bo198214 Wrote:jaydfox Wrote:Random speculation, but what if the coefficients look like the real part of a logarithmic spiral because one of the complex solutions based on the continuous iteration from the fixed point has a complex spiral for its coefficients? By dropping the imaginary parts, we recover a solution that yields real results...So you think that Andrew's solution is the real part of the regular fractional iteration at a complex fixed point?!
What are then the real parts of the other solutions at a fixed point?
And is this then Kneser's solution at all (he started by regularly iterating at a certain fixed point and then by some transformation I did not fully understand yet he came up with a real solution)?
Questions over questions ...
See my post here:
http://math.eretrandre.org/tetrationforu...php?tid=59
Essentially, I think that Andrew's solution may be a sum of complex-valued functions which come in conjugate pairs, thus cancelling all the imaginary parts if we create a power series at a real point.
I've already demonstrated that there is a singularity in Andrew's slog at the primary fixed point (and, logically, at its conjugate). I've already numerically confirmed that the logarithms at the fixed points give an excellent approximation of all the terms in Andrew's slog beyond about the 10th or 20th, so the singularities appear in fact to be exactly due to these logarithms. This both confirms the approach of hyperbolic iteration from a complex fixed point, and confirms that Andrew's slog is quite probably "the" solution to tetration, at least for base e.
There is a residue to Andrew's slog, if you subtract out the logarithms at the fixed points. I'm currently studying the nature of this residue, hoping to find the non-fixed-point singularities.
~ Jay Daniel Fox