07/01/2010, 03:31 PM
(This post was last modified: 07/01/2010, 04:10 PM by sheldonison.)

(06/29/2010, 06:53 AM)bo198214 Wrote: I think it is not "equivalent" in the sense of "equal", because complex logarithm does not satisfy log(zw)=log(z)+log(w). I guess my formula lands on a completely different branch of slog, hm.

Well I will try to reproduce your curves, with your formula. And dont think that my understanding is that much deeper, perhaps we indeed can develop a secondary fixed point real analytic slog, if we dont start with a "plain" initial region, but with an overlapping initial region/manifold. I am really interested in this unitude/multitude topic. (For example what happens with Kouznetsov's method if we just plug in the secondary fixed point pair?)

The path from the real axis to the secondary fixed point is much less direct. Kouznetsov's method requires three consecutive exponents, (he uses 0, 1, e), and their vertical (along increasing imaginary) paths to the fixed point. The path to the primary fixed point is fairly direct. But not the path to the secondary fixed point. For example, consider the path from real=ln(0.5), 0.5, and exp(0.5) to the secondary fixed point (via the i*3pi contour). These are the real values near the maximum of the i*3pi contour, (see graph in earlier post). Here is a graph of the first two. The third, exp(0.5) path to the secondary fixed point is already very chaotic, with img varying in the hundreds, so I left it off the graph.

Notice how much more direct the path is to the primary fixed point is. Using this path to the primary fixed point as a seed, I would imagine Kouznetsov's method would converge very nicely. But it is difficult to imagine Kouzenetsov's method converging to anything meaningful for the path to the secondary fixed point (especially considering the even more chaotic exp(0.5) path). I also looked at the path to the secondary fixed point from i*pi contour, but that appears to be even less well behaved than the path via the i*3pi contour.

- Sheldon