I reread old posts and the Trappmann/Kousnetzov[1] - article about the tetration with base sqrt(2). The point around which I'm fiddling is the imaginary-height iteration, which allows to relate real values from the interval (-oo to 2) with values from (2 to 4), so for instance gives

where is

It is now interesting, that the point , which is usually assumed in tetration as "at -oo", has as well an existing relative to it, which can simply be computed if we compute the relative

to by

and do integer Tetration to get

Here is a picture, where I graphed the trajectories for the imaginary heights from in steps of, say or starting at some example points on the line below +2.

Even more interesting is now the question what happens for the points in the interval because they are related to somehow values larger than the negative infinity. There is the asymptotic vertical line at ; the trajectories starting from values greater than , say from have a trajectory to the right side of that asymptote.

WHen I simply repeated my procedure for the computation of that trajectory from it looked as if it arrives at some value with positive imaginary part and then, when the imaginary height goes over towards it does a jump to the conjugated values - with a discontinuity at exactly. This is the thicker red line at the right hand side of the image. But what is the exact limit-point where the jump occurs?

Increasing the precision, with which I compute the tetration shows, that I can extend the trajectory towards some expected limit. Such improved computations allow the grey circles which extend the red trajectory.

But here my problem occurs: I need to double the precision to increase the length of the trajectory, and the last point was already computed with precision of 1600 internal decimal digits... at (and further approximations of do not change the computed values with that given precision)

So that calls for a better analytical consideration of that trajectory. How can we express the limit depending on the where ? (Remark : in Henryk's/Dmitrii's article[1] at page 13(pg 1739 of the printed journal) there is a much nicer and richer picture of that trajectories, but I cannot relate anything in their pictures to *that* trajectory).

Updated: Ahh, I should also mention that I use the powerseries around the lower (attracting) fixpoint .

[1] PORTRAIT OF THE FOUR REGULAR SUPER-EXPONENTIALS TO BASE SQRT(2)

MATHEMATICS OF COMPUTATION

Volume 79, Number 271, July 2010, Pages 1727–1756

(Article electronically published on February 12, 2010)

where is

It is now interesting, that the point , which is usually assumed in tetration as "at -oo", has as well an existing relative to it, which can simply be computed if we compute the relative

to by

and do integer Tetration to get

Here is a picture, where I graphed the trajectories for the imaginary heights from in steps of, say or starting at some example points on the line below +2.

Even more interesting is now the question what happens for the points in the interval because they are related to somehow values larger than the negative infinity. There is the asymptotic vertical line at ; the trajectories starting from values greater than , say from have a trajectory to the right side of that asymptote.

WHen I simply repeated my procedure for the computation of that trajectory from it looked as if it arrives at some value with positive imaginary part and then, when the imaginary height goes over towards it does a jump to the conjugated values - with a discontinuity at exactly. This is the thicker red line at the right hand side of the image. But what is the exact limit-point where the jump occurs?

Increasing the precision, with which I compute the tetration shows, that I can extend the trajectory towards some expected limit. Such improved computations allow the grey circles which extend the red trajectory.

But here my problem occurs: I need to double the precision to increase the length of the trajectory, and the last point was already computed with precision of 1600 internal decimal digits... at (and further approximations of do not change the computed values with that given precision)

So that calls for a better analytical consideration of that trajectory. How can we express the limit depending on the where ? (Remark : in Henryk's/Dmitrii's article[1] at page 13(pg 1739 of the printed journal) there is a much nicer and richer picture of that trajectories, but I cannot relate anything in their pictures to *that* trajectory).

Updated: Ahh, I should also mention that I use the powerseries around the lower (attracting) fixpoint .

[1] PORTRAIT OF THE FOUR REGULAR SUPER-EXPONENTIALS TO BASE SQRT(2)

MATHEMATICS OF COMPUTATION

Volume 79, Number 271, July 2010, Pages 1727–1756

(Article electronically published on February 12, 2010)

Gottfried Helms, Kassel