Uniqueness summary and idea

[bo198214]

Nice, I'm definitely learning about this uniqueness/convergence/analiciy thing. I guess the series expansions of a flow based on fixed-points does not need to converge for the flow to be analytic elsewhere.

Exactly. A well-known example is \( \sqrt{x} \). It is an anlytic function on \( \mathbb{R}_{>0} \) and has a fixed point at 0, but there is no series expansion in this fixed point. However it has the reason that the first derivate is infinity at 0.
Now there are other on \( \mathbb{R}_{>0} \) analytic functions such that all derivatives converge at 0, so we could make a series expansion, but that would have convergence radius 0. Taking this as step further (in analogy the step from \( C^\infty \) to analyticity), there are asymptotically developable function, that still have convergence radius 0, at 0.
And such a case is \( (e^x-1)^{[t]} \), the series does not converge (i.e. has convergence radius 0) in 0 but for \( x>0 \) it is analytic.

If this is true, and an analytic iterate can be found by some other means, then can we perhaps derive an accuracy function?

These are two methods.
1. Practical mathematicians have developed a quite interesting theory how we can despite successfully use non-converging power series. However I am not familiar with theory, but the main point is that most non-converging series have a certain index k at which they are quite near the actual value of the function. If one can determine this k, one has an ultrafast approximation for the function.

An accuracy function would be a function E(n) associated with the flow of a particular function f(x) such that:

The accuracy function is probably a means to determine this k. (However I dont know whether such a function is known for \( (e^x-1)^{[t]} \), probably not.)

2. Yes, the function can be found by other means, as is written in Szekeres and in more generality in Ecalle. Mainly as limit of a sequence of functions. However the computations involved are quite complicated and I can not realiably describe them in the moment. (I have also the excuse of never having learned French )

This would allow us to use the series to compute the flow of a function instead of whatever other methods we might use to compute the more "accurate" version. We could use this even when the series for the flow does not converge (which appears likely for many functions).

Yes.

Since it is stated in terms of a function of t instead of a polynomial of t, this method should work for hyperbolic iteration series as well as parabolic iteration series.

1. AFAIK the hyperbolic iterate has always cr > 0 (of course for base function with cr>0). 2. That cr=0 for the iterate can only happen for the parabolic case, i.e. for that case we need the other approximation formula or the error function for the non-convergent series. 3. as function in t the iterates of \( e^x-1 \) are entire, but as function in x at 0 they have cr=0 for non-integer t.