bo198214 Wrote:I wouldn't go so far as to say "no meaning", but certainly heuristic analysis can impute more meaning in a situation than is warranted at times.jaydfox Wrote:Actually, Henryk, if you graph even a rough mockup of the regular slog developed for the fixed point z=2, the corrugation effect as we wrap around the fixed point at 4 appears, and furthermore, it looks like it should be there.Mm, "should" has perhaps no meaning in the mathematical context. But good illustrations of a fact are highly welcome. So any pictures?

As for pictures, I'm working on some. I was going to have a go at making some pictures with SAGE, because they look better than those in Excel, but also take longer to get the code set up. I was doing my initial analysis in Excel, until I had a clear enough mental picture to extrapolate the more of the details in my head, much as I had done for base e.

And the quick and easy predication is that, in addition to the singularities at 2 and 4 and their images at imaginary offsets, there would be additional singularities stretching out towards positive infinity, with a relatively calm strip between them, much like the ribs off the backbone of the base e slog. And, as with that slog, there would be additional branches off those branches, between each set of singularities, though without the alternating "logarithmic" and "exponential" types. I still need to create a decent graph for this, because as yet I've only been able to picture it in my mind.

Quote:My conjecture is that whenever there is an analytic function for which the regular iterations at two fixed points coincide, then this function is already the identity.

Not quite sure what you meant by that. At any rate, I can already tell from a series of thought experiments that the rslogs at the upper and lower fixed points (4 and 2 for base sqrt(2)) are quite different and introduce the "corrugations" as you called them in different places, and yet I would imagine that both satisfy the infinite matrix solution. I suppose this is an example where a non-first order differential equation can have multiple solutions.

~ Jay Daniel Fox