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

[sheldonison]

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.

Hey Gottfried,

I should've used approximate. I've since changed the equation to put an adjust(b) term in there. The graph showing the estimate along aside the actual shows the adjust(b) required, although it is base 10. Here the adjust term is base e.
\( \log(\text{amplitude})=\frac{\pi^2}{\log(\log(b))}+\text{adjust}(b) \)

Below there is a graph, showing the required adjust(b) term. Notice that when b=e, the log(asum_amplitude) has an arbitrarily large negative singularity, but the adjust term is still defined. I actually plan to calculate the asum and the adjust term for base e, iterating exp(z)-1. It has an infinite period, but the asum from the upper super function is defined at \( \frac{\pi i}{3} \), which I think turns out to be equivalent. If I have time, I'll post more later. Good luck with your matrix and derivative calculations.
- Sheldon