To be honest, I'm stumped on how to squeak more accuracy out of Andrew's solution. I've gone through the analysis, and I know where all the singularities are that can be "seen" from the origin. I can drill into any branch and locate singularities. And I just don't see any close enough to explain the radius of convergence of the residue.

What I can imagine being the case is that, from the point of view of the origin, if I subtract out the two primary singularities, then as I pass the singularities on my way out from the origin, I'll see a split. If I were sitting at the origin, looking at the upper primary fixed point (0.318+1.337i), I would see a discontinuity to the left and to the right of the fixed point. At the fixed point, I would see the start of a "tear" or "rip" in the fabric of my graph. As I look farther and farther beyond the fixed point, the gap between the values would increase.

This would explain a few features of the power series, as I'm observing it. Very near the fixed point, the difference is small, but farther away, the difference is large. So the root test starts out rather low, because the tear is hardly perceptible near the fixed point. As I look further into the power series, the root test climbs, very slowly, up towards 0.67, 0.68, perhaps 0.69. I conjecture that after a very large number of terms, perhaps thousands, I'll climb up to 0.70, and then 0.71. Eventually, of course, if I could get tens of thousands of terms, I would begin to approach the 0.727, which would put my radius of convergence back at the primary fixed point. After all, the tear starts there, so I can't really get past it. But the tear is so subtle that it doesn't really affect the first few hundred derivatives very much at all.

Another feature of the power series of the residue is that it has a cycle pattern to it, with a period of just under 7, which puts its direction from the origin at an angle of just under 1.346 radians, pretty much where we'd expect it to be if it were in fact at the fixed point itself (1.337 radians).

Now, removing a point singularity, especially one as simply as a logarithm, is trivial. I was proud I figured it out, but in hindsight it was actually pretty obvious. But how do I remove a "tear" in the fabric of my graph? It's not a point, and it's certainly not going to be as trivial a function as a logarithm. I have no "trick" for squeezing more precision out of my slog. For that matter, I had hoped to come up with a simple definition of the slog in terms of a combination of (perhaps an infinite number of) more basic functions. Now I'm not so sure we'll be so lucky.

At any rate, I have enough precision with my 700x700 solution (and I think I have enough memory to go to about 750x750, since I'm no longer using rational math), so I will start working on the pretty graphs I've been promising. It may be some time next week before I get some good ones generated.

What I can imagine being the case is that, from the point of view of the origin, if I subtract out the two primary singularities, then as I pass the singularities on my way out from the origin, I'll see a split. If I were sitting at the origin, looking at the upper primary fixed point (0.318+1.337i), I would see a discontinuity to the left and to the right of the fixed point. At the fixed point, I would see the start of a "tear" or "rip" in the fabric of my graph. As I look farther and farther beyond the fixed point, the gap between the values would increase.

This would explain a few features of the power series, as I'm observing it. Very near the fixed point, the difference is small, but farther away, the difference is large. So the root test starts out rather low, because the tear is hardly perceptible near the fixed point. As I look further into the power series, the root test climbs, very slowly, up towards 0.67, 0.68, perhaps 0.69. I conjecture that after a very large number of terms, perhaps thousands, I'll climb up to 0.70, and then 0.71. Eventually, of course, if I could get tens of thousands of terms, I would begin to approach the 0.727, which would put my radius of convergence back at the primary fixed point. After all, the tear starts there, so I can't really get past it. But the tear is so subtle that it doesn't really affect the first few hundred derivatives very much at all.

Another feature of the power series of the residue is that it has a cycle pattern to it, with a period of just under 7, which puts its direction from the origin at an angle of just under 1.346 radians, pretty much where we'd expect it to be if it were in fact at the fixed point itself (1.337 radians).

Now, removing a point singularity, especially one as simply as a logarithm, is trivial. I was proud I figured it out, but in hindsight it was actually pretty obvious. But how do I remove a "tear" in the fabric of my graph? It's not a point, and it's certainly not going to be as trivial a function as a logarithm. I have no "trick" for squeezing more precision out of my slog. For that matter, I had hoped to come up with a simple definition of the slog in terms of a combination of (perhaps an infinite number of) more basic functions. Now I'm not so sure we'll be so lucky.

At any rate, I have enough precision with my 700x700 solution (and I think I have enough memory to go to about 750x750, since I'm no longer using rational math), so I will start working on the pretty graphs I've been promising. It may be some time next week before I get some good ones generated.

~ Jay Daniel Fox