11/06/2007, 01:34 PM
Hmm, well, I solved at the origin, so I can only speak for the singularity at 2, though I had already assumed singularities at 2 and 4. Prior to discovering that the rslogs at the two points were different, I would have expected both singularities to be simple logarithms in the limit as we approach the fixed points. But now I suspect that the singularity at z=4 will be a little more screwed up.
Anyway, I can't very well shift the series to be centered at z=3, since I'd have to go around the singularity at z=2, and moving a distance of 4 or 5 units, given a radius of 2, would strip off way too much precision (I'd need maybe 6 to 10 small shifts).
I haven't yet tried shifting the system before solving, though I suppose it's possible in principle, and I'd be curious to see if it worked. The mental gymnastics involved to derive the system prior to solving are a bit beyond me this morning (I hardly slept), so I'd have to think about it this evening.
Anyway, I can't very well shift the series to be centered at z=3, since I'd have to go around the singularity at z=2, and moving a distance of 4 or 5 units, given a radius of 2, would strip off way too much precision (I'd need maybe 6 to 10 small shifts).
I haven't yet tried shifting the system before solving, though I suppose it's possible in principle, and I'd be curious to see if it worked. The mental gymnastics involved to derive the system prior to solving are a bit beyond me this morning (I hardly slept), so I'd have to think about it this evening.
~ Jay Daniel Fox