jaydfox Wrote: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.

Oh, shifting is another interesting topic. What happens if we develop the series for (or more simple for ) not at 0 but at some other point , then derive the slog and compare the back shiftet slog with the original slog. To see whether the method is independent on the development point.

The easiest way is by considering the shifted conjugate:

as its iteration is .

So if we solve this translates to

This means whenever we have a solution of then we have a solution to the equation . And the question is whether if is the natural Abel function (Andrew's method) for the first equation, whether then is the natural Abel function for the second equation. Or in other words if is the natural Abel function for the second equation whether then is the natural Abel function for the first equation.

The power series development of is just a linear combination of the powerseries development of .