The AIS-method for finding the fractional iteratives seems to be still insufficient. In a discussion in mathoverflow (http://mathoverflow.net/questions/201098...p-from-the ) I study the two-way-infinite alternating iteration series on the base function r(x)=x^b, so the left-associative power tower. I applied the scheme as I did here in this thread where the base function is decremented exponentiation (r(x)= b^x-1) and also got an oscillating function asum. However, that base function allows a natural explicte fractional iterate, so we can compare the asum-computed and the naturally computed funcion values. I found that the asum for this function differs from the sine-wave by tiny differences, in the sixth digit or even below. So the consequence is, that I should distrust the apparently nice solution for the determination of the fractional iterate for the tetration based on the AIS.
Maybe someone else here of the forum is able to determine the characteristic of the d(x)-function in my linked MO-post, perhaps by some fourier-decomposition, with the hope, that such an axpression could then also be introduced as a correction for the asum() for the tetration, but I don't really know whether it shall lead to somewhere.
Gottfried
Maybe someone else here of the forum is able to determine the characteristic of the d(x)-function in my linked MO-post, perhaps by some fourier-decomposition, with the hope, that such an axpression could then also be introduced as a correction for the asum() for the tetration, but I don't really know whether it shall lead to somewhere.
Gottfried
Gottfried Helms, Kassel
