correct animals.Hi Gianfranco -

But, perhaps, I don't understand what you precisely said. I didn't sleep well, last night. I shall improve! Tomorrow is another day.

GFR

I've begun a short consideration of the multivalued log(1+x)-function in the context of matrix-operations (extended version of ContinuousIteration, which may of interest.

It seems interesting, but it exhibits, that we also need a more general notion of divergent summation, especially for the complex case. Since Euler-Summation (although principally able) is not well suited (and apparently not much studied) for the complex case, I'm always at the edge of possibilities. At least I cannot proceed much more without finding a reliable base for such summation-concepts. It seems, that all (or nearly all) fractional iterations of tetration, (if based on powerseries) produce hypergeometric divergent powerseries with convergence radius zero, not only the x->exp(x)-1 version. These series cannot be Euler-summed (principally) and thus we need this concept of assigning valued to such divergent series.

The powerseries for log(1+x) = x - x^2/2 + x^3/3 - + ... may be configured for multivaluedness by log_k(1+x) = k*2 Pi i + x - x^2/2 + x^3/3 - + ... and the matrix-operator is then square and has all the nasty properties of divergent series.

Disclaimer: all this is merely more or less speculation, and only motivated to find any usble entry-point to access the problem, which I focus in this thread.

Well - have a good night, I'll stop soon, too; I've to be prepared for an exam of my statistics-class tomorrow. I'll need hawk's eyes...

Kind regards -

Gottfried

Gottfried Helms, Kassel