# Tetration Forum

Full Version: fractional iteration/another progress
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Hi -

I am currently reviewing the problem of divergence of powerseries (for the fractional iteration of UtÂ°h(x), but real bases).

First I produced some plots with few different bases to get a better overview over the general characterisitcs of the divergence of the powerseries.

Today I found a start to overcome that nasty type of divergence: using a Stirling-transform (as I call it at the moment) of the powerseries I get nicely convergent series even for the fractional case (just had only time to check this for one (divergent) base b=exp(2)). This is a big progress for my implementation of diagonalization!

Look at
http://go.helms-net.de/math/tetdocs/Coef...Height.htm

Gottfried
Gottfried Wrote:Look at
http://go.helms-net.de/math/tetdocs/Coef...Height.htm

Just updated the text. The said "Stirling-transform" seems to stretch the radius of convergence for x from zero - perhaps to infinity, at least (and surprising) for bases b(="t") > 2 . It seems to be not useful for classical bases like b=exp(1) and for the range below 2 (I'll have to investigate that more)

Gottfried
Gottfried Wrote:
Gottfried Wrote:Look at
http://go.helms-net.de/math/tetdocs/Coef...Height.htm

Just updated the text. The said "Stirling-transform" seems to stretch the radius of convergence for x from zero - perhaps to infinity, at least (and surprising) for bases b(="t") > 2 . It seems to be not useful for classical bases like b=exp(1) and for the range below 2 (I'll have to investigate that more)

Gottfried
Another update. Some more explixite example and improving of plots.

Hellgate -
in summer...

Gottfried