06/23/2009, 02:57 AM

(06/22/2009, 09:46 PM)bo198214 Wrote: I asked you som time ago to apply your algorithm to other fixed points, but you somehow did not follow that path.Assume that for another tetration , we have some probe function for real ; id est, along the imaginary axis.

Assume, Function is suposed to go to at infinity and at minus infinity; , and is integer constant.

In order to adjust the probe function to some tetration, we need to evaluate the contour integral. If we use the same contour as in the paper http://www.ams.org/mcom/2009-78-267/S002.../home.html

then we need the values at and for real .

It is easy to estimate values at ; use estimate .

As for ; we use estimate for positive and for negative . Either we have singularity (jump) at , or .

This explains, why I did not construct such "another tetration", but this is not a proof that this is impossible. Suggest the holomorphic probe function to begin with. Such a function should have some smooth kink of the phase, in order to avoid the jump;

but allow such a jump for the principal branch of its logarithm.

Then we can run the same algorithm, with additional control of the branch of the logarithm. Such a control should recover ,

adding unity or minus unity to each time when passes through negative real values. We need the holomorphic kinky probe function, then we can run the algorithm to recover the kinky tetration.

I think about something like