[AIS] (alternating) Iteration series: Half-iterate using the AIS?

[Gottfried]

Hi Sheldon -

your post shows really a great deal of work!
I'm currently not yet able to follow that all, I'm still concerned (and stuck) with a much more basic question about the possibility to define an optimized procedure to compute the derivatives using powerseries instead of the iterationseries itself. I'll describe this (and'll ask for help) in another post.

I have an equation for the approximate log(amplitude), for the alternating sum, iterating b^z-1. The approximation works for bases<e, and becomes more accurate as the base gets closer to e. The constant is about 0.73.
\( \log(\text{amplitude})=\frac{\pi^2}{\log(\log(b))}+\text{constant} \)

Well we can see the slight change of slope in the left of your reproduced graph for the amplitude. I suspected thus, that your proposed formula for the amplitude is too simple; simply look at the smaller bases. To improve the visual impression I took for instance base b=1.05 and got an amplitude of 4.388... Smaller bases give values again greater in absolute value, however changing in sign (possibly reflecting the "phase"-graph in my older article, I'll see to extract it and put it here for the current discussion).

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I'm extremely triggered because of your comments concerning the complex-valued arguments and various bases - I'll look at it more deeply after I got my derivative problem solved/optimized: I'm not yet able to express the computation of the derivatives d asum(x) /dx in terms of my (Bell-)matrix ansatz to speed up computations (and complete the matrix-formalism). I'll insert such a posting later...

Gottfried