Funny pictures of sexps

[jaydfox]

Edit: Ah now I get what you mean with recentering. If you have already a truncated powerseries and want to know the powerseries development at a different point then of course this different point must lie inside the convergence radius of the powerseries. But see Jay here it is different I dont recenter a (truncated) powerseries, but a function and *then* I compute its powerseries. I can do that at any point of the function without regarding convergence and convergence radiuses.

I'm not so certain. Again looking at the logarithm, there are branches, so if we recenter, how do we know which branch we are within?

Perhaps this isn't an issue, because the branches are identical except for a constant difference.

But what about log(log(x+0.5))?

Here, the branches do matter. Recenter to from x=0.5 to x=0.5i. Then recenter to x=-0.5. Then to x=-0.5i. And finally to x=0.5 again.

We have completed a winding around a double logarithmic singularity. The function is now quite different, not just due to a constant difference. The only way to get the correct answer is by analytic extension, which involves the very same "shift and truncate" approach I outlined. Shift the power series, then truncate it because the last fraction of the terms have become inaccurate. Repeat as necessary.

Even though it might not be explicitly obvious, I suspect the same problem is hidden in the steps you have taken.

Edit: By the way, back when I had tried converting my accelerated slog solution into a sexp function, I had to shift the slog at x=0 to x=1. When it came time to revert the series, I found I could only keep at most half of the terms before reverting. So my 1200-term slog solution yielded about a 600-term sexp solution. And mind you, this was with my accelerated solution, which is vastly more accurate than the unaccelerated intuitive solution.